A radar detector using quantized envelope samples

by C. C. LEE? C&G.

Department

J. JAW

of Electrical Engineering and Computer Science,

Northwestern ~niversity~ Eu~nston, IL 60201, U.S.A.

ABSTRACT: The detection of a sequence of echo pulses with random carrier phase is an important problem in radar. It is well known that, to a first approximation, the optimum receiver is a quadratic detector for small signal-to-Norse ratios and is a Iinear detector for large signal-to-noise ratios. In this paper, a detector based on scaling and rounding envelope samples into digital words of only a few bits is presented and analyzed. This detector is essentially a generalization of a suboptimum detector known as a binary detector which performs postdetection intqqration by digital counting and is particularly useful in automatic detection systems. The asymptotic reiative eficiencies of this detector relative to the quadratic detector and the linear detector are obtained. Also, nob-asymptotic performance comparisons based on relative efficiency and on probability of detection are made. It is shown that this detector is more eficient than the other two detectors in most signal-to-noise ratio ranges. In addition, this detector is o~pI~~able when the noise environment is somewhat dl~erent from Gaussian.

The detection of a sequence of echo pulses of incoherent phase is an important radar problem (l-3). In terms of statistical hypothesis testing, this problem is to test

Ho:r,(t) = n,(t)

i= 1,2,...,n

against HI :rj(t) = A sin (~~t+~i)+~i(t)

tE(tj, ti4T)

where rl(t), rz(t), . . . , r,,(t) are n statistically independent received waveforms, and nr(0, ~(0, . . . , .q(t) are sample functions of additive white Gaussian noise with power spectral density N,,/2. Also, A, o, and 7’ are the constant amplitude, frequency and duration of each signal pulse, respectively, and each & is unifo~ly distributed (0, 2~). For this detection problem, it is known that the optimum receiver has a structure as given in Fig. 1. Theoretically, the nonlinearity g( *) in

TThe research of the first author was supported by the National under Grant ECS-8307264.

Science Foundation

C. C. Lee and G. J. Jaw postdetection

rice

matched

compared to threshold

filter

i=1,2,...,n

FIG. 1. Optimum receiver for detection of multiple signal pulses with incoherent phase.

the figure is such that

where I0 is modified Bessel function (4) and d = 2A/N0. Because the Bessel function and the logarithm

are difficult

to implement,

approximations

to this nonlinearity

have been employed in practical applications. Specifically, g(a) is replaced by a square-law gl(*) such that zi = gl(ri) = v,J (quadratic detector or square-law detector) for a small signal-to-noise ratio and by a linearityg*~*) such that zi = g&J = yi (linear detector) for a large signal-to-noise ratio. In the intermediate range of signal-to-noise ratios, however, it is easy to verify that both approximations involve substantial errors, although this does not imply that the resulting detectors perform much worse than optimum. Both the quadratic detector and the linear detector require a postdetection integration whose implementation requires some form of video signal storage and are thus inconvenient in automatic detection systems (1). Figure 2 shows a suboptimum detector called a binary detector, also known as a double-threshold detector, which quantizes the envelope samples into a sequence of zeros and ones ; the number of ones is then counted and used as test statistic. This detector performs postdetection integration by a binary counter and is very useful in automatic detection. However, since much information is lost by the two-level quantization, the efficiency of the binary detector is substantially below optimum. For example, the asymptotic relative efficiency (ARE) of the binary detector compared to the optimum detector or to the quadratic detector is only 64% (=2/z). In this paper, we present a detector which, like the binary detector, is digital beyond the envelope detector but, unlike the binary detector, performs very well. We assume in particular that the target is nonfluctuating and that the signal has a known amplitude. Our intention is to generalize the concept of the binary detector so that its efficiency can be improved. We present an approximation to the optimum receiver based on rounding each envelope sample yi into a short digital word of only a few bits. The accuracy of this approximation is insensitive to the signal-tonoise ratio. When a random variable is digitized, so also is the statistical model governing it (5). In our case, a Rayleigh distribution and a Rician distribution are

r,(t) 1=1,2,...,*

matched

envelop

filter

detecto

FIG.

80

quantizer 2-1eve1 )s:
2. The binary detector. Journal ofthe Franklin Institute Pergamon Journals Ltd.

