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DETECTION PERFORMANCE OF AN FM CORRELATOR
JTR-79-03
March 1979
Prepared for
The Office of Naval Research
Statistics and Probability Program
Under Contract N00014-77-C-0056
Prepared by:
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4. TITLE (mnd Subtitle)
I
Detection Performance of an FM Correlator.
pr author^
J. S.Jlee^L. E. ^Miller
>. PERFORMING ORGANIZATION NAME AND ADDRESS
J. S. LEE ASSOCIATES, INC. ^
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Statistics and Probability Program f II
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IS. SUPPLEMENTARY NOTES
19. KEY WORDS (Continue on reverse elde It nece emery end Identify by block number)
Frequency modulation, detection, FM correlator, false alarm rate,
detection probability
20. ABSTRACT (Continue on reverse elde It nece emery mnd Identity by block number)
—^Detection of a carrier frequency-modulated by a Gaussian random process is
studied for a detector consisting of the filtered product of two FM limiter/
| discriminator outputs. Results show that high detection probabilities can
be achieved for appropriate values of receiver parameters.
DD ,5STW 1473 j COITION OF I NOV AS IS OBSOLETE
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DETECTION PERFORMANCE OF AN FM CORRELATOR
Table of Contents
page
I. INTRODUCTION 1
A. Background 1
B. Summary 5
II. MODELS 7
A. Channel Inputs 7
B. Channel Outputs 8
C. Filters and Correlator Output 10
D. Detection 11
III. ANALYTICAL RESULTS 12
A. Mean of Correlator Output 12
B. Variance of Correlator 13
IV. NUMERICAL RESULTS 16
A. Receiver Operating Characteristics 16
B. Parameter Variations 22
V. CONCLUDING DISCUSSION AND RECOMMENDATIONS 28
APPENDICES 29
REFERENCES 41
DISTRIBUTION LIST 42
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DETECTION PERFORMANCE OF AN FM CORRELATOR
I. Introduction
A. Background. Developing the capability to detect and track signal
energy contained in selected narrow spectral windows continues to be a sig-
nificant challenge to the undersea warfare community. Potential targets can
be characterized acoustically by spectra featuring one or more narrowband
emissions of uncertain or random bandwidth, whose center frequencies are
subject to doppler shifting as the targets move. Therefore, detection of
such emissions and estimation of their frequencies as they vary in time
(tracking) allow both target identification and localization. In many ways
the dynamic behavior of these emissions resembles conventional frequency
modulation.
The methods presently employed in spectral estimation for the purpose of
detecting and tracking undersea targets for the most part are based on com-
putation of discrete Fourier transforms (DFT). In some applications, these
methods are not practical from the point of view of complexity and cost.
For example, the design of an expendable sensor array is usually constrained
by a cost figure, and thus it is desirable to use a spectral estimation
scheme which is inherently simple and relatively inexpensive.
The investigation reported herein is aimed at evaluating alternate methods
proposed for spectral detection and estimation which do not require DFT
\
processing. Representative of these methods is the FM correlator il lustra tec
in Figure 1, in which waveforms from two sensors with the same spectral band
(or in two spectral bands from the same sensor) are each processed as if
they were FM signals, and the results correlated.
1
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The FM detectors depicted in the diagram are assumed to be conventional
employing a limiter and a discriminator such as the Travis type shown in
Figure 2. Use of an integrated circuit version would also be in harmony
with the analysis employed, since the modelling is based on the idea of
frequency-to-amplitude conversion in which the multiplier inputs z ^ and z 2
are, over a given bandwidth, linear functions of the instantaneous frequencies
of the inputs to the two channels. For convenience, the conversion is under-
stood to be
zi(t) = g^u.U)] ) 1-1,2
= o^(t) - u>0 | | u),j - u>0 1 < W/2.
= Aw.. (t) )
Thus the correlator filter output at time T is
z(T) = j^dt Zj (T - t)z2(T - t)h(t)
= ^ dt Ab>i (T - t)Aa>2(T -t)h(t).
an estimate of the cross-correlation between the frequencies in the two
channels. When a target is present, the output of the multiplier is
(a<*»s + Aa)n^)(Au>s + Au>n )
p
= (Aw$) + noise
and the correlator provides a smoothed estimate of the mean square target
frequency deviation from the center of the band when the relative time delay
between the channels has been compensated for.
