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'1T/GE/ENG/93-D-38
An Analysis of FM Jamming and
Noise Quality Measures
THESIS
Presented to the Factdty of the Graduate School of Engineering
of the Air Force Institute of Technology
Air University
In Partial Fulfillment of the
Requirements for the Degree of
Master of Science in Electrilc^ Engineering
Timothy Nathan Taylor, BSEE
Captain
December, 1993
Approved for public release; distribution unlimited
Best
Available
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AFIT/GE/ENG/93-D-38
An Analysu of FM Jamming and
N<^ Quality Meaauies
THESIS
Timothy Nathan Thylor
Ci^tain
AFIT/GE/ENG/93-D-38
93-31532
IIIIIIIM
Approved for public releaae; distribution unlimited
l2 29 01^
The views expressed in this thests ate those of the author and do not reflect the official policy or
position of the Department of Defense or the U. S. Govonment.
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Preface
This thesis was conceived to meet three primary objectives. The first was to explun the
behavior of FM jamming, particularly in the case where the relationship between the bandwidth
the modulating noise and the bandwidth of the jamming barrage is such that the modulation could
not be characterised as Wideband FM. The second was to critically cmtsider (both theoretically
and experimentally) three proposed methods of measuring noise quality in jamming scenarios and
determine under what conditions they are valid and useful. The third was to design and demon¬
strate a valid technique for measuring the noise quality of operational jammers which could be
implemented using commercially available equipment.
The first objective led to the development of terminology describing four possible types of FM-
by-noise (FM/N) jamming baaed on the relaticmships that must exist betsreen the three bandmdths
involved: the bandwidth of the modulating new, the bandwidth of the FM/N barrage, and the
bandwidth of the victim receiver. The general characteristics of the noise produced in the victim
receiver by each type of jamming are carefully considered srith analyses of both the shape of the
noise spectrum and the univariate probability density of the noise. The predicted characteristics
of each type of januning were experimentally validated using a simulated jammer and a simulated
receiver.
The second objective led to the discovery d strengths and weaknesses in each of the nmse
quality measures proposed to date. Only the two noise quality measures which measured noise
at the output of a victim receiver were considered in depth because the effectiveness of a given
jaimner is highly dependent upon the system it is trying to jam. Of these tsro, one was found
to be inadequate under certain specialised conditions, and the other was found to be generally
theoretically adequate, given some obvious modifications which srere made.
The third objective led to the design of a set-up consisting d a digital oscilloscope, a PC
contrdler and a set of programs srritten in C and Matlab, which were used to measure the noise
ii
quality of an operational jammer in the sponsor’s laboratory. It is bdieved that the ability of the
sponsor to make noise quality measurements o( operational januners has finally beoi restored.
In all this work I owe much to the members of my thesis committee. Dr. Vital Pyati provided
direction and the necessary technical background. Mr Eugene Sikora put together the necessary
components for measuring noise quality on the commercial jammer. Major Mark Mehalic helped
overcome initial difficulties with the laboratory equipment, and C^t. Joseph Sacchini gave me
the idea that was central to the improvement of one of the noise quality measures. Additionally, I
am indebted to Mr. Marvin Potts for his aid in programming, and to Ciq>t. Charles Daly for his
insights, support and the invaluable work which he did in this area previously. Lastly, I thank my
wife, Christa, and my children for their sacrifices and encouragement.
Timothy Nathan Taylor
Table of Contents
Page
Preface . u
List of Figures . vii
Liat of T^Ies . ix
Abstract . xi
I. Introduction . 1
II. Background . 8
2.1 Introduction to Noise Jamming . 8
2.2 Noise Quality in Jamming . 12
2.3 Current Techniques in Measuring Noise Quality . 16
2.3.1 Woodward’s Theoron . 18
2.3.2 IF and RF noise quality . 18
2.4 Summary . 21
III. Review of the Literature on Noise Quality and FM/N and FM/S+N . 23
3.1 Early Articles in the Open Literature . 24
3.2 Declassified Documents . 31
3.3 Tech Reports, Texts and Articles on EW . 36
3.4 Summary . 42
IV. Theory of Noise Quality in FM/N . 44
4.1 Ideal Noise . 44
4.1.1 Ideal spectral characteristics . 45
4.1.2 Ideal probability density fimetioa . 48
4.2 Theory of FM/N . 50
iv
Page
4.2.1 Theory <^FM . 50
4.2.2 Spectrum of FM/N . 52
4.2.3 Behavior of FM/N Jamming . 60
4.3 Measuring Noise Quality . 70
4.3.1 IWner Noise Quality . 70
4.3.2 IF Noise QuaUty . 73
4.3.3 FFT-IF Noise Quality . 75
4.4 Summary . 78
V. Experiments . 79
5.1 Verification and Use of the Daly Simulati<m . 79
5.1.1 Equipment . 81
5.1.2 Summary of experimental procedure for Daly Simulation .... 86
5.1.3 Experiments Using the Daly Simulation . 91
5.2 The Pathology of NBFM/LFN . 93
5.3 Measurements on an Operational Jammer . 94
VI. Results . 96
6.1 Measurements Using the Daly Simulation . 97
6.1.1 Narrowing Bif to Illustrate the CLT . 98
6.1.2 The Four Cases of FM/N Dlustrated . 99
6.1.3 NBFM/LFN . 109
6.1.4 Abnormal Readings . 113
6.1.5 Problems with the Chi^uare Test . 116
6.2 Measurements to Explore NBFM/LFN Effect . 117
6.3 Measurements of Operatkmal Jammer . 118
V
Page
VII. Ck>nclusioaa and Reconunendationa . 120
7.1 Concluaicns About the Theory oi FM/N . 120
7.2 Conclumons About Noiae Quality . 121
7.3 ConcluaioDa About the Daly Simulation . 122
7.4 Concluaiona About Operational Meaaurement of Noiae Quality . 122
7.5 Recomendationa . 123
Appendix A. Programa . 125
Appendix B. Data . 152
B.l Data from Daly Simulation . 153
B.1.1 Effecta of a Small Sample of Unintentionally Correlated Data . . 153
B.1.2 Effecta of a Small Sample of Uncorrelated Data . 153
B.1.3 Anomoloua reaulta poor choice of Volt/div netting . 156
B.1.4 Increaaed aample aet . 157
B.1.5 The Central Limit Theorem . 157
B.2 Pathological NBFM/LFN Meaaurementa . 159
B.3 Operational Jammer . 162
B.4 Final Notea on Data . . 162
Bibliography . 164
Vita . 166
vi
List of Figures
Figure Page
1. Probabilitiee of Detection and Fake Alarm . 9
2. J/S versus Noise Quality with fixed Pd . 15
3. Variable Gaussian Test Source Calibratkm - FM/N . 37
4. FM/S+N Effect rm Radar Output . 39
5. Baseband noise, RF spectrum and IF output for WBFM/LFN . 65
6. Block Diagram of Equipment setup . 82
7. Central Limit Themrem Illustrated (TNQ vs Bir) . 99
8. Central Limit Theoron Illustrated (TNQ vs Btp) Averages . 100
9. Time Samples of Baseband Noise . 101
10. FM/WBN Time Samples at RF . 102
11. FM/WBN Histogram at RF . 102
12. Histogram of Samples of Baseband Noise . 103
13. FM/WBN Spectrum at RF . 104
14. WBFM/WBN Baseband Noise (time samples) . 105
15. WBFM/WBN Output of IF Filter (time samples) . 106
16. WBFM/WBN Histogram of Samples at Output of IF filter . 106
17. WBFM/WBN Spectrum of Output cX IF Filter . 107
18. WBFM/LFN Baseband Noise (time samples) . 108
19. WBFM/LFN Output of IF Filter (time samples) . 108
20. WBFM/LFN Histogram of Time Samples . 109
21. WBFM/LFN Spectrum of Output of IF Filter . 110
22. NBFM/WBN Baseband Noise (time samples) . 110
23. NBFM/WBN Output of IF FUter (time samples) . Ill
24. NBFM/WBN Histogram of Time Samples . Ill
25. NBFM/WBN Spectrum 6t Output <rfIF filter . 112
26. NBFM/LFN Baseband Nose (time samples) . 113
▼ii
Figure Page
27. NBFM/LFN Output of IF Filter (time aamplea) . 114
28. NBFM/LFN Histogram of Time Samples . 114
29. NBFM/LFN Spectrum of Output IF Filta . 115
30. Pathological NBFM/LFN (TNQ vs A/,) 118
31. Pathological NBFM/LFN (Histogram at Peak TNQ) . 119
vui
List of Tables
Table Page
1. Table of Equipment for Daly Simulation . 81
2. Exploratory FM/N scenarioa . 84
3. Variationa of Parameters in Daly Simulation . 91
4. Table of Equipment Used to Measure Noise Quality of Operational Jammer .... 95
5. Settings to Unintentionally Correlate Data . 153
6. Noise Quality of 3277 Unintentionally Correlated Samples . 154
7. Noise Quality of 1490 Unintentionally Correlated Samples . 154
8. Settings to Decorrelate Data . 155
9. Noise Quality of Uncorrelated Data . 155
10. Noise Quality of 1490 Unintentionally Correlated Samples . 156
11. Mismatched Volt/div Setting . 156
12. Noise Quality With Mismatched Volt/div Settingl . 157
13. Noise Quality With Mismatched Volt/div Setting2 . 157
14. Increased Sample Set Setting . 158
15. Increased Sample Set Data Settingl . 158
16. Increased Sample Set Data Setting2 . 158
17. Increased Sample Set Data Settings . 158
18. Increased Sample Set Data Setting4 . 159
19. Settings to Demonstrate CLT . 159
20. Central Limit Theorem Datal . 160
21. Central Limit Theorem Data2 . 160
22. Centrid Limit Theorem Data3 . 160
23. Central Limit Theorem Data4 . 160
24. Centrd Limit The(»em Data5 . 161
25. Settings to Demonstrate Pathdogical NBFM/LFN . 161
26. Pathological NBFM/LFN Data . 161
u
Table Pa^e
27. Operational Jammer Datal . 162
28. Operational Jammer Data2 . 162
X
AFIT/GE/ENG/93-D-38
Abstract
This thesis attempts to address three related problems. The first is to provide a complete
description of the operation and characteristics of FM-by-noise (FM/N) jamming both at RF and
at the output of the radar receiver, in terms of spectrum, time domain waveforms and univariate
probability density. Particular emphasis is give to the case where the peak frequency deviation
the FM modulator is on about the same order as the bandwidth of the modulating signal since
this case has been largely neglected previously. The second problem has to do with measuring
noise quality in a jamming scenario. Noise quality measures which have been used in the past are
analysed theoretically and experimentally. The third problem has to do with designing a technique
for making practical noise quality measurements on operational jammers.
The first problem is addressed by considering four cases: Wideband FM by wide frequency
noise (WBFM/WFN), Wideband FM by low frequency noise (WBFM/^FN), Narrowband FM by
wide frequency noise (NBFM/WFN), and Narrowband FM by low frequency noise (NBFM/LFN).
The characteristics of these four cases are explored theoretically, and experimental results demon¬
strating each of the cases are presented.
The second problem is addressed by suggesting a new measure of noise quality at IF based
on two measures. The gauaianity of a n<^ signal is measured by forming a histogram of the
amplitudes of uncorrelated samples as suggested previously, and, in addition to this, the whitneas
of the signal is measured by taking the FFT of correlated samples of a waveform, dividing p<wt-
by-point by the discrete frequency transfer function of the IF filter, and comparing the result to a
flat spectrum.
xi
The third problem is addressed by the developmeot of a hardware and software setup which
has measured the noise quality of an operational noise jammer. The hardware used is described
here, and the sctftware is included with a brief description.
The theoretical and experimental analysis of NBFM/N lead to the conclusion that a measure
of noise quality which incorporates both the spectral whiteness as well as the gaussianity of the
probability density function of a jamming signal should be adopted. A noise quality measure which
does this is presented here. Also, it is reconunended that the setup which was designed to measure
operational jammers be used.
xii
An Anniysis of FM Jamming and
Noise Quality Measures
/. Introduction
The wwk outlined in this thesis is the result ongrang research to better understand FM-by-
noise (FM/N), a frequently used, but theoretically complex, method ot noise jamming, and to attach
well-defined, quantitative measures to the charactoisties of wavefcwms produced by different FM/N
systems. As such, it does three things. First, it gives theoretical consideration to the case of FM/N
where the peak frequency deviation of the frequency modulator is smaller than or on the same
order as the bandwidth of the modulating noise. This conditkm is referred to here as narrowband
FM/N (NBFM/N). This case is analysed in order to ccmqilement the descriptions of FM/N which
consider primarily the effects of wideband FM/N. Second, it details a series of experiments which
were performed in order to determine the validity and usefulness ot three proposed measures of
"nc^ quality,” an indicatkm of the jamming effectiveness of a given signal. It also makes an
additional recommendaticm about the theoretical determination of noise quality. Thirdly, it makes
recconunendations for a standardised method of measuring new quality on operational jammers.
In order to accomplish these objectives, this then has been divided into seven ch^>tas. The
first ch^ter is the introduction, intended to give an overview of the entire work.
The second chapter provides a background for the following chapters. First, it gives a brief
overview of the importance of new jamming in the area ot Electronic Warfare (EW). Secondly, it
explains the the basic concept behind FM januning, both FM/N and FM by sinusmd plus noise
(denoted FM/S-I-N). Lastly, the second ch^ter introduces and summarises two previous efforts
which have a direct bearing on the subject of this current thesis.
1
The first effort was a group of experiments commissioned by the USAF and carried out by
a team of scientists and engineers at Stanford Research Institute from the mid 1960’s through the
1970’s. These experiments measured different characteristics of operational radar jammers and
correlated those characteristics with the jammers’ abilities to effectively jam. One impMtant result
of these experiments was a measure of noise quality which has been used for the last two decades.
This noeasure of noise quality has sometimes been referred to simply as Quality, (20). In other
places, it is referred to as Gausnan Noise Quality (GNQ), since it is based on comparisons between
a histogram of samples of the noise waveform being measured and the probability density function
of an ideal gaussian random process having the same mean and variance (14). Since TWner was
the first member of the team to explain this measure of noise quality in the open lito'ature, it has
also sometimes been referred to as Tlimer Noise Quality (TNQ), (8) and throughout this thesis,
the term Thmer Noise Quality will be most often used. Because many of the noise jammers tested
by the Stanford team were FM jammers, their work was foundational to the concepts being studied
in this present work, both in terms of the function of an FM jammer and in term measuring the
difference in effectiveness between one type of FM/N jamming scheme and another. Therefore, a
brief fanuliarization with th«r results is an important prelude to the them as a wh<^e.
The second effort was some solid theoretical work on the nature of wideband FM/N, and a
small group of experiments which simulated an FM/N jamming system, demonstrated the charac¬
teristics of wideband FM/N (WBFM/N), and measured the TNQ of two basic types of WBFM/N
systems. This work was reported in Ah Ans/yficsf sad ExperimeHtHl /nvesfiysfton of FM~hf~Noi»t
Jamming which was writted by Captain Charles Daly in 1992 (8). The expoimental set-up devel¬
oped by Daly was duplicated in the research rep<»ted in this thesis, and many of the recommen¬
dations for future experimentation which were made by Daly are carefully considered here. Thus,
a brief outline of Daly’s work is also an important part of the general background of this thesis.
2
The third chapter is a literature review. In the context at the background provided in Chapter
2, the third chapter k>oks at relevant articlea and technical reports from the last four decades which
touch on FM jamming and the measurement of noise quality in noise jamming. Two recurring
ccmcepts are particularly significant in these works: 1) In the works that admit the purpose of th«r
investigation is the analysis of noise jamming, it is generally shown that the optimum noise should
have a flat spectrum in the paasband of the receiver being jammed, and a gauanan first ord«
probability density function. However, the emphasis in quantitatively measuring nc^ quality is
almost always focused on the gauasianity of the pdf rather than the flatness of the spectrum. 2) In
many of the articles, it is shown that the spectrum of the FM jamming signal generally conforms to
Woodward’s theorem (described in more detail in Chapter 2 and Chiq>ter 4 of this thesis), provided
that the peak frequency deviation of the modulator is sufficiently large. This implies that und«
the best of circumstances, the FM/N spectrum will never be perfectly flat over any bandwidth, and
may in fact deviate quite a bit from ideal flatness. These two facts sparked an interest in defining a
new standard <d' noise quality that quantitatively measured whiteness as well as gauasianity. They
also lead directly to the discussion in Chapter 4.
The fourth chapter is divided into three general parts: 1) it gives theoretical consideration to
the concept of ideal noise, 2) it describes the time and frequency-domain behavior of FM/N both at
RF and at the output of the IF filter of a victim receiver, and 3) it gives theoretical consideration
to four methods of measuring the conformity of a sampled noise waveform to the characteristics of
ideal noise.
The first part merely demonstrates that ideal noise is both white and gaussian. The second
part is, in part, a reit^ra icn of the theory explored in (8); howevCT, this thesis focuses on the sh^>e
of the RF spectrum of the FM/N signal when the peak frequency deviatkm of the modulator is too
low to meet the requirements for the ^plication of Woodward’s Theorem. Throughout the sequel.
3
this caw will be referred to aa NBFII/N beeauw of the cofteapoadeace to the rough definitioa of
narrowband FM (29).
A full discuaaion of the apectrum of the wideband FM/N aignal ia found in (8); howevw, the
NBFM/N apectrum was not deacribed in detail tb«e beeauw it ia not intentionally uaed fee noiw
jamming in practice. The noiw produced by NBFII/N ia conaidoed to be of a poorer quality
than that produced by WBFM/N beeauw cf ita diatinct non-flatnew. Nevertbeleaa, there are
circumatancea (which will be deacribed below) in which TWner Noiw Quality (which haa been
the atandard definition of noiw quality (30)) actually ahowa an incteaw in noiw quality with
a decreaw in peak frequency deviation from WBFM/N towarda NBFM/N. An explanation for
thia obaerved behavior ia developed by conaidwing four poaaible caaea ot FM/N jamming baaed
on the relationahipa between the three bandwidtha which muat be involved: the bandwidth of the
modulating noiw, the RF bandwidth of the FM/N aignal, and the bandwidth of the victim receiver.
'nimer noiw quality ia deacribed in detail aa wdl miite IF nawc fua/tljr developed in (8). It
ia noted that IVimer noiw quality doea not depend quantitatively on the whitenew of the jamming
aignal while IF noiw quality doea. RF noise ptalHf, alao developed in (8), ia commented on briefly.