Radar Detector Using Quantized Samples r,(t) i=l.Z,...,a

matched

and cam?utaticn

filter

FIG. 3a. The multinomial

detector.

reduced to two sets of multinomial probabilities from which a simple optimum decision function is obtained. Since any physical equipment has limited precision, the input digitization represents a natural approximation to reality. In principle, the number b of bits used for each digital sample determines the amount of information lost in A/D conversion. On the other hand, the complexity of the resulting detector increases rapidly with b (see Fig. 3b). Fortunately, as indicated in the literature (M) concerning statistical decisions based on quantized measurements, a “rough” quantization is usually adequate for a nearly optimum performance, In particular, it was shown in (5) how digitization of input samples is related to a sampling of input statistics and how the Nyquist sampling theorem can be applied to continuous probability densities and their characteristic functions. Little of the required statistical info~ation required for decision making is lost by a rough quantization, even though the exact signal itself may be distorted. As a result, despite the information loss by digitization, with only 4 bits for each envelope sample, the resulting detector is shown to be more efficient than the linear detector and the quadratic detector when the signal-to-noise ratio (SNR) lies between -6 and

FIG. 3b. A parallel

architecture

Vol. 322, No. 2, pp. 79-92, August 1986 Primedin Great Britain

for input quantization statistic T,.

and computation

of the test

81

C. C. Lee and G. J. Jaw 0 dB. More surprisingly, the quadratic detector and the linear detector are superior to this detector for SNR < -6 dB and for SNR > 0 dB, respectively, by less than 1%. In addition, if those required multinomial probabilities are measured directly at the envelope detector’s output instead of being computed from Rayleigh and Rician distributions, it is clear that this detector is applicable even though the real noise distribution may deviate from Gaussian.

II. Input Digitization

and the Muitinomial

Detector

Assume that each envelope sample yi is coded (quantized) into a b-bit binary digital word ui. Then, the statistical distribution of zij is a discretized version of Rician or Rayleigh, depending on whether signal is present or not. Let C, (k = 1, 2 , ‘..> 2’) be those codes (i.e. quanta) and pk = Prob (ui = C, j f3,)

qk(a) = Prob {v, = C,lH,>. These discrete probabilities tributions :

can be computed

Pk 4d4

from Rayleigh and Rician dis-

=

~dr&)-FO(rk-

1)

=

F~(rk)-F~(rk-

11

(2)

where F,, and F1 are the cumulative distribution functions of Rayleigh and Rician dist~butions whose density functions are given respectively by f&J

= 0-2yi exp (-y,2/202),yi & 0

(3)

and .fi(~i) =fo(~,) exp (-a2/2~*)Z0(ayjl~*),

yi 3 0

(4)

where a = AT/2 and o2 = N,,T/4. The digitization thresholds (rk) are such that Ui= C, when JJ,E(rk_ i, rk). Also, let flk = &(Z’r,u2, . . . , u,) be the number of digital samples 0; assuming quantum ck. Then, the joint probability of z),, u2, . . . , ZJ,can be expressed in terms of a multinomial distribution using the sufficient statistic fz1,&, . . . I ntft (II? = 2b) : (8, 9, Ch. 8) : m A%,%,

. *., %lff,)

= Pbl,

fl2,.

. .,

%llHo)

=

n!

n k=I

P”k*hk!

and

The optimum statistical decision function (the log-likelihood ratio) is thus given by T,=

5 nkbk k=

I

Joumai

82

of the Franklin Institute Pergamon Journals Ltd.

Radar Detector Using Quantized Sampbs where bk = log [~~(~)/~~Iare known constants. The optimum detector is then the one that compares T, to some preset test threshold to decide between No and H1. The first and the second moments of T,, are computed in the Appendix and will be used later in the performance study. Figure 3a shows the block diagram for this detector which we will call a “multinomial detector”. Detectors of this kind have often been studied in the literature of nonparametric detection (6-8). We encounter them here for a parametric detection problem. Like the binary detector, the multinomial detector does not need an analog postdetection integration. In particular, it is shown below that the computation of the test statistic T, including the required quantization can be performed by a desirable architecture of simple digital circuits. Define the following : nz , =