The present effort is intended to prove the concept of detection using
an FM correlator with quantitative results. Numerical results presented assume
that the relative time delay between the target waveforms received In the two
channels has been removed, although in the analysis and in computer programs
this delay can be specified as a parameter. Also, in the present analysis
f
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frequency modulation due to the target is modeled as a zero-mean Gaussian
random process; the non-zero mean case, including doppler effects, can be
treated in a simple extension of the present results.
Previous efforts [l] have sought to calculate the detection performance
of the FM correlator model (shown in Figure 1) by finding the probability
distribution at the output of the multiplier. In the continuation of this
approach, the exact distribution was found, and it is quite difficult to
compute. An approximate method describe^ also in [1] , which used a common
expression for the FM detector outputs valid under high carrier-to-noise
power ratio (CNR) assumptions, also yields a rather complex probability
density function (pdf) for the multiplier output. These latter results were,
however, computable and some performance calculations were given in [l] which
do not include the filter.
The previous performance (for no filter) was shown to be poor. The ques-
tion remaining was whether "integration" or filtering of the multiplier out-
put would improve the performance to an acceptable level.
B. Summary. In the present work, the output of the lowpass filter in
Figure 1 is assumed to be Gaussian because of the integrating or summing
effect of the filter. Therefore, the performance--receiver operating charac-
teristics (ROC)— can be calculated using only the mean and variance of the
filtered multiplier output as functions of the various bandwidths and CNR's
involved. Analytical expressions are developed in Section III, based on
the models and assumptions presented in Section II.
Using this approach, numerical results were computed and are displayed
graphically in Section IV. Although further study is desirable in connection
with direct system applications, the performances calculated so far indicate
that the integrating filter does indeed improve the performance of the FM
5
i
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correlator to an acceptable level, even for zero-mean Gaussian modulation.
More detailed discussions of the effect of the various system parameters on
the probability of detection accompany the figures in Section IV. I
Recommendations for future work are given in Section V.
Acknowledgement: the authors wish to thank R. H. French for the develop-
ment of the computer programs used to achive the numerical results, with the
assistance of Y. K. Hong.
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II. Models
A. Channel Inputs. The inputs to the limiter/discriminators in the two
channels are assumed to be of the form (A^ constant)
si ( t) + ni(t) = A^sin jw0t + 4>mi(t)J + nci(t)cosio0t + n$^ (t)sinu>Qt
= R. (t)sin[(i)Qt + *mi(t) + ^(t)] ; 1-1,2, (1)
in which the narrowband noise components n^ , ngi ^re assumed to be independent
zero-mean Gaussian random variables from random processes with the correlation
functions
E{nci(t)nci(t+T)} = E{nsi(t)nsi(t+T)}
(2)
= Ooip.(T).
In this work the noise spectra are assumed to be of Gaussian shape, that is.
p 2
2°oi
s (f) = SL exp
! t.i r~
W.^n
f>0.
(3)
The noise bandwidth Wi here defined corresponds to a 4.34 dB roll-off of the
spectrum. Correspondingly, we have
ao
>,•(1) = I df s.(f) C0S2irfx
» ft
I -("W^t)2 j .
0
= exp
(4)
The angle function 4>mi ( t) indicated in (1) is assumed to be given by
*mi(t) = f (5)
7
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where the modulation m..(t) is assumed to be from a zero-mean Gaussian process
with the correlation function
E{m.(t)m.(t+T)} « PmPm(T) = Pmexp{-(irWmt)2}. (6)
where W is defined as the 4.34 dB bandwidth of the modulating process,
m
Further, it is assumed that
mj(t) = m(t)
n^t) = m(t-At) , (7)
so that in (6) no channel subscript (i) is required.
In (1) also we have
\
(8)
where
"lift) * "c1(t)sin*mf(t) + ns].(t)cos,mj(t) (9)
n21 (t) - nci(t)cos*m.(t) - ns.(t)sin^m.(t).
The transformed processes n^ft), n2^(t) are also independent, zero-mean
Gaussian.