Additionally, a fourth meaaure of noiw quality ia devebped and preaented.
The moat important pointa to be found in Chapter 4 are the further clarification of the
behavior of FM/N from a theoretical atandpoint and the proviaimi of further motivation for a
standard definition of noiw quality which includes a quantitative examinatimi of the flatneaa of the
apectrum.
The fifth chapter deacribea the experinmital aetup. Thrw goieral groups experimenta
were performed. The first group included the experimenta which were uaed to compare and analyw
the pn^Kwed measures of noiw quality. This group of experimenta frdlowed directly from the
recommendatkms in (8). Specifically, numerous meaauranents were made using essentially the
same setup described there, but var3ring some the experimental parameters and, in some caaea.
4
the computer programs which computed noise quality. One purpose in these experiments was the
attempt to find some consistency in a proposed noise quality measure which included a quantitative
measure of the fiatness of the spectrum.
Also included in this group was a demonstration of the increase in the gaussianity of the noise
at the output of the IF filter of the receiver which occurs when the bandwidth of the IF filter is
successively narrowed. This phenomenon occurs as a result of the Central Limit Theorem as has
been noted by Thrner and others (30), (6). The general consensus of scientists and engineers in the
field of electronic warfare seems to have been that the bandwidth of the IF filter should be smaller
than the bandwidth of the modulating noise; yet experimental results from T^irner seemed to
indicate that once the IF bandwidth was made ss smmll ss the bandwidth of the modulating noise,
there were no benefits (in terms increased gaussianity) in further decreasing it. Furthermne, a
cryptic remark in the work by Daly: (8:3-13)
FM-UBN seems to behave much like FM-WBN.
indicates that this was Daly’s experience also. Therefore, it seemed worthwhile to explore this
phenomenon thoroughly from an experimental standprmt.
The second group experiments used a setup similar to that used in the first group, * but
was designed primarily to demcHistrate the central problem with Turner Noise quality as a noise
quality measure, which is that 'Dimer Ndse Quality does not quantitatively measure the fiatness of
the spectrum of the noise. Specifically, it shows that under certain circumstances, when the peak
frequency deviation of the frequency modulator is decreased, the spectram of the jamming signal
becomes increasingly non-white, yet the 'Dimer Noise Quality actually increases. It is reasonable
to expect that a robust measure of noise quality would not do this.
The third group of experiments used a somewhat different setup than that used by the first
two groups. It employed the same commercial sanqiling osciUoscr^ and modified vonons of the
’The eanw -*•—*-»-** iamoier and receiver ware eaad, bat the ccwuiwtetioiiel hardware and eoftware had been
changed ae a result of leaa<»is learned while performing the first group of esperhaents.
5
cfHnputer pn^anw and algorithms used in the first two groups of experiments; however, whoeas
the first two groups of experiments measured the noise quality (d a simulated jammer operating in
frequMKies ordinarily used for conununications, this third group of experiments measured the noise
quality of an operational radar jammer. Tlie purpose of the third group experinaents was, first of
all, to demonstrate that the noise quality a radar jammer could be measured using commercially
available equipment, and, second, to develp a reliable valid technique for making the measurement.
The sixth chapter records the results produced by the experiments described in the fifth diap-
ter. Some of the trivial results are explained verbally, but an effort is made to present characteristic
waveforms, spectrums and probability density functions griq>hically, either by reproducing oscil¬
loscope and spectrum analyser displays or by showing the results of computer generated sample
histograms. General trends are also supported with graphs.
The seventh chapter presents conclusions and reccmimendations for further work. Some con¬
clusions about the natures of the different types oS FM/N systems were obviously supp<»rted by
both theory and experiment. Others are offered as tentative observaticms which may be verified or
explained more adequately by future researchers. Several questions were raised in the course of the
research and experimentation reported in this thesis which could not be adequately answered un¬
der the time constraints which were imposed, and it is hoped that future researchers may c(»sider
them.
The appendices at the end are provided to facilitate further research in this area. The first
appendix includes listings of the computa programs written in C and Matlab to measure ncm
quality on both the FM/N system simulation and the (q>erational jammer which operated at fre¬
quencies typically used by radar. Listings of the programs which were used to make measurements
on the simulated jammer can be found in Appoidix A of (8). Such modificatums at those programs
as are suggested as a result of lessons leaned from the experiments reported here ate found in
Chapter 6 ai this thesis.
6
The aecond appendix lists the raw data which was collected frtMn all three groups of experi*
meats in both tabular form. Some comments, conclusions and gr^hical analysis of this data are
found in Chapter 6 of this thesis.
7
11. Background
The purpoee of this ch^ter is to introduce nnd explain the proposed research into the op¬
timisation of noise quality measurements in radar nt^ jamming. There are two aspects to this
research. The first involves a mathematical analysis of currently used and proposed noise quality
measures. The second involves experimental measurements o( noise quality using the different noise
quality measures.
Before describing the specifics of either the mathematical analysis or the experimentation,
a short introduction is provided on radar jamming and noise quality measurements. Particular
attention is paid to Turner noise quality, a measure of noise quality which was developed in the
late 1960’s to classify the noise quality of operational jammers in the United States Air Force
invenUwy (30).
Following the introduction, there is a discussion at the current methods of measuring noise
quality. Recent research has provided two new proposed messures of noise quality which need to
be explored analytically and experimentally. The specifics of these two new measures, how they
are defined and how they can be measured in a laboratory, will be given some attentkm. These
measures will also be compared briefly to T\imer noise quality, and an explanation of how they are
sufiBciently different from IWner n<m quality to warrant investigation will be offered.
g.J Introduction to Noise Jamming
The subject of noise quality in radar noise jamming techniques might best be introduced
by defining noise januning in radar systems. All radar B]rstems consist of a radio trannnitter (or
transmitters) and a receiver (or receivers). Specific receivers vary widely in design depending on
the intended <q>plication of the radar system, but ail radio receives, in radar systems or othowise,
are designed to detect meaningful electromagnetic radiatkm (signal) in the presence of meaningless
radiation (noise). A griq>h depicting probability of detectum and probability of false alarm in a
8
0
Figure 1. Piobabilitiea of Detection and Falae Alarm
radar ayatem in the preaence of gauaaiau noiae ia ahown in figure 1 above, to illuatrate the basic
problem. As the amount energy becomea large with reapect to the aignal energy, the peak
of the noiae energy probability distribution moves cloaer and closer to the peak of the distribution
of the ngnal plus noiae until they beocmie virtually indistinguishable, in a statistical sense.
Ordinarily, the greatest source oi the meaningless radiation is the radar receiver itsdf. Re¬
sistive elements inside the radar receiver produce random dectrical currents because of thermal
vibrations of the component electrons when the radar ayatem is at any temperature above abso¬
lute aero (29). The first order probability density of these electrical currents is gauanan, and they
generate a power spectrum with constant power in all frequencies from DC up to about 10^’ Hs.
This nearly constant power spectrum is flat enough ovct enough frequencies that f<» all practical
purposes we will refer to it as ‘Svhite.” To the radar system, these random electrical currents gm-
erated by the radar receiver itself are indistinguishable from the currents gen^ated by the incident
electromagnetic radiation which the radar system is dengned to receive.
A certmn level of ndse power can be tolerated, so long as the signal power is sufilciently large
by comparison, but, for each receiver, there exists a kwa limit in the ratio signal power to nmse
9
power beyond which the output of the receiver cannot be characteriied as having any correlation
to the desired signal. This limit is known as the signal to noise ratio threshold, and although it
can be lowered somewhat, at the expense of more sophisticated signal processing techniques, or
longer observations of a relatively consistent signal, it cannot be made arbitrarily small. Radar
noise jamming, then, is an electronic countermeasure that seeks to deny meaningful information to
the operator of an opposing radar system by delivering sufficient noise power to the radar system’s
receiver to drive the receiver below its signal to noise ratio threshold. This is a simple concept,
but because noise is such a fundamental physical limiter in the operation of receivers, radar noise
jamming is still the most common electronic countermeasure used (22).
Often noise jamming is referred to as “masking jamming” because the purpose of the jamming
is to obscure or “mask” any signals which the victim radar receiver is interested in intercepting.
Masking jamming is fundamentally different from deception jamming which U used to confuse the
victim radar receiver operator. Deception jamming is more sophisticated than noise jamming, but
for that very reason is not always as reliable.
The concept behind noise jamming may be simple, but the iq>plication is more complex.
First of all, not all noise is equivalently useful in noise jamming. In order to be effective against
a particular radar system, the noise must be received by the targeted system. This requires that
the noise contain frequency components that lie within the bandwidth of the receiver. Secondly,
in order to be conservative of power, it is desirable that a radar jammer employ noise that is
frequency limited to the bandwidth of the targeted receiver. Thirdly, the noise must be generated
within the constraints of relatively deterministic circuitry, yet it must not be deterministic (if it
were predictable it could be eliminated at the targeted receiver by signal processing).
This third problem has an immediately suggested solution in that all resistive elements in
a circuit produce noise with characteristics as mentioned above. If noise is commonly generated
inside the targeted radar system by rerative circuit dements, then it may be generated by nmilar
10
elements outside, and then broadcast as electromagnetic radiation directly into the radar receiver.
In actual practice, the noise power of resistive noise is so small, even when greatly amplified, that it
is difficult to use it as a noise source, and other circuit elements such as back-biased diodes (13) are
often employed to produce low-level noise with the appropriate characteristics; however, this brings
us to the fourth problem: the noise must be broadcast at radio frequency with a sufficient power
level to drive the targeted receiver below threshold, but both the broadcasting and the power
amplification have some deterministic effects on parameters that characterize the noise. These
deterministic effects may or may not change the noise sufficiently so that it becomes predictable
enough to be canceled at the receiver through signal procesung.
More than a few techniques for generating radio frequency noise targeted to specific radar
receiver bands, amplified to high levels and sufficiently non-deterministic to defeat any signal pro¬
cessing counter-counter measures have been designed and used since the advent of radar, although
almost all of them function along the lines of the general principles outlined above: simple low-level
noise is produced by a circuit, it is filtered to some frequency band, it is amplified to sufficient
power levels, and then it is directly broadcast (if it is tdready at the appropriate radio frequency)
or used to modulate (either AM or FM) an RF carrier. Occasionally, the noise is amplitude clipped
before broadcasting. Sometimes a sinusoid is tulded to the noise to increase the noise bandwidth. A
recent innovation in electronic countermeasures is the Digital Radio Frequency Memory (DRFM)
which Lb most often used for deception, but which can be used to broadcast noise at RF, if the
memory is loaded with the appropriate radio frequency noise.
When noise is generated in the appropriate frequency band and merely amplified and broad¬
cast, this technique is known as Direct Noise Amplification or DINA. When the noise is used to
modulate a carrier at RF, then the technique is referred to as either AM-by-noise (AM/N) or
FM/N, as appropriate. In the early decades of radar noise jamming, all three techniques were
iinplemented in operational radar januners; however, DINA and AM/N were found to have some
11
drawbacks in terms of efficient use of the microwave amplifiers or efficient use of the bandwidth, so
that in 1983, Golden writes (11:192):
There are basically two ways of constructl::.^ a noise jammer: direct noise amplification
(DINA) and frequency modulation by sine wave plus noise (FM/S+N).
And he goes on to explain why the FM jammer makes more efficient use of the microwave
amplifier. But at this point, rather than discussing how to produce noise under the constraints
listed above, the discussion will turn to what constitutes "good” noise under those constraints.
2.2 Noise Quality tn Jamming
The foundation for the theory of noise quality is found in information theory developed by
Shannon (24). A fuller development, based on the concept of information entropy, is presented in
Chapter 4. The essential result is that the worst possible noise, in terms of destroying information
in a communications channel, is noise which has a flat pow' r spectral density and a gaussian
probability distribution function (25). This noise is commonly referred to as ‘Svhite gaussian
noise.” This result has an intuitive appeal. As was pointed out above, the noise which every radar
system must endure in the course of its normal operation is, to a large extent, resistive noise, which
is gaussian and essentially white up to very high frequoicies. Thus, the noise which should be
iqjected into a communication channel in order to most degrade the information passing through
that channel is noise which is indistinguuhable from the noise generated internally by the receiver
itself.
The next question to be answered then, in terms of noise quality, is, how should power spectral
densities and probability distribution functions of a non-deterministic wave-form be measured? This
question was and is important, not only for the purpose of analytically verifying a noise-jammer
design, but also for the purpose of testing the finished product. Before sending an operational
jammer to perform a specific mission, it is useful to know whether the jammer actually produces
noise that will jam effectively.
12
Eogineering a setup to measure the probability distribution functions of noise produced by
noise jammers was the subject of a study commissioned by the United States Air Force in the late
1960*8 (20). A team at Stanford M>proached the problem by januning a receiver with the noise
jammer under investigation and then making measurements on the time series of the output of the
victim receiver’s intermediate frequency filter.
Now it is important to note that noise was not measured at radio frequency. There were two
reasons for this. The first was that most receivers at that time did not bandlimit their inputs until
after heterodyning to an intermediate frequency, and the relative spectral whiteness of the noise is
only relevant in relationship to the frequency band that is being jammed (i.e. the bandwidth of the
victim receiver.) The second reason is that while DINA produces an RF signal that is bandlimited
white gausnan noise, as would a DRFM whidi was loaded with distal noise and used for noise
jamming, the AM/N and FM/N signals at RF are neither gaussian nor white. Whether or not an
AM/N or FM/N signal will produce bandlimited white gaussian noise in the output of the victim
receiver is highly dependent on the precise characteristics of the noise jamming system in relation
to the characteristics of the victim receiver. Thus in comparing one noise source with another it
was important to normalize out the factors whidi were related to how the jamming was done, and
focus instead on the ultimate effect of the jamming in a given receiver.
The first order probability density of the victim receiver output time series was computed,
and that was compared to the probability density of an ideal gaussian random variable scaled to
have the same mean and variance as the measured time series. Several error measures were then
computed. Average error, rms error, and summed error; kurtosis and skewness; and the relative
entropy of the measured output were originally used as the basis of three individual measures. Later
they were combined by 'Dirner and others to form the measure known as Thmer noise quality in
1977(30).
'Dirner Noise Quality is defined as:
13
N.=» = 5 . !L:|±W)
(1)
where e« is the suimned error, ta is the average mor, e, is the rms error, k is the kurtosis,
and s is the skewness,
0 < noise quality < oo. (2)
In actual practice, a very good gaussian noise source will have a large noise quality (some have
been measured as high as 70) while a non-gauaaian source should have a relatively low noise quality
(for example: a perfect sinusoid has a theoretical noise quality of 1.5).
The practical value of the Thmer Noise Quality measure was demonstrated by a series of
experiments carried out by Turner, Ottoboni and Imada at Stanford, and reported on in 1977 (30).
A white gaussian noise source was amplified and then lowpaas filtered. The output of the filter
was then used as the input to a adid state voltage controlled oscillator and FM modulator. The
RF signal at the output of the FM modulator then had characteristics which were similar to those
of signals produced by many of the most common operational jammers, and these characteristics
could be modified (by being more or less white in a pwticular bandwidth) by varying the bandwidth
of the lowpaas filter, the amplitude of the white gaussian noise source, or the parameters of the
FM modulator. Furthermore, when the RF rignal was broadcast into an operational radar s]rstem
that was hooked up to it, it was mixed down to an IF frequency and passed through a bandpass
filter. By altering the bandwidth of the baseband noise with respect to the bandwidth of the IF
filter of the victim receiver it was noted that the output of the IF filter could be made more or leas
gaussian. The unit as a whole was rderred to as a GMiasianity Test Source.
The output of the Gaussianity Test Source was monitored by a ctunputer contrtdled proba¬
bility density analyser which sampled the output signal and then computed the various parameters
of the Thmer nt^ quality measure in order to report the Ihmer noise quality in near real time.
14
reduction in jamming power.
Figure 2. J/S versue Noiae Quality with fixed P4
Once the noiae quality of the signal waa determined, it was used to jam conventional pulse radar
eystems in the presence of signals generated by a sophisticated radar simulator. Experienced radar
operators then monitored the ^hq>lay scopes of the radar tecavos and attempted to accuratdy
determine the presence or absence of a real target in the midst of the noise. A long series of
experiments in which several thousand operator detection decisions were made dononstrated that
there was a rough inverse relationship between the jamming power required to obscure a target (as
measured by J/S, which is the ratio of jamming power to target rignal power) and the 'Ihmer ndse
quality. As the Ihmer noise quality increased, a smaller J/S was required to significantly decrease
the operators’ probability of target detection. As shown in Figure 2 taken from (30), a very good
noise source requited roughly five times less power than a very poor noise source to eifectivdy jam
a conventional receiver.
The correlation between noise quality, as measured by Thmet, and jammer effectiveness, as
demonstrated by the experiment outlined above, is a strong argument in favor of the validity of
the Ibrner n<m quality measure.
15
U is interating to note that all theae meaaurea were baaed on the probability diatribution; none
of them dealt with the uniformity of the power apectral denaity. That the membera of the team were
aware that the optimal n<^ ahould be white as well aa gauaaian ia well documented (20). However,
becauae of the limitationa of the meaauring equipment available at that time, their ^>proach to
meaauring the whiteneaa of the noiae waa entirely qualitative rather than quantitative. The ahape
of the apectrum was observed, using a frequency analyzer, and then a decision was made, baaed
on its conformity, or lack thereof, to the theoretical shape of the IF filter, aa to whether the n<^
seemed to be sufficiently fiat in the paasband of the IF filter to be called white. If it waa, thoi
the noise was considered acceptable and the Tbrner noiae quality was ccHnputed. If the noiae did
not aeem to be suflSciently flat, then it was rejected, and its gauasianity or lack thereof was not
considered. Documentation of this procedure was included in the technical report in the form of
Polaroid pictures of the frequency analyzer displays pasted in next to the results produced by the
probability density analyser.
