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1. kGEHOX USE OHli (Leave b/ank/
3. HEPORT TYPE AND DATES COVERED
2. REPORT DATE
I 4 August 1998
4. TITLE AND SUBTITLE
AN ENVESHGATION INTO THE PASSIVE DETECTION OF POINT TARGETS
USING FM BROADCASTS OR WHITE NOISE
6. AUTHOR(S)
Chadwick D. Lindstrom
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESSIES)
University of Washington
8. PERFORMING ORGANIZATION
REPORT NUMBER
98-038
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THE DEPARTMENT OF THE AIR FORCE
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17. SECURITY CLASSIFICATION
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19. SECURITY CLASSIFICATION
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20. LIMITATION OF ABSTRACT
Standard Form 298 (Rev. 2-89) EG)
Prescribed by ANSI Std. 239.18
Designed using Perform Pro, WHSfDKHt, Oct 94
AN INVESTIGATION INTO THE PASSIVE
DETECTION OF POINT TARGETS USING
FM BROADCASTS OR WHITE NOISE
by
Chadwick D. Lindstrom
A thesis submitted in partial fulfillment of the
requirements for the degree of
Master of Science in Electrical Engineering
University of Washington
1997
Approved by
Chairperson of Supervisory Committee
Program Authorized
to Offer Degree _
Date
19980810 089
Master's Thesis
In presenting this thesis in partial fulfillment of the requirements for a Master's
degree at the University of Washington, I agree that the Library shall make its
copies freely available for inspection. I further agree that extensive copying of this
thesis is allowable only for scholarly purposes, consistent with "fair use" as
prescribed in the U.S. Copyright Law. Any other reproduction for any purposes
or by any means shall not be allowed without my written permission.
Signature 0.
Date 0. 1
TABLE OF CONTENTS
LIST OF FIGURES . iii
LIST OF TABLES . vi
INTRODUCTION . 1
CHAPTER 1: System Overview and Requirements . 3
1 . 1 ; Receiver Topology . 3
1.2; Basic Signal Processing Algorithm . 4
1.3: Receiver Geometry . 5
1 .4; Sensitivity Requirements . 7
CHAPTER 2: The Scattering Model and the Ambiguity Function . 13
2.1: The FM Radio Broadcast . 13
2.2: Scattering Models . 14
2.3: The Ambiguity Function . 18
2.4: Discrete Computation of the Ambiguity Function . 20
CHAPTER 3; The Bias of the Self Ambiguity Function . 21
3.1; Notation . 21
3.2: The Bias of the Self Ambiguity Function . 22
CHAPTER 4: The Probability Distribution of the Self Ambiguity Function . 27
4 .1: Derivation of the Probability Density of the Clutter Floor . 27
4.1.1 Comparison to Simulated Data . 34
4. 1 .2 Model Comparison to Experimental Data . 38
4.2:Determination of Target Distribution . 51
4.3: Ideal Operating Charactersitics of the White Gaussian Signal . 57
4.4: Adjustments for the FM Waveform . 63
CHAPTER 5: Adaptive Beamfoming/Directional Antennas . 70
5. 1 ; System Overview . 70
5.2: Analysis of Adaptiv Beamforming Effects on Sensitivity . 73
5.3: Experiments results . 76
CHAPTER 6: Results and Conclusions . 79
6. 1 : Experimental Results . 79
6.2: Conclusion . 81
BroUOGRAPHY . 82
11
LIST OF FIGURES
Figure Description Page
1 . 1 The Direct Conversion Receiver . . • ■ 4
1.2 The Multistatic Receiver Geometry . 5
1.3 The Bistatic Radar Geometry . 6
1.4 The Bistatic Triangle . 8
1.5 Plot of ovals of Cassini about the receiver as a function of scattered to
direct signal ratio (SDR) . 9
1.6 Plot of ovals of Cassini about the receiver as a fimction of baseline length . 10
2. 1 Geometry for a single moving target showing the directions of the incident
wave, scattered wave, and motion of the particle from a reference point . 15
2.2 An illustration of the discrete computation of the ambiguity function at
range . . 20
4. 1 Mesh plot of the ambiguity function, |X(r,f)p, for a White Gaussian Signal ... 34
4.2 Mesh plot of the clutter floor of the ambiguity fimction plotted in Figure
4.1 . 35
4.3 Mesh plot of the clutter floor for |X(r,f)p for a White Gaussian Signal
Computed with 128 Fourier frequencies instead of 64 as in 4.2 . 35
4.4 Histogram of the clutter floor of ambiguity fimction displayed in Figure 4.2
vs. a xi .
4.5 Histogram of the clutter floor of the ambiguity fimction in Figure 4.3 vs. a
. 37
4.6 Plot of the theoretical vs. empirical cumulative distribution fimction for the
data of figure 4.5. The two curves are essentially indistinguishabel . 38
iii
L
4.7 Mesh plots of signals A1 = 106.9 Mhz & A2 = 93.3 Mhz sampled at
different times . 41
4.8 Mesh plots of |X(r,f)| of signals A3 = 94.9 Mhz & A4 = 94.9 Mhz sampled
at different times . 41
4.9 Mesh plots of |X(r,f)| of signals A5 = 97.3 Mhz & A6 = 97.3 Mhz sampled
at different times . 42
4.10 |X(r,f)|^ for A3&A4 for ranges 16.5 km - 50 km and frequencies -500 Hz
to -15 Hz . 42
4.11 |X(r,f)|^ for A3&A4 for ranges 16.5 km - 50 km and frequencies -500 Hz
to -15 Hz . 43
4.12 |X(r,f)p for A5 & A6 for ranges 16.5 km - 50 km and frequencies -500 Hz
to -15 Hz . 43
4.13 Histograms of |X(r,f)|^ for ranges 16.6 - 50 km and frequencies -500 Hz to
- 15 Hz . 45
4.14 Empirical Cumulative Distributions corresponding to the histograms of
figure 4.13 plotted against the theoretical distributions . 46
4.15 Plot of |X(r,-f)| - |X(r,f)| for the White Gaussian Signal . 48
4.16 Histogram of figure 4. 1 5 vs. the theoretical Gaussian curve . 48
4.17 Empirical Cumulative Distribution corresponding to the histogram of figure
4. 16 plotted against the theoretical distribution . 49
4.19 Histogram plots of |X(r,-f)| - |X(r,f)| for signals A1 - A6 . 50
4.20 Empirical Cumulative Distributions corresponding to the histograms of
figure 4.19 plotted against the theoretical distributions . 52
4.21 Plot of noncentral chi-square with k = 102.4, 2M = 16, versus a chi-square
of 2M degrees of freedom . 55
4.22 Plot of Gaussian with mean » 6.84, variance « 1.5 versus a Gaussian with
mean « 0 and variance « 1 . 57
IV
4.23 Plot of Pc vs. Pf for the White Gaussian Signal with M = 8, N = 64, S =
250, and 201og(a) . 59
4.24 Plot of Pd vs. a for constant Pf = lO"® and A = 128,000. M is indicated on
the plot . 60
4.25 Pd vs. Pf for the White Gaussian Signal with using Gaussian densities and
M = 8, N = 64, S = 250, and 20 log(a) . 61
4.26 Pd vs. Pf for As = 128,000 with M = 8 and p = 0.067 . . 66
4.27 Plot of lX(r,-f)-X(r,f)| for T = 0.5, M = 8, Pf = 10■^ and Pd = 0.5 for a
White Gaussian Signal . 68
4.28 Plot of |X(r,f)P for T - 0.5, M = 8, Pf = 10'^ and Pd = 0.5 for a White
Gaussian Signal . 68
4.29 Plot of |X(r,-f)| - |X(r,f)| for T = 0.5, M = 8, Pf = 10•^ (3=0.15, and Pd =
0.645 for signal A2 with target at 21 km and -125 Hz.
4.30 Plot of |X(r,-f)i - |X(r,f)| for T = 0.5, M = 8, 3= 0.15 for signal A2 with the
target injected at the same position as figure 4.29 except whose magnitude
was decreased by 3 dB . 69
5 . 1 Geometry of adaptive beamforming array . 71
5.2 Plot of |X(r,-f)| - |X(r,f)l for the cross ambiguity plot of the adaptive
beamformed signal with signal A2. T = 0.512 s, M = 8, and the amplitude
level is the same as 4.28 . 78
5.3 Plot of |X(r,-f)| - |X(r,f)| with the same parameters as figure 5.2 except the
scattering to direct ratio is 3.3 dB less . . 78
V
L
LIST OF TABLES
Table Description Page
4. 1 KolmogrofF-SmimofF Statistics for A1 -A6 for the . 44
4.2 KolmogrofF-SmimofF Statistics for A1-A6 for the Gaussian . 5 1
4.3 Minimum detectable scattering amplitude for the White Gausssian Signal
sampled at 250 kHz . 62
4.4 List of P for signals A1-A6 . 65
4.5 Minimum detectable scattering amplitude for FM signal sampld @250 kHz 66
vi
L
ACKNOWLEDGMENTS
I wish to thank John Sahr, my thesis advisor, for encouragement and insight during this
research project. Also, I would like to thank Frank Lind for providing the data sets that
were used throughout this thesis, and for providing a practical perspective on radar
receiver operation.
Vll
INTRODUCTION
Over the past three years, researchers at the University of Washington have
been developing a multistatic radar system that uses commercial FM radio broadcasts to
detect plasma waves in the ionosphere. This research has shown that the short
autocorrelation times of FM broadcasts make it possible to achieve good range and
doppler resolution despite the continuos operation of the transmitter However, because of
the great distances of ionospheric targets (>600 km) most research has focused on a
multistatic system consisting of a reference receiver and another receiver isolated from the
direct broadcast signal by the Cascade Mountains. Although some research was done on a
bistatic system, it was quickly seen that the clutter limited dynamic range would make the
observation of plasma waves very difficult if not impossible. This analysis though did not
rule out the detection of aircraft at distances close to the receiver-transmitter baseline. In
fact, the first detection using FM broadcasts was that of a bistatic detection of an aircraft
displaced 1.5 km from the baseline [1].
The primary problem with the detection of aircraft with the bistatic radar is
discriminating the scattered signal(s) against the strong direct broadcast. This is similar to
the detection of weak signals against strong interfering white noise sources as encountered
in sonar and radar environments with strong ECM (electronic countermeasures). The
optimal solution is known to be matched filtering combined with spatial filtering technique
such as adaptive beamforming [2]. However, because the signal we are seeking to identify
is not known before reception it is impossible to use the traditional matched filtering
technique. Instead, we compute the self-ambiguity function of the signal. This is still
optimal if the signal itself is white noise, but it requires that the scattered signal compete
with the clutter of the direct signal. A performance analysis of this technique is done for
the case of white noise which is then phenomenologically extended to FM radio
2
broadcasts. The key results of this analysis are that detection of aircraft will probably not
be possible at FM broadcast bandwidths. However, the theory developed for white noise
processes shows that a spread spectrum radar which uses 10-20 Mhz of bandwidth
could be developed with a range radius of 4 - 5 km from the receiver. Although not
interesting by itself, a network of the receivers could be formed whose presence would be
impossible to detect since they would be completely passive. Also, because the spectral
density of the transmitted signal would be close to the noise floor, it would even be
difficult to determine the radar’s presence. This would allow an active transmitter to be
used.
CHAPTER 1 : SYSTEM OVERVIEW AND REQUIREMENTS
As with any system analysis problem, before we begin the detailed analysis
of the capabilities of the radar it is necessary to ensure that the reader understand at the
broad system(s) level the radar operation. A radar is a system which generally consists of
a transmitter, receiver, antenna(s), and either analog or digital equipment which
implements a signal processing routine on the raw receiver data [3]. The purpose of the
signal processing routine is to use the principles of electromagnetic theory along with that
of statistics to detect and determine characteristics such as range, direction, or velocity of
a target(s).
The major differences between a system which uses commercial FM radio
broadcasts and that of a conventional radar come from the operator not controlling the
transmitted waveform and from the separation of the receiver and transmitter. The
inability to control the waveform requires that the signal must be modeled in order to
analyze its performance. We will be modeling the transmitted signal as white gaussian
noise which is primarily done for analytical convenience, but we will also show how the
model can be extended to colored gaussian noise which more accurately describes the FM
broadcast. The separation of the receiver and transmitter means that the geometry and
location of the receivers will be very important. It also has the effect that the receiver
topology can be simplified to direct conversion.