B. Channel outputs. The limiter/discriminator operations diagrammed in
Figure 3 are assumed to be ideal so that their outputs are
zi(t) = m.(t) + ^(t), (10)
neglecting any constant factors. It should be noted that and m are not
independent.
R-(t) - [A. + n^t)]2 + [n2i(t)] 2
^ (t) = tan
-1 f n2i^
A,+n..(t)
FIGURE 3 FM DETECTOR MODEL
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This fact is understood from
(Ai*n11)n2,-nlin21
*M ?
/a j > \ »
(ID
(A.+nlir+(n2ir
and from (9), in which it is evident that the derivatives of the noise terms
contain i . = m . . Thus we can also write
mi i
(Ai+nli)n4i"n3in2i
zi(t) = mi (t)
Ai(Ai*"li)
(A1+nn)Z+n|,
(12)
using
n3i = "cisir%i + nsicos*mi
n4i = "cicos*mi - "sisin*mi
(13)
Under this representation, n^, n2i , n3i , n^^ are independent.
The probability density functions for z^ and z2 are derived in the appendix,
with m^ and m2 as parameters, in order to calculate their means.
C. Filters and Correlator Outputs.
Using h(t) for the filter impulse response, the correlator output is
given by
oo
z(t) = j dt h(T)z1(t-T)z2(t-T).
(14)
It is assumed that every T seconds the filter output is sampled (to be com-
pared to a threshold) and the filter is reset. Thus the samples are
written
J
(15)
z(T) = f dx h(T)Zj(T-T)z2(T-T).
10
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In this work three types of filters are to be considered:
Filter 1:
Filter 2:
Filter 3:
h.(t) = <
1 i
( j, 0<t<T
( 0, otherwise
(Integrate and dump)
(16)
h2(t) = <
, 1 -t/ RC . n
RC e • t>0
( 0 , t<0
(Singled-tuned LPF)
(17)
h3(t)
( sin
\ 0
(^t/^), t>o
, t<0
(18)
(2-pole Butterworth LPF)
The 3-dB bandwidths of filters 2 and 3 are
B - -i-
“2 2nRC
and
B3 = V2n-
(19)
D. Detection. The Neyinan- Pearson model of detection will be used. Under
the null hypothesis,
Z<TIH0) ' N(P0.o|0)
wi th
H_: CNR, = CNR, = 0.
ol 2
The alternative hypothesis is
Hji CNRj , CNR2 t 0
for which it is assumed z(T|Hj) N[y(T) ,o2(T)]
11
V
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Probability of false alarm is therefore given by
vfdip(2|t) '?'?erf
A *n I/20 J
For a given value of PF , then, we can compute the threshold r>. The probability
7\
of detection becomes
QO
PD = J dz p(z|H1) = \ - ^-erf T
n L
where P„ is a function of the CNR's.
D
n~p(T)
a2(T)/?_
III. Analytical Results.
A. Mean of Correlator Output. From (13) we have
e{z(T^ w(T)
J
= J dr bNE^T-Oz^T-t)}.
Now, the expectation is taken over both modulation and noise; and for stationary
processes.
e{Zi(T-t)z2(T-t)} = E|Zl(t)z2(t)|
* Em{En|.{2l(t)l2<t)}
From the appendix
:n(|m,{21(t)} ""i'*’'1
h2
-e"hi ]
where
h* h CNRi = A*/2
12
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Therefore,
mu ciui j
u(T) = E{m1(t)m2(t)}(l-e-h?)(l-e~h2) fdrh(T)
■'o
= Pmom{4t)(l-e‘hl)(l-e'h2) f dx h ( t )
Jo
using the notation of (6).
B. Variance of Correlator. First, the mean square is written
rT r1
e{z2(T)} = I dvl dTh(v)h(T)E|z1(T-v)z2(T-v)z1(T-T)z2(T-T)J
•'A » A
(26)
'0 "0
0 70
dt h( v)h( t)Rx( t-v) ,
(27)
where Rx(r) is the correlation function of the multiplier output. In the
appendix it is shown that (27) can also be written
e{z2(T)} = 2 f
•'A
dx Rx(T)g(t)
(28)
with g(x) the filter autocorrelation function:
fT'T
g(t) = I dv h( v)h(v+x) .