As the members of the team pointed out, almost all of the jammers which they tested had
power spectrums which were nearly fiat over almost every fluency that was tested. (20)
2.S Current Techniques in Measuring Noise QuaUtg
Tbmer noise quality was consistently used as the primary theoretical measure of the effec¬
tiveness of operational jammers up through the 1980’a. As evidence of the universal acceptance
of the ad hoc measure, we note that in 1985, a nmulation program based on the 'Dimer noise
quidity measure was proposed as a method for optimizing n<^ jammer design. (14) The test setup
designcHI by the team as Stanford has long since been disassembled; however, with new jamming
techniques there has been a renewed interest in measuring noise quality and in 1991 a study of
FM/N jamming was sponsored by Wright Laboratory (WL/AAWA) and carried out by Obtain
16
Chwlet Daly. Because this present thesis k partly a continuation ci the effort begun there it will
be appropriate to give a brief summary that recent work.
The focus of i4a Experiment^ aai An^piicnl /asesliyalioa «/ FM-Bp-Noixe Jnmmimp (8) was
a thorough explanation and denoonstration of FM/N. It accomplished thk by reviewing the literar
ture on FM/N, deriving or citing the equations which describe the behavior of FM/N, and, lastly,
constructing a laboratory simulation of an FM/N jammer and a victim receiver and measuring
the noise output of the receiver. The last part the thesk would have successfully demonstrated
the operation of FM/N if it had merely produced reproductions of the oscilloscope and q>ectrum
analyser traces showing the time and firequoicy domain characteristics of FM/N signak. Daly’s
work did thk; however, it also produced a new setup for duplicating the Turner nmse quality
measurements.
There were some differences between the Daly setup and that used at Stanfmd. For example,
the Stanford setup builds a histogram of voltage levek baaed on five million saiiq>les, while the
Daly setup uses mily a few thousand samples. The Stanford setup sorts vdtages into either 512
or 1024 bins while the Daly setup usually uses around 30 voltage bins. The Stanford setup was
designed to measure the ndse quality of radar jammers at frequencies wdinarily used by radar, and
actually measured operational radar while the Daly setup (q>erated at frequencies of less than 1
GHs, and was primarily intended as pure simulatkm to demonstrate a concept. However, it should
be noted that the Stanford setup was the result specially engineered equipment, while the Daly
setup exclusively used equipment which k commercially available.
Two important things came out of the Daly investigation into FM/N which have a direct
bearing on thk thesk. The first k a proof of Woodward’s theorem which k given in Chi4>ter 4 (8).
The second k a practical recommendation that two other measures of noise quality should also be
investigated.
17
t.S.J Woodward’s Theorem. Woodward’s theorem has beoi mentioiied previously in this
thesis but has not yet been defined. Because it is an important theorem in describing the spectra
of a WBFM signal in gmeral, and particularly important to the description of the WBFM/N
spectra, an informal description of it will be included here. In an FM system, the frequency
output of the system is directly related to the voltage level of the input. Thus if an input is a
constant offset voltage, one would expect the RF spectrum of the FM system to be dominated by
a single frequency component at a constant offset from the nominal carrier frequency. And, for a
general modulating signal, we would expect the RF spectrum to have large frequency comp<»ents
corre^Kmding to voltage levds that the modulating signal visited frequently, and small frequency
components corresponding to voltage levels that were rarely visited. Or, to state it another way,
it is anticipated that the shape of the RF spectrum will generally correspond to the shape of the
univariate probability density function of the voltage level of the baseband noise (28:307).
g.S.i IF oud RF Hoioe quotitt. Of the two new noise quality measures proposed by Daly,
the first of these measures is called IF uoioe fsslify, and, like TWner ndse quality, it measures the
gaussianity of the output of the targeted teodver directly after the IF filter. Unlike Tbmer noise
quality, it also makes a quantitative measure of the whiteness of the spectral density of the noise
and imposes a penalty for a power spectral doiaity in the passband of the IF filter which deviates
from absolute flatness. It seons intuitively appealing that the IF nc^ quality would be able to
distinguish more clearly betweoi a good n<m source and a poor noise source than the Ibmer noise
quality because it contains informatkm about the frequency domain characteristics of the nmse ss
well as the time domain characteristics. However, as was observed by Tiimer and the research team
at Stanford, under most conditions, the spectral densities of most noise jammers are reasonably
white, leading to the result that IF noise quality and TWner noise quality are most <dlen relatively
equivalent.
18
IF noiae quality calculates a “time-domain penalty* (pt) by essentially calculating tbe error in
the difference between a histogram of the voltage samples in the IF noise and a theoretical gauasian
histogram • a technique very similar to that used in a portion of the TVimer noise quality measure.
The value pt has a maximum value of one. IF noise quality also calculates a “frequency domain
penalty* (p/) by taking the ratio of the measured jamming power to the ideal power over the range
at the 3dB bandwidth of the IF filter of the receiver. Thus, (pj) should be between sero and one.
The two penalties are then combined to form the IF noiae quality pip by the formula:
PtF = (1 - P«)(P/) (3)
Perfectly white and gauasian noise then would have a pit ot 1 (or 100%) while any nmae that
deviated from perfection would have an IF nmse quality that was some fraction of that, or, in some
extremely poor cases, a negative value (8).
RF noise quality is the second of the new prt^Kised noise quality measures, and it differs
dramatically from TUmer noise quality. It is a meamirement made at radio frequency (RF) rather
than at IF and it is based <» fiiequency domain measures alone rather than on time domain
measurements (as does IXimer noise quality) or time and frequency domain measurements (as does
IF noise quality). It represents a rather clever analym from a mathematical perspective, but its
results are only valid under the specific tjrpe ct noise jamming (FM-by-noise jamming) which was
the primary focus of the investigation sponsored by Wright Laboratory.
In FM/N jamming, a low-frequency noise source is used to frequency modulate a carrier
centered on the receive band <ff the targeted recdver. Under these circumstances. Woodward’s
Theorem states that the power spectral density of the RF signal will have the same shape as
the probability density function of the modulating signal. Therefore, a chi-square goodness-of-fit
test can be applied to the power spectral density of the RF signal to determine whether the low-
frequency noiae was truly gauasian, and so, wheth« the noise at the output of the IF filter of the
19
Urget lecmvet b gauaaian. Since meMuring the power ipectraJ density of n wnvef<wm is much easier
than computing the first order probability doiatties, this is a great advantage.
However, the problem with this portion of the RF noise quality measure, ss a genoal noise
quality measure, is that the shape of the pown q>ectral density of an RF noise waveform that
was not generated using FM/N should not necessarily have a gaussian characteristic (for examine,
DINA noise or noise produced from a DRFM may be of a high quality, but would not receive a
good RF noise quality figure).
Furthermore, the efficacy of an arbitrary signal in producing gaussian noise in a victim receiver
cannot necessarily be established simply by looking at the RF signal. DINA noise has a gaussian
pdf at RF and produces gaussian noise in a victim receiver at IF. But, as noted above, although the
RF signal generated by FM/N jammer has a gaussian spectrum, it is not a gaussian process and
does not have a gaussian probability denwty function. Rather, it is more properly charactmsed ss
a sinusoid of varjring frequency, and it tends to have a voltage histogram (a rough naeasure the
true probability density function) that is characterised by a local minima at the mean voltage and
two local maxima (one being the absolute max), one above and one below the mean voltage, in
sharp contrast to the voltage histogram of a gaussian noise signal which has an absolute maximum
at the mean voltage. It is only when the RF signal is heterodyned to an IF frequency and passed
through a filter of sufficiently narrow bandwidth that the output ci the filter resembles gaussian
noise.
Furthermore, RF n<^ quality is unique among noise quality measures in that the measure¬
ments taken are independent at any parameters of the victim receiver which is being jammed.
Because the efficacy of any noise jamming system is heavily dependent on what victim qrstem it
is trying to jam, this again makes the measure problematic as a genual measure of nc^ quality.
This dependence at the efficacy of a noise januning qrstem <ni the parameters of the victim receiver
20
being jammed is one of the reasons why the team at Stanford avoided using any such measure, as
noted above.
Thus RF noise quality is a potentially useful measure of noise quality in FM/N jamming when
noise quality is not normalised to the parameters of a specific victim recdver being targetted, but
it cannot be blindly applied to an arbitrary noise source.
s
RF noise quality is based on measurements made at RF, and it uses those values to compute
time and frequency domain penalties and then combines them via the same equation used for IF
noise quality.
SummMrjf
In concluding this chapter, it is important to note the ftdlowing points: First, that noise
jamming, because of the fundamental physically limiting nature of noise in radar receivers, is an
important technique in electronic warfare. Second, that ‘Hdeal noise” for noise jamming is noise
which is both white (in the frequency range being jammed) and gaussian. Third, that the noise
employed in noise jamming nonist be generated in some mm-deterministic fashion and broadcast in
a specific frequency range and thus is unlikely to have the i(feal charactoristics of purdy resistive
noise merely by accident or coincidence. Thus, some jammers will have characteristics that are
more ideal than the characteristics of others, based on their design.
Fourth, recall that jammers which produce more ideal noise (primarily in terms of more
gaussian first order probability densities) have been experimentally shown to mask targets better
at lower J/S ratios. And, last, that there are currently three quantitative measures of noise quality.
One of these measures (Timer noise quality) has been rigorously experimentally verified and was
universally used for the last 20 years or so; however, it only gives a quantitative measure the
gausnanity of the noise, and addresses the question of whiteness (mly qualitatively. The other tsro
21
meMuies of none quality «cfe piopaaed in the laiA two yean, and they attempt to employ the
better equipnmit available today to improve on Tumer’a work.
22
III. Review of the Literature on Noise Quality and FM/N and FM/S+N
The purpose of this chapter is to review the EW literature which relates to FM noise jamming
and to the measurement of noise quality in noise jamming. Much of the early material in the field
of EW in general is either indirect or classified because of security considerations. However, the
basic theory of FM noise jamming was developed and experimented with sufficiently long ago
that most of the relevant classified documents are now unclassified, and the more indirect articles
have been clarified by more direct later material. This leaves us with several distinct categories
of material to look at. Firstly, there are articles in the open literature from the 50’s and into
the 60*8 which speak indirectly about the spectra of signals which are frequency modulated by
low frequency noise. Secondly, there are technical reports <m experiments and theoretical work
done with FM/N and FM/S'fN jartuning which were classified at cme time but which have since
undergcme declassification. Thirdly there are mme recent articles, EW texts and theses which deal
with the theory behind FM jamming in a forthright manner.
For a very good review of the literature on FM/N from the perspective of a theoretical and
experimental description of FM/N at both RF and IF, one should consult Ch^ter 2 of the work by
Daly (8). Although only one declassified technical report is alluded to there, the research of articles
and EW texts is very thorough and it covers the essential topics of what categories of jamming
FM/N can be placed in, what kind of spectra it generates at RF, and how it is used to jam the
IF filter of a receiver, and even touches briefly on the material surrounding the rather obscure
topic of FM-by'erfer noise. (Ordinarily when FM/N is spoken of in EW literature, the noise at the
input of the FM modulator is assumed to be white gaussian noise, bandlimited to some baseband
frequency.)
The emphasis in this present thems, however, is not so much on the theory of the function of
FM/N (with the exception of NBFM/N) but rather on the characteristics of the noise produced by
FM nrm jamming and the concern of the authcm of the various materials to produce noise that had
23
certain characteristics. Additionally, there is emphasis on the techniques that were suggested and
implemented to determine the degree of presence or absence of these characteristics. .Accordingly,
this chapter deals with three different types of material. The first type is material that describes
the optimal characteristics of a noise jamming signal. The second is material that describes the
characteristics at IF and RF of signals arising from FM/N and FM/S+N, and the third is material
that deals with the subject of noise quality measurement.
Because some articles and almost all of the technical reports and texts deal with more than
one of these subjects, if each subject were considered separately, there would be multiple citations
of many of the sources. (For example. Golden spends some time discussing optimal noise (p 92),
then later turns his attentions to types of active jamming systems (p 199) (11).) In order to avoid
this kind of redundancy, this ch^ter is composed of sections that first consider early open literature
articles and papers that refer to FM jamming indirectly, then look at declassified technical reports,
and more recent articles, texts and theses that deal openly with FM/N and FM/S-fN as EW
techniques. However, in each of these sections, the material will be looked at as it relates to each
of the subjects mentioned above, and it is hoped that certain recurring themes will be noticed. A
section summarising the important points is found at the end of this chapter.
3.1 Earlf Articlea ta ike Open Litentnn
The earliest papers that address the concept of FM/N are almost all concerned with the
shape of the spectrum produced by FM/N or phase modulation by noise (^M/N), and have little
or nothing to say about the statistical characteristics of the FM/N signal. The most probable
reason for this is that from the perspective of the radar jammer designer, the probability density
function of the RF januning signal is not really all that important; he is more concerned about the
probability density function of the detected signal that comes out of the victim radar’s IF filter.
For security reasons, the writers of the early articles avoid direct mention of noise jamming, but
24
the fact that much of their research was sponsored by military organisations makes it more than
likely that they were working on the radar jamming problem and, therefore, if they were interested
in the statistics of the FM/N signal, they would only have been interested in the statistics from
the perspective of a radar jammer designer. Thus, in terms of probability density functions, the
problem they were interested in was precisely the problem that they were prohibited from discussing
in the open literature.
In 1951, David Middleton presents a paper on the spectra of carriers amplitude, phase and
frequency modulated by gaussian noise (17). His presentation begins by assuming ergodicity, finding
the autocorrelation function of the RF signal, and applying the Weiner-Khintchine theorem. His
approach is a little unusual in that he assumes a modulating noise with a gaussian spectral density
as well as gaussian amplitude characteristics, while most noise models assume the power spectrum
of the noise to be either white or rectangular between two fluencies. While the final expression
which he obtains is the result of a rather unwieldy MacLaurin series expansion that does not, by
itself, readily give much insight into the shape of the FM/N spectrum, he offers some observations
in addition, such as explaining under what conditions there will be a discrete amount of power
in the carrier frequency or in certain harmonic frequencies, and when (as is more common) the
carrier power is distributed throughout the continuum. He also is the first to distinguish between
modulating gaussian noise which has spectral compmients at sero frequency and that which has
spectral components which are merely close to sero frequency. His most important statement from
the perspective of this thesis is (17:699):
Note that as the mean intensity of the modulating noise ( o't) becomes very small, or
as the r.m.s. deviation ( U4 or $4) becomes very great, the other parameters of the
system remaining constant, one always approaches a paussian modulation spectrum,
quite independent of the precise power distribution of the modulating wave . . .
Although the point of the quote may seem a little obscure without a thorough understanding
of the symbols Middleton is using, or the precise context, it is essentially a specific iq>plication of
Woodward’s Theorem. In essence, Middleton is saying that when the peak frequency deviation
25
of the modulator is sufficiently greater than the highest frequency components of the modulating
noise, it doesn’t really matter what the precise shape of the spectrum of the modulating noise is:
all that matters is the univariate probability density o{ the noise, which in this case happens to be
gaussian. Under those circumstances, the shape of the RF spectrum will approach gaussianity.
The technical report produced by Stewart in 1953 is more lengthy and more narrowly focused,
but it begins with a similar statement (26):
It should not be inferred that a knowledge of the power spectrum of a frequency- or
phase-modulated carrier tells a great deal. In fact, the power spectrum of FM or
yields much less (relative) data than that of an amplitude-modulated wave. For
example, if the frequency deviation of an FM signal is larger than the bandwidth
the modulating voltage, the shape of the spectrum is essentially independent of the
spectrum of the modulating signal.
Stewart’s presentation differs from Middleton’s in several respects. Firstly, it deals with
gaussian noise with a rectangular power spectrum rather than the gaussian spectrum dealt with by
Middleton. Secondly, it produces a set of asymptotic closed-form expressions for the shape of the
RF spectra (i.e. the spectra conforms to these expressions asymptotically as certain parameters are
made arbitrarily large) which are presented graphically, and thirdly, it deak exclusively with phase
and frequency modulation and does not touch on the subject of AM/N (which has been shown to
be less effective as a jamming technique).
Because of the simplicity and clarity of the closed-form expressions derived by Stewart, they
are shown below:
Wf{^) =
(4)
WV(Aw) =
AV2
nB (jri?2/2fl»)»-»-(Aw/B)»
(5)
where:
26
A — peak amplitude of cartiei vcdtage,
Dl = mean-squared instantaneous radian frequency
deviation (proportional to the mean-squared
modulating voltage),
Am = radian dilTeraice frequency from the unmodulated
carrier frequency,
B = radian bandwidth of the modulating voltage
and the power spectrum of the modulating signal is unUorm from sero to B radians, and seto above
that point (26). The frequencies have been shown in radians rather than Hi so that the first equation
becomes easily recognizable as a gaussian shape with variance Di, which perfectly exemplifies the
expectations of Woodward’s theorem. The seccmd equation, of course, is distinctly non-gaussian.
Note that the first condition is what we would call WBFM/N and the second condition is the
extreme case of NBFM/N where the peak frequency deviation of the RP signal is actually less than
the bandwidth of the modulating signal.
After deriving the closed-form asymptotic expressions, Stewart also spends some time dealing
with the corrections which should be applied when one is not at either the WBFM or the NBFM ex¬
tremes. Specifically, he states that there are two distinct cases when the baseband nc^ bandwidth
is on the same order as the rms frequency deviation of the modulator; 1) when the modulating
noise spectrum extends all the way to DC and 2) when the modulating noise extends down cmly to
some lower cutoff frequency. In the first instance he produces a correction expression for the tails
of the RF spectrum. When he gives some attention to the questimi of how the spectrum is changed
when the modulating noise has a spectrum which is rectangular but does not extend all the wiqr
to DC, he finds that the major difference is that the FM/N spectrum gains a delta function at the
carrier frequency; however, he concludes that the effect of the delta function is slight as long as the
lower cutdf frequency is small in comparison to the bandwidth of the modulating signal.
27
Between 1953 and 1957, there is a continuing discussion in the literature about the exact
nature of the FM/N spectrum (the ^M/N spectrum having become leas of a concern) where ques¬
tions were raised and responded to concerning the applicability of Stewart’s expressions for the
case where the modulation index is moderate or low (the modulation index being the ratio o( tlw
modulating bandwidth to the rms modulated bandwidth) (16) (27) (12).
In 1957, Middleton and Mullen respond to Stewart’s work and to this entire discussion in a
letter in the proceedings of the IRE (18). They agree that Woodward’s theorem generally holds
when the modulation index is high, and also that there are two separate cases when the modulation
index is low. However, they go into much more detail explaining a “suitable expression” for the
tails of the RF spectrum through the derivation of an approximating series. In this letter, they use
a band-limited white spectrum for the modulating noise rather than the gaussian spectrum which
Middleton worked with previously. Additionally, they use an analytical technique which assumes
a complex modulating wave and discovers the real RF spectrum through Hilbert transforms. The
equations are too lengthy to be included here but they indicate a spectrum that is more peaked,
roughly more triangular than gaussian in shape. It should be noted that they seem to have satirfied
the other participants in the discussion.