1 . 1 RECEIVER TOPOLOGY
The direct conversion receiver can be used in this radar since there is no
transmitter sharing the same antenna as the receiver. Hence, the traditional interference
problem which requires that the signal first be mixed with an intermediate frequency is not
required. Thus, the receiver topology for this radar is given by figure 1.1 [3]:
Antenna.
iBPy I co5(wt)
J
-my-
-ra —
Digital Signal
Amp'Ll
>
sm(wt)
-111-
—
Processing
Figure 1.1: Direct Conversion Receiver
1 .2 BASIC SIGNAL PROCESSING ALGORITHM
The optimal linear detection algorithm in radar and also for other
applications which must detect signals in noise is matched filtering [2], In the time
domain, because the impulse response of the matched filter is the time reversed conjugate
of the signal, matched filtering is equivalent to correlating the received signal with the
signal that was transmitted. This intuitively makes sense because we would expect the
algorithm to compare the transmitted signal to what was transmitted. The matched
filtering allows us to quantitatively determine the degree of correlation between the signal
that was transmitted and what has been received and it does this in the optimal sense of
maximizing the SNR (signal to noise ratio).
The bistatic FM radar uses just one receiver which records a vector sum of
both the direct and the scattered signal as shown in figure 1.3. To be able to record both
signals simultaneously requires sufficient dynamic range. This is primarily determined by
the number of bits of the analog to digital converter. The matched filtering is implemented
by correlating the received signal with itself There will be a large spike in the ambiguity
function at zero delay and doppler shift because the signal perfectly matches itself
5
Targets must now compete with the noise plus the clutter of the direct signal.
Characterizing the clutter of the direct signal is the main problem that must be understood
to determine the sensitivity of the radar.
1.3 RECEIVER GEOMETRY
The location of the receivers is important in understanding the operation of
the radar. Several options are open to the designer each of which offers its own
advantages and problems.
Figure 1.2: Multistatic Receiver Geometry. The direct signal is received at RXl and
correlated with the scattered signal received at RX2. The direct signal is blocked by the
Cascade Mountains which allows RX2 to receive only the scattered signal.
The first design is given in figure 1.2. This is the multistatic radar which is examined
extensively in Paul Hall’s thesis [4]. It has the advantage that the mountain range blocks
the direct signal to the second receiver. This alleviates the djmamic range and some of the
clutter problems which are present in the bistatic geometry. Unfortunately, this method of
implementation requires suitable geographic features which limits its use to a few select
regions of the world. It also has the problem that a relatively high speed data link be
6
established between the two receivers. Finally, this geometry has been extensively
investigated by researchers at the University of Washington.
Figure 1.3: Illustration of Bistatic Geometry. In both cases, the receivers must receive a
vector sum of the direct and scattered signals.
Figure 1.3 shows the single receiver bistatic radar. It requires the least amount of
hardware with only a single receiver. However, the simplicity in hardware components is
offset by the requirement to distinguish targets against the strong direct signal whose self
clutter will obscure targets.
This problem may be partially dealt with by using two directional antennas
one pointed at the receiver and the other in the direction of the target. This can also be be
done by using two or more closely spaced receivers (within several wavelengths of each
other). It is then possible to use the phase differences due to the difference in the path
lengths to cancel the direct signal which can be implemented through the spatial filtering
technique called adaptive beamforming. Of course this requires two or more receivers to
be built, as well as an increase in the computation burden which are the systems main
drawbacks.
7
be built, as well as an increase in the computation burden which are the systems main
drawbacks.
1.4 SENSITIVITY REQUIREMENTS
The sensitivity requirements can be determined from the bistatic radar
equation. This is given by (ignoring polarization or impedance mismatch) [5] :
where Pr, Pt are the power at the receiver and transmitter respectively. The two gain
factors, Gt(i ) - transmitter antenna gain and Gr(o) - receiver antenna gain, are dependent
upon direction patterns which is the reason for the vector arguments. The bistatic cross
section, is the parameter determined by the target and is also dependent upon
direction. The factor is the wavelength of the carrier frequency and results from the
receiver cross section. The final two factors, Ri and R2 are defined in the figure 1 .4.
Similarly, the received power of the direct signal can be written as;
(1.2)
where the prime indicates the directions will often be different from the scattered signal.
The two quantities which are pertinent to the problem of interest is the
ratio of 1.1 to 1.2, and the ratio of 1.1 to noise power. First, we write the ratio of the
received power of the scattered signal relative to the direct signal (1 . 1 to 1 .2):
GXbGAo)
GXhGAo')
i47!:)R^R^
(1.3)
8
Target
Transmitter L Receiver
Figure 1.4; The bistatic triangle.
We will call this quantity the scattered to direct ratio (SDR). It will often be the case that
we will be operating with Pr » N where N is the noise power. Thus, it is useful to
determine the performance of the system for a minimum SDR and specific values of L, p,
and crj, (6,7") . The requirements on Ri and R2 are then that they satisfy the law of cosines
with L being the third side of the triangle as shown in figure 1.3, and that they satisfy
equation 1 .3 for the given values. It is then easy to show R2 must obey the following
quartic equation with 0 being defined as indicated in figure 1.4:
- ILRl cos(^) + I'i?' - =0 (1.4)
(4^)«nim
Notice that for small R2 that 1.4 reduces to the familiar equation of a circle in polar
coordinates. This is one possible equation that can be used to describe the Ovals of
9
Cassini that result, and other solutions that use a different origin or coordinate system
exist [2], This equation although difficult to solve analytically can easily be plotted with
Figure 1.5: Plot of ovals of Cassini about the receiver with L = 20 Km, <Tj,;(o,/)= 40 m^,
p = 1, and c)r^„ as given on the curves.
the implicit plot function of Maple. The result for various values of is shown in
figure 1.5. This indicates that we must be capable of detecting scatter -55 dB below the
transmitter signal to be able to detect a target with a cross section the size of an
Figure 1.6: Ovals of Cassini for indicated baseline lengths and — 65 dB, p-1
and (Tj,, (6, i) = 40 m^.
11
airliner, 40 at 1 km from the receiver without directional antennas. However, a similar
mirror image of the ovals occur about the transmitter which means it is possible to detect a
target 21 km away. It is also important to see as shown in figure 1.6 that the ovals
described by equation 1 .4 are quite close for varied baseline lengths. The reason for this is
that both the scattered signal and the direct signal depend upon the baseline length.
Hence, when the power ratio is taken the change due to the baseline length changing is
typically small.
Next, we use the model that the noise is white and hence its power can be
written as [5]:
N = kBTfB (1.4)
kB = 1.38 X 10'^^ J/K is Boltzman’s constant, T is the noise temperature of the receiver
plus the cosmic background radiation, and fe is the bandwidth of the receiver. This means
that the ratio of the scattered signal to the noise (SNR) is given by;
P, _ X^G,{i)GXo)<^Alo)Pt .15)
Normally T is taken close to the effective cosmic temperature which can obtain a
maximum of 10,000 K, however, in this case it was found the measured noise power was
closer to -60 dBm because of the loud radio environment in the Seattle area due to factors
such as FM sidebands [1]. This means that T should be taken as 2.89 x 10* K. However,
this still only results in an SNR of about -25.5 dB for a baseline length of 80 km and 100
kW transmitter power which is much higher than the corresponding SDR. Indeed, if we
can handle the SDR then it will be possible to easily deal with the noise at almost all
baseline lengths of interest.
12
We have overlooked the improvement in gain that may be possible with
directional antennas/adaptive beamforming and the effect of physical clutter. We will
examine those effects as degradation or improvement factors in the signal processing.
This means that problem that we must pursue is determining if it is possible to achieve
through signal processing the detection of a target whose SDR is near -60 dB if we wish
to pursue passive radar nets and even smaller if we wish for a passive radar to have the
range of a short range surveillance radar. It is obvious that pursuit of passive radar nets
will be easier to pursue. Thus, a SDR of about -60 dB is what we should be looking for in
the analysis that follows.
CHAPTER 2: THE SCATTERING MODEL AND THE AMBIGUITY FUNCTION
This chapter discusses the scattering models that will be used in this thesis
and their relation to the ambiguity function. The emphasis is on developing the correlation
properties of the fields which are incident upon the receiver, and not upon detailed analysis
of scattering cross sections. These properties will then be used to describe the output of
the ambiguity function.
2. 1 THE FM RADIO BROADCAST
The FM Radio broadcast uses voice and music signals to modulate the
carrier signal. This is done through the use of instantaneous frequency. Suppose the voice
and music signal is given by a(t) and that the carrier frequency is given by fo. Then the
modulation for a monophonic radio would be given by [6]:
t
v(t) = cos(2^o^ + Kja{t')dt' + (2.1)
— 00
where k is a constant called the frequency sensitivity which is used to spread the signal out
over the given bandwidth. Now differentiating the argument of the cosine gives us the
instantaneous frequency which yields:
1
= (2.2)
Thus the instantaneous frequency is indeed modulated by the voice or music signal. Now
real FM radio broadcasts are done in stereo so that a(t) is replaced by a multiplexed
version of the sum and difference of the left and right speaker channels [6].
14
However, for the purposes of this thesis the importance of the FM radio
modulation is that the transmitted signal can be expressed as a modulated time-harmonic
wave as follows:
v(0 = Re{M(OexpO'2^oO}
t
u(t) = Qxp(j[K J aiOdt'+Oo ]) (2.3)
— 00
The modulation function, u(t), in the following analysis will be assumed to be a Gaussian
random process with a Gaussian autocorrelation function. This assumption may not be
valid in all cases, however, experimental evidence presented by Hall [4] suggests it can be
made.
2.2 SCATTERING MODELS
We describe two scattering models. The first is the simple model which
assumes one target. The second model will include multiple targets and can be extended
to distributed targets.
First, we start by considering the single target model. We know the field at
the transmitting antenna is given by equation 2.3 which means that we can solve for the
harmonic amplitude terms at zero range;
1
^ j E(0,(i))e\p{j(a)d(i) = u{t)exp{j(o^t)
£(0,m) = C/(a>-u>o)
(2.4)
Next, since the radio antenna can be modeled as a collection of oscillating dipoles with
field pattern /,(i) the plane wave approximation can be used. This allows the time-
harmonic wave to be written at the target a distance Ri fi'om the transmitter as indicated
15
Figure 2.1: Geometry for a single moving target showing the directions of the incident
wave i , scattered wave d , and the motion of the particle from a reference point with
velocity V [7],
in figure 1.4 as;
X,. .exp(-#7?,)
E{R^,co,i) = U{co-o)f,) - - - /,(i) (2.5)
^1
The wave is then scattered from the target as indicated in figure 2.1 with scattering
amplitude /,(©, i) . This allows the field at the receiver to be written as;
E{R„R^,coXo)^U{(o-co,)
exp(-y^(7?i +R^))exp(-jk^r)
/,(i)/,(o.i)/,(6) (2.6)
where R2 is the range from the target to the receiver, ks = A:(i - 6) , r is the displacement
vector from the reference point as in figure 2.1, and X(d) is the receiving pattern [7].
Now since U(co-coo), the baseband modulation, is approximately gaussian with a width of
30 kHz for FM broadcasts, the narrowband approximation that the scattering amplitudes
and antenna patterns are the constant can be made [7]. Next, since k= o/c, we separate
16
2.6 into three parts;
E{R^,R2,co,i,o) = C/(<»-6?o)exp
JO>\
^R, +R2 +f,-r'
a
(2.7)
Where a is the scattering amplitude divided by the range and = i - 6. Now we take
the inverse fourier transform to get the field as a function of time at the receiver;
t-
i?, + 7?2 + fs ■
exp(j6>oOexp(-A(^i +'^))exp(-Ar. ' 0 (2.8)
Now mixing this signal and making the approximation that • r « i?, ,/?2 allows it to be
written as;
oa4\
t-
7?1 +i?2
exp(-A(7?i +/?2))exp(-Ar. -r)
(2.9)
Since r is the time- varying displacement vector from the reference point in figure 2.1, it
follows that if we let t= 0 be the time when r = 0 then if the objects velocity is V it is
possible to write the third term as exp(-7Aof^ V/) which is the bistatic doppler shift.
With this change and the addition of the direct signal and the receiver noise the single
scatter model is given by;
L R
x{t) = Pu{t - — ) exp(-yA:o A ) + «*/(/ - — ) ex^irjk^R^ ) exp(-yA:of , • \t) + n{t) (2. 1 0)
c c
f (i') f (-i')
where |3 = — - - - with i' being the unit vector of the direct line of site between
transmitter and receiver, L being the baseline, R* = Ri + R? called the range sum, and n(t)
being the receiver noise.
17
The scattering by multiple targets is simply the sum of the scatter by the
targets given by equation 2.9, the direct signal, and the receiver noise since the scattered
fields are added as vectors. This means that the measured field can be written as:
T
x{t) = Pu{t — ) exp(-yAo ^) + Z ~ ~) exp(-A^« ) exp(-Ar« • VO + «(0 (2.11)
C j c
An important assumption that will be made in this thesis that has been shown
experimentally to be true is that the correlation between scattered fields at different ranges
is zero.