•/o
The square of the mean (26) is
(29)
p2(t) = PK(At)(1“e’h^)2(1-e‘h^ f dvfd^ h(v)h(r)
J0 j J0
■(At)(l-e"hlJ(l-e-h^2 2 J dT g(r).
P2p2(
m mv
(30)
13
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In this expression the double integral was reduced in the same way as it was
in the transition from equations (27) to (28). Thus the variance is
o2(T) = e{z2(T)} - y2(T)
JT 2 2
diS(T){Rx(i) - P^(4tm-e-hl)2(l-e'h2)2}. (31]
Now Rx(t) is found to be
Rx(t) = E|z1(t)z2(t)z1(t+x)z2(t+T)|
• Em{Eni |4Il<t)zlU+T)}En2l">{z2<t)z2(t*T)} }
(3i
where the R . ■ (t) are the (conditional) correlation functions of the FM
zi |mi
detector outputs. The correlation function of an FM detector is known to be
of the form [2, chapter 13]
Rz |m.(T) = m^(t)m^ (t+x)(l-e h^)2 + f^x)
where
f^(t) = f[P.j(T), Pj(t),Vj(t); h^]
f = _e_
' 2
2pZ
l+£ -2h /(1+p)
,-h2/p
. .> ■ .. .
*•*
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f
IV. Numerical Results
After the necessary analytical expressions for the mean and variance of
the filter output were developed, detection probabilities were calculated
via numerical integration, according to the model of detection shown in
Section II-D.
In setting up the calculations, the following basic parameters were
identified:
carrier SNR (CNR)
ratio of mean square frequency
modulation to square of channel
bandwidth
channel bandwidth-time product
ratio of modulation bandwidth
to channel bandwidth
filter bandwidth-time product
In all of the computed cases, the modulations in the channels were
assumed to be aligned in time (at = 0). The reference case for the calcula-
tions to be shown was chosen to be
Y = 1, WT = 5, BT = .3, e = 1 . (40)
A. Receiver operating characteristics.
The probability of detection (Pp) at the filter output, as a
function of CNR, is displayed in Figures 4-7 for probabilities of false
O O a
alarm equal to 10 , 10 , and 10 . Because of the rather steep slope
with the linear PQ scale of Figure 4, the expanded, probability scale of
Figures 5-7 is to be preferred for discussion. Several interesting
features of the figures invite comment.
y = Pm/(2*W)2
WT
c = W^W
BT
16
L
jM
SNR (dB)
FIGURE 4 FM CORRELATOR RECEIVER OPERATING CHARACTERISTICS
(LINEAR SCALE)
17
SNR (dB)
FIGURE 5 FM CORRELATOR RECEIVER OPERATING CHARACTERISTICS
(INTEGRATE -AND- DUMP FILTER)
18
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FIGURE 6 FM CORRELATOR RECEIVER OPERATING CHARACTERISTICS
(RC FILTER)
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iwHmimnnn«nM«nm mntnwiinnumiir ■
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FIGURE 7 FM CORRELATOR RECEIVER OPERATING CHARACTERISTICS
(2-POLE BUTTERWORTH FILTER)
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Limiting values. The value of Pp for very small CNR, of course,
has an immediate interpretation:
PD -*• PpA as CNR + 0 (41)
However, the fact that PQ approaches a value less than one for large SNR is,
at first sight, somewhat unusual because we are accustomed to thinking of
SNR as a location parameter for the pdf of the decision variable. In this
FM situation, though, the SNR involved is actually the CNR, and what is
"happening" in Figures 4-7 can be described very simply. At high CNR the
correlator output approaches that of the noiseless case, in which (see (26)
and (39))
u(T) = C^lm2}
(42)
= cipm
I m
and
°Z2'T> - C2Pm
(43)
Thus for Gaussian frequency modulation, both the mean and standard deviation
of the correlator output are directly proportional to the mean square of the
modulation. For all values of P , there is always a portion of the distri-
bution which falls below the threshold; thus PQ < 1 as CNR -+■ ®.