While the issue of the RF spectrum of the FM/N signal was fairly settled at this point, there
were a number of other issues which had some indirect bearing on the topic which continued to
be raised. Blachman presents a short article on Fourier Series representations for gaussian noise
which discusses the independence (or lack thereof) of the Fourier Series coefficients, depending on
the fundamental period of the series (3). Later Blachman writes another article which deals only
peripherally with FM/N, but deak primarily with the spectrum of FM/S-(-N. (4) He begins by
discussing Woodward’s theorem and the places where Woodward’s theorem does not hold. First
of all, he notes. Woodward’s theorem does not hold when the modulation index is low, although it
may be a first step towards an approximation. Secondly, Woodward’s theorem does not hdd when
28
the baaeband modulation ia deterministic in such a fashion that it leads to spectral lines in the
RF spectrum (modulation by a sinusoid being a prime example). Why is this? The answer can be
explained in terms of the “resolution” of Woodward’s theorem.
Blachman refers to a non-rigorous but intuitive ptoat of Woodward’s theorem which was
understood within the community prior to Woodward’s paper on the subject. If we think of a
filter-bank in the RF spectrum being connected to the output of an FM modulator, we can imagine
the baseband signal sweeping into a particular voltage region and causing the RF signal to sweep
into a particular frequency range and then measuring the power that is passed by the filter that
covers that frequency range, and thus, by measuring the power passed by each filter, over time,
develop a power spectral density for the RF signal. The question is, how narrow should each filter
be made in order to have accurate results from this technique? Blachman ’s response:
. . . the duration of the transient response of the filter must be small OMnpared to the
ratio of the filter bandwidth to the rate of change of frequency. Since the duration of the
transient response is of the order of magnitude of the reciprocal of the filter bandwidth,
this means that the filter bandwidth must be large compared to the geometric mean
of modulation bandwith and frequency excursion ...Thus Woodward’s theorem can
resolve only those spectral details whose widths are much greater than the geometric
mean of the modulation bandwidth and the fluency excursion.
(3)
Fortunately for us, it is possible to increase the resolution of Woodward’s theorem by including
more terms of an infinite aeries of which the probability density function of the modulating signal
is only the first term. This is essentially what was done by Mullen and Middleton in 1957. Also it
is possible to generalise Woodward’s theorem to cover deterministic modulating signals as well as
random signals, and combinations of random and deterministic signals. This leads into the main
thrust of Blachman’s article, which u the consideration the spectrum of FM/S+N. Blachman
does not say so, but classified reports of about this time were studying the effects of FM/S-l-N
jamming, and finding that adding a sinusoid to narrowband gaussian noise was one way to increase
the bandwith of a jaimning signal without losing significant gaussianity in the signal at the output
29
of the IF filter of the victim receiver. Blachnun’t development shows this increase in bandwidth,
but, consistent with other articles in the open literature of this time period, does not touch at all
the issue of the statistical characteristics of the output erf an IF filter receiving the FM/S+N signal.
Norman Abrahamson offers a paper in 1963 on the bandwidth and spectra of FM and dM
waves which considers sinusoidal carriers and gaussian random processes as modulating signals. Its
purpose is not accuracy in developing an expression for the FM/N spectrum, but rather simplicity
and generality. He also has some very informative graphs which show how altering parameters of
the modulating noise can change the RF spectrum. The approach is really nothing more than a
clarification or perhaps a clever implementation of the ^proach used by Middleton more than a
decade earlier, but because of the simplicity and generality of the Abrahamson’s implementation,
the equations he presents will be expanded on in considerable detail in Chapter 4, when the theory
of the shape of NBFM/N signal is derived.
The last paper in this category is by Bladiman, 1969 (S). It again deals with Woodward’s
theorem and Blachman’s filter>baok proof. This proerf is finally made rigorous, and, as a result,
an upper bound on the error of the approximation indicated by Woodward’s theorem is found. He
uses gaussian noise with a butterworth (rather than rectangular or bandlimited white) spectrum
as his modulating noise, and he generates a series of paphs showing the difference between the
calculated RF spectrum and the Woodward approximati(»i for various modulation indices. As the
modulation index is high, the Woodward ^proximation is very good, as it is lower, the actual
calculated spectrum is not gaussian at all, but looks very much like a spike at the carrier frequency,
as predicted by all the previous work. The fundamental contribution of this work is the new upper
bound on the error associated with the Woodward ^>proximation.
30
S.t Deetuaifiei DceumenU
Probably the most imp<»tant declaaified document on electronic warfare in general is the
massive compilation Eleeinuie Coaatermessarej (6). In Chapter 12, Morita and RoUin discuss
types of “masking” jammers (as opposed to deception janunets. Spectra are shown for FM/N and
FM/S+N as well as for the other jamming techniques (DINA, AM/N where the noise is clipped,
AM/N + FM/S), and, although specific equatkms an not given, it is ^>parent that the FM/N
RF signal has a gaussian spectrum and the FM/S-fN has the desirable characteristic of a broader
bandwidth. Comments are made about the relative effectiveness of different jamming techniques,
but these comments are fairly general. The whiteness or lack thereof ot any of the spectra is not
dealt with except to say that noise power should be distributed over the bandwidth being jammed
rather than concentrated in a discrete carrier. However, the following quote concerning clipping of
DINA:
If the receiver bandwidth is narrow compared to the clipped noise bandwidth, the
noise signal will appear to be gaussian to the receiver and will have the same effectiveness
as gaussian noise.
(6:12-5) shows that there was a concern about the gaussianity of the time-statistical characteristics
of the noise coming out of the IF filter of the receiver.
Also, in chapter 14 of Electrvmie Conniermeanru Benninghof, Farris, Lauderdale and others
discuss the effectiveness of different jamming ngnals. Thdr discusnon begins by calling attention
to the fact that different types of jamming (deception, noise, spot, barrage) may be better or wotm
depending on the different circumstances. However, suppomg that it has been decided that a
noise jammer is what is needed, the question is, which noise jammer is most effective at producing
noise at the output of the IF filter of the receiver. To answer this question, they consider field
testing, simulations and mathematical analym. They choose DINA as their baseline noise for the
simple fact that DINA is white and gaussian. As was pointed out earlier in this thera, DINA does
not make the best use of the microwave amplifier in the noise jamnm; however, it is white and
31
gaiuiian, aad a aiaiple I ud Q analyaii of Uie nanowband proceas coming out of the IF receiver
with gauMian noise as its input will show that ptocsss to he gaussian also. It is not here prosed
that gaussian noise is optimal for noiae jamming, hut it is assumed.
When Benninghof and the rest turn their attention to FM/N as a jamming technique, they
divide it into two categories: FM hy wideband noise (FM/WBN) and FM by low frequency noise
(FM/LFN). * The distinctirm between the two categories is based on the ratio of the bandwidth of
the modulating noiae to the bandsridth of the IF filter of the victim receiver, and this distinction
is made because of the two distinct types of outputs of the victim receiver.
Even if the bandwidth of the RF q>ectrum tit the FM/N signal is wider than, or at least as
wide as, the bandwidth ot the IF filter til the victim receiver, if the bandwidth of the moiuUtiȤ
noise is smaller than the bandwidth of the receiver filter, this will result in an RF signal that moves
slowly in and out of the paasband al the receiver filter. Each time it moves into the receiver filter,
it will *^t the filter to ringing” or, in other words, produce a sinusoid at the output of the filter of
limited time duration. If this rignal is that iq>plied to an oivelope detector, the output will be a
pulse of random shape, and duration equivalent to the time-constant of the IF filter (approrimately
the inverse of the bandwidth of the IF filter.) Omr time, we should expect to see a series of these
distinct pulses at the output of the envdope detector. This type of signal may be effective as a type
of deception jamming, but it is not optimal as a n<^ jammer (in the smse of a masking jammn)
and its statistics are non-gaussian.
On the other hand, if the bandwidth of the modulating noiae is sufficiently wider than the
bandwidth of the victim receiver’s IF filter, the RF signal will quickly sweep back and forth through
*lt Aoutd be noted here that this disthictioii is bsseiiow the laUe of the b—dwidtb of the moduUtiag noise to the
receiver bandwidth, while the dietinctioo between NBFM/N and WBFM/N is bseed on the ratio of the bandwidth
of the noise to the fireqaencj deviation of the FM nwdnlator. Thoa, in order to get an entire picture
of the FM/N possibilities, we are forced to cosnpare three separate bandwidths: the bandsridth of the nstiilwli>ing
noiae, the bandwidth of tk RF spectrum of the FM/N sipiai, and the bandwidth of the victim receiver’s IF filter,
and we thus have four distinct categories: NBFM/LFN, NBFM/WBN, WBFM/LFN and WBFM/WBN. Pmctical
Boise jammers faU into the last category, but this thesis will consider, in Chapter 4, an analysis of the first categoay,
(NBFM/LFTf) and produce some intereating results mnreming the thne stetistirs of such a signal.
32
the paMband of the IF filter producing "ovalapping pulaea” at the output of the envelope detector.
The aignal reaulting from thia overlapping ia the reault of the addition of a number of random
pulaea, and the atatiatica of the aignal ia thua the convolution of the atatiatica of the individual
pulaea. (9) Aa thia numb« increaaea, the Central Limit Theorem holda and the atatiatica the
aignal become gauasian, giving piedaely the aame reault aa if the input to the victim receiver had
been DINA rather than FM/N. Thia development leada Boyd to the following equation:
Ir< In < Si («)
where /r ia the receiver bandwidth (we aaaume it to be the bandwidth of the IF filter the
receiver), Sn i* tbe bandwidth tA the baaeband modulating noiae and Si >* bandwidth of the
januning barrage, i.e. the RF bandwidth of the FM/N aignal (6:14-22).
Again, there ia no proof that gauaaianity ia iiiq>ortant in effective noiae jamming, but thia
fact ia aaaumed.
The queation of the whiteneaa A the FM/N jamming q>ectrum ia conaidered peripherally.
No proof ia given to ahow that a white apectram ia aupetior to a non-white apectrum, but a abort
derivation, taken from Middleton, ia uaed to ahow bow gauaatan noiae can be paaaed through a filter
with an error function ahape prior to being uaed to frequency modulate a carrier. Thia reaulting
RF apectrum ia then uniform over a range. Such noiae ia called “erfer” noim becauae (ff the uae
of the erm function (6:14-24). FM/S+N ia alao mentkHied briefly at the end of thia aection aa
another way of whitening the apectrum of a broad FM/N barrage.
The next aet of the p^>era of fundamoital importance to the aubject of noiae quality in
FM/N jamming comes from a aeriea of clnaaified experiments carried out by a team at Stanford
Research Institute in the 1960’s and 1970’a. At least one of the technical rep<»ts from this time
period has now been declassified (21) and an article based cm that work was presented in Electronic
33
W*rf*re/Defeu»e Eleetro»ie$ in 1977 f(30). Thi* work ha* already been touched on in Chapta 2
of thk theatt, ao only the major prwta will be explored here.
Two diatinct typea ot experimenta were carried out by the SRI reaearch team. The firat
uaed actual operational jammera and a aophiaticated aetup that allowed experienced radar aoope
operatora to obaerve the aignala diq>layed on A-acopea, B-acopea and PPI’s and determine the
preaence or abaence of real targeta. Two thinga came out of thia exptfimoit: 1) The jammera which
were moat effective were aeparated from thoae whidi were leaa dfective and 2) It was diacovered
that there waa a very high correlation between gawaaianity aa meaauted by TVirner noiae quality
and the J/S ratio required to produce a 50% probability of target detection (30).
The aecond type <ff experinnenta uaed a aimulated jammer and a aimulated radar receiver
and merely meaaured the ‘nimer noiae quality of the aignala produced by altering the different
parametera of the aimulated aetup. The aimulated jammer utilised a high quality baseband gauaaiaa
noiae aouroe feeding a voltage controlled oacillator with a bandwidth which was held conatant. The
baaeband noiae could be filtered to different bandwidtha, allowing the jammer to operate in both
WBFM/N and NBFM/N categories. Also, it waa possible to add a sinusoid to the modulating noise,
allowing the jammer to operate as aa FM/S+N jammer. The simulated receiver had aa adjustable
IF baadwidth which could be made either broader w nartown thaa the baaebaad modulating ntm,
thus giving rise to either the FM/LFN or the FM/WBN cases, either with or without an added
sinusoid. Concerning the case of FM/N when there is not an added sinusoid, only two conclusions
are drawn about the various bandwidths:
Brf > Bif (7)
and
Bm > Bif (*)
34
where Bm is the bandwidth of the modulating baseband noise (cotte^Kmding to Benninghof’s fs)t
Brf is the the RF bandwidth of the FM/N signal (corresponding to Benninghc^’s fj) and Bip is
the bandwidth of the IF filter of the victim receiver (corresponding to Benninghof’s fa.) If both
these suggestions are followed sad the RF bandwidth is several times greater than the modulating
noise bandwidth (as would be logical), it is easy to see that the remit will be WBFM/WBN. It
should be noted that these requirements are a little mote stringent than those offered by Benninghof,
but still somewhat ambiguous. A third conclusion is offered concerning FM/S-fN: if the sinusoidal
waveform has a period with frequency greater than the IF filter bandwidth, the Central Limit
Theorem can again be invoked and the result will be similar to the case of FM/N with an extended
RF bandwidth.
The case of WBFM/LFN is conaidoed and found wanting because the output of the envelope
detector is a aeries of discrete random pulses rather than a continuous random wave. The pdf of
the WBFM/LFN was found to be characterised by a delta function at sero volts (corresponding to
the ‘‘dead” time between random pulses) and thus the noise quality was low (TNQ < 4). When a
Dicke fix receiver is used with an A-scope, it is found that the WBFM/LFN can be easily screened
out and the real targets are not masked. However, a note is made that on a B-scope or a PPI,
the WBFM/LFN is actually more effective than the WBFM/WBN because it produces a great
multitude of false targets. However, this function might be better described as deception jamming
than noise januning, so this increase in performance does not indicate that the signal was more
power efficient as a noise jamming signal.
It was difficult for the research team to explore the case of NBFM/N because the bandwidth
of the modulating noise was limited to less than the bandwidth of a nwmal noise jantuning barrage,
and there was a desire to keep the bandwidth of the barrage constant; however, there were cases
where the ratio of the RF bandwidth to the modulating bandwidth was as low as 1 .5. If a ratio of 3 is
taken as the arbitrary cut-off between WBFM and NBFM, then this could be ermsidered NBFM/N.
35
It waa noted that aa the IF filter and RF FM/N bandwidtha were held constant, the Turner noiae
quality decreaaed with increaaing baaeband modul^ing noiae bandwidtha, thua demonatratinf that
tranaitioning (rom the WBFM/WBN to the NBFM/WBN ia aometbing that ahould be avoided,
although an explanation for thia behavior ia not cdTeted. The NBFM/LFN waa not apparently
explored at all.
One laat note ahould be made about theae experimenta. Although TWner atatea that the
bandwidth of the baaeband noiae ahould be yreafer than the bandwidth of the IF filter of the
victim receiver for FM/N, (and, indeed, the analyaia leading to the application of the Central Limit
Theorem would aeem to require it) the following figure 3, taken from hia own work, (30) indicatea
that a mote accurate statement would be that the bandwidth of the modulating noiae ahmild be
comparaUe to the bandwidth of the victim IF filta. In fact, it ia aeen that the maximum noiae
quality occurs when the modulating noiae bandwidth ia slightly leas than the bandwidth of the IF
filter (B„ s SMBs, B,f = 6.7MHt).
S.3 Tech Reports, Texts and Articles on EW
Several textbooks mention or allude to the noise produced by FM/N or FM/S+N jamming
although the earlier references tend to be more obscure. Introduction to Radar Spstems written by
Skolnik in 1962 speaks briefly of *impuliaive that can sbock-excite the narrow-band radar re¬
ceiver and cause it to ring,” and he suggests the counter-countermeasure of the Dicke fix. Although
he does not use the term “FM/N” there can be no doubt that this is what he has in mind, since
the Dicke fix is not terribly effective against any other noise jamming scheme.
In 1983, the scene has changed somewhat, and Golden not only describes the function of the
FM/S-f-N noiae jammer in Radar Electronic Warfare, but presents block diagrams and suggests
parts to build your own noise jammer (11). He presents two ideas which are of interest here. The
first is a brief explanation of why the FM noise jammer is preferable to DINA. Although DINA
36
Amplitude
Control
FM-by-Slnusold-plus-Noise — FM/S+N
is much easier to analyse in its effects on the victim receiver, and its efficacy is leas dependent
on the relative paraiiKters of the jammer and the victim receiver, the RF gaussian noise signal
makes much leas efficient use of the high powered microwave transmitter than does the FM signal.
The amplitude of an RF gaussian noise signal is most often close to sero, but can theoretically be
infinitely large, thus the microwave transmitter must operate at a fraction of its maximum power
capacity most of the time or else clip the gaussian noise drastically, causing it to be non-gauasian.
The FM jamming signal, on the other hand is at a more nearly roughly constant amplitude most of
the time which allows it to take full advantage of the power capacities of the jammer transmitter.
The second idea is presented graphically in the figure reproduced here as Figure 4. Since the
baseband signal FM modulates the jammer’s carrier, it is valid to think of the top signal as being
either the amplitude of the baseband signal with time or the frequency of the jammer signal with
time. What is clear from the figure is that if the bandwidth of the RF jamming signal is much
much greater than the bandwidth of the filter of the victim receiver, then the modulating noise will
have to fluctuate more rapidly to produce constant ringing in the output of the victim receiver.
In 1985, Knorr and Dimitrios publish a paper describing work that exactly paralleled the
work performed by the SRI research team, with the important exception that this work was done
through computer simulation rather than laboratory simulation (14). The same tendencies were
noted concerning FM/WBN, FM/LFN and FM/S-fN. It is discovered that if the RF spectrum
of the barrage is not centered on the victim receiver (so that the receiver picks up mie of the
“tails” of the barrage) the noise quality is reduced. In addition to sinusoids, periodic triangular
and sawtooth waves are added to baseband noise and the results are found to be virtually identic^
to the FM/S+N case. Also, Knorr and Dimitrios offer their opinion that a noise quality of 10 or
greater represents good noise quality.