The previous model does not account for targets which are distributed in
range and it does not account for the changing environment. The first of these problems
can be handled by changing the summation to an integration. The second requires that the
coefficient of the modulation signal, u(t-R«/c), in equation 2.11 be replaced by a quantity
called the scattering amplitude, <I>(Rs,t), which is a random variable. Its value at any given
time at particular range is given by the targets that are there. Now this means it is possible
to write the measured field as:
x{t) = +n{t) (2.12)
The variation in phase due to doppler shift means that over long time periods the
scattering amplitude goes to zero. Also, it will go to zero because of range migration.
These properties along with those given above can be summarized as follows:
{<^R.,.iyt>\R,„t- T)} = ^ exp(-Af. • Vr)
(2.13)
18
Notice that the time-correlated scattering amplitude is the radar equation 1.1 normalized
by the transmitter power times the bistatic phase shift due to the target’s motion. The
bistatic phase shift is the result of the bistatic doppler shift of the target.
2.3 THE AMBIGUITY FUNCTION
The ambiguity function is the output of a filter matched to a delayed,
doppler shifted version of the transmitter wave. It is a function of range, r, and doppler
frequency, v, with the ideal shape of the self-ambiguity function being a delta function at
the origin. The following is a slightly modified version taken from Levanon [8]:
\z(.r,v)\ =
I y(t)x*{t - r) Qxp(-j2^vt)dt
(2.14)
Notice that it looks nearly like the fourier transform in doppler, except it is a fimction of
range. Indeed if we define the yx sequence as being:
yx^(t)=y(t)x\t-r) (2.15)
then for a constant range the ambiguity fimction is the fourier transform of the yx
sequence. Now if we square the ambiguity function then at a constant range we obtain the
power spectrum of the yx sequence. Then it can be shown that the inverse fourier
transform of the ambiguity fianction at a constant range results in the autocorrelation
sequence of the yx - sequence [4]. This can also be shown through the Wiener-Khintchine
theorem in the case of stochastic transmitter waveforms. The result is that the ambiguity
function in the correlation domain is given by:
Q(r, t) = E[yx^ {t)yx*{t - r)} = E[yit)y\t - T)x{t - r - r)x* (t - r)} (2.16)
19
The yx sequence provides a natural way of interpreting the ambiguity
function. The question then is what is the physical meaning of the yx sequence. The
following will provide the answer in the case of a traditional pulse radar. Consider the
scattered wave,
CX)
y(t)= j^(R,t)u(t--)dR + nit) (2.17)
0 ^
and the transmitter waveform x(t) = u(t). Take the time average of the yx sequence;
Ryxif) = -r)) = ( j - ^)dR + «(/)
= + (2.18)
» J {<t>(R,l))L(t - -)»•(/ - r)W = I {^R,t))R..(r - ^)dR
where we have used the independence of the noise and transmitter waveform along with
their zero mean value. Also, the independence of the transmitter waveform and the
scattering amplitude was used. The average of the scattering amplitude is zero mean over
long time intervals for moving targets because of the bistatic doppler shift. However, if
the averaging is done only for relatively short intervals it is constant. The averaging over
short intervals is coherent averaging. Since the noise and transmitter waveforms average
to zero much quicker than the scattering amplitude, coherent averaging is useful. The
sequence that results from coherent averaging at a particular range is samples of the
convolution of the autocorrelation of the transmitter waveform with the scattering
amplitude(s) of the target(s) at that range. Ideally, it is desired that this autocorrelation
would be a delta function centered at the origin. This would mean the resulting sequence
consists of samples of the scattering amplitude at the desired range. The power spectrum
of these samples could then be computed to find the spectrum of the scattering amplitudes
20
of the target(s) at that range. This is what the ambiguity function does. In an ideal sense,
the ambiguity function at a given range should give us estimates of the power spectrum of
the target(s) at that range.
2.4 DISCRETE COMPUTATION OF THE AMBIGUITY FUNCTION
The computation of the square of the ambiguity function is easy to infer
from the discussion above. First, we form the yx sequences from the data at the ranges of
interest. Next, we perform coherent averaging which can also be thought of as low pass
filtering of the yx sequences. This is then followed by computing the periodogram using
block averaging. The block averaging step is called incoherent averaging in the radar
community because it involves averaging power quantities which are phase independent.
Figure 2.2 illustrates the discrete computation.
x(t)
= ^S
. I I I I I I I i-i 1. 1 I I M M mca:
x*(t-r)
. Ill I~rm
izxiixnnnnQxiixnnnniim^
N \ \ \ / /
N
I
I
^ 'A ^
I I : izn
I I I 1 M=2
Output of Ambiguity
Function at Ranee r -
N
N
> I I I 1 . 1
Figure 2.2: An illustration of the discrete computation of the ambiguity fimction at range r.
As, is the length of the sequence (also will be defined as time-bandwidth product later), S
is the number of coherent averages, M is the number of incoherent averages, and N is the
number of Fourier frequencies.
CHAPTER 3: THE BIAS OF THE SELF AMBIGUITY FUNCTION
In this chapter we evaluate the bias of the self ambiguity function of the
FM radio broadcast in the correlation domain. This can then be Fourier transformed to
the frequency domain to give the bias of the self ambiguity function. By doing this, we
learn about the general shape of the self ambiguity function of signal composed of a direct
plus scattered signal. It is found that because the bias is concentrated origin of the self
ambiguity function that it does not limit the sensitivity of the radar.
3.1 NOTATION
It is quite inconvenient to write out all the terms in equation 2.16 since 81
terms result when the received signal is described by equation 2.12. Instead, the following
convention from section 2.3 will be adapted;
x(t) = fiuit - —) = direct signal
c
CO
y{t) = f , t)u{t - -)dR^ = scattered signal (3.1)
n{t) = noise
In the analysis of multiple receivers a subscript (1,2,. .n) will denote the receiver. A capital
X(t) means the sum of all three signals: X(t) = x(t) + y(t) + n(t). It too will have a
numbered subscript in the case of multiple receivers.
For four-product correlations, it will be necessary to expand them out and
use a simple convention for the four products which result. Consider the fourth-order
22
moment;
X(t-r-T)
"»ini = {x(t)x*(t - r)x*(t - T)x{t-r- 1))
/w,2,3 = ix{t)y\t - r)x\t - T)n(t -r-r))
(3.2)
The two terms mnii and mms are the examples of the notation used for the four product
terms. Notice that an index being one indicates it is the direct signal, two indicates the
scattered signal, and three means the noise term. The position of the index indicates the
time shift of the term with the first index having no shift, the second being shifted by the
range r, the third by autocorrelation lag t, and the fourth by both range and
autocorrelation lag.
The final simplification is to represent an autocorrelation as Ruu(r) where
the subscript is the random variable which is being correlated.
3.2 THE BIAS OF THE SELF-AMBIGUITY FUNCTION
To evaluate the bias of the self-ambiguity function in the correlation
domain we must evaluate;
^r, r) =< {x(t)+y{t)+n{t)){x{t-r) +y{t-r)+iit -r))\x(t - r) +y(t - T)+n{t - r))’
(x{t -T-r) +y{t - r-r) +«(/ -T-r))>
Instead of writing down all 81 four products it is easier to first identify terms which go to
zero. To do this let us review the assumptions being made;
1) The noise and the transmitter waveform are zero mean random
variables. This means the gaussian moment theorem may be applied
23
[4], namely the second moment of the same variable without
conjugation on one goes to zero, third moments are zero, and the
fourth moment with two conjugated variables may be expanded using
equation 2.29, otherwise they are zero as well.
2) All three random variables: the noise, the transmitter waveform, and the
scattering amplitude are independent of each other.
3) The averaging time is long enough so that any product of the scattering
amplitude involving an odd number of terms or an odd number of
conjugations will go to zero. This is a result of the doppler shift of the
moving target. This does not apply to stationary targets. However, we
ignore these since the power spectrum should allow us to distinguish
moving targets from them.
We use these assumptions to eliminate terms in equation 3.3. Notice the third term of the
received signal is noise. Using this combined with the assumptions above means the
following 48 terms average to zero:
miii3, mii23, mii3i, mii32, mi2i3, mi223, mi23i, mi232, mi3ii, mi3i2, m^i,
mi322, mi33i, mi332, mi333, m2U3, m2123, ni2131, ni2132, ni2213, m2223, Itl2231,
m2232, m23n, m2312, m2321, m2322, m2331, m2332, m2333, m3iii, m3ii2, m3ii3,
m3l21, m3i22, m3i23, m3i33, m3211, m3212, m3213, m3221, m3222, m3223, m3233,
m33i3, m3323, m333i, m3332
The scattering amplitude appears only in the scattered signal so we can eliminate 18 more:
niiu2, mu2i, mi2n,mi22i, mi222, mi233, mi323, m2m, m2n2, m2i22, m2i33, m22i2
m2221, m2313, m3i32, m3231, m33i2, m3321
This reduces the problem to the following 1 5 four products; notice that every subscript
appears two or four times:
mini, mii22, mms, mi2i2, m^n, m2i2i, m22ii, m2222, m2233, m2323, m3i3i,
m3232, m33ii, m3322, m3333
24
Unfortunately, all these terms must be evaluated. First, let’s look at the terms not
requiring the use of the gaussian moment theorem. These all involve the autocorrelation
of the noise times either a correlation of the scattered or direct signal. The terms are:
"^1133 ="^*311 ={x(Ox\t-r)){n\t-T)nit-T-r))=\a\\^(r)R^(-r)
^313 =^*31 ={x(Ox\t-T)){ri\t-r)n(:t-r-T)) = \a\^R^^(T)R^(-T)
^2233 = = {y(0y\t -r))(n{t - T)n(t -
= "432 = (yit)yit - r)){nit -r)n(t -
T-r))= \R^{R„r)R^{r)dR^
ILi-r)
r-'r))= \R^{R,,t)KuiT)dR^
^Rd
JLi-r)
(3.4)
Since the autocorrelation at a positive lag is equivalent to the complex conjugate of the
autocorrelation at the negative lag we see that the terms above appear in conjugate pairs.
This means that when summed the bias due to these terms only occurs in the real part of
the estimate of the autocorrelation function of the scattering amplitude. When the
autocorrelation lags are Fourier transformed to the frequency domain this clutter should
be symmetric about the range axis.
The last seven terms: mmi, mii22, nti2i2, ^2121, 1112211, 1112222, 1113333 all
require the use of the gaussian moment theorem. The first six terms require that it be used
on the four product of the transmitter waveform and the last term requires its use on the
noise. The use of the scattering amplitude being uncorrelated at different ranges will also
be used. Finally, there is no need to calculate m2222 since it is the four product of the
scattered wave so the small value of the scattering amplitude to the fourth power
eliminates it. We are left with:
25
"*1111 =iarK«(^)r +i«rK(^)r
^1122 ~ 1^1
Rd
00
^1212 “1^!
Rd
CO
^^2121 ~ 1^1 J^OO 5 ^)
Rd
00
'”2211 =i«r
^3333 “|^«n(^)| "^|^Mn(^)|
Kwf
«..(r-^)|
\dR
i«..('--^^-^)i
^K(4 dR,
K(r)f
Kui.'^-
R-L.
m.
(3.5)
The terms which correspond to the target are mi2i2 & ni2i2i. If the transmitter waveform
is nearly a delta function, then m2i2i should yield an estimate of the autocorrelation lags of
the range of interest and mi2i2 yields the complex conjugate of the autocorrelation lags at
the negative range of the target. This is the hermite symmetry of the self ambiguity
function [8]. The terms m22ii & mii22 result from the interchangability of the range and
the lag terms. Notice there is a hermite symmetry present here as well. This will be easy
to notice in the following plots of ambiguity functions. The final terms are the self¬
correlation of the direct signal and the noise.
We can now determine the bias of the self ambiguity function. Because of
the bandwidth of the FM radio broadcast (~ 30 kHz), we know that the autocorrelation
function of the baseband modulation is a sharp peak. This means that all terms whose
arguments are either x or r must be concentrated either on the lag or range axis
respectively. From equations 3.4 and 3.5 it is apparent this is every term except those
involving the target(s) or its hermite counterpart. When Fourier transformed, the bias due
to the clutter on the lag axis results in a constant being added to the clutter floor, and the
bias on the range axis results in the spike of the self-ambiguity function at zero range and
Doppler. This means that the bias of the self-ambiguity function will not affect the
26
detection of a target which is not on the receiver transmitter baseline. We verified this
result by carrying out simulations which injected a target into sampled FM broadcasts and
trying techniques which can be derived from the analysis above to reduce the bias. In all
cases, the results showed that there was no difference between biased self-ambiguity
function estimate and the estimate whose bias we had reduced. This means that the
sensitivity of the radar is determined not by the bias, but instead is determined by the
variance of the self-ambiguity function. We discuss this problem in the next chapter.