Transition values. For CNR neither very large nor very small, we may
think of the correlator output pdf as the outcome of a "battle" between the
noise only case and the noiseless case:
noise only transition noiseless
mean »n * 0
2 2
variance a„ a*\
n ntm
As so often occurs in nonlinear systems, one effect captures or suppresses the
other; therefore the transition from a low Pp to a high value takes place over
a relatively small interval of CNR values.
(
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What is intriguing is that there is an interval over which PQ <
It seems that, as CNR increases from zero, for a while the net (noise +
modulation) distribution begins to have a sharper peak (smaller o) before
the mean value begins to shift.
B. Parameter variations.
Some calculations were designed to explore the effect of depar-
tures of the various parameters from the reference case (40) for CNR = 10 dB.
In Figure 8. y is varied. As anticipated in the previous discussion,
increasing y (increasing P ) causes the limiting value of PQ to increase, but
this effect saturates due to the fact that both mean and variance depend upon P .
Therefore, increasing "modulation power" beyond a certain point is not productive.
WT is varied in Figure 9, indicating that, for W fixed, increasing T
improves detection performance by integrating longer the (nonzero) multiplier
mean when there is modulation. For fixed T, the interpretation is less clear,
since postulating a variation in W and holding y constant involves increasing
P also,
m
In Figure TQ e, the ratio Wm/W, is decreased from its reference value
of unity. The effect is to decrease the value of Pn (i.e., Pn“£). If a
modulation index be defined as
= /y/e
A ^7
2itW,
(44)
m
the bandwidth of the modulated carrier may be approximated by
B. W. = 2Wm(l + B) = 2W(e + (45)
For fixed y, then, bandwidth of the modulated carrier is proportional to e,
and the effect shown in Figure 10 can be attributed to variation in correlator
output SNR, as demonstrated in [6].
22
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BT = 0.3
e = 1 l
At- 0
Y = 1
SNR = 10 dB
INTEGRATE AND DUMP FILTER:
——————————a
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FIGURE 9 EFFECT OF BANDWIDTH-TIME PRODUCT
0 0.2 0.4 0.6 0.8 1.0
c
FIGURE 10 EFFECT OF MODULATION BANDWIDTH ON PROBABILITY OF DETECTION
25
J. S. LEE ASSOCIATES, INC.
A most interesting effect is seen in Figure 11 , in which the filter
bandwidth-time product (BT) is varied for the case of the 2-pole Butterworth
filter. Evidently there is an optimum value of BT in the neighborhood of BT = .4.
A similar phenomenon was reported in [7j and attributed to a tradeoff between
noise rejection and signal rejection; as the filter bandwidth decreases, at first
more noise is rejected than signal, causing improved output SNR and Pp. Reduc-
tion of filter bandwidth beyond a certain point, however, rejects more signal
than noise. Thus there is an optimum value of BT which in some sense matches
the signal (in this case, the modulation), and which will depend upon CNR.
FIGURE 11 EFFECT OF FILTER BANDWIDTH ON PROBABILITY OF DETECTION
(2-POLE BUTTERWORTH FILTER)
. J ' ■
r
J. S. LEE ASSOCIATES, INC.
V. Concluding Discussion.
The question which has motivated this work - whether an FM correlator
(with integrating filter) can perform acceptably as a detector - has been an-
swered in the affirmative. For, as demonstrated numerically in the figures of
the previous section, probabilities of detection close to unity can be achieved
for arbitrary false alarm rates by manipulation of receiver parameters. Having
answered this basic question (with much effort initially), only a sampling of
parametric investigations have been carried out so far. For example, with the
computer programs now in use, it is a straightforward effort to determine the
minimum detectable signal, both in terms of CNR and modulation power (Pm),
required to assure a given PD for fixed Pp^. Several studies of this type,
characterizing the basic performance tradeoffs associated with FM correlation
detectors, would be quite interesting and, if widely disseminated, would provide
system designers with fuel for thought.
One basic study which seems to have a significant potential for system
application is the case of random modulation (treated herein) but with nonzero,
time-varying mean. This model corresponds to narrowband emissions of uncertain
or varying center frequency, and subject to doppler shifts.
Further investigations should begin to focus on system applications,
taking into account specifically doppler (motion) models and details of fre-
quency conversion circuitry, in order to assess accurately potential system
performance. The future study should address itself to the problem of de-
tecting multiple narrowband spectra as an extension of the current effort.