In 1979, Cassara, Muth and Getty’s publish a paper which proposes to apply the error function
to gaussian noise prior to using it to frequency modulate a carrier in order to get a uniform RF
38
RAUAK
VIDtO
M AM AM A/AA
riMt
Figure 4. FM/S+N Effect on Radar Output
spectrum (7). Thomas Weil responds later in the same year explaining that the same idea had
already been proposed by Middleton in 1955 (as had been noted by Benninghof) but was not
subsequently employed because, essentially, the statistical characteristics of the RF signal were
such that the output of the IF filter of the victim receiver was less gaussian (31). When one is
forced to pick between gaussianity and whiteness in such a situation, it is found that while whiteness
is more efficient at putting power into the passband of the victim receiver, the implication seems
to be that the power that enters the receiver is not as efficient at masking the real targets.
Three other radar texts were investigated (19) (15) (10), but in each of these, all noise jamming
was considered to be more or less equivalent. The J/S ratio was substituted into the radar range
equations to demonstrate the effect of januning, but the assumption in all cases was that the noise
power entering the radar receiver was white and gaussian, while such is not the case, and, in fact
noise jamming which deviates substantially from gaussianity has been shown to be far less effective
at jamming than the amount of power delivered would indicate. From the standpoint of the radar
39
designer, this assumption will only lead to reasonable conservativeness, but there is a possibility
that the radar jammer designer would overestimate the effectiveness of his jammer on the basis of
these sorts of equations.
The most recent work of importance on the subject of FM/N is An Analftical and Experi¬
mental Inveaiigaiion of FM-By-Noiae Jamming written in 1992 (8). The primary emphasis of the
research reported there was to thoroughly describe and demonstrate the FM/N effect. The rela¬
tionships between the modulating noise bandwidth, the RF bandwidth of the FM/N signal, and the
IF bandwidth of the victim receiver are analyzed more thoroughly than was done by Benninghof
or the Stanford research team, and a new category: FM by unity bandwidth noise (FM/UBN)
in addition to the categories of FM/LFN and FM/WBN, is suggested. The term refers to the
relationship between the modulating noise bandwidth and the victim receiver bandwidth and it
suggests that there may be some advantage to keeping these bandwidths reasonably close. This is
a slight departure from the suggestion by Benninghof that the modulating noise bandwidth should
be greater than the IF filter bandwidth, but it agrees with the experimental results produced by
the SRI team.
Also a ratio called the sweep to victim ratio (SWR) is developed which quantifies the notion
of a sweep rate: how often a noise signal is being swept across the frequencies of the victim IF filter.
A criteria for determining the difference between a fast swept jammer and a slow swept jammer is
then offered. This quantification is directly applicable only to an FM modulation scheme (it can
be applied to FM/S-fN as well with some modification) but it is, ^parently, entirely new in the
field of noise jamming or masking jamming.
The secondary emphasis of the research was the re-establishment of a technique for measuring
noise quality on an operational januner. Although this goal was not entirely achieved because of
the limitations imposed by using only commercially available oscilloscopes and spectrum analyzers
together with a fairly slow personal computer, it was demonstrated that 'Himer Noise quality could
40
be measured on a physical laboratory simulation. An important reconunendation that came out
of this secondary emphasis was the idea that a new standard of noise quality should be proposed
that would measure the whiteness of the spectrum as well as the gaussianity of the univariate
probability density. Two new measures were proposed which are mentioned in Chapter 2 of this
thesis, examined theoretically in Chapter 4, and one of them was treated experimentally in Chapter
5.
The research reported in An Analj/tical and Experimental Inveatigation of FM-Bg~ Noise Jam-^
ming (8) included two parts: 1) a theoretical investigation of FM/N which included some math¬
ematical analysis and 2) the laboratory simulation which has been described earlier in Chapter
2. Other than the development of the SWR and the more rigorous analysis of the relationship
between the three bandwidths which are fundamental to describing the FM/N behavior, there are
no startling results in the mathematical analysis. When the experimental setup is considered, it is
found to be similar in many respects to the setup used by the SRI team in that it produces noise of
varying degrees of gaussianity at the output of the IF filter of a simulated victim receiver. It does
not have an FM/S-I-N capability and it used only two different IF bandwidths. It duplicated the
SRI result, showing an increase in gaussianity as the bandwidth of the baseband noise is increased
from the WBFM/LFN to the WBFM/WBN case, but did not measure noise quality in either of the
NBFM/N cases. In some respects it did less than the Stanford setup did, which is likely due to the
amount of time and the cost and availability of equipment; however, what it accomplished which
the Stanford setup failed to accomplish, was to produce a series of oscilloscope traces of baseband
noise signals and corresponding IF filter outputs, which greatly facilitate an understanding of the
FM/N behavior.
41
S.4 Nummary
Although the articles, EW texts, technical reports and theses covered in this chapter have
covered a lengthy span of tinM and represent a fair number of authors, it seems that with reqtect
to the emphasis of this present them, two important concepts are amsistently repeated. Firstly,
it is always either assumed or stated that optimal noise for the purposes of noise jamming in tlw
presence of a conventional receiver should have both a flat spectrum in the passband the receiver
being jammed and a gaussian first order probability density function as demanded by Shannon’s
work. Nonetheless, in the noise quality measure proposed by TWner, theoretically verified and
apparently universally accepted by the EW community, the emphasis in quantitatively measuring
noise quality is almost always focused on the gaussianity of the pdf rather than the flatness of the
spectrum.
Secondly in the articles that deal with the shape of the FM/N spectra (Middleton, Stewart,
Blachman, Abrahamson, Mullen, Turner) it is shown that the spectrum of the FM jamming signal
generally conforms to Woodward’s Theorem, provided that the peak frequency deviation of the
modulator is sufficiently large, but becomes increanngly characterised as having a q;>ike at the
carrier frequency as the peak frequency deviation is decreased. Since Woodward’s Theorem states
that the shape of the RF spectra of a carrier frequency modulated by a baseband signal will take
on the shape of the first order probability dennty of the baseband signal, under the situation of
FM/N where the modulating noise is gaussian, this implies that under the best of circumstances,
the FM/N spectnun will have a gaussian shape. This has been repeatedly proven and demonstrated
by researchers in the area. Thus the FM/N spectra will never be perfectly flat over any bandwidth,
and may in fact deviate quite a bit from ideal flatness.
That this realization caused some concern among researchers is indicated by the articles which
deal with the investigation into erfer noise, a technique specifically designed to increase the flatness
of the RF spectrum of an FM/N signal by changing characteristics of the baseband modulating
42
noise. The consensus seems to be that the increase in flatness obtained by erfing was not worth
the loss of gaussianity which accompanied it; however, it is undeniable that increasing the flatness
of the FM/N spectrum was seen as a desirable goal which was not perfectly attainable through the
most common simple FM/N modulation scheme.
These two facts: the desirability of a noise signal in the receiver with a flat spectrum as
well as a gaussian pdf, but a lack of quantitative measurement of the flatness of the received noise,
coupled with the theoretical impossibility of perfectly flat noise when using a simple FM/N jamming
scheme, seem to point to a need for defining a new standard of noise quality that quantitatively
measures whiteness as well as gaussianity. Two such measures were proposed by Daly (8). These
measures are considered theoretically in the next chapter, as well as a third measure proposed here
for the first time, and all three measures are examined experimentally in the last three chapters.
43
IV. Theory of Noise Quality in FM/N
The problem of quantifying the properties of a jamming signal must deal with three compo¬
nents: 1) the analysis which shows a noise jamming signal of certain statistical characteristics to be
ideal, 2) the analysis which shows what statistic^ characteristics a particular wav^orm generated
in a particular way is likely to have and 3) the analysis which demonstrates how measurements ot a
particular jamming waveform may be used to estimate its statistical characteristics and reasonably
compare them to the statistics of the ideal. In the particular case being discussed here, FM noise
jamming, these three components are 1) demonstrating that ideal masking noise is gaussian in
univariate probability density and iq>pears white to the input of the victim receiver, 2) analysing
the theoretical whiteness and gaussianity of the four FM/N cases, and 3) showing the statistical
validity of a proposed method for measuring whiteness and gaussianity.
4.1 Ideal Noise
There are two signal characteristics that are ot primary importance for the purpose of de¬
termining the optimal noise jamming signal. The first is the “entropy” the random process
which characterises the noise, and the second is the autocorrelatkm function of the random pro¬
cess. Entropy, in this context, is the concept introduced by Shannon, which is, precisely, a measure
of the unpredictability of the random variables which compose the signal. A jamming ngnal with
maximum entropy in a particular communication channel is optimally destructive of information
in that channel (25) Entropy is defined as:
= - [ p(x)\np{x)dx (9)
sF— OO
where X is a continuous random variable and p(x) is the probability density functicm of X.
Now, if the radar jamming ngnal is characterised by the random process X(t), composed of
random variables X, then the autocorrelation functimi of the randmn process shows how the value
44
of the random signal at cme instant in time is related to the amplitude at any other instant in time
and is defined as:
R..(ti.h) = EX(ti)X(t3) (10)
when X(t) is a real random process (23:122). If the value of the jamming signal at any given instant
of time is completely unpredictable baaed on the knowledge of the signal at all other inrtances of
time, this, again, is most destructive of information in any signal it is added to, and is likewise
difficult to counter.
In order to avoid any confusion, it must be reiterated here that “destroying” informatkni
in a communication channel, in the sense of information destruction developed by Shamum, is
not necessarily the best jamming technique. Often, deceptive jamming techniques which ate less
destructive of the desired signal than noise jamming would be are more effective at producing a
specific desired jamming result. However, in terms of information destruction in a givoi channel,
the signal with the highest entropy and the lowest correlation is optimal.
In the analysis that follows, the optimal autocorrelation function will be found first, and then
that information will be used as a constraint on the maximising oi the entropy of the univariate
pdf of the jamming signal.
4.1.1 Ideal apeciral ckaracieruiica. First, it is important to note that X(i) should be
taken to be stationary. This simply means that the pdf’s of the random variables composing X(t)
are time invariant. As ff(x) is independent of any time variable, it obvious that the pdf found by
maximising ff under any constraints will be found to be independent of time. Thus, f<» optimal
jamming, we should use the signal with the maximum entropy tor all time. Stationarity implies:
£{X’(f)} = itg^ a constant (11)
46
and
P(-^(0) = viX(t + T),r€ (-00,00)
(12)
This then implies that
= r{X(tt)Jf(«,)} = £{,^(11 + T)X{ti + r)} (13)
which implies that Rgg depends only on the difference between and <3. If we let r = jt] — ti|
then we can write:
Rmai^tt^i) ~ Rmm('^) (14)
If we now apply the criterion that X(<) should be completely unpredictable based on knowl¬
edge of any other values of the jamming signal at any other times then we are implying that A’(f 1)
is statistically independent of Ar(t3) for all ft ^ € (—00,00) which implies:
^{X(<i)A^(<2)} = 0,l| ^ 6 (-00,00) (15)
which implies:
ff„(r)s:0,r#0 (16)
This leaves us with two posrible types ci Mitocorrelation functions. The first is the function
which is identically sero, but this will hardly sme as an optimal jamming signal, as it ctmtains no
power. The second is:
Jl„ = ^S(r) (17)
where w’ is a cmistant, which is the autocorrelation function of the <^timal jamming signal and
will be denoted by Rig(r).
46
The Fourier tranafonn of A,, ia known aa the power apectral denaity function of X{t) when
A’(t) ia atationary and ia given by:
S„(f) = r
*/— OO
which ia known aa the Wiener-Khinchine rdation. For optimal jamming:
(18)
S:.(f) = r iT’i(r)e->»'/^4r (19)
*/ — 00
5:.(/) = <r» (20)
which impliea that our optimal jamming aignal haa equal power in all fiequenciea. Thk kind of
signal ia known as a white aignal.
The average power in a stationary aignal is given by:
f?{.v»(t)} = a:.(o) * r s:mw (21)
sF-OO
= (22)
which, unfortunately, is unbounded, implying that we should need to generate a signal with infinite
average power. However, it will be later shown to be sufficient if the jamming signal merely
appears white to the input of the IF filter of the victim receiver, a much less stringent ccmditicm.
This condition of white “appearance” is not necessarily perfectly achievable, but at least it is not
physically impoaaible.
Also note that for a white process, A{A(f)} = ^ = 0. This can be easily seen by observing
that, for a stationary random signal, the mean of the signal coneaponda to the dc, or sero frequency,
power in the signal. The power over any range of frequenciea of a statkmary random signal may
47
be found merely by integrating the power epectral density over that range. Thus, to find the de
power a white signal, we take:
= /ttn«_o J
s:.
(23)
which goes to sero because of the absence oi any delta function at 5]^«(0). This makes sense,
intuitivdy. If the mean of a signal is aol sero, then a constant (i.e. predictable) nonsero amount
of its power must be located at dc and it therdbre esaaot be (q>timally unpredictable. In a white
process, no energy is located at any discrete frequency.
4-1-i IdaU probabUH]! ietuUp function. Now connder the entropy condition. In ordtt
to find the signal with the maximum entropy for a ^ven signal energy, we must maximise H(*)
subject to the following constraints:
p(*) > 0, * € (-00, 00) (24)
r P(*}dx = 1 (23)
y— 00
/I. = 0 (26)
f x*p(x)dx = oi (27)
y-00
The ciHistraints follow naturally from the fact that p{x) is a pdf and therefore cannot be
negative, must integrate to one and must have first and second roorooits. That the first momoit
is sero is a direct result frtxn our contention that the optimal jamming signal be white.
48
At this point we will introduce two Lagrange multipliers to redura the problem from one
of finding a ccmditional maximum on H{x) to one of finding an unconditional max on H{p(x)) -
Adi(l>(')) P^OK')) where di and da ve the constraining functions arising from the fourth and
second conditions. Thus we are to maximise / where:
J=f p(x)\np(x)dx - X f x^p(x)dx + ti n p{x)dx ' (28)
•/.QO y»00 J-~oo
= / p(*)(-lnp(*)- Ax’ + /i]d* (29)
Taking the partial derivative with respect to p and setting it equal to lero then yields;
-lnp(*)-^-A*» + ,i = 0
(30)
which implie
p(x) = e***+'-‘ =
(31)
substituting this expression back into the condition that p{x) must integrate to unity, we
have:
e"-^ / e-***d» = 1
which implies:
‘■‘ = 1/;
(32)
(33)
and then applying the constraint that oi be the variance of p(x), we have that
As
2ol
(34)
49
Thus yielding:
which is the familiar gaussian distribution.
The conclusion then, is that optimal noise, from the standpoint destroying information in
a channel, is characterized by hoik a white spectrum and a univariate probability density that is
gaussian.
4.g TkeoiyofFM/N
4-t-l Tkeorfi of FM. The concept behind frequency modulation of a carrier is intuitivdy
simple. The idea is that the instantaneous frequency deviation of the carrier from its nominal
frequency should be directly pn^rtional to the an^>litude the modulating baseband signal.
Mathematically, if the nominal frequency of the carrier is /«, and the instantaneous frequency of
the FM rignal is /<, then the instantaneous frequency deviation is:
A/(<) = /«-/<(0 = /<m(0 (36)
which implies
+ (37)
where is some proportionality constant which will be referred to as the frequenq/ ievioiion
constant with units of Hz per volt, and m(t) is the baseband modulating signal.
In general, the expression for any angle-modulated caniet can be written as:
»(t) = i4cos(di(0) (38)
SO
where A is the amplitude of the signal and 4i(i) i< the instantaneous phase, usually at the form:
^(0 = 2»/.l + tf(0
(39)
If we then use the fact that the instantaneous phase of an angle-modulated carrier signal is equal
to the integral of the instantaneous frequency we can write:
^i(t) = 2x /
Mt)dt
(40)
which implies for the FM case:
vruif) - Aca^2t j U(t)dt) (41)
y.oo
and then, substituting,
VFM(i) * A oos(2x(/et + f 4 f m(t)dt)). (42)
which is the commonly found expression for FM.
Some terms associated with FM need to be defined. Now the theoretical q>ectrum of an
ideal FM signal contains en«gy at frequencies which are potentially infinitely removed frmn the
carrier frequency as the result of the fluctuating instantaneous frequency. Thus the widtii of the
theoretical spectrum of the FM signal depends to some degree on the bandwidth of the modulating
signal. If the bandwidth of the modulating signal increases, then fluctuations in the amplitude of
the baseband signal, and hence, in the instantaneous frequency of the FM signal will increase. The
increase in the rate of the fluctuations of the instanataneous frequency of the FM signal will result
in a wider FM spectrum.
However, if the amplitude of the modulating signal is limited, this implies that tlM tnsfsats-
neons frequency deviation of the carrier from its nominal value, /« will be limited. Thus there will
51
be aoine maximum instantaueous frequeocy deviation attainable. This value will be consistently
referred to as the peak frequency deviation and denoted A/^. In the case of a random modulating
signal which is not necessarily bounded (and in the gaussiao case is theoretically not bounded) it
is sometimes more practical to speak of the rms frequency deviation denoted as: A/naa-
Even in the case where the random modulating signal m{t) is unbounded, it is always assumed
that it has a bandlimited spectrum which extends from around 0 Hz (not necessarily including any
power at dc) up to some maximum frequency /m- This implies that there is a bandwidth associated
with m{t) which we will denote as Bm sod it will generally be assumed through the sequel that
Bm = /m unless some other definition of bandwith is specifically referred to.
Typically the relationship between the spectral behavior of a baseband signal and the spectral
behavior of the corresponding FM signal is spoken of in the FM literature in terms of the modsfstton
constant, 0, which is defined only for the case where a carrier is frequency modulated by a constant
amplitude sinusoid. However, in the case of FM/N there is a greater direct application in using
the terms developed above to introduce the concept of a itviution ratio D to compare the spectral
behavior of the baseband noise to the spectral behavior of the FM signal when the modulating
signal is arbitrary. The deviation ratio D is defined as:
„ peak fluency deviation A/,
- - - ftf <"•
4.2.2 Sptetrum of FM/N. The preceding section laid the groundwork for the discussion
of frequency modulation in the general case. When our attention turns to the spectrum of the FM
signal, however, there are no general closed-form expressions that cover all situations, so the focus
will be restricted primarily to the FM/N case from this point forward.