CHAPTER 4
THE PROBABILITY DISTRIBUTION OF THE SELF AMBIGUITY FUNCTION
The last chapter indicated that the sensitivity of the bistatic FM radar is
independent of the bias of the self ambiguity function. Instead, it depends upon the
variance of the clutter floor of the self ambiguity function. If we were to pursue the
variance through the technique of last chapter, the combinatoric blossoming of terms
would quickly overwhelm us. However, if we make the additional assumption that the
signal’s spectrum is white it is possible to arrive at a useful description of the self
ambiguity function’s probability distribution. We can then make phenomenological
adjustments for the case of the FM radio broadcast. In addition, knowing the self
ambiguity function’s probability distribution also allows a performance analysis of the
radar to be made, and it provides a way to determine the threshold for automatic detection
systems.
4. 1 DERIVATION OF THE PROBABILITY DENSITY OF THE CLUTTER FLOOR
Calculating the probability density of the self ambiguity function seems a
daunting task considering the steps needed to determine the bias of the ambiguity
function for a gaussian signal. However, if we make the following two assumptions
about the received signal then it will be possible arrive at a theoretical model that can be
compared to experimental results.
1) The process is white which means E{x(t)x*(t-r)} = 0 for r 0.
2) The real and imaginary terms of x(t) are uncorrelated and Gaussian
distributed with zero mean.
28
These two assumptions mean the in-phase (real) component x.(t) and the quadrature
(imaginary) component x^(t) are independent of each other and are independent of
themselves except for r=0.
To calculate the probability density of the clutter floor of the we first
consider just the direct signal. Form the new sequence a(t) = x(t)x‘(t-r), where r is the
range variable which is greater than zero. Consider the mean of the new sequence:
E {x. (t)Xi (t-r) + x^ it)x^ (t-r) + j^x^ (t)x,. (t - r) - x^ {t)x^ (t - r)]} = 0 (4.1)
which follows from the independence of the zero mean gaussian random variables. Next
consider the variance of the in-phase component of a(t):
va:{aXt)}=E{(x,(t)Xi{t-r)+x^{t)x^{t-r)f} =
E{xXtfxi{t-rf+2xXt)xXt-r)xXt)x^(t-r)+x^(tfxXt-rf}= (4.2)
E[xXtf}E{xXt-rf}+2E{xXt)}E{xXt-r)}E{x^(t)}E{x^(t-r^^^^
Where the last two steps follow from independence and the assumption the random
variables are N(0,ct). It can be similarly shown that:
var{a^(0} = 2(7" (4.3)
Next, we show that the real and imaginary components of a(t) are uncorrelated:
corr{aXt),a^(t')} = E{aXt)a^(t')} = E{x,.(0JC,(?-r)x^(f)x,.(f-r)-
X, (r)x,. (f - r)Xi (/’ )x^ (t'-r) + x^ {t)x^ (t - r)x^ (f )x,. (t'-r) - (4.4)
(^' )^, (^'“'■) } = 0
Where the last step follows from the independence and zero mean of the random
variables and holds for all sets of (t,t') including t = t'. Finally, we show that real and
29
imaginary components are uncorrelated with respect to themselves except for t = t'. We
show the proof for the real part only since its similar for the imaginary component.
E{{Xi{t)x.{t-r) + x^ {t)x^{t - r))(x,. (t' )x,. (t'-r) + (f {t'-r)) } =
E{Xi {t)x. {t - r)x. it')Xi (t'-r) + x^ (O-r,- (t - r)x^ {t')x^ {t'-r) + t^^t' (4.5)
(0^, it - r)x. {f )x. (t'-r) + x^ (t)x^ (t - (f ' )x^ (t’-r) } = 0
The result follows from the requirement that r^^O. In this case the only way there could be
a nonzero term is if the following two equations are satisfied:
t = t'-r
-t'=t-r
t-t'=t'-t=>t = t'
which is what we wanted to show.
With these properties of a(t) in mind we proceed in a manner quite similar
to the computation of the probability density of the spectral density function given in
Percival pages 220-223 [9]. First, we note from chapter 2 that the ambiguity function is
the power spectrum of a(t). This is estimated through the periodogram which we define
as [10]:
S(f) = Z«(0expf^^^^) (4.6)
n=0 V iV /
We expand the inner summation into its real and imaginary components:
30
This means S(f) = A(f)’+B(f)l Now we wish to show that A(f) and B(f) are independent
Gaussian random variables over the set of fourier frequencies (f an integer between 0 and
N-1). To do this we first show that A(f) and B(f) are uncorrelated over the Fourier
frequencies.
corr{A{flB{r)} = £{A(/)B(/')} = £:|z«/(0cos[^]
^N-\
V A J;
N-\ N-1
•=^zz
1 r’=0 r=0 L
3,(0a,(Oco{^]sin(^)+a,(0a,(Osin(
N-1
=z
a,(Oa^(Ocos(— Jco^ ^
(24' f
N J
(4.8)
(24'] (24' f
«,(0«,(Osir(-^jcos(—
"■"2.‘4¥1sA^
\ N J
+
f=0 _
N
= 0
The third step follows from 4.3, 4.4, 4.5, and the linearity of the expectation operator.
The final step involves the use of the relation derived in exercise 1.4 of Percival [9]:
N-1
N-1
X[cos(2;?f0sin(2;zf' r)] = — X[exp(;2;S(/'+/)) -exp(-;2;zr(/'+/))
f=0
r=0
exp{-j27aif - /')) - exp(;2;7tr(/ - /’)] = 5^ (/ - /’) + Sfj{f + /').
N sin( NttF) N
S, (/) = Y^^4^^sin((A - l);zf ) = jD, (/)sin(( A - \)4)
(4.9)
where D^(f) is one form of Dirichlet’s kernel. Since we are interested in the frequencies
between 0 and N-1 it follows from 4.9 that 4.8 is zero. This is for all except f = 0, but we
are not interested in it since we expect our targets to have some motion. This shows that
that A(f) and B(f) are uncorrelated for any combination of fourier frequencies except for
f,f = 0.
31
Next, we show that A(f) and B(f) are Gaussian distributed using the
Central Limit Theorem. The Central Limit theorem states that the sum of n independent
random variables tends towards a N(p,a) where p=p,+...+|j,„ and aW',+...+a\ as n
approaches infinity [10]. This requires that we show the variables in the summation are
independent. In the case of 4.7, this means it is necessary to justify that ai(t) is
independent of ajO') and similarly for a^(t). Consider aj(t):
a. (r) = X,. (r)x. (r - r) + x (t)x (t - r)
’ ^ (4.10)
a^{t’) = Xi(t')Xi(t'-r) + x^(t')x^(t'-r)
We offer the following heuristic justification based on conditional probability. It is
necessary to show that knowledge of aj(t) does not increase our probability of guessing
the value of a(t'). Since Utt' it means Xj(t')Xj(t'-r) is independent of Xj(t)Xj(t-r) and
x^(t)x^(t-r) is independent of x^(t')x^(t'-r). This follows from the fact that multiplying two
independent random variables by a constant does not change their independence. Since
this is true it quickly follows that aj(t) must be independent of aj(t’). This reasoning can
be extended to showing that all the random variables in the set {ai(0)...ai(N-l)} are
independent. We use the following Bayesian theorem of probability which states that
[10]:
P(An...Aj=P(An|An.i ... AJ...P(A2 | A,)P(Ai) (4.11)
where An ... Ai are events which in the case of a random variable Ai = Xi<Xi which means
P{Ai} is simply the cumulative distribution of x;. Consider the case of N = 3. We have
already shown that the random variables ai(0), ai(l), ai(2) are pairwise independent, but
we must show:
P({ai(0)<xo}n{ai(l)<x,}n{ai(2)<X2})=P({ai(0)<xo})P({ai(l)<x,})P({a.(2)<X2}) (4.12)
This follows easily through adding an additional term ai(t") to the two already present in
4.10. Notice that this third term contains a new random variable ai(t') which will be
32
independent of ai(t) and ai(t’). Using equation 4.1 1 and this independence between terms
it follows that 4.12 holds. This process can be continued inductively to show that the set
{ai(0)...ai(N-l)} is composed of independent random variables. Similarily, this can be
shown to be true for aq(t).
Now we may apply the Central Limit Theorem to the four sums in 4.7.
This means we have A(f) ~ N
+ N
V A
V
N-l
0,JX2cr^ sin'
r=0
N
and B(f) ~ N
cos^
(=0
2#
N
+ N\
where we have used
the fact that the taking the negative of a zero mean gaussian random variable does not
change its distribution. Combining the distributions of A(f) requires that the correlation
between the two normal distributions be computed. However, this is easily seen to be
zero since equation 4.4 shows that ai(t) and aq(t') are uncorrelated for any combination
(t,t'). This obtains A(f) ~ A(0,cr^V^)and similarly it can be shown that B(f)
NiO,C7^^/2N).
Finally, we can determine the probability density of the self ambiguity
function of a white Gaussian signal away from r = 0. First, we note that because A(f) and
B(f) are uncorrelated and Gaussian they must be independent. Next, we scale A(f) and
B(f) as —A==^A(f) ~ N(0,1) and — t-7=B(/)~ N(0,1). Now squaring the
cr V2A cr V2A/'
normalized variables results in two independent chi-square random variables with one
degree of freedom [10]. Next, summing two independent chi-square random variables of
one degree of freedom results in a chi-square random variable of two degrees of freedom.
This gives:
_
INa^
|X(r,/)f
1N(t
2Na
■A(/f +
2iVo-
-Bi/y ~ X2
(4.13)
33
where we have used capital chi squared to represent the ambiguity function squared and
the lowercase chi squared represents the chi-square distribution. Next, if incoherent
averaging (block averaging with no overlap) is used it follows that the chi-square random
variables from each block should be independent of each other, since the data from block
to block is independent. This means the ambiguity function squared will have the
following distribution:
|X(r,/)f=
INo-
2N(7
M 1
(4.14)
Finally, we consider the effect of coherent averaging. In this case, ai(t) and aq(t) are being
pre-averaged before being summed in 4.7. As long as the blocks of a(t) that are being
pre-averaged do not overlap the independence relations hold, but the variance of ai(t) and
aq(t) changes. The averaging is done as follows:
I S-l I 5-1
(0 = + s) = -^,^1 (tS + s) + ja^ {tS + s) (4.15)
where S is the length of the average. It follows that the semivariances of as(t) are both
equal to 2a^lS. This means that to account for coherent integration 4.14 should be
modified as follows:
(4.16)
34
4. 1 .2 COMPARISON TO SIMULATED DATA
Theory must agree with experimental evidence in order for a model to be
useful. We start by examining whether 4.16 actually predicts the true behavior of the
square of the ambiguity function. To begin, a complex sequence whose real and
imaginary components are uncorrelated and gaussian distributed with zero mean and unit
variance is generated in Matlab. Although it is not necessary to attach a sampling
frequency, we use a frequency of 250 kHz since this is the sampling frequency of the
experimental data that is presented later. The length, L, of the sequence that is analyzed
is 131,072 points which means it is approximately a half second of data. The following is
the plot of |X(r,/)|^using a coherent averaging factor S = 250, block size N=64, and the
incoherent averaging factor M = 8. The incoherent averaging factor can be determined as
M = Ll/(SN)J (L J is the greatest integer less than or equal to the quantity in the brackets).
White Gaussian Signal
X lO""
Figure 4.1: Plot of |x(r,/)|^of a White Gaussian Signal. Created from 0.52 seconds of
data sampled at 250 kHz. Block size (N=64) and coherent integration factor (S=250).
Doppler Shift - Hz
U
Range - Km
Figure 4.2: Same as Figure 4. 1 except range 0 is not displayed. Shows clutter floor.
Doppler Shift - Hz Range - Km
Figure 4.3: Plot of the clutter floor for same white gaussian signal except this time
|X(r,/)l was computed with N=128 and S= 250.
36
Figure 4.1 is the desired thumbtack appearance of the white gaussian signal. The next
figure on the following page, figure 4.2, is the same plot except for all ranges greater than
zero. This is the clutter floor that equation 4.16 should describe. Figure. 4.3 is the plot of
the clutter floor except this time the ambiguity function was computed with a different
block size N=128. The next step is to compute the normalization factor and multiply it
times the square of the ambiguity function. Then the histogram of the clutter of figure
4.2 is plotted versus the predicted chi-square random variable of 16 degrees of freedom in
figure 4.4. Similarly, the clutter floor of figure 4.3 is plotted against a chi-square of 8
degrees of freedom in figure 4.5. The results appear to fit the distributions very closely
which gives us confidence in equation 4.16.