Also, labaratory simulation should be considered in future efforts.
28
J. S. LEE ASSOCIATES, INC.
APPENDICES
A. pdf for the Output of the FM Limiter-Discriminator
In this derivation a slightly different form for the FM channel out-
puts is used. Instead of (10), let us write
z.(t) = e^t)
where the channel input is written
A-jSin [“0t + *si(t)] + nci(t)cosa>0t + n$. (tjsin^t
■ Ri (t)sin[a>0t + ei(t)].
(A-l)
(A-2)
Since, under this representation.
e^t) = tan
■l("ci + A15l"9i<t)l
("si + AicosVt )(•
we have
5,(0 = z(t) -
u + v
using u = n$i + A^cos^ and v = nci + A^in^. .
Input Distribution
At the same instant, (n • , n . , n . , n . ) are mutually independent
w I j I L I o I
Gaussian variates [3] with zero means and
A 9
var(nc) = var(n$) * oQ
var(nc) = var(n$) 5 o*
where, for a flat noise spectrum over the passband b (in Hertz), [4],
2 _ 2.. _ . 2 2.2.,
Oj - oq/K = 4ir oQb /3.
(A-3)
(A-4)
(A-5)
(A-6)
(A-7)
.* .
J. S. LEE ASSOCIATES, INC.
Thus the variates (u, v, u, v) have the probability density function (pdf)
j( u ,v ,u , v) = (21ro0o1)'2exp|-
( 2a
u)2 + (v-v)2 (G-u)2 + (y-v)2
-»<(u,v,G,v)<“.
(A-8)
Mean values are taken to be
u = Acos^,, v = Asin*ml
u = -mAsin^ + Acos$mi
v = m A cos 4 . + A sin $ .
mi mi
(A-9)
Output Distribution
Consider the transformation of variables (dropping the subscript i)
u = R cos e 0*R<®
v = R sine
u = 5 cos e - n sin e
0ses2ir
(A-10)
v = £ sin e + t\ cos e
-0°<n<co
for which
z - - n/R.
(A-U)
u + v
The variable £ can be interpreted as ft, the derivative of the
envelope, and n, as Re, e being the phase of the input waveform as given by
(A-2).
30
The Jacobian of the transformation is R, so that the pdf of the new variables
P^R.e.C.n) = Rp (Rcos 0, Rsin 0, S cos 0 - nsin e, Csin 0 + ncoso)
= R(2no0o1)'2 exp|- ^[r2 + A*- - 2RAcos (0-^)]
- y? [t2 + n2 + m2A2 + A2 - 2(mAc-nA)sin(0-^)
-2UA+nmA)cos(0-4fo)j> . (A-12)
Eliminating £ by integration, we have
P2(R,0,n) = I <U P^R.o.t.n)
•'•oo
- «[(?,|3,vi] 1,xpj-^- ‘wf}
x exp | ^ cos (0-4^) - T
I °o 2ol L
2 2
a cos (e_*m+^) _ 2nacos(o
using
a2 = m2A2 + A2, 4> = tan_1(A/mA).
The terms in the second exponential may be written also
a2 a2
R b cos (e-4 +4-4.) =r - — p cos2(0-4 +♦ )
m a b 4o2 4o2 m a
( A- 13)
(A-14)
(A-15)
*“1 fcts,M*)/(3cos*> * 3)]
' (A-16)
= tan
*> w s
J. S. LEE ASSOCIATES, INC.
Defining a second transformation of variables (with Jacobian = w)
n = zw , Osw<®
R = w , — <z<®
results in the new joint pdf
P3(w,e,z) = [(2’r)3/2°Qo1]"1w2e'h exp | - (1 + kz2)J
2oo
x expjw b cos(e - *m + ^ - ^|cos2(e ~ +m +
The integration of (A-18) with respect to the variable w involves an integral
.1
of the form
f
dw w2 e"T”+2 cos B
Y' y | cose + (ucos2b + H)eocos ^1 + erf(»^7cos B)]|
/n -3/2
' TT
where
jocose + (ucos2b + >s)eucos 8
x + ^cos B jFj^s;3/2;u cos
1+kz
Y = — 2“
2o!