As has been noted several times in Chapter 3, the most conunon descriptions of the FM/N
spectrum have employed Woodward’s theorem in the WBFM/N case to approximate the RF spec-
52
trum u being gauasian, and have generally been complicated in any other case. A proof of Wood¬
ward’s theorem can be found in Daly’s thesis and it is applied to the WBFM case (8). That
work will not be repeated here. Instead the approach developed by Abraharoson which covers
the WBFM/N case, but which, more importantly, yields very nice results over the range of values
where Woodward’s theorem does not hold, which may or may not fall within a strict ddinition of
NBFM/N will be considered here (1).
Rather than speaking in terms of the relationship between the size of the modulating band¬
width to the size of the FM/N bandwidth, as has been traditional, Abrahamson chooses to consider
the rms value of the amplitude of the modulating signal (1). In order to appreciate the importance
of Abrahamson’s approach to the particular case of FM/N, it will be useful to contrast it with the
traditional approach to the concept of bandwidth and spectra in FM.
Closed-form expressions for the FM spectrum cannot be found in general, but they can be
found for particular special cases. A ubiquitous example, which will not be derived here, but
which has done much to color thought on the notion of bandwHths and RF spectra, is that of a
sinusoidal baseband modulating signal. When m{t) is chosen to be a sinusoid, this gives rise to
Bessel functions in the Fourier transform of the FM signal and thus a spectrum of delta functions
(sometimes referred to as “sub-carriers”) located at /< ± n/m, where fm is the frequency of the
modulating sinusoid and n is a positive integer. If we restrict ourselves to looking only at the
spectral lines which carry 90% or more of the power in the FM spectrum, we will find that the
number of spectral lines which meet that criterion is directly related to average magnitude of the
expression:
Acoe2irfmidt
(44)
In other words, the number of spectral lines which contain a significant amount of power can be
said to depend on the frequency deviation constant, sad on the amplitude of the modulating signal.
53
The traditional approach is to focus on the frequency deviation constant. If is very small,
then we find that we have power in the carrier frequency, /«, and in the first two spectral lines
located at /« + fm and fc — fm- Thus the bandwidth of the RF spectrum is roughly twice as wide as
the bandwidth of the modulating signal, and, for low values of fd, the RF bandwidth will increase
with increasing baseband bandwidth as is held constant. This condition is found to have some
things in common with the large carrier AM spectrum (specifically, the size of the bandwidth and
the fact that a large proportion of the energy of the signal is found at the carrier frequency) and is
known as narrowband FM (NBFM).
However, as fd increases, the FM signal deviates further in frequency from /« and thus power
is is no longer located at the carrier frequency but is pushed into additional spectral lines at the
subcarrier frequencies. Thus for large fd, the peak frequency deviation of the carrier may be many
times the maximum frequency component found in the modulating signal m(f), and thus the RF
bandwidth will be found to depend more on fd than on Bm. When this condition holds, this is
known as wideband FM (WBFM) because the FM bandwidth is fairly wide in comparison to the
baseband bandwidth.
Carson’s rule, which is based on these kinds of general observations rather than specific
theoretical considerations suggests that the bandwidth of an FM signal is;
BFM=2iD + l)B,n (45)
For large D, this expression becomes roughly 2DBm or 2^/p (the approximation for WBFM). For
D less than 1, this expression becomes roughly 2Bm (the NBFM approximation). It is obvious
that in order for Bpu to be roughly equal to 2Bm, D must be much less than 1. Which is on the
same order as the bandwidth of the RF spectrum of an AM signal. As D is much much less than
one, the FM signal will have many of the same characteristics as a large carrier AM signal in the
time-domain as well as in the frequency-domain, and in the FM literature, a system referred to as
54
an NBFM system is a system characterised by a deviation ration much less than one. However,
it is only necessary that D be on the order of 1 in order for the AM like spectral behavior to be
present. And in the case of analyzing FM/N this spectral behavior which is significantly different
from the WBFM/N spectral behavior is of the greatest significance. Thus, an FM/N system with
a D as large as 2 will still be referred to as a NBFM/N system in this thesis.
In addition to Carson’s rule, traditional FM analysis also refers to a null-to-null bandwidth,
which may occur if there are distinct nulb in the RF spectrum, a 3 dB bandwidth (measured
between the points where the magnitude of the spectrum falls 3dB below the peak magnitude) and
a power bandwidth, based on the criterion that a certain high percentage of power be contained
within the bandwidth.
For the purposes of an analysis of a sinusoid modulating signal, (such as might be used
to transmit a baseband OOK signal at RF), or for the case where not much is known about
the modulating signal other than its average power and its bandwidth, this preceding traditional
analysis with the two categories of WBFM and NBFM is sufficient. However, when we know the
precise spectral characteristics and the probability density function of the modulating noise we can
find a much more precise characterization of the FM spectrum, particularly in the area which is
neither precisely NBFM nor WBFM. Abrahamson does not develop a closed fwm expression in
this intermediate range, but he does develop an analytical technique that allows the estimation of
the FM/N spectrum to an arbitrary degree of accuracy mth surprisingly little computation.
Going back to the example of the sinusoidal modulating signal, it is easy to see that if we talk
about the amplitude of the modulating signal and assume a fixed frequency deviation constant, for
some small amplitudes there will be only three spectral lines in the FM spectra {Brf = 2Bm)-
However, as the amplitude increases, the frequency of the FM signal will deviate further and further
from /e, yielding a wider spectrum.
55
If we depart from the sinusoidal modulation case and consider the case where the baseband
modulation is some bandlimited white gaussian process, then we have a roughly analogous situation.
The gaussian process can take on say value from — oo to oo; however, based on its variance or, to
put it another way, based on the power in the process, there is a mean square value which gives an
indication, on the average, of what its amplitude will be limited to. Thus for very low rms values
of the modulating signal, the bandwidth of the FM spectrum will be roughly the same size as the
the baseband bandwidth shifted in frequency {Brf = 2Bm). However, as the rms value of the
modulation signal increases, the FM signal will deviate further and further from /«, on the average,
until, at some point, the bandwidth of the FM spectrum will be independent of Bm-
This fact leads Abrahamson to define an rms bandwidth, which has particular significance
when speaking of random signals:
BrUrm,
where Pm is the rms amplitude of the modulating ngnal m(t). Abrabamson’s comment on this
result is significant: (1:408)
Note that in the FM case, the rms bandwidth of the modulated wave does not
depend upon the bandwidth of the modulating wave, but only upon its rms value.
It will be helpful at this point to introduce the rest of Abrahamson’s notation. As noted
above Pm is used to denote the rms value of the modulating signd. In general, Abrahamson uses
P^ is used to denote the mean square value of a random process, i.e:
p^ = m=r (47)
s/— oo
where i2(r) is the autocorrelation and 5(/) the spectrum of the random process, and the usual
Fourier transform relationship links them (1):
S(f) = r R{r)e-i^^dT
s/— OO
(48)
56
R(t)= r
J^OO
where
u = 2r/.
R and S may be normalised to yield p(r) and w(r):
(<r) =
Rjr)
<S) =
S(f)
It follows directly that p(r) and <r(/) also f<»m a Fourier tranaftmn pair. Also note that the
spectral density, ff{f) is a non-negative function of unit area, and thus has the properties required
of a probability density function.
Furthermore, if p(r) is the normalised autocorrdation of some bandlimited process centered
around a frequency /«, then it makes sense to talk about the baseband autocorrelation po{t) where:
P(»’) = /»o(r)cos2»/e
and it is well known that the principle of heterodyning insures the bandlimited spectrum corre¬
sponding to p{t) is merely the spectrum of the baseband process shifted to center around ±fe:
Now, to solve for the statistical form of the spectrum, we will find the Fourier transform of the
autocorrelation function of the FM signal. If we define the FM signd to be vpitit), befme, and let
57
*{t) reprcMiit the angle modulation cauaed by our measage aignal m(t) (i.e. r(() = 2irf4 f* m(t)dt)
then the autocorrelation function, iZ«(r) of the FM aignal haa been found to be: (17)
Rv (r) = coa 2ir/eT (54)
and there ia a Fourier transform relationship between A«(r) and the spectrum of vj^iu (t), 5«(/).
If we now replace iZ«(r) by the normalised autocorrelation function p«(r) and let i4 = 1 we
have;
p,(r) = e“^^*<“^*e~^^*^’')coe2»/er
-* P,(t) = e“^ cos 2ir/eT (55)
The baseband autocorrelation function associated with the FM signal is then (by eq 52):
p,o(r) = e-'2e-*^MO. (56)
We can now make use of the series expansion for the exponential to obtain:
p,o(r) = e-^* [l + Pjp,(r) + ^p2(r) + ^pj(r) + - ■ ] (57)
And, if is finite, we may tranform eqn 57 term by term to obtain the baseband normalised
spectrum <r«o(/)- Note that the transform of p’(r’) is merely the convolution of «’(/) with itself,
and similarly, for p^(r), we have <r(/) * ff{f) indicating the double convolution of <r s tr s w:
<r.o(/) = [«(/) + P^<r.if) + * «r,(/) + I <r.if) + • • ] (58)
An expansion similar to this was found in Middleton’s work (cited in Chapter 2) so Abra-
hamson refers to eqn 58 as the Middleton expansion; however, Middleton did not express it in a
58
form that could be easily used to calculate the FM/N spectrum. If we return to the case where
our modulating signal m(t) is a bandlimited white gauasian process, with maximum frequency fm,
then it is readily apparent that <rg(f) is a rect function with unit area in the frequency domain,
extending from — fm to fm. The first convolution of ffg{f) with itself will give rise to a tri function,
also with unit area, extending from —2/m to 2/m- And the double convolution and all higher
convolutions will give rise to functions that are very nearly gaussian for all practical purposes, with
increasing variances.
Thus, the FM/N spectrum wilj look (excluding the small amount of power present in a q>ectral
line at /«) like a constant, multiplied by a rect of bandwidth 2Bm, plus a tri of bandwidth 4Bm,
plus a number of bell shaped curves of bandwidths 6Bm,6Bm, ■ ■ - When Pg is much less than 1,
its higher powers (P^) will be very small and the rect function will dominate, giving us an FM/N
bandwidth of roughly 2Bm. As Pg increases to 1, the tri function will become larger, giving us a
pointed, but still narrow, spectriim. As Pg becomes much larger than one, the gaussian-like terms
will dominate giving the overall appearance of a gausnan spectrum, actually becoming gaussian in
the limit.
It is important to remember that is a measure of both the frequency deviation and the
power present in the modulating signal, thus it can be increased either by amplifying the baseband
modulating signal or by increasing fd-
The parameter which could most easily be controlled in the experiments demonstrating the
various types of FM/N was the peak fluency deviation, therefore it was desirable to develop an
expression relating Pg to A/p. This has no theoretical significance, but is useful for relating the
shape of the spectrum shown on the frequency analyser to the shape the spectrum predicted by
Abrahamson’s analysis.
59
Stewart has shown that for wideband FM:
Bwbfu = A/ri»j(8ln2)^
m
which, f<» values of P, > 1 is equivalent to the remit oi 46. I.e.:
Bwbfu =
(60)
Now it is known that for a gausnan process, less than 99% of the samples of the process will
have amplitudes exceeding three times the standard deviation. Therefore, if the rms value ot the
modulating signal causes a fluency deviation Afrmt then the peak frequency deviation is likdy
to be no greater than
(61)
which inqiUes:
3
(62)
Additionally, we can find an expressi<m for H in t«ms of Bmi
D =
2ir>/8ln2^
(63)
4.t-S BehBvior of FM/N Jmmming. At this p<wt, the discussion turns to the effects
receiving an FM/N signal, specifically to what the output of the IF filter of the victim receiver will
look like. Having obtained a spectrum for the FM/N signal, for both NBFM/N and WBFM/N, it
is ea«y to find the spectrum for the output the IF filter of the victim receiver if we know the IF
frequency and the transfer function the filter. Since we know that the RF FM/N q>ectrum will
60
be a bandpaae procew, we can write it as:
SpM(f) = 5[5*(/ - /.) + 5»(/ + /.)) (64)
where 5)(/) is the unnormalised baseband vetsiaa of the q>ectrum as found in the previous section
(it is merely crg(f) scaled by a OHutant.) Similarly the transfer function of an IF bandpass filter
centered at some frequency ftp can be written as
JTipif) = [Htif - ft) + Htif + ft)] (65)
If we assume that the FM spectrum is centered on the receiving band ot the victim receiver, and
that the victim recover onpkqrs the principle of heterodyning to bring the rignal at /« down to ft,
then it is obvious that the spectrum at the output of the IF filter can be written as:
SifU) * \liItiS,)(f - ft) + (H»5,)(/ + ft)] (66)
Since we have shown in section 4.1 earlier in this diopter that ideal noise has a white spectrum,
it is obvious that if the input to the victim receiver is ideal noise, then the output of the IF filter
of the receiver will have a spectrum that precisely matches the transfer function of the IF filto.
Furthermore, if the input to the victim receiver is isadfimtfed white noise centered on the victim
receiver, with a bandwidth wider than the bandwidth of the IF filter, the output of the filter will
be precisely the same. Thus it is shown that ideal n(^ wilk rupect to s psrtica/sr vtdim receiver
need not be absolutely white, but mody white in the passband of the victim receiver.
Uring the terminology introduced in Chapta 3, it is posrible to examine four different possible
FM/N jamming schemes in terms of the required whiteness. If a WBFM/N jamming scheme is
used, then the shape of the RF spectrum will be gaussian. If a relatively small central pwtion of
61
thia •pectrum is intercepted by the IF filter of the victim receiver, then the output of the IF filter
will roughly ^proximate the shape of the IF filter, giving a result similar to that oi a purdy white
noise input. If it is also true that the bandwidth of the modulating noise is as wide or wider than
the bandwidth of the victim receiver, then this condition is known as WBFM/WBN, and it is nearly
optimal from the standpoint of the spectral analysis. If the bandwidth of the noise is mot quite
as wide as the bandwidth of the IF filter (for example, Bm = 100 kHs, D = 10 implies B/tp = 1
MHz, but Bif = 200 kHz) then the radar is operating as a WBFM/LFN jammer. From the
standpoint of analysis of the magnitude of the spectrum alone, this situation is exactly equivalent
to WBFM/WBN.
If, on the other hand, the RF spectrum is narrower than the bandwidth of the IF filter, then
the output of the IF filter will contain a bell-ahaped hump at the center frequency, but will be
largely untouched at higher and lower frequencies. It can be reasonably deduced that this is the
WBFM/LFN condition because we know that in WBFM the RF bandwidth of the signal is much
wider than the bandwidth of the modulating nc^, thus, if the RF bandwidth of the WBFM/N
signal is narrower than the bauidwidth of the victim recaver, then it f(Jlows directly that the
bandwidth of the modulating noise is much smaller than the bandwidth of the victim receiver.
This situation is highly undesirable from the standpoint of masking jamming, because it allows the
operator of the victim receiver unrestricted use (from a theoretical standpoint) of those higher and
lower frequencies.
It is this analysis, baaed on the assumption of WBFM, which leads us to the recommendation
made by IWner and others that in FM/N jamming, the RF bandwidth cf the FM/N signal should
be much much larger than the IF bandwidth of the victim receiver.
If a NBFM/N jamming scheme is assumed, we have two other possibilities, neither of which
are very desirable. If, again, the RF spectnun of the jamming signal substantially wider than
the bandwidth of the victim receiver, then only a small central portion of the RF spectrum will
62
be intercepted, and the output of the IF filter will be the product of the RF spectrum and the
transfer function of the filter. This condition is most likely NBFM/WBN because we know that the
RF bandwidth of the NBFM signal is very well approxiinated by 2Bm- Thus, if a NBFM system
produces a signal substantially wider than the baseband bandwidth 61 the victim receiver, unless the
three bandwidths are very closely matched, it is likely that the modulating noise has a bandwidth
as wide or wider than the bandwidth of the victim receiver. The important characteristics oS the
output signal are that in ail NBFM cases, more power is concentrated at the center fluency of
the spectrum than at the edges by comparison to the WBFM case. This is not necessarily easily
countered by signal processing in the victim receiver, but it is, theoretically, leas than optimum. This
situation, on the basis of spectral analysis alone, would seem to be better than the WBFM/LFN
when the RF bandwidth is narrower than the bandwidth of the victim receiver because it at least
puts $ome noiM power in all frequencies used by the victim receiver.
If NBFM/N jamming is used and the RF bandwidth is substantially smaller than the band*
width of the victim receiver, it can easily be seen that this condition must be NBFM/LFN. If
we agnin approximate the NBFM/N RF bandwidth as 2Bm, then a victim receiver IF bandwidth
greater than the RF bandwidth directly implies an IF bandwidth at least twice as wide as the
bandwidth of the modulating noise. Ftem the standpoint of spectral analysis, this situation suffers
from being less than ideal in the same sense as does the WBFM/LFN case mentioned above.
However, as has been constantly reiterated in this thesis, the spectrum of the noise (in the
sense of the magnitude of the spectrum) is only half the equation. The other half is the probability
density of the output of the IF filter of the victim receiver. Determining the theoretical probability
density is far more complicated than determining the theoretical spectrum of the FM/N signal
either at RF or after being passed through a filter. Nonetheless, a few generrd observations can be
made.
63
Referring to figure 5 we see the baseband modulating voltage passing through a voltage range.
As it does so, the FM/N signal passes through a corresponding range of frequencies. A range ot
these frequencies correspond to the frequencies in the pass band oi the IF filter of the victim
receiver. As the modulating noise enters these frequencies, a narrowband response is generated
at the output of the IF filter which has some characteristics of linear FM in frequency, a random
envelope, and a duration that is the greater of the the time the baseband signal lingers in the IF
pass band, and the time constant of the filter. (The time constant of the filter is approximately
I/fifF-)
If the modulating noise is bandlimited white gaussian noise limited to some maximum fre¬
quency Bm, then it will have a sero crossing about once every l/Bm seconds on the average. Again,
assuming that the RF spectrum of the FM/N signal is centered on the passband of the victim re¬
ceiver, this implies that the FM/N signal will sweep through the passband of the victim receiver
about once every l/Bm seconds.