However, an even better test for determining the closeness of a probability
distribution to a theoretical model is to use the Kolmogroff-Smimov (K-S) test [10]. The
K-S test is a hypothesis test with null hypothesis that the distributions are equal and the
alternative hypothesis is that they are different. The test statistic is;
q = max|F(x)-Fo(x)| (4.17)
where Fo(x) is the hypothesized cumulative distribution and r(x)is the empirical estimate
of the cumulative distribution. Estimating the cumulative distribution is done by first
sorting the data into ascending order and then assigning the value of i/N to F(x, ) where i
is the position of the data point in the sorted sequence. N is the length of the sequence.
Obviously, it follows from equation 4.17 that if the two distributions are equal then q
should be near zero if a sufficient number of points, N, are used. This is expressed as
[10]:
- 1 a
2
Accept H„ iff q <
(4.18)
38
Figure 4.6: Plot of theoretical vs. empirical cumulative distribution function for data of
figure 4.5, N=2,240. The two curves are essentially indistinguishable.
where a=P{q>clH(,} « is the significance level. A value of a near one indicates
that we have correctly identified the distribution. Figure 4.6 shows that the empirical
cumulative distribution of the data in figure 4.5 is essentially indistinguishable from the
theoretical cumulative distribution of a xl ■ this case, q = 0.00825. A value so small
that for all values of a<l it easily passes 4.18. This means we should accept the null
hypothesis, and it gives strong justification that equation 4.16 correctly models the
simulated data.
4. 1 .2 MODEL COMPARISON TO EXPERIMENTAL DATA
The data that will be used is six different samples each of length 0.5
seconds of FM radio broadcasts taken at a sampling frequency of 250 kHz. The
broadcast stations that were sampled were A1 =106.9, A2=93.3, A3=94.9, A4=94.9,
A5=97.3, and
39
A6 = 97.3. A3 and A4 were sampled on different days as well as A5 and A6. The three
sequences Al, A4, and A5 were sampled at approximately the same time on the same
receiver, and the sequences A3 and A6 were also taken at nearly the same time on the
same receiver. The purpose of choosing these samples was to ensure that any
characteristics that could be due to receiver error could be identified by showing up in
sequences taken at nearly the same time.
Before we begin the comparison to empirical data, we give a brief
justification why we should expect similar behavior. The assumptions that the radio
broadcast is white Gaussian noise is obviously incorrect. However, it is well known that
the autocorrelation time of the broadcast is less than 10 ps [4]. Also, since the spectrum
of the FM broadcasts are known to be approximately gaussian which implies symmetry
about the carrier frequency, it can be shown that the in-phase and quadrature components
are uncorrelated [2]. Thus, we appeal to the same argument that Percival uses to justify
the density of the periodogram for non-white/non-gaussian processes which requires
certain higher-order moments to be finite [9]. It applies because that analysis is being
used here to determine the probability distribution of the ambiguity function, and because
the FM waveform has finite moments of all positive orders [11].
First, we look at the ambiguity plots of the six sequences. The plots are all
of IX(r,f)l not its square since this makes the clutter problems more apparent. By range
we mean the difference between the direct and scattered path. Also, the Doppler shift is
given in Hz instead of velocity since the carrier frequencies change from station to
station.
Out to about the range of 10 km there appears to be quite a bit of
unexpected clutter in all the signals. Signals Al, A4, and A5 all have a strip of apparent
clutter out to 50 km at zero doppler. Since the data was taken at different frequencies, it
is probable that this is due to a constant DC offset in the receiver. However, if we look
on either side of zero doppler the plot appears relatively flat. The flatness of the clutter
plane also appears in A2 and A6. However, signal A3 appears to have considerable
40
clutter. A possible explanation for this is that it is known that the station at 94.9 MHz has
its transmitter located on Capitol Hill which is only 1-2 km from the University where the
data was collected. In contrast, the rest of the stations are known to have transmitters
located further than 20 km away. However, we also know A4 is a sample of the station at
94.9 MHz, but does not appear to have the clutter at ranges greater than 10 km. One
important difference is that A3 was generated at approximately 4 pm opposed to A4
which was sampled at 1 am. It is possible that the change in daily activity along with a
possible change in the transmitting mode at night could make the difference.
With the clutter problems of the signal, we would not expect our model to
hold for ranges less than 10 km. Additional inspection showed that there was still some
clutter in the ranges out to 16.5 km. That the waveforms of the IX(r,f)l‘ resembled the
clutter floor of the white Gaussian signal for this set of ranges can be seen in figures 4.10
- 4.12. Notice that there is still a bit of clutter in Al, A2, A4, A5, and A6 with a
considerable amount present in A3. However, we continue by plotting the histograms of
the data versus xfe • O*'® problem is proper normalization of the data. In the simulation,
we controlled the variance of real and imaginary components of the signal, but here there
is no control. This means it is quite likely that the real and imaginary components will
have a different variance. Also, we know it is not a white noise process. This means the
normalization constant calculated in 4.16 is incorrect because when summing the
variances the cross correlation between terms must be included. Instead, we compute the
normalization constant by forcing the mean of the data to take the same value as the mean
of ajfg. Since, the mean of a chi-square is equal to its degrees of freedom this is a
simple procedure. The histograms can be seen in figure 4.13 on the following page. The
number of data points used in each histogram is 806. Notice that there is definite
mismatch for signal A3. However, Al, A4, and A5 also appear to have a problem with
their means being located in the wrong place. The general shape of the histograms
though appear close to the chi-square density. We gain confidence by looking at A2 and
A6. Notice that
Doppler Shift - Hz
Range - Km
Figure 4.7: Mesh plots of IX(r,f)l of signals A1 = 106.9 Mhz & A2 = 93.3 Mhz sampled at
different times.
X 10
5
Signal A3
3-n
Doppler Shift - Hz Range - Km
Figure 4.8: Mesh plots of IX(r,f)l of signals A3 = 94.9 Mhz & A4 = 94.9 Mhz sampled at
different times.
44
these two data sets nearly match the chi-square distribution. Indeed, when we compute
the cumulative distribution and plot it versus the theoretical cumulative distribution of a
Xis A2 and A6 appear quite close to the theoretical distribution. This is shown in figure
4.14.
However, when the significance level of the Kolmogroff-Smimoff test
statistic is computed we find that it indicates that we should reject the hypothesis that the
distributions are the same for all the signals.
Table 4.1: Kolmogroff-Smimoff Statistics for A1 - A6
Signal;
Ai
A2
A3
M
A5
A6
q
0.1603
0.0429
0.3783
0.1603
0.1438
0.0795
a
00
1
o
(N
0.103
1.3- 10-'“
2.05- 10-'*
o
1
7522- 10'’
The extremely small values of the K-S significance level definitely
indicate that despite looking quite similar the distributions are indeed different. The
reason for the difference is that the clutter introduces points which lie far outside the
region of probability for the chi-square distribution. For example, the probability of a
point being greater than 45 for the chi-square distribution is 1/7205, but we see in figure
4.13 that all the signal’s histograms contain points beyond this value. This shows that this
theoretical model is unable to adequately describe the characteristics of the self ambiguity
function of the FM waveform. This stands even if A3 is considered an anomaly.
However, a technique does exist that can help eliminate some clutter
variation at the expense of increasing the noise variance. The idea can be seen by
recalling from section 3.2 that the clutter due to noise is symmetrical about the range axis
in the frequency domain. There is strong evidence of this in the Figures 4.7, 4.8, and 4.9.
It is also known that the target signal is a complex sinusoid and should therefore appear
asymmetrically. Normally, subtracting two random variables is not advised because it
increases the variance due to noise. However, in this case it is apparent that the clutter
45
Signal A1
Signal A2
lOOi - ^ -
300
200
100
0
0 50 100 150 200
Signal A5
150
100
50
0
0 50 100
100
50
0
0 50 100
Signal A3
Signal A4
Signal A6
Figure 4.13: Histograms of the IX(r,f)P for ranges 16.6-50 km and frequencies -500 Hz to
-15 Hz.
contribution is dominant so the overall effect is to decrease the variance. This can be
seen in the following calculation:
EiiSif) - S{-f)f - E{S(f) - S(-f)f} = E{S{fy } - 2E{Sif)S(-f)] +
E{S{-ff } - E{S{f)}^ + 2E{S(f)}E[S{-f)}- E{S{-f)}^ = (4.19)
(jf +0-2 -cov{5(/),5(-/)}
46
A1
A2
A6
1
0.5
0
0 20 40 60
Figure 4.14: Cumulative Distribution corresponding to data and theoretical curves of
figure 4.13.
where a, is standard deviation of S(f) and a^is the standard deviation of S(-f). Notice that
if the covariance is quite large, then the overall variance of the resulting random variable
is reduced.
Since the distributions are correlated, it may seem quite difficult to
calculate the resulting probability density function. However, it can be shown that if we
take the square root of the IX(r,f)l^ then the result is a chi density of 2M degrees of
freedom. Its mean, variance, and distribution are given by [10]:
2M + 1
r(M)
Xi
2M-\
M
exp(-y)t/(x) (4.20)
47
where r(x) is the gamma function and U(x) is the unit step function. The chi density can
be approximated by a Gaussian with mean and variance given by equation 4.20 [12]. It is
well known that if two gaussian random variables are summed then even if they are
correlated the result is a gaussian [10]. This means IX(r,-f)l - IX(r,f)l will be a Gaussian
with a mean close to zero and variance given by equation 4.19.
First, we verify that this works with the simulated data. Figure 4.15 shows
the result of subtracting the positive frequencies from the negative frequencies for the
white Gaussian signal. Notice that it appears to have increased the variance while
making the mean near zero. Figure 4. 16 shows the histogram from the data in figure 4. 1 5
and it shows the curve representing the theoretical gaussian model with mean zero and
variance given by 2a = 2(2M-p^) where p is given by equation 4.20 and M is the number
of incoherent averages (M=8 in this case). Notice that the data in figure 4.15 has to be
properly scaled by the square root of the factor given in equation 4.16 for the model to be
applied correctly which is the reason for the different scale in figure 4.16. Next, figure
4.17 shows the plot comparing the theoretical vs. the empirical cumulative distribution.
The distributions appear to nearly identical from the plot. The q = 0.0157 which results in
a significance level for the 806 points of a «1. Let’s see if this procedure achieves the
desired result with experimental data. Figure 4.18 shows the negative frequencies minus
the positive frequencies for ranges 16.5 - 50 km for signals A1 & A2. The key fact is that
the large hump is gone in A1 and that both data sets look more like identically distributed
noise than in Figure 4.7. Figure 4.19 shows histograms for the negative minus positive
frequency data sets for all of the experimental signals that we have looked at. Notice the
definite improvement with all except A3 being closely approximated by the theoretical
Gaussian curves. The parameters for the densities were estimated from the data since we
already know that the normalization of equation 4.16 is not adequate for the
8
Figure 4.15: Plot of the negative frequencies minus the positive frequencies for the white
Gaussian waveform.
Figure 4.17: Cumulative distribution function of the theoretical (smooth curve) vs
empirical cumulative distribution of figure 4.16.
Frequency - Hz Range - Km
Signal A2
Figure 4.18: Mesh plot of IX(r,-f)l - lX(r,f)! for signals A1 & A2
50
A1
jj
A
h
Q| . . . . . . I
-5 0 5
A3
A2
1
ttw.
Ql . . Ill . . . I
-5 0 5
A4
jI
ikbfw _ _
5 0 5
A6
_
Ql . . LI . . - _ I
-5 0 5
Figure 4.19: Histogram plots of IX(r,-f)l - IX(r,f)l for signals A1 - A6.
experimental data. Figure 4.20 shows the cumulative distributions that result from the
histograms and the theoretical densities. Notice that all except A3 are indistinguishable
from their theoretical distributions. Indeed, the K-S statistics are all close to one except
for signal A3. However, the values of the significance of the K-S statistic should be
taken rather heuristically since we estimated the parameters of the theoretical distribution
from
Signal
Al
A2
A3
A4
A5
A6
q
0.0266
0.0309
0.0791
0.0266
0.0221
0.0238
a
0.639
0.429
8.33E-5
0.639
0.910
0.803
the data. This seems to indicate that the distributions are close to being Gaussian. Also,
the histograms of figure 4.19 show that the histograms of the data are quite similar to that
of the simulated white Gaussian signal. For these reasons, and because the Gaussian
distribution is simple to calculate we use it as the basis for establishing the detection
threshold.
4.2 DETERMINATION OF THE DENSITY WITH A TARGET PRESENT
The next step is to determine the distribution of IX(r,f)l" when a target is
present at a given range and doppler frequency. In the case of high SNR, the value of
IX(r,f)Pis relatively easy to compute for a single target. First, we rewrite the signal, x(t),
as jc(0 = u(t) + ae\p(j2;rut)u(t -r^). This means a(t) becomes more complex.