8 5 6 ' + \ - *b
17
Using [5], #'s 3.462.7, 9.236.1
32
' i « . ■
(A-17)
(A-18)
(A-19)
• (A-20)
(A-21)
J. S. LEE ASSOCIATES, INC.
and
(l+kz2)u = ^ = h2 +^-^k2z2 + 2(kz)^-^h cos*a;
(A-22)
jFj denotes the confluent hypergeometric function, and h^ = A^/2cjg .
Note that for A = 0, a = mA and ^ = 0; also.
u(z)|. = U+mkz)_ h2
A=0 1+kz
(A-23)
Using (15) , we have
A
P4(e,z) =
2"372 e exp
2*(l+k z2)3/2
{- ^ cosZ(6VJ
u(z)cos e
cosb + |^u(z)cos2b + %JeL
[u(z)cos2p + »5jeu^z^cos B1F1[js;3/2;u(z)cos2b]|.
(A-24)
Only the second term of (A-24) survives integration with respect to e, leaving
p5(z) = 7 T7T
(1+kz )
2n
37? exp |-h2 -
± J dej^f- + >s + ^cos[2(e-V^a)]|cos[2n(e-V*a+,s*p)]
2(l+kz
A ( u2 a2 A u(z) )
-^e.pj-h --^+ 2 J
X j[u(z) + 1] I0[p(z)] + u(z)I1[p(z)]cos[*p(z)]|
(A-25)
33
y v ‘ • i -%*•**’
J. S. LEE ASSOCIATES, INC.
Unless the effect of (incidental) amplitude modulation is being studied,
this case is sufficiently general for most purposes. If the detector model
itself is to be studied in more detail, one can consult the appropriate chap-
ters in Middleton's book [3).
Further specializations of (A-28) and (A-29) are the following sub-
cases :
(a) For no signal (A=0),
P5(z|A=0) = ^ (l+kz2)'3/2.
(A-30)
(b) For no modulation (m=0), from (17) and (20)
2 / h2 \
P5(z|m=0) - y e'h (l+kz2)‘3/2 ^^3/2,1; — -~A (A-31)
= f (l+kz2)"3^2exp
ft •£)&]•£ •■&])■
(A-32)
In the next section, the mean value is calculated to be
J.
z = m(l
e_h );
(A-33)
as the carrier SNR (=CNR) increases, the mean approaches the noiseless case,
as expected. In Figure A-l, this effect is demonstrated numerically for constant
(frequency shift) modulation- -that is, when m = 2*fd; the bandwidth b was
selected such that m/i( = 1 (b = fd/l). Also, the figure displays the pdf of
the scaled variable v = z/fd, so that asymptotically the mean approaches 2*.
35
. y: • . - '
J. S. LEE ASSOCIATES, INC.
Another effect we anticipate is that, as the bandwidth of the post-
limiter filter (b) is decreased, the output SNR decreases. Also, for fixed
b, if the modulation or frequency shift f^ increases, the effective output
SNR should increase. Both these effects are evident in Figure A-2.
B.
Calculation of the Mean Value.
The mean value of the FM channel output for a given value of the mod-
ulation m is obtained from (A-29) by
— 2^°° ^ 2n
E jz ;mj = ^ e‘h r(n + 3/2)1F1(n + l/2;n+l;-m2kh2)
n=0
zQ+mkz)
2n
X / dZ (Wa2)"*3'2 ’
(B-l)
where the integral equals
a
*/.
dx(l+mvirx)2n
F | (1+x2}n + 3 71 ~ W
n-1
- . . . */2
)(m^F)r j de(cos0)2n_r(sine)r+1
f5G“)
^(2'"‘)
*/2
(m/k)2r+1 f de(cose)2n2r_1(sine)2r+2
-’o
(m/k)2r+1B(n-r,r+3/2) .