If the bandwidth of the modulating noise is greater than the bandwidth of the IF filter of
the victim receiver, this implies that, on the average, the filter’s response to one noise sweep will
not be over before the next sweep occurs; thus the responses will overly) and there will generally
always be a consistent amount of total power at the output of the IF filter. If the bandwidth of the
modulating noise is equal to the bandwidth of the IF filter, the response from one sweep will be
ending just as the next one begins, with the consequence that there will probably be some small
gaps left between responses. If the bandwidth of the modulating noise is less than the bandwidth
of the IF filter of the victim receiver, there will definitely be gaps between one response and the
next, of average duration l/Bm — l/Bip-
When the pulses overlap, it has been contended by almost all the authors who have written
on the subject (20) (6:14) (21) that it is not necessary to know the probability distribution of any of
the individual filter responses: the univariate probability density of the total waveform can be found
64
VBW-lOHz RBWalOkHz
SWP>30t
Figure 5. Baseband noise, RF spectrum and IF output for WBFM/LFN
(8:4)
65
to be gaunian merely by obaerving that the total waveform is the sum of the individual pulses, the
sum ot a number of random variables has a pdf which is the convolution of their individual pd£i,
and if you convolve any pdf enough times you will obtain a gaussian pdf. This general concept
is presented explicitly as the Central Limit Theorem, which holds when the random variables are
independent and have pdfs which are bounded.
This line of reasoning might lead one to conclude, falsely, that the pdf of the output of the
IF filter of the victim receiver would be gaussian even when the pulses do not overlap. However,
if there are dead spaces between filter responses and samples are taken during those dead spaces,
the correlation between one sample and the next is likely to be very high. Thus the Central Limit
Theorem fails because of the lack of independence between samples. Likewise there is a failure of
the Central Limit theorem when NBFM/N is employed. Because of the predominance of power at
fe in the NBFM/N signal, the response of the filter tends to look more like a sinusoid at the central
frequency plus gaussian noise, rather than purely gaussian noise.
Again, it will be useful to look qualitatively at the characteristics of the probability density
functions associated with each of the four possible FM/N jamming schemes.
When WBFM/WBN jamming is employed, we are now virtually guaranteed that the Central
Limit Theorem will hold and that the output of the IF filter of the victim receiver will be gaussian.
There will be no dead spaces between filter responses, nor will there be any carrier frequency
component. This noise should have the same quality as DINA noise, in a univariate statistical
sense, and the fact that it seemed to do as well as DINA experimentally is what led 'Dimer,
Ottoboni and others to suggest that B/f < Bm-
WBFM/LFN is obviously 1&;8 gaussian than WBFM/WBN. Even if the RF bandwidth of the
WBFM/LFN is wide enough to cover the bandwidth of the victim receiver with fairly white noise,
there will be gaps between one filter response and the next and the univariate pdf of the output
66
of the IF filter will have a delta function at zero volta, indicating the certain probability that the
signal will take on the value of zero for a certain percentage ot the time.
NBFM/WBN is also less gaussian than WBFM/WBN, although it is more gaussian than
WBFM/LFN. It has rapid enough sweeps through the pass band of the IF filter that there is a
consistent noise power generated at the output of the IF filter; however, the presence of the carria
invalidates the application of the Central Limit Theorem, and the pdf of the output of the IF filter
will tend to be ‘Tatter” than a gaussian pdf. As is well known, the pdf of a pure sinusoid is given
by:
iw(*) =
(67)
y/1 - {*/Ax)2
where Ax is the max’nrium (and —Ax is the minimum) value taken on by the sinusoid. This
function is characterized by two sharp peaks: one each at Ax and —Ax. When this kind of
function is combined with an essentially gaussian function where the gaussian function dominates,
the result is to make the gaussian function more ‘Svide-shouldered”: flatter across the top and mcwe
steeply descending down the sides.
Lastly, consider the NBFM/LFN case. It might be supposed at first that the signal at the
output of the IF filter would suffer from “dead spaces” on a regular basis, but in the case of true
NBFM, this is not so. The entire frequency excursion of the RF signal is either less than or on the
order of the bandwidth of the IF filter, thus the RF signal is always causing a response in the IF
filter of the receiver. A true NBFM/LFN signal will mainly resemble nothing so much as AM/N;
that is, it will look much like a single frequency with a randomly modulated envelope. The pdf
then looks much like the pdf of a sinusoid.
However, if one starts with a WBFM/LFN system (which is characterized in the time domain
by these “dead spaces”, and in pdf by the delta function at zero) and, rather than increasing the
baseband noise, begins decreasing the peak frequency deviation of the FM modulator, thus moving
away from WBFM/LFN and toward NBFM/LFN, two things will happen: 1) the dead spaces
67
will become shorter, &nd some will disappear, because the RF signal is not wandering out of the
passband of the victim receiver so often, and 2) the “wide-shoulderedness” of the excess carrier will
bring up the rest of the pdf. Thus, when the two problems one commonly encounters in FM/N
are carefully combined in moderation, a few dead spaces combined with a slightly wideshouldered
function, can, ironically, give the appearance of a more gaussian pdf than that obtained from either
NBFM/WBN or WBFM/LFN. This fact demonstrates one of the problems with trying to measure
the quality of a noise signal using only univariate statistical data, since a whiteness test would
certainly screen out this kind of pathological behavior.
To summarise the mostly qualitative discussion of the effects at IF of the various FM/N
jamming schemes, it is asserted here that: 1) the FM/N jammer should operate in a WBFM mode
in order to insure a spectrum that is as ‘Srhite” as can be achieved, and 2) the FM/N jammer
should have a modulating noise bandwidth that is at least as wide as the bandwidth of the IF filter
of the victim receiver. In other words, (using Stewart’s criterion for WBFM (26):
D = ^> 2.253 (68)
and
> Bjf (69)
If we substitute Bm = Bif, and A/p = 2.2S3Sn> iofo fba equation for the 3 db bandwidth of the
WBFM signal (equation 59), we obtain:
Bwbfm — •751BmV81n2 = 1.768Rin — 1.768Bif (70)
It is obvious that the 3 dB bandwidth of the FM/N signal will be somewhat wider than the
bandwidth of the victim receiver, and this implies that some jamming power will be “wasted” in
that it is being broadcast, but will not be received by the victim receiver. However, it is contended
68
that this is the absolute minimum bandwidth that can be broadcast and still be an optimal FM/N
jamming signal in terms of not having significant "dead spaces” in between filter responses as a
result of FM/LFN, and not having a vrideshouldered (i.e. non-gaussian) univariate pdf because of
the typical NBFM/N excess carrier power.
To complete the description of the behavior of FM/N, some terms associated with the concept
of a "sweep rate” should be explained. Benninghof et. a/.(6:14) introduce the concept of a "fast
swept” as opposed to a "slow swept” signal by talking about the sweep rate of a linearly swept
signal
«,(0 = ^icoe^^^ (71)
moving with increasing frequency through the pass band of a gaussian filter with transfer function:
^(«) = >i’exp[^^i^j (72)
and they define a ratio a where:
(73)
It is obvious that as a increases, the signal will "sweep through” the frequencies passed by the
filter more rapidly. An FM/N signal which sweeps too slowly is likely to have the "dead spaces”
characteristic of FM/LFN. Avoiding this problem is as simple as adhering to the criteria already
given above that B/f < Bm‘, however, it is possible to calculate a statistical sweep speed for an
FM/N system and place appropriate constraints on it, and Daly has done so (8:3-10). He defines
two ratios that get to the heart of the FM/N issue, the noue~to~victim ratio (NVR) which is defined
as:
NVR =
bandwidth of baseband noise
bandwidth of victim receiver
(74)
69
and the dematton-to-vtctim ratio (DVR) which is defined as:
DVR =
peak frequency deviation _ Af,
bandwidth victim receiver Bip
(75)
The NVR determines whether the FM/N jamming scheme is FM/WBN or FM/LFN, while the
DVR determines whether or not the pass band of the victim receiver is being completely jammed.
He then defines a third ratio, baaed on the previous two, the $wetp-to-metim ratio (SWR)
defined as
SVR = NVR ■ DVR = (76)
°IF
which indicates how often a noise pulse will be generated in the victim receiver and how long the
baseband modulating noise will linger in the IF pass band, on the average. As Daly points out, an
FM/N sjrstem cannot operate efiSciently if it has an SVR below a certain threshold (it is suggested
in this thesis that NVR can be no leas than 1 and DVR can be no leas than 2.253, thus the SVR
can be no leas than 2.253); however, an SVR higher than any given threshold does not necessarily
insure the proper functioning of the FM/N scheme. The NVR and the DVR must be mixed in the
proper proportions.
4.S M tanring Noise Qnalitg
Knowing what kind of noise is ideal and what kind of noise an FM/N system reliably produces,
in general terms, the question arises, how can the superiority of one noise source to another be
qnantiiaiivelg determined? So far three suggestions have been offered (30) (8). Two of them will
be examined here from a theoretical standpoint, and a fourth one, which perhaps combines some
of the best aspects of the preceding nneasures, will also be offered.
4-3.1 Tamer Noise Qaalitg. IWner noise quality, briefly defined in Chiq>ter 2, is baaed
on the similarity between the properties of a histogram samples of the output of the IF filter
70
and the corresponding properties of the univariate pdf of a theoretical gaussian noise signal. The
group at Stanford which produced the initial noise quality measurements and formulated IWner
noise quality formed a histogram using on the order of 1 million to 5 million samples while Daly’s
implementation of 'Dirner noise quality used only on the order of 1000 to 10000 samples. Again,
the Stanford group sorted sample points into either 512 or 1024 voltage bins, while Daly used
voltage bin widths of 0.2<t, where a is the standard deviation, which typically generated 30 voltage
bins. (If we find maximums at around +3<r, and minimums at around —Za as expected, then
+3<r — (— 3o’)/0.2<r = 30 ) These details aside, once the samples were processed and sorted into
voltage bins, the error measures used were consistent.
Assuming N samples Vi,i= and K voltage bins, where N, K are positive integers,
the number of samples in the tth bin can be denoted as p«[t]. If the mean and variance of the
samples are computed as:
= (77)
* ial
and
^1 = ^ E(^ ■ *'•)* (7®)
isl
then the number of samples associated with the tth bin predicted by the ideal gaussian distribution
can be easily estimated as:
p,[{\ = IVAn—e^n^ (79)
where Av is the bin width and v, is the average voltage associated with the tth voltage bin found
as:
Vj = -tAw - Vmtn
(80)
where Vmtn is the minimum voltage chosen to be ^3<r.
71
Three error terms are computed by directly comparing p,[t] with p«[t]- The summed error t.
is simply:
tsl
The rms error Cr is found as:
And the svei^e error ie found as:
Ca
|P<{«1 - P«[«1l
pit
(81)
(82)
(83)
Three other measures of the sample histogram are computed and compared to the ideal
gaussian. The relative entropy in bits Hk is the absolute value of the difference between the entropy
of the sample histogram and the ideal entropy of a gaussian with the same variance, calculated as:
The kurtosis, k is found as:
and the skewness s is, similarly:
* = “ »)VW- (*«)
It is known that as a sampled function has a nnivariate probability donnty function approach¬
ing the ideal gaussian pdf, the three error measures will become increasingly small, the rdative
entropy will approach zero, the kurtous will approach a value ot 3, and the skewness (which gives
72
an indication of the symmetry of the pdf) will approach lero. IWner noise quality combines these
measures in an ad hoc manner as:
1
TNQ
1
(87)
and it is easy to see that as the gaussianity of the curve increases, T\irtter noise quality will increase
without bound. Turner indicates that high quality baseband video noise sources in the laboratwy
have noise qualities ranging from 10 to 70, and he suggests that a TNQ of 4 is acceptable in jamming
applications (30).
The team at Stanford indicated that the whiteness a[ a noise jammer, in the passband the
victim receiver, was also important to effective jamming, but they applied a pass-fail whiteness test
rather than measuring the whiteness quantitatively. If the display of a spectrum analyser connected
to the output of the IF filter of the victim receiver displayed a trace that was roughly the same
shape as the transfer function of the IF filter, the noise was considered to be “white”.
4-S-2 IF Noi$e Qualitn. As has been suggested by the previous theoretical work in this
chapter, noise from an FM/N jammer will never be perfectly white, and, in the case of NBFM/N,
may be significantly “colored” . Furthermore, it seems that in some situations, there is a trade-off
that can be made between whiteness and gaussianity. A noise source that is somewhat whiter
than another source of the same gaussianity should theoretically be a better jammer for the same
amount of power. These considerations caused Daly to introduce two new noise quality measures:
IF noise quality and RF noise quality.
RF noise quality has been described qualitatively in Chapter 2 of this thesis, and it is suggested
there that RF noise quality is not universally applicable as a noise quality measure. Thetefne it
will not be considered further here. IF noise quality, on the other hand, is similar to ’Dimer noise
quality, in that it measures the gaussianity of the signal at the output o( the IF filter (ff the victim
73
receiver, and thus includes consideration of the parameters <d' receiver being jammed, as well as
being independent of the particular method used to iiyect noise into the victim receiver. However,
IF noise quality also makes a quantitative measurement of the whiteness of the output of the IF
filter. How it does this warrants some attention.
IF noise quality is based on the product of two penalties, one associated with the flatness
of the fiequency domain: pj and one associated with the gaussianity of the univariate pdf of the
signal measured in the time domain: pt- These numbers each have a maximum value of 1, thus the
product has a maximum of 1, and specific values associated with particular FM/N signals may be
multiplied by 100% in order to obtain a percentage noise quality.
The penalty pt is calculated by in a manner somewhat similar to the first three error measures
used in Turner noise quality, in that it is baaed on a histogram composed of equal width voltage
bins. However, instead of making a direct comparison, it converts the histogram into a sequence of
sample pdf estimates, and compares these estimates with the ideal gauasian pdf at corresponding
points. Using the same notation which was introduced above, this could be written as:
where Vd is the midpoint of the tth bin and pg(vc<) value of the Gaussian pdf at that point.
Daly states that this penalty was chosen as the measure of gaussianity, based on an algorithm given
by Shanmugan and Breipohl (23:497-500) and he finds the results that it gives consistent with the
theory of FM/N and well correlated with the ‘Duner noise quality (8).
The frequency domain penalty p/ is a little more problematic. The manner in which it is
assessed is straightforward. It is calculated <» the basis of a trace of the spectrum of the output of
the IF filter, as displayed on a fluency analyser which has been set to cover the 3dB bandwidth
the IF filter. The power underneath the trace is then ctMnpared to the power underneath a
constant theoretical trace having magnitude equal to the maximum magnitude of the actual trace.
74
The frequency-domain penalty is a great success in terms of simplicity, but it suffers fr<»a three
drawbacks 1) it does not take into account the shape of the filter, 2) it is very heavily influenced by
the processing which the spectrum analyser can perform, and 3) it is subject to wild fluctuations
because of a single spurious data point.
Daly was aware of all these problems, and he comments on them for several paragraphs:
Note that this penalty is conservative because it is based on the erroneous assumption
that the ideal trace can be uniform across the 3 dB bandwidth . . . the 3 dB bandwidth
of a filter is, by definition, non-uniform.
(8) He also notes that pj should have increased as the WBFM/N bandwidth increased for a
constant Bm ! however, this did not take place because of “trace averaging provided by the video
bandwidth selected on the HP 8566B spectrum analyser ...” (8).
However, despite the drawbacks, the concept of IF noise quality as a whole has some com¬
pelling features. It does, to a degree, measure the flatness of the spectrum in a quantitative sense,
as well as measuring the gaussianity of the univariate pdf; furthermore, the fact that it is a per¬
centage of unity lends it to use in jamming versions of the radar range equation. Daly gives a
brief example of how it could be incorporated into Barton’s equation for jammer temperature (a
measure of the increase in effective input temperature produced by a jammer) (2:139) (8: 6-11).
'Dirner noise quality, on the other hand, is completely unsuited for this type of insertion.
4.S.S FFT-IF Noiat Q%alUf. These factors led to the developoMnt (ff a modified IF noise
quality which will be referred to here as FFT-IF noise quolHp because it makes use of an FFT
algorithm to find the whiteness of the spectrum, instead of relying on a spectrum analyser trace.
As in IF noise quality, two penalties are assessed for deviations from gaussianity and whiteness, pt
and pj. The penalty pt is calculated by a simple transformation of 'Dimer Noise Quality:
Pt =
(89)
75
Thus pt retains the strong measure of gaimsianity that was developed and experimentally verified
by the team at Stanford, but also has the pr(^>erty that increasing noise quality gives us a value
increasingly close to one, making it suitable for inaerti<m into a jammer power equation.
For extremely low values of noise quality, a INirner Noise Quality of less than 1 will give us a
negative pt, which has questionable meaning, but noise jammers very rarely produce noise that is
that poor in practice (recall that a perfect sinusoid has a theoretical TNQ of 1.5). More reasmiable
values of TNQ for operational jammers range from 4 (giving us a pt = .75) up to 10 or higher
(giving us p, > .90).
This method for calculating pt does not produce significantly different results from the method
Daly suggests for calculating pt; however, it has the advantage of being easily computed if TNQ
for a system is already known.
The method for calculating p/ is where the significant theoretical difference between pip and
Pfpt u found. For pfft, PJ u calculated by using a diptal oscilloscope to take correlated data
samples (sampling at higher than the Nyquist rate), taking the FFT of the data samples to find
an estimate of the spectrum of the noise process, and then dividing point-by-point by the discrete
frequency transfer function of the IF filter.
Essentially, the problem that must be solved is mot the determination of whether at not the
process coming out of the filter is white; it is already known that it isn’t. The real problem is to
determine how closely the output of the filter conforms to the ideal output if the input to the filter
were perfectly white. If the transfer function of the IF filter is again taken to be H{f), then the
Fourier transform of the post-filter samples may be taken to be ^»»(/). efhere ^wwif) ^ estimate
of the spectrum Pgyif) obtained by pasring some signal z(f) through H and:
Pw(/) = ^*(/)-P~(/) (»0)
76
This implies that an estimate of the spectrum of the input signal, Pgx(f) may be found as;
(91)
The question now is, how closely does Pmm(f) approximate a white spectrum the same
average power, denoted as Pmw(f) ? At this point, the question is answered by fining the absolute
point-by-point difference from the mean of P«s(/) and dividing this by the number of points in
the spectrum. This quantity is the average error power. The normalised error power is found by
dividing the average error power by the average power in Pxx(f), mid the frequency domiun penalty
is then the difference between this normalised error power and one. In other words, if there are N
points in the spectral estimate P««(/), and the mean of Pm»(f) is Mxt then:
This retains the advantage of approaching unity as the estimate of the spectrum of the input
to the IF filter becomes increasingly white, and it avoids the drawbacks associated with spurious
data points, the specific shiqie of the filter being used, and processing performed under different
settinp on a spectrum analyser.