However, as long as T^r„ the probability density of the clutter floor remains quite similar
to equation 4.16 with only the variance changing. Now when r = r„ and we assume the
energy of the scattered signal is much larger than the variance of the clutter floor of the
ambiguity function it is possible to approximate value of the ambiguity function at r„.
This assumption allows us to write the ambiguity function at the point of the target as:
, w-i 2 ^
|X(r„,u)|" « rj| e\p{-j27tt(v-f / N)) (4.21)
1=0
which evaluates to 4a^a''N^ if vwf/N. Next, this value must be scaled by the quantity
given in equation 4.16. The result is 2a^NMS. However, we recognize that NMS as the
length of the sequence. We rewrite this as NMS = Tf3 where T is the time length in
seconds and
52
A1 A2
Figure 4.20: Cumulative distributions corresponding to figure 4.19.
fjis the sampling frequency. We will call this the time-bandwidth product of the signal
processing algorithm, A^. Now in the case of a white noise signal the time-bandwidth
product of the signal processing algorithm is equal to that of the transmitted signal, A^..
This means that for a white noise signal the value of the self-ambiguity function when a
target is present is the ratio of the energy of the scattered signal to the power per hertz, N^.
= P/fj, of the direct signal, 2E^/Nc. This shows that there are two basic ways to increase
the sensitivity of the radar. The first is to increase the averaging time. The second is to
increase the bandwidth of the transmitted signal. In fact, it will soon be shown that the
time-bandwidth product of the FM broadcast signal is insufficient to realize the gain
required to detect aircraft.
53
To show this for the bistatic FM radar we compute an approximate form of
the probability density of IX(r,f)P when a target is present. To do this we reconsider
equation 4.21 and rewrite the inner term of the summation
yv-i
= Yj {[“(Om* (t-ro) + cai{t)u {t - 2ro ) •
^ 2
exp(-;2;n/(t - r^)) + a expU'2^^)\u(t - ro)| + a' exp(j2;rvt)txp(-j2;rv(t - Tq)) •
u{t - r^)u(t - 2ro)]exp(^-
- r^f] = A(f) + 2aa^N + jB(f)
r=0 ^ **
N-\
<=o
/
u(t)u(t-ro)txp\-j2m —
(4.22)
N-\
x(t)x*(t ■
f
■ro)exp\-j2a —
where A(f) and B(f) are given by equation 4.7. It has been assumed here that the
scattering amplitude is sufficiently small that the variance of A(f) and B(f) is not affected
by the extra terms involving a. Now when we square the real and imaginary parts the
result is a noncentral chi-square density [2]. The noncentral chi-square density is defined
as the distribution that describes the random variable:
(4.23)
/I=I /2=1
where the x„ are independent zero-mean Gaussian variables with common variance ol
The C„ are the constants and we have also defined the quantity Q which is the parameter
along with a and N which characterize the distribution of y given in the unnormalized
form as:
fiy)-
2a^ VQ
N-2
y \ 4
exp[-(y + Q) / (2(7^
<J
(4.24)
where In^.^Y) is the modified Bessel function of first kind and order N/2-1. The case of
interest is the noncentral chi-square that results when the value of the square of the
54
ambiguity function at the range and doppler frequency of the target is computed with both
coherent and incoherent integration. Coherent integration scales the variance of A(f) and
B(f) as described earlier by a factor of 1/S. Incoherent integration requires that we
rewrite the step as:
1 M-\ 1 M-\ 1 M-\
«/) = 17 Z S,(/) = ( A (/) + Z(fl, (/)) =
M /=0 /=0
A,(/) lacr^N , ^(B,{f)V
h^sfM Vm ^ 1^1 VmJ
(4.25)
This means the variance is <3^ =2cr''N/MS and that Q = 4a'a'‘N\ Next we normalize the
S(f) by its variance (notice it is the same as the normalization parameter in 4. 1 6). This
results in the normalized noncentral chi-square distribution with noncentrality parameter
X = = 2a^NMS = This results in the standard form of the noncentral chi-
square density with noncentrality parameter X [2]:
fiy) = \^yl exp(-(y + X) /
//,. =A + 2M (4.26)
0-2 =4;i + 4M
where N has been substituted by 2M because we summed 2M independent noncentral
chi-square densities. Figure 4.21 is a plot of the normalized noncentral chi-square density
with a=0.02, N = 64, M = 8, and S = 250 contrasted with the chi-square density of 2M
degrees of freedom. Notice that there is almost no overlap between the densities which
means probability of detection, Pp » 1, and probability of false alarm, PpA ® 1- Next, we
consider the density of the modulus of the ambiguity function, IX(r,f)l, at the target. The
55
Figure 4.21: Plot of noncentral chi-square with 1=102.4, 2M = 16, versus a chi-square of
2M = 16 degrees of freedom.
density is simply the noncentral chi density given by [2]:
fiy)^y
2M-2)
(4.27)
where X is the same noncentrality parameter as before and I^ iCy) is still the modified
Bessel function of the first kind. The mean and variance of the noncentral chi density can
be approximated as [from equation 26.4.38 found in reference 12]:
^ = + a = A + 2M b = X/a
CT- ^{^)-^{%b + {\ + b){\-lb)\ + 0{a-^)
L 0^3
(4.28)
56
However, if the noncentrality parameter is moderately greater than the degrees of
freedom, 2M, then we may take b»l. This means we can approximate the mean as
pa-\//i- + 2M-l and a’«l. Although, first order correction could be completely made
this will be quite close if the condition above is met (this will be the case for any targets
we wish to detect because the mean must be greater than the mean of the clutter). In
addition this approximation lowers the estimate of the mean and increases the value of the
variance. This is good because it means the true values will yield a better probability of
detection than the approximation will.
Because the chi density can be approximated as Gaussian, it follows that
the noncentral chi density can be approximated as the Gaussian density,
N(VA + 2Af - 1 ,1). This means that when the positive frequencies are subtracted from
the negative frequencies that if a target is present its distribution is given as a Gaussian
with mean and variance of:
// = Vl + 2M-l-V2-
r(M)
= 1 + 2M-
V2-
r(M)
(4.29)
However, just as the noncentral chi density’s mean and variance can be approximated, we
can do the same with the chi density. Abramowitz and Stegun (equation 26.4.34) give
the approximations as [12]:
/i = {l + [32M(2M - 1)]-' }^2M - ^ + 0{{2My^'^ ) «
1
1
1
(4.30)
cr = — -
2 16M 64M^
+ - ^ + G((2M)'")«-
512M-’ 2
Where we have used the assumption that M>2, which means that we are overestimating
the mean and variance of the clutter. This good because when subtracted from the
noncentral chi density it makes the mean smaller than it should be and the variance larger.
57
Figure 4.22: Plot of gaussian with mean «6.84, variance »1.5 versus a gaussian with
mean »0 and variance «1
Thus, it will generate a probability of detection which is slightly less than the true value.
Using 4.30 we may rewrite 4.29 as:
//»V>l + 2M-l-V2M (4.31)
4.3 IDEAL OPERATING CHARACTERISTICS OF THE WHITE GAUSSIAN
SIGNAL
The previous two plots give us an intuitive feel for how the operating
characteristics of the radar can be determined from equations 4.16, 4.20, 4.26, and 4.29.
The figures show the detection of a target can be posed as binary hypothesis-testing
problem with hypotheses: Hg - No target present and H, - target present. The performance
of the radar is determined by finding the P^, the probability of detection, and Pp the
probability of false alarm.
58
Although many possibilities exist, we choose to analyze the performance
of the radar through a simple threshold system. The system is simply:
\X{r,ff > X, or
X(r,-/)|-|X(r,/)||>X,
(4.32)
where the second equation is a two sided threshold which allows detection of a target at
either a positive or negative frequency. The probability densities that describe the first
test are given by equations 4.16 and 4.26. The mean and variance of the Gaussians that
describe the second test are given by 4.20 and 4.29. The probability of false alarm for the
simple threshold system is given by = P{ |x(r,/)|^ > X,l H,,}, and the probability of
detection is given by P^ = P{ |x(r,/)p > X, 1 H,}. These can be easily determined after
X, is chosen by integration of the appropriate probability density. However, in radar it is
desirable to chose a maximum value of P^ and then to choose the threshold level to
maximize P^ which is called the Neyman-Pearson strategy [2]. In this case, it is obvious
that the threshold value(s) that will achieve this can be determined either from the
percentage points of a chi-square or the standard deviation of a Gaussian.
First, we consider the performance characteristics using the square of the
ambiguity function. The distribution when no target is present is the Zim a
target it is the noncentral chi-square of equation 4.26. Figure 4.23 is the plot of the P^ vs.
Pf with values of 201og(a), the ratio of the scattered signal’s power to the direct signal,
given in the plot. Notice that for values of 201og(a) < -38 dB that detection with a
reasonable value for P^ is unlikely for the given values of M, N, and S. A good question
to ask is how does changing M, N, or S affect the probability of detection. First, we
notice that changing N or S does not change the density of H^, however, changing M
increases the mean and variance of the Zim which is the density of H^. The effect of
increasing the values N and S for the noncentral chi-square can be seen
59
Figure 4.23: vs. P, for the White Gaussian Signal with M = 8, N = 64, S = 250, and
201og(a).
to directly increase the noncentrality parameter X = 2a’NMS. Notice that the sampling
frequency, f^, divided by NS gives the frequency resolution of the ambiguity function. If
we keep the same frequency resolution, then any changes of N or S which are governed
by the equation NS = f^ will not change the probability of detection. Now the length of
the time series from which we generate the ambiguity function is NMS. This means for a
constant time-bandwidth product the only interesting tradeoff to investigate is between N
or S and M. It is obvious that since the time-bandwidth product is constant that it is ideal
to pick M as small as possible because this will decrease the variance of the clutter.
Figure 4.24 shows this for = 128,000, and P^ = 10 ^
Next, we determine the performance characteristics for the threshold
described by the second equation of 4.29. Notice that it requires a two sided threshold.
However, because of the symmetry of the gaussian random variable we only consider the
one sided case of the target being located at a negative frequency. This test requires that
60
Figure 4.24: Plot of vs. a for constant P, = 10'*' and = 128,000. M is indicated on
the plot.
M>2 because of the approximation of the chi density by a Gaussian. However, it is
useful for two reasons. First, the Gaussian probability density is simple to work with
especially in determining the Pf. Second, section 4. 1 .2 showed that the actual FM radio
data appeared to be more accurately characterized by a Gaussian density if the
frequencies were subtracted. Figure 4.25 shows the same plot as figure 4.22 except the
densities are now Gaussians with means and variances described by equations 4.26 and
4.29. It is useful to determine a formula which can be used to determine the minimum
detectable scattered signal as a function of P^ P^ M, N, and S. This task can be
accomplished by considering the two Gaussian densities. First we set the false alarm rate
by setting the threshold level at a multiple of the standard deviation of the density that
corresponds to no target present. Next, we set the minimum probability of detection by
subtracting a multiple of the standard deviation from the mean of the distribution
corresponding to the target. We then
61
Figure 4.25: vs. for the White Gaussian Signal with M = 8, N = 64, S = 250, and
201og(a).
equate the results. This results in:
li-nG'=k<y (4.33)
where and a' are given by equation 4.3 1 and a is approximated as one from equation
4.30. We can then substitute these values into 4.33 which gives the following result:
A = ] + 3.464nVM + 2.828jfe Vm + + 2A49nk + k ' (4.34)
However, we may rewrite X = 2a^NMS = 2a^A5. This means we can now solve for the
minimum scattering amplitude that is detectable as:
M>2
=
1 + 2>A64n^[M + 22>l%k^ + \5n^ + 2.449«il: + k‘
2A.
(4.35)
The probability of detection is given by Pj = erf(x)+0.5 and the probability of false alarm
is given by P, = 0.5 - erf(k) where we are using the definition [10]:
The following table shows how the minimum detectable scattering amplitude decreases
as the observation time increases for a single target:
Table 4,4: Minimum detectable scattering amplitude for WG sampled at 250 kHz
T (seconds)
M
Pa
Pr
201og(a„.„)
0.5
8
0.841
10'"
-34.71
0.5
4
0.841
-35.50
1
16
0.841
-36.808
1
4
0.841
-38.51
2
32
0.841
10''
-38.79
2
4
0.841
10'"
-41.52
4
64
0.841
10^
-40.66
4
4
0.841
lO*"
-43.742
8
4
0.841
-46.75
16
4
0.841
-50.6
Notice doubling the observation time T doubles the sensitivity of the radar because it
doubles the time-bandwidth product when no change in the number of incoherent
averages occurs. This is what we would expect because this doubles the energy of the
scattered signal while the increase in NS appropriately scales the clutter power to keep its
contribution constant. This table also indicates that even at a bandwidth of 250 kHz that
the time length of the series must extremely long in order for the SDRs of -60 dB to be
detected.