(B-2)
Here B(x,y) = r(x)r(y)/r(x+y) is the beta function:
B(n-r,r + 3/2) i >
(B-3)
37
J. S. LEE ASSOCIATES, INC.
Substituting (B-3) in (B-2) yields
n-1
1 nlr n + 1/2
V r(n +
1/2) Jit V'
(mA)2^1
r! r(n-r + 1/2)
(B-4)
With this expression for the integral, (B-l) becomes
2 2n
E|z;m| = — e"h r( n + l/2)1F1(n + 1/2 ;n+l ;-m2kh2)
n=0
r! r|n
r=0
A)2r+1
r + 1/2)
jl y r(n+r + 3/2)1F1(n+r + 3/2;n+r+2;-m2kh2)
/k n=0 r=0
2r+l
(mA)
r!r(n + 3/2)
(B-5)
in which was used the progression
OO OO
—2° n-1 «» «o
2^Z/(n,r) = S^-/f(n,r)
n=0 r=0 n=r+l r=0 n=0 r=0
(B-6)
Now, the summation over the index r may be recognized as a Taylor's series:
E/h2m2klr (n + 3/2) ? ?
*->'} ' ~Cn + 2) iFi^ + 3/2 + r;n+2+r;-nrkh2)
r=0
= 1F1(n + 3/2;n+2;m2kh2-m2kh2) = 1.
This fortunately simplifies (B-5) to
E{Zim} ■ ' m(1-e'h2)-
n=0
(B-7)
(B-8)
38
J. S. LEE ASSOCIATES, INC.
Filter Integrals. The transition from equations (27) to (28) in the
text can be shown as follows. The integral is
yT -T
J " dvj" Jt h(
v)h(x)Rx(T-v),
(c-1)
0 0
in which h(v) is a filter impulse response and Rx(t) is a correlation function.
Since Rx(x) is even, there is a symmetry about the line x=v; thus rotating
the coordinates v,t by 45° gives
I T
=2 I dvf<
HD •'v
dt h(v)h(x)Rx(T-v)
2 dv'Rx(-v
(C-2)
Making use again of the even-ness of Rx(x) and rescaling the variables results
T T-v
1 = 21 dv Rx(v) I duh(u)h(u+v)
J0 J0
2 I dv Rx(v)g(v)
(C-3)
where g(v) is the filter autocorrelation function for the case of h(t)=0,
t<0 and t>T :
• T-v
g(v) * j du h(u)h(u+v) * I du h(u)h(u+v) ,
•'o
(C-4)
J. S. LEE ASSOCIATES, INC.
A corollary to this result occurs for Rx(T) = 1:
| f dv h(v) | = f dv f dr h(v)h(T) = 2 f dv g( V) .
' ‘'n ’ ■'o J0 •'o
For the various filters given in ( 16)-( 18) , we have (T>0)
T
j dt hj( t)=l , g j ( t ) « ;
dth2(t) = l-e-T/RC, g2(r)
-t/RC
[-
-2(T-t)/RC
J
J dth3(t) = 1 - [sin^T//?) ♦ cos^T/^]
93(t) = ^|e"“bT/^[sin(u,bT/^) + cos(«bT//f)l
/8 I J (C-8
+ e-«b(2T~r)//? [cos(u>b(2T-T)//2)- sin(o.b(2T-T)/^)- 2cos(u>bT/^
J. S. LEE ASSOCIATES, INC.
;
i
REFERENCES
1. J. S. Lee Associates, Inc., "Some Analytical results obtained during the
current period under contract N00014-77-C-0056," presented to Probability
and Statistics Program office, ONR, 21 July 1978.
2. J. L. Lawson and G. E. Uhlenbeck, Threshold Signals, McGraw-Hill, New
York, 1950.
3. D. Middleton, Introduction to Statistical Communication Theory, McGraw-
Hill, New York, 1960.
4. P. C. Jain, "Error probabilities in binary angle modulation," IEEE
Transactions on Information Theory , IT-20, pp 36-42 (January 1974) .
5. Gradshteyn and Ityzhik, Table of Integrals, Series , and Products, Academic
Press, New York, 1965.
6. H. Osawa, N. Morinaga, and T. Namekawa, "Output signal -to-noise ratios of
FM correlation systems," IEEE Transactions on Information Theory, IT-17,
pp 32-36 (January 1971 ).
7. M. C. Austin, "Wide-band frequency-shift keyed receiver performance in the
presence of i ntersymbol i nterference , " IEEE Transaction on Communications
(concise paper), pp 453-458 (April 1975T
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