The final value for the FFT-IF noise quality is denoted prrr mid is found as before;
Prrr —Pj-Pt
(93)
and it can be used in jammer noise calculations or to modify linked budget calculations based on
the jammer ngnal power needed just as has been suggested of pip-
77
Summary
In this chapter, ideal noise was found to be white in the frequency-domain and also to have a
gaussian univariate probability density function. The theory of FM was used to develop the upec-
trum of the FM/N signal and to define four types of FM/N jamming: WBFM/WBN, WBFM/LFN,
NBFM/WBN and NBFM/LFN. The characteristics at these four types erf' jamming in terms of spec¬
trum of received signal and pdf of received signal were examined qualitatively, and it was concluded
that WBFM/WBN was the only type of FM jamming which is good for masking jamming both
spectrally and in teriiu of the gaussianity of its pdf. However, it was found that the pdf of the
NBFM/LFN system may appear gaussian under certain pathological conditions. Thus it was con¬
cluded that merely looking at the pdf of a signal was insufficient to conclude that it was “good”
noise. An expression was found for the minimum baseband bandwidth and RF bandwidth required
for a WBFM/WBN system, based on the bandwidth of the victim receiver, and this was converted
into a minimum sweep-to-victim ratio (SWR).
Lastly, three noise quality measures were discussed. Timer noise quality was presented and it
was pointed out that Turner noise quality does not measure the spectrum of the noise quantitatively
for whiteness. IF noise quality as defined by Daly was examined and found to be excellent in crmcept
but lacking in implementation as regards the assessment of a penalty for spectral deviation from
ideal flatness. A new noise quality measure was introduced which modifies IF noise quality in order
to make it a more consistent and more accurate measure.
78
V. Experiments
This chapter discusses the experiments which were carried out to support the theory given in
the preceding chapter, to demonstrate the concepts which were mentioned, and to both verify and
suggest modifications to the noise quality measurement techniques proposed and implemented in
the preceding thesis (8). The experiments are grouped by the experimental setup employed rather
than by their function, and each experimental setup performed more than one function in terms of
supporting theory, demonstrating concepts and responding to the suggestions of the previous work.
The first group of experiments used generally the same setup that was employed by Daly
in 1992, the exceptions being some simple changes to computer programs and some changes in
the band widths and peak frequency deviations chosen. The second group used an FM/LFN setup
designed to demonstrate the behavior of NBFM/LFN as theoretically described in Ch^ter 4, and
to demonstrate the specific failing of Turner noise quality and any other measure of noise quality
which only considers the whiteness of the noise spectrum in a qualitative sense. It also featured the
use of new computational hardware, and a new program written to demonstrate the new measure
of noise quality which was introduced in Ch^ter 4. The program is written in Matlab and a listing
is included following the C programs in Appendix A. (The new hardware and the C programs
were used to increase the speed of acquiring and processing data from the oscilloscope in order to
reduce the variance of the data samples.) The third group of experiments made measurements of a
commercial FM/N radar jammer in order to demonstrate how the techniques of noise quality mea¬
surement developed in the initial laboratory setup could be extended to a more practical situation.
This group of experiments used the C programs and the Matlrd> code exclusively.
5.1 Verification and Use of the Dalf Simulation
In response to the conclusions and recommendations made by Daly (8:7), and in support of
the theoretical results found in Chiq>ter 4 of this thesis, roughly 200 noise quality measurements
79
were made, uaing what will be referred to in thia ch^ter aa the Daly Simulation. Detaila of
the simulation are provided in Daly’a work, but slight changes in parameters that Daly does not
mention (such as the effect that the number of sample bins has on the chi-square calculation, or
the gaps in sample histograms that result from specific voltage/di vision settings on the digital
oscilloscope) require a short summarization of the equipment setup and some explanation of the
computer programs used.
It should be noted that although a reasonable understanding of the basic techniques and con¬
cepts of the Daly Simulation can be easily conveyed here, an experimenter interested in reproducing
the results found here should consult (8) for the full range of specific details. Equipment setup is
explained thoroughly in Chapter 4 of that document, and listings of the HP Basic programs used
are found in Appendix A of that document.
Along with the description of the Daly Simulation, a critique of some aspects of the simulation
is offered. In general, the simulation was good. Specifically, it gave reasonable and correct readings
of Turner noise quality, IF noise quality and RF noise quality for the noise sources being measured,
when it was set up correctly. However, it was concluded in the course of the experimental work
reported on here that two portions of the processing programs employed in the simulation need
modification. Firstly, the original software of the DsJy nmulation sampled at greater than the
Nyquist and then rejected a number of samples because they were cwrelated. The reason for this
is explained here, and an alternative approach is offered. Secondly, the Daly simulation adds a chi-
square test to the computation of IVirner noise quality. Within the context of the Daly Simulation,
there are some circumstances where the chi-square test provides a good measure of gaussianity, but
as a general rule, it does n'>t. In part, this has to do with quantization error introduced by the
oscilloscope. Some of this will be addressed here, and some of it will be covered in more detail in
Chapter 6 where results of the experiments in general are discussed, and an alternative approach
to this is also suggested.
80
Table 1. Table of Equipnaeat for Daly Simulation
ITEM
COMPANY
MODEL
Simulated Jammer
Noise Generator
Hewlett Packard Co.
HP3722A
Signal Generator
Hewlett Packard Co.
HP8640B
Simulated Receiver
Signal Generator
Hewlett Packard Co.
HP8640B
Dual Hi/Lo Filter
Waveteck Rockland
Model 852
Mixer
Anzac
MD 141
Meaaurement Equipment
Oscilloscope
Hewlett Packard Co.
HP54111D
Spectrum Analyzer
Hewlett Packard Co.
HP8566B
Computer
IBM
286 PC
Coprocessor
Hewlett Packard Co.
HP82324A
5.1.1 Equipment. The Daly Simulation consists of three parts: 1) A simulated FM/N
jammer composed of a white gaussian noise generator and an FM modulator, 2) A simulated
receiver composed of a signal generator and mixer used to heterodyne the jammer signal down to
an intermediate frequency and a bandpass IF filter, and 3) a noise quality measurement system
composed of a programmable digital oscilloscope, a programmable digital frequency analyser, and
a personal computer. A block diagram of the equipment setup is shown in figure 6.
For reasons of practicality and manageability, all equipment was chosen to be commercially
available and of a fairly generic nature, and it should be made quite clear that any similar system
should produce reasonably similar results in terms of general trends in noise quality figures based
on the conditions of NBFM/N and WBFM/N, and relationships between the bandwidth of the
modulating noise and the bandwidth of the IF filter. However, because some of the observations
made in verifying the Daly Simulation are peculiar to the specific equipment used and specific
settings on that equipment, a table identifying the particular pieces of equipment is included in
table 1.
The specific capabilities of each piece of equipment can be discovered in the appropriate
manual or by contacting the company. The limitations which led Daly to pick the specific pieces
81
HolieGenenior
OMpot
Q
Sigpil OeDcniw
FM
RF
i
lopai Ooqiw
:
a
a
a
a
a
■“^0
- SLJ
I
t
t
t
•
I
t
Figure 6. Block Diagram of Equipment setup
or equipment which he did, baaed on the choicea he had, are deacribed in hia thesia (8:4-4). For the
purpoaea of verifying the Daly aimulation and meaaurement of noiae quality, it waa only nercaaary
to inaure that: 1) the equipment did not preclude the inveatigation of all four typea of FM/N
jamming, and 2) the oacilloacope and frequency analyser were able to operate at the frequencies
used by the FM signal and the output of the IF filter, and they produced sufficient data samples
of sufficient quality for accurate processing.
The absolute values of of the three bandwidths which interact in an FM/N jamming scenario
are, of course, important as factors in any system design, but for the purposes of investigating
the phenomenon of FM/N and measuring noise quality, it is only the relative values of the three
bandwidths which are important. In light of this then, we will describe the bandwidth limitations
briefly.
The baseband noise produced by the HP3227A noise generator was bandlimited white gaus-
sian noise with a 'Dirner noise quality of about 10 when measured directly. The maximum band¬
width Bm which it could produce was 50 kHz. Successively narrower bandwidths could be produced
with Bm of Ifi kHz, 5 kHz, 1.5 kHz, .5 kHz, etc.
The peak frequency deviation, A/p, which determined the RF bandwidth, Bfm, was limited
by the HP8640B signal generator to 1% of the lowest frequency in a tuning range. Thus the
maximum Bfm could be, where Bfm is now an absolute rather than a 3dB bandwidth, was
2(.01)/e, and generally it was less. To insure wideband FM, the deviation ratio D was generally
chosen to be at least 3. Thus Daly chooses A/p = 3-50 kHz = 150 kHz, and this was a commonly
used value during the verification of the simulation. This value implies an /« then of at least 15
MHz. In actual practice a commonly used value was /< = 250 MHz, so obtaining a sufficiently wide
RF bandwidth to insure WBFM was not difficult.
The bandwidth of the IF filter of the victim receiver was a little more restrictive. The
bandpass filter was composed of a lowpass filter with cutoff frequency /a,- followed by a highpass
83
Table 2. Exploratory FM/N acenarioe
1^
kHs
125
kHs
150 kHs
WBFM/LF
BIS
kHs
150 kHs 1
Bom
150 kHs
i NBFM/WBN 1
El]
kHs
50 kHs
1 40 kHs
50 kHs
60 WTl
25
221
1 NBFM/LFN I
filter with cut-off frequency /i«, giving a 3 dB bandwidth of fu — /(•• The filters were configured
with a maximally flat paasband response and a roll-off of 24 db/decade. The minimum frequency
available for /i« was theoretically dc; however, in order to avmd frequoscy foldover, it was decided
not to go any lower than 10 kHx. The maximum frequency available for /m was 111 kHs. So the
maximum bandwidth available for Br was roughly 100 kHs, and Br could be chosen successively
smaller in bandwidths down to almost sero. When Daly used the simulation, he chose Br = 25, 50
kHs. In the use of the simulation presmted here, bandwidths were explored from 10 to 100 kHs,
in increments of 10 kHs.
It is easy to see that with these frequency ranges, all four types tff FM/N jamming schemes
could be explored. Assuming that /« is kept constant at 250 MHs, that typical * WBFM/WBN,
WBFM/LFN, NBFM/WBN and NBFM/LFN scenarios for the purposes of exploration using the
Daly Simulation might be as given in table 2
The capabilities of the spectrum analyser and oscilloscope far exceeded tlM requirerooits
placed on them as far as bandwidth is concerned, but the memory capacities of each device had
*The t«nn “typical” ia used loosdy. Bach of the caaea aperffifally wieBtiowBd hare, dong with a wide variety of
other cMea, waa explored. Some caaea are bctderiine; other caaea ware dioaen becanae they were more extiame, and
thue more daariydenioiiatrated the diaracteriaticipocaliar to their category. The heportaiit thing to note ia that yew
can move from any one acenario to any other aoenario by hotding any ain^ bandwidth conatant and appropriately
varying the other two.
the potential of imposing limitations. The oscilloscope was able to hold 8192 samples at a time per
channel, but for the purposes of developing a histogram to estimate the pdf the IF filter output,
it was necessary for all the samples to be decorrelated, thus nothing was lost by capturing 8K
samples, downloading them for processing, obtaining a second 8K, downloading that, etc. Thus
the practical number of samples available for processing is much greater than 8K when the samples
were intended to be decorrelated, as is always the case in the Daly Simulation when the 'Dimer,
IF and RF noise qualities are calculated. If one channel is used to record the baseband noise and
a second channel is used to record the noise in the victim receiver, records of both channels may
be obtained for purposes of comparison.
The last characteristic of the oscilloscope which had some bearing on the measuring of noise
quality was its amplitude resolution. The oscilloscope display had graticules separating it into eight
divisions in voltage amplitude. The maximum and minimum voltages recorded by the oscilloscope
were set indirectly by specifying a number of volts per division (a typical value was 10 mV/division,
giving a range of 80 mV total). This voltage range was divided by the oscilloscope into 254
quantization levels so that the amplitude of a sample falling within the tth voltage range, v, where:
“ Vmin) < OCr'insx ~ Vmin) (®4)
would be recorded as a one-byte integer with value between 0 and 255. (The values of 0 and 255
were reserved for recording "holes” and values that exceeded the range being considered. 'These
digital values could be easily converted to their analog equivalents, and, in fact, this was done in
the Daly Simulation to find the true mean and true variance of the signal, although this conversion
has no theoretical effect on the calculation of the componenet of noise quality measures that focus
on gausrianity. (A histogram does not become more or less gaussian by adding or scaling by a
constant.)
85
The memory capacity of the spectrum analyser had only an indirect bearing on the resolution
of the data gathered for processing. The spectrum analyser would only return 1000 data points at
a time; however, by changing the video bandwidth, it was possible to get it to average a greater or
smaller number of passes before producing a given set of 1000 points.
This somewhat tedious explanation of hardware details is given solely to explain two points
which should be noted here; 1) The number of voltage bins which could be chosen for the purposes
of forming a histogram was absolutely limited to 254, with the result that quantization error in the
histogram could not be arbitrarily reduced, and 2) Choosing the minimum and maximum voltages
carefully had an important impact on the calculation of noise quality. If the voltage range was
too great, only a few voltage bins around the centa of the histogram would be filled. If, on the
other hand, the voltage range was too narrow, the tails of the hisU^am would be clipped. Both
problems virtually guarantee a noise quality reading which is inaccurate. This resolution limit did
not seriously hamper the measurement of noise quality, it merely necessitated that a certain amount
of care be taken in the use of the hardware and software in order to obtain valid results.
S.J.i Summary of exyerimtuial procedure for Daly Simulation. Three general types
of measurements can be made using the equipment set-up described above and the HP BASIC
programs written by Daly: 1) A time domain or frequency domain sample of the baseband or RF
noise may be taken from the oscilloscope or spectrum analyser and stored as a data file on the PC
for further processing, or perhaps to generate a display 2) RF noise quality may be measured, or
3) Tiimer noise quality or IF noise quality may be measured. In mder to carry out the first type
of measurement, it was necessary to control the three bandwidths involved, and the amplitude of
the IF signal.
The modulating noise bandwidth could be controlled by means of the switch on the noise
generator. As noted before, it could be chosen from 50 kHz down to 5 Hz, at powers of 10
multiplied by either 5 or 15 Hz.
86
The RF bandwidth could be controlled by means oi the peak frequency deviation knob on
the HP signal generator configured as an FM modulator as follows. The peak frequency deviation
control is set to a certain range (for example: 2.56 MBs). The meter on the SCALE pand then
indicates the actual peak fluency deviation as a value lem than or equal to the peak frequency
deviation indicated by the position of the knob. (For example, if the position of the control is set to
2.56 MHz, then the light on the SCALE panel indicating *^3” will light up, and a meter reading
on the 0-3 scale of “.15” would indicate an actual peak frequency deviation of 150 kHz.) The actual
peak frequency deviation may be fine-tuned to a particular value by using the fine-tuning control
located in the center of the peak frequency deviation control. Additionally, the RF bandwidth
could be controlled by changing the amplitude of the output the noise generator. If the settings
on the signal generator (aka FM modulator) were held constant, then the peak frequency deviation
could be increased by increasing the rms voltage of the output of the noise generator.
The IF bandwidth was controlled solely by dual hi-b filter which could be varied by as little
as 1 Hz. For purposes of the verification of the Daly Simulation, the IF center frequency was always
chosen to be at the center of the IF filter. In oth« words,
fiF = - M
It can easily be shown that the IF center frequency may be varied by holding the center f^
quency of the FM modulator constant at some /« and varjring the frequency ot the signal generator
in the nmulated receiver as:
fttoam = /e + fiF
87
or by bolding the frequency of the signal generator in the simulated receiver constant at some /«
and varying the center frequency of the FM modulator as:
/fM modulator = /c + StF-
Either method is equivalent. In the experiments done in the context of this thesis, the signal
generator producing the reference signal for the receiver was generally held constant at 250 MHs,
and the center frequency of the FM modulator was varied as 250 MHs + ftr-
The amplitude of the IF signal could be controlled by varying the rms voltage output of the
noise generator, the output of the signal generator used as an FM naodulator, and by the output
of the signal generator used as a mixer in the rimulated receiver. Varying any or all of these
parameters has no effect on the noise quality of the output of the IF filter other than indirectly: if
the rms value of the output of the IF filter is increased, then the voltage range of the oscilloscope
must also be increased in order to meet the requirement that clipping of the signal not be too severe,
as discussed above in section 5.1.1. Similar cate must be taken if the rms value of the output tA
the IF filter is descreased.
Once the amplitude and the various bandwidths of the givoi setup are fixed, it is only
necessary to ensure two things: 1) that the aq>pR^riate signal (i.e. baseband noise, RF signal, or
IF output) is directed to the oscilloscope or spectrum analyser that data is desired from, and 2)
that the measuring device is connected by HP-IB bus to the HP coprocessor in the PC bring used
for signal processing and data storage. The settings of freqiwncy q>an on the spectrum analyser
and sampling rate on the oscilloscope should be set to whatever values seem ^propriate to c^ture
the essential elements of information. It is suggested that if the time-domain data is to be used
to plot a sample noise waveform, then the sampling rate should be at least twice the mximiini
frequency of the noise waveform; preferably several times the maximum frequency. If, however,
88
the time-domain data is to be uaed to form a hiatogram, it is necessary that the noise data be
uncorrelated so "slow” sampling is mandatory.
When these considerations are met, then the appropriate program may be run (either TIMEOMN.BAS
or FREQDMN.BAS when using a PC with the HP coprocessor.) Details on the use of the programs
and program listings are found in (8:4), although one is advised to beware of minor errors.
In order to make either of the other three measurements (RF noise quality or IF or 'Dirner
noise quality), it is important to pay attention to all the suggestions made so far about bandwidths
and amplitudes, and, in addition, for IF noise quality, ioih the oscilloscope and the frequency
analyser nnist be connected to the HP coprocessor in the PC via HP-IB cables, and care must
be taken that the output of the IF filter be connected to the input of the spectrum analyser and
channel 2 of the oscilloscope. For the measurement of 'Dimer noise quality, only the oscilloscope
need be connected to the HP coprocessor in the PC, but care must still be taken that channel 2
is used to display the output of
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