63
4.3 ADJUSTMENTS FOR THE FM WAVEFORM
The previous section discussed the ideal operating characteristics for a
radar with a white Gaussian signal with no clutter besides itself and a single scattered
signal. Unfortunately, this is not an ideal world and neither the signal of interest, FM
broadcasts, is not a white Gaussian process and of course there are buildings, cars, and
the earth all of which introduce clutter into the detection problem. Finally, there are
receiver noise sources which will also increase the variance of the clutter floor. In this
section we discuss how the non-ideal world effects our calculations and we propose a
crude phenomenological correction to our equations which should allow order of
magnitude estimates to be made.
The non-ideal factors of the FM broadcast can be broken into three
categories:
1) The FM broadcast is not White Gaussian noise. However, the Central
Limit Theorem can still be applied because over time intervals longer
than approximately 20 microseconds the data is uncorrelated. The
difference is that the calculation of the variance must be modified to
account for correlation. This can also be viewed as bandwidth
inefficiency. If the broadcast were sampled at a slower rate of 30 kHz,
then it would be much closer to being a white process.
2) The FM broadcast is not just the direct broadcast but also includes
scatter from the earth, buildings, and cars as well as the airplane we are
interested in. This scatter will be mostly concentrated near zero range
so it should not introduce much bias. However, it will increase the
overall variance of the process.
3) The target of interest is an aircraft and in general is moving (possibly
accelerating). This means the scattering amplitude is best modeled as
a Rayleigh random variable with uniformly distributed phase [8].
First, we examine the increase in variance due to using a signal with nonzero correlation
for lags other than zero. It can easily be shown that the variance of the sum of two
random variables which are correlated can be given as:
64
E{{x + yY} = E{x-}+2E{xy} + E{y-] =
<7] + Ircr^cr^. + = 2cr^ (1 + r)
where r, the correlation coefficient, can be inferred from 4.37 and is between 0 and 1. The
correlation this introduces can be represented as follows:
Ict^N
MS
■+r =
2N
MS
+
MS
2N
r) =
(4.38)
where cr^ is the rescaled value of the semivariance of the signal. Notice that this can also
be expanded to include the increase in variance due to physical clutter such as buildings,
cars, ect. This does not change the value of O that is generated by equation 4.25. This
means that these two effects can be accounted for by adjusting the noncentrality
parameter as follows:
X=Q.Ig‘1 =2a^As{a* I (j‘1) = 2a^ (4.39)
where P is the ratio of the true transmitted signal’s variance, to the rescaled variance
given by 4.38. We suggest here that it is best to empirically determine P from repeated
observations of radio broadcasts. We can do this in two parts. First, we approximate the
increase in the semivariance caused by the clutter. This can be done by using a
directional antenna or adaptive beamforming to isolate the scattered signals. Then the
semivariance due to physical clutter only can be estimated. This can then be added to the
direct signal’s variance to give a in equation 4.38. Another way is to use a worst case
scenario estimate for the physical clutter.
^ cr"* 2
Now we can estimate the ratio B - —r = P{\ + ^ where ^ represents the
ratio of the physical clutter to the broadcast signal’s variance. Next we use the linearity
of the expectation operator to determine p. We do this by first estimating a" from either
65
the real or imaginary component of the complex data. Then we use this estimate to scale
the ambiguity function by equation 4.16. Here it is now possible to estimate p either
from the mean of the chi-square distribution or from the variance of the Gaussian density
which is empirically generated from the clutter floor. This means we have:
XlM
2M 2(2A/-/y;)
P
(1 + ^^
(4.40)
The following table lists the estimates for P of signals A1 - A6 for K= 128,000 and M =
8.
Table 4.5: List of p. the bandwidth inefficiency factor, for signals A1-A6
Signal
B _ __
A1
0.12
A2
0.16
A3
0.07
A4
0.12
A5
0.23
A6
0.22
Average
0.15
This suggests P is probably close to 0.15 and that if we take a conservative value of ^ =
0.5 this shows that p is close to 0.067. It is interesting to note that the ratio of the true
bandwidth occupied by the signal, 30 kHz, to the sampling frequency, 250 kHz, is 0.12.
This seems to indicate that yff is a parameter which determines bandwidth inefficiency.
This value can now be used to estimate the corrected value of the noncentrality
parameter. No change occurs to the chi-square and chi densities of the clutter. Figure
4.26 shows the vs. P, for the same case as 4.24 except the noncentrality parameter is
corrected for the value of P=0.067.
66
Figure 4.26: Pj vs. for T = 0.5 seconds and M = 8 with p=0.067.
The correction to X means that equation 4.35 is:
M>2
oc„
1 + 3A64n^fW + 2.828i^ Vm +\5n^ + 2A49nk + k "
(4.41)
The results for p=0.067 appear in table 4.6 on the following page. From this table, it
appears that it would be possible to detect a target whose power relative to the direct
signal is -38.8 dB. However, we must recall that the aircraft will move over 4 km during
this time period and could change its velocity by 157 m/s if its limited to a one g
maneuver. This means table 4.6 and equation 4.41 should still only be taken as lower
bounds for the minimum detectable scattered signal.
67
Table 4.6: Minimum detectable scattering amplitude for FM signal sampled @ 250 kHz
T (seconds)
M
P,
P,
0.5
8
0.841
10'^
-23.0
0.5
4
0.841
10'
-23.8
1
16
0.841
10'
-25.1
1
4
0.841
-26.8
2
32
0.841
-27.1
2
4
0.841
10'
-29.8
4
64
0.841
-28.9
4
4
0.841
10'
-32.8
8
4
0.841
-35.8
16
4
0.841
-38.8
However, there exist multi-threshold tracking algorithms that could possibly be used to
approach this lower bound. Also, for short time periods table 4.6 should be close to the
real situation because most aircraft are limited in velocity and acceleration.
Figure 4.27 is a plot of the minimum detectable signal for the white
gaussian signal where the plot is IX(r,-f)l-IX(r,f)l with a target located at 21 km and -125
Hz (a fourier frequency). The length of the time series was 0.512 s with = lO ’ and =
0.5. Figure 4.28 is the plot of IX(r,f)l^ for the same signal which shows that is important
to have the statistics in mind when making judgments about detectability. Figure 4.29 is
the plot of signal A3 with an computer generated scattered signal injected at the level of
minimum detectability, P^ = 0.645 and P^ = 10 ’ where we have taken p=0.15. Figure
4.30 is signal A3 with a scattered signal whose SDR, a, is 3 dB below the signal injected
to make figure 4.27.
68
Frequency -Hz -500 10 Range - Km
Figure 4.27: Plot of IX(r,-f)l-IX(r,f)l for A3 = 128,000, M=8, P, = lO', and P, = 0.5 for a
white gaussian signal with target at 21 km and -125 Hz.
Frequency -Hz -500 10 Range - Km
Figure 4.28: Plot of IX(r,f)|- for A3 = 128,000, M=8, P, = lO', and P, = 0.5 for a white
gaussian signal with target at 21 km and -125 Hz.
69
Figure 4.29: Plot of IX(r,-f)l-IX(r,f)l for = 128,000, M=8, = lO’, p=0.15 and P, =
0.645 for signal A2 with target at 21 km and -125 Hz.
Figure 4.30: Same as figure 4.29 except a was decreased by 3 dB.
CHAPTER 5: ADAPTIVE BEAMFORMING/DIRECTIONAL ANTENNAS
The last chapter discussed the limitations of the monostatic FM radar with
one receiver. It is obvious that isolating the scattered signal from the direct broadcast
signal should improve the sensitivity of the radar. Indeed, this is the impetus for using
the Cascade Mountains as a barrier to shield the receiver at Manastash Ridge from the
direct FM broadcast. However, it is possible to eliminate through spatial isolation the
direct signal without having to use two receivers which are physically separated by a
barrier. This can be done either through using antennas with directional gains or a
number of omnidirectional dipoles whose signals are phase shifted and then summed. In
the case of dipoles, the phase shifting can adapt over time to provide a time-varying
optimal response to the radio environment. Hence, the name adaptive beamforming.
However, this differs from the use of a physical barrier because a physical barrier will
also attenuate random scatter of the direct signal by buildings, cars, and the earth. The
question of interest is how does physical clutter limit a directional antenna/adaptive
beamforming monostatic FM radar. We will limit our discussion to adaptive
beamforming because limited experimental data was available to explore this option.
Also, the results of using directional antennas should be quite similar.
5.1 SYSTEM OVERVIEW
The basic idea of adaptive beamforming can be seen in figure 5.1. An
array of more than one antenna is arranged in linear fashion which causes the phase shift
of the incident radiation between antennas to be given by (t)=kdsin(0), the electrical angle
[13]. It is also possible to use two dimensional arrays which would of course allow
cancellation of interference in two dimensions. We assume that the direction of the
interference is unknown, but is stationary over the convergence time of the adaptation
A.
H - d - ►
Figure 5.1: Geometry of adaptive beamforming array
interest. This means that if we let the signal vector be defined as x(t) = [x,(t) Xj(t), ... ,
x„.|(t)]^ then we should be able to attenuate the interference by minimizing the variance of
e(t) = x„(t) - w"(t)x(t) (5.1)
where w’^(t) = [w,(t), w^Ct), ... , w„ ,(t)] is a set of weights which can slowly vary in time,
Xo(t) is viewed as the desired response, and e(t) can be viewed as the error signal.
Traditional adaptive beamforming also places a constraint on the weight vectors to
preserve a unity gain in a predetermined look direction. However, this also requires a
minimum of three receivers to implement. Hence, we will concern ourselves only with
minimizing modulus square of e(t).
In general, there are two traditional approaches of approaching the
minimization problem. The first is to view the squared error le(t)l^ as a multivariate
72
function of the adaptive filter’s weights. The instantaneous gradient can be formed which
is then used to update the filter weights so that they follow the path of steepest descent.
This is the reason the oldest version of the algorithm is called the method of steepest
descent [13]. We will use a more widely used algorithm which is recursive and is easy to
use called the least mean squares algorithm (LMS). The second general approach to this
problem is to recognize that this problem is the same as a linear least squares problem.
This can be implemented using algorithms such as recursive least squares, QR-
decomposition, ect [13]. However, these algorithms are more computationally complex
and are harder to implement than LMS.. The reason the recursive least squares and QR-
decomposition algorithms exist is because they offer better convergence, but it was found
the normalized LMS algorithm (a close variant to LMS) was sufficiently convergent for
this problem. For these reasons, and because the experimental data that will be used only
comes from two antennas we chose only to examine the LMS algorithm and its close
relative the normalized LMS algorithm
The complete derivation of the complex LMS algorithm is given in
Haykin where it is shown that it results in three basic steps [13]:
1) Filter output: g(t) = w"(/)x(r)
2) Estimation error: e(t) = x^Ct) - g(t)
3) Tap-weight adaptation: w(r -i- 1) = w(/) + fjs.{t)e* {t)
This is simple to program especially in the case of two antennas because in that case the
vectors reduce to scalars. The quantity p is the step-size parameter, and determines the
convergence properties of the LMS algorithm. Picking a value of p which is to large will
cause the algorithm to diverge and picking a value which is to small will make the
adaptation time to long to be useful. It was found through trial and error that using this
form of the algorithm on the set of radio broadcasts that were sampled did not yield
useful convergence properties. However, if we modify the LMS algorithm by
73
normalizing the step-size parameter by the squared magnitude of the data vector, x(t),
then the algorithm satisfactorily converged. Thus, the normalized least mean squares
algorithm that was actually implemented in the following analysis modifies step 3 by
[13]:
3) w(r -h 1) = w(r) +
||x(r)|| + k
x(0e*(0
where ju is the normalized step size and k is constant which insures that the denominator
cannot become to close to zero. The values of ^ and k that were used were 1 and 100,
respectively.
In contrast to most adaptive filtering problems, the result that we want is
not the filter output. Instead it is the error sequence that is of interest. The filter adjusts
its tap weights so that its output closely models the main components of x„(t) which
should be the direct signal and any strong clutter sources. Subtracting the two should
leave scattered signals which arrived from different directions than the direct signal. This
is of course the error sequence of equation 5.1, and is where we will start the analysis of
how adaptive beamforming will affect the sensitivity of the FM radar.
5.2 ANALYSIS OF ADAPTIVE BEAMFORMING EFFECTS ON SENSITIVITY
To analyze the effects that adaptive beamforming has on the sensitivity of
the FM radar system we limit ourselves to the simple case of two receivers. We do this
because the direct signal is the major source of “interference” in this problem and because
it should be easy to generalize from this example to more than two receivers. Also, the
currently available experimental data was for only two receivers.
First, we write the two signals without including the noise terms because
we will assume the clutter, direct, and scattered signal are all much greater than the noise.
From chapter 2, we have:
74
XQ{t) = u(t)+ j^^(r,t)u(t-r)dr
00
x^(t) = e^^u(t)+ j(i>2(r,t)u(t -r)dr
R<,
(5.2)
where ^ is the electrical angle, and 0,(t,r) and <l>2(t,r) are the scattering amplitudes of the
clutter and targets which are different because of the phase shift which is presumed
unknown but different from the direct signal. If u(t) is significantly larger than the
clutter, then we can assume the single spatial weight should converge to approximately e'
, but there will be some error, e. This means the resulting error term e(t) which now we
will call y(t) is given as:
y(t) = su(t)+ (5.3)
Next, we form the product y(t)x*(t-rQ) as is done in the previous chapter’s analysis. This
results in:
CO
y(t)x\t - To) = m{t)u (t - To) + £ jd)*(r'-ro,OM(OM*(^ - r'-rQ)dr'+
Rj
00
Jm* (f - ro)M(t - r)[<I), (r,0 - (r,r)](ir + (5.4)
Rd
00 00
J \u{t - r)u {t - 2(^d)]dr
^d
This can now be compared to the sequence a(t) as defined in the previous chapter. Notice
that if we assume l£l«l, then the dominant terms in the sequence are:
75
(5.4)
OO 00
j ju(t - r)u (t - r'-r^ )<t>^ {r'-r^ , 0[<I> i (r, 0 “ 2 ^
This obviously shows that the clutter contribution due to the direct signal with itself is
dramatically reduced if e is small. Instead, now the important contribution to the variance
of the probability distributions is the physical clutter. This shows that in the case of near
ideal adaptive beamforming that the FM radar will be limited by the physical clutter in
the environment. Also, since the target(s) of interest is implicitly included in the
scattering amplitude, we see that another effect is that it can reduce the scattering
amplitude of the target of interest by:
= (5.5)
where <1).^ is the electrical angle of the target of interest and (|) is the electrical angle
corresponding to the direct broadcast. The value of k ean be between 0 and 2. This also
occurs for any physical clutter in the problem and shows that adaptive beamforming
could actually increase the contribution to the variance by some of the physical clutter.
Now we use the factor ^ from before to represent the ratio of the physical clutter to the
broadcast signal’s power as given by:
^ = std[RQ{y{t)x*{t-rQ)]]l (5 6)
V2crf = St J[Re {x(/)x*(t-ro)}]
then we may write the variance of the unnormalized chi-square as:
MSr 2ecT:N
MS MS ^ MSj3
(5.7)
Then we can rewrite the noncentrality parameter as:
76
X = m a] =2K-a^Tpi (5.8)
where k is the loss factor due to the distortion of the beam in the direction of the target, ^
is the gain due to cancellation of the direct signal, and P is the loss factor as defined
before. This means we can rewrite equation 4.46 for the case of adaptive beamforming
as:
, 1 + 3.464nVM + 2.828jt Vm +I5n^ + 2AA9nk + k- .
M>2 aL= - (5-9)
Notice that if < 1 for most electrical angles then adaptive beamforming should be
useful barring a dramatic change in p.
5.3 EXPERIMENTAL RESULTS
The previous section showed that the key to understanding the
improvement offered by adaptive beamforming or directional antennas is the
characterization of the typical physical clutter. The best way to do this is simply to turn
on the receivers and make measurements of the clutter environment using adaptive
beamforming. It should then be possible to estimate
Estimating ^ requires that we first find the standard deviation of the direct
signal plus the physical clutter and then use adaptive beamforming to estimate the
standard deviation of the physical clutter only. This results in the following estimate:
direct __ ^ scatt+direct
direct
k - 1
(5.9)
where is the standard deviation of the u(t)u*(t-r(,) where u(t) is the direct signal.
77
Only one set of data was available to determine whether adaptive
beamforming improves the sensitivity of the radar. The results were encouraging. The
data is signal A2 and a new data set B2 which was synchronously sampled by a different
receiver. The baseline was 1.5 meters. After the data was passed through the adaptive
beamforming algorithm the standard deviation of the real and imaginary parts were
computed. It was found that they were both approximately 10. In contrast, signal A
which was used as the reference signal had a standard deviation that for both in-phase and
quadrature was 30. This gives a ratio of 1/3. This meant that ^ should be 1/9. However,
a calculation of ^ for r^ = 21 km showed that it remained about 1/3. This indicates that
the contribution of the physical clutter to the standard deviation is more than is predicted
and probably occurs because of more correlation due to physical clutter.
However, ^ being close to 1/3 shows that some improvement is possible.
Notice that this means it could improve the sensitivity by a factor of 1/9 which is about
9.5 dB. To test this we injected the same target into both A2 and B2 that was used in
figure 4.28. Figure 5.1 shows that it is plainly distinguishable. Figure 5.2 shows a target
3.3 dB below the one used in figure 5.1. Notice it is still visible. This seems to indicate
at least a 6 dB gain in this case. It was also known that the signals used were quite noisy.
With better receivers it may be possible to improve the gain of the adaptive beamforming
system.
78
Frequency -Hz lO Range - Km
Figure 5.2: Plot of IX(r,-f)l - IX(r,f)l for the cross ambiguity plot of the adaptive
beamformed signal with A2. Ag =128,000, M = 8, and the amplitude level the same as
4.30.
Figure 5.3: Plot of IX(r,-f)l - IX(r,f)l with the same parameters as figure 5.2 except the
scattering amplitude is 3.3 dB less.
CHAPTER 6; RESULTS AND CONCLUSION
Throughout this thesis we have been exploring the sensitivity of the
monostatic FM radar. The key issue in determining the sensitivity is the ratio of the
scattered signal to the direct signal. The introduction shows that to be useful the bistatic
system must be able to detect signals whose SDR are less than -54 dB. Unfortunately,
because the bandwidth of the FM broadcasts is insufficient it is probably not possible to
us FM radio broadcasts to detect aircraft. However, if control of the transmitting
waveform is possible then a white Gaussian noise signal with sufficient bandwidth should
be possible.
6. 1 WHITE GAUSSIAN SIGNAL RADAR NETWORKS
Equation 4.35 which determines the minimum SDR shows the strong
dependence of sensitivity on bandwidth. Throughout the thesis we have constrained
ourselves to processes with bandwidths on the order of 250 kHz. Notice that if the
bandwidth is increased to 10-20 Mhz, the radar now becomes sensitive to targets with
SDRs in the range of -60 dB. Because of the wide bandwidth of these signals, it is
probably desirable to move the carrier frequency into the 3 - 10 Ghz range so that the
narrowband scattering approximation still applies. However, it obvious there are two
problems that are encountered with the practical implementation of the above system.
The first is the increase of noise power with bandwidth, and the second is the increase in
computational burden as the bandwidth increases.
First, we examine the increase of noise power with bandwidth. If we
increase the carrier frequency to 3 - 10 Ghz, then the noise floor can be approximated by
the cosmic background radiation if we use good receiver design [14]. This means that
equation 1 .4 can be use to approximate the noise floor with T being approximately 5 K.
This means that with a 10 Mhz system that the total noise power is only 6.9x10 '^ Watts
80
which is still far below the signal power at distances of interest. In fact, increasing
bandwidth will result in gain versus noise because from equation 4.46 the increase in gain
is linear with a slope of approximately one. In contrast, the noise power increases
linearly with bandwidth but with a slope of = 6.9x10'^^ W/Hz. The way to think about
what is occurring is to realize that by increasing the bandwidth of the signal its power is
being spread in frequency because we are forcing it to be white. However, the scattered
signal’s energy still remains the same. From this it follows that as the bandwidth is
increased the system will eventually reach the point of being the noise limited matched
filter output with SNR given by:
Where Eg is the energy content of the scattered signal for the integration time. Notice that
N(, is not dependent on bandwidth, and therefore determines the ultimate sensitivity of the
radar system for a fixed transmitter power.
The second problem is the issue of computation of the ambiguity function
for bandwidths of 10 - 20 Mhz. The computation requirement of the ambiguity function
can basically be broken down into two steps. First, we perform coherent integration to
reduce the bandwidth of the signal. The bandwidth of the targets of interest depend on the
carrier frequency and the maximum radial velocity they obtain. For man-made objects,
we can take the maximum radial velocity as about 1000 m/s. This means that at a carrier
frequency of 3 Ghz we can expect a maximum doppler shift of 63 kHz. This means our
complex sampling frequency should be 63 kHz. Hence, if we are operating with a
bandwidth of 2^“* « 16.78 Mhz then we can decimate by a factor of D = 256. We must do
that for each range of interest with 225 ranges required if we wish to use the maximum
range resolution for 0 to 4 km. Following this, each range must be fourier transformed to
give the frequency spectrum of the scattering amplitudes. This means it requires 4.01 x
10'* complex multiplies per second which since each complex multiply requires four real
81
multiplications translates into 16x10*’ real multiplications. The general expression for
algorithms with iJD = 2" where n is an integer is given for multiplications as:
/«/?(! +
log2(A/-P)
D
(6.2)
where R is the number of ranges to examine. Notice that the decimation step, fgR,
dominates if D is even moderately large. Thus, although the operations count is at first
apparently daunting it should be possible because of the simplicity of the decimation step
to effectively implement the algorithm in most cases.
6.2 CONCLUSION
The investigation of the possibility of using FM radio broadcasts in a
monostatic version of the radar was shown to be ineffective primarily because the
bandwidth of the FM broadcast was not wide enough to give sufficient signal processing
gain to detect point targets such as aircraft. However, the results of the investigation have
shown that monostatic radars which use wide bandwidth white gaussian signals approach
the maximum SNR available from a signal processed by a matched filter in the
background of the cosmic noise floor. This shows promise in developing passive radar
networks and even passive radars which would be capable of operating without detection.
It has also been shown how to determine the probability density functions of the
ambiguity function which is used to detect targets. This is important because it allows
the performance of the radar to be determined as a function of the false alarm probability
and detection probabilities.
As with any complex problem, there is still much that remains to be
explored. The power of the physical clutter relative to the direct signal has not been
adequately estimated. The geographic location of the transmitters and receivers with
relation to the target of interest can also have a profound effect on the radar’s
performance. An analytical model of the effect of the noise being colored instead of
82
white could also give much insight. Finally, an actual experiment using wide-bandwidth
white gaussian signals could be conducted.
List of References
[1] Personnel Correspondence with Frank Lind a Ph.D. candidate in the Department of
Geophysics at the University of Washington who was currently building the radar
receivers at the time this report was prepared, August 1996 - September 1997.
[2] Whalen A. D., McDonough R. N., Detection of Signals in Noise, 2"^* ed..
Academic Press, San Diego, 1995.
[3] Skolnik, M.I., Introduction to Radar Systems, 2"‘* ed., McGraw-Hill, New York
1990.
[4] Hall, P. W. “Correlative Range-Doppler Detectors and Estimators in Bistatic
Radar using Commercial FM Broadcasts,” Masters Thesis, Dept. Electrical
Engineering, University of Washington, Seattle, WA, 1995.
[5] Ishimaru, A., Electromagnetic Wave Propagation, Radiation, and Scattering,
Prentice Hall, New Jersey, 1991 .
[6] Hansen, J. M., “A New Radar Technique for Remote Sensing of Atmospheric
Irregularities by Passive Observation of the Scattering of Commercial FM
Broadcasts,” Masters Thesis, Dept. Electrical Engineering, University of
Washington, Seattle, WA, 1994.
[7] Ishimaru, A., Wave Propagation and Scattering in Random Media, IEEE Press,
New York, 1997.
[8] Levanon, N., Radar Principles, John Wiley & Sons, New York, 1988.
84
[9] Percival D. B., Walden A. T., Spectral Analysis for Physical Applications,
Cambridge University Press, New York, 1993.
[10] Papoulis, A., Probability, Random Variables, and Stochastic Processes,
McGraw-Hill, New York, 1991.
[11] Lind F. D., Sahr J. D., “The Manastash Ridge Radar: A Passive Bistatic Radar
for Upper Atmospheric Radio Science,” Accepted for publication in Radio
Science, 1997.
[12] Abramowitz M., and I. A. Stegun, eds.. Handbook of Mathematical Functions,
Applied Mathematics Series-55 (AMS-55). Washington, DC: National Bureau of
Standards (1964). Paperback edition. New York: Dover (1964).
[13] Haykin, Simon S., Adaptive Filter Theory, Prentice-Hall, New Jersey, 1986.
[14] Hagen, Jon B., Radio Frequency Electronics: Circuits and Applications,
Cambridge University Press, 1996.