n-n , = No. of samples Xi that exceed

r,

m2 = r~----y1~ -n2 = No. of samples Xi that exceed r2

m,_j

=n--nj-

. . . -n,_

, = No. of samples Xi that exceed r,,__ ,

m, = 0. Then we have nk = rnk_, -mk for k = 2,3,. . . , m. It is straightforward that

k=

1

k=

to prove

I

where hk=bk_l-bk, k= 1, 2, . . . . m- 1. We thus obtain the block diagram of Fig. 3b which shows that a parallel connection of 2’ identical comparators and binary counters perform the quantization as well as the computation of T,. The output of the comparator is either 0 or 1 depending on the relative magnitude of its inputs. The multinomial detector here is basically an approximation to the theoretically optimum detector which involves Bessel functions as well as logarithms. This approximation is a natural one since any physical equipment has a limited precision (i.e. finite word length) and thus a quantization of input data is unavoidable. In addition, this approximation eliminates the difficulty in realizing the Bessel function and the logarithm. The accuracy of the approximation certainly increases with the number &-the digital word length. On the other hand, note that the complexity of the multinomial detector is of the order 2b. Therefore, we are particularIy interested in a small b as long as the resulting detector performs satisfactorily. As will be seen later, the performance of the multinomial detector is comparable to those of the quadratic detector and the linear detector with b as small as 3. IlIe Performuuce

of the ~~u~ti~omiu~ Detector

We now study the performance of the multinomial detector developed using the quadratic detector and the linear detector as references. For convenience, let D1 denote the multinomial detector, O2 the quadratic detector and D3 the linear Vol. 322. No. 2, pp. 79-92. August 1986 Printed in Great Britain

83

C. C. Lee and G. J. Jaw

detector. All the numerical results obtained in this section are based on the quantization thresholds {rk} selected in the following manner: the (envelope) sample space (0, co) is truncated to (0, D) which is then uniformly quantized into 2b levels. Since it is assumed that the signal has a known amplitude, the number D is selected so as to optimize the performance of the multinomial detector. a. The asymptotic

relative eficiency

(ARE)

The ARE of a detector Di relative to another detector Dj will be denoted by ARE(i,j) and is defined as the ratio of sample sizes (i.e. the numbers of pulses in the present case) required by Dj and Di to achieve some fixed false alarm rate a and probability ofdetection fi when signal strength approaches zero. It is often a number (10) that represents a compa~son of detector performances under vanishingly small signal situations. It is known that ARE(i,j) can be computed as a ratio of efficacies (11, 12) of Di and Dj. Let (pi,Cf) and (,~{,a;~) be the mean and the variance of the test statistic of Dj under Ho and H,, respectively. Then, the efficacy of Dj is defined as

(6) where s is a measure of signal strength. For the detection problem under consideration, it is convenient to use the signal energy as the signal strength : .s = E. For the quadratic detector, it is easy to verify that &2= l/N;

(7)

which is also the efficacy of the Neyman-Pearson optimum detector which uses (1) as test statistic. This is because the quadratic detector is a locally optimum detector for the detection problem under consideration (12) ; a locally optimum detector is known to be asymptotically as efficient as the Neyman-Pearson detector as signal strength approaches zero. For the linear detector D3, it is easily shown that s3 = n/[(16-47c)N;].

(8)

Finally, for the multinomial detector, we show in the Appendix that

cl; = n

$I4da) log M4l~kl

(9)

and

0: = ~15 pk[logkkG$ld-n k=

1

5 Pklog

(10)

k=l

It is easy to verify from Eq. (2) and Eq. (4) that q;(o) = q;“(o) = 0. 84

Journal

of the Franklin Institute Pergamon Journals Ltd.

Radar Detector Using Quantized Samples

series expansion for qk(a) is

Thus, the ~aclau~n f&(a) =

pk

+

(a2/2!)qi(0)

+

(a4/4!)qi”‘(0)

+

O(oS)

=p~+(TE/4)q;I(0)+(T2E2/96)q;“‘(0)+0(E2~5)

(11)

since a = AT/2 and E = A2TJ2, where 0(x’) satisfies lim [O(x’)/x”] = 0

x-0

for any 0 < u < t. It follows that

$$?=(T/4)q;(O)

-t-O(E)

(12)

and that

cmdq&> dE

o

(13)

-=

*

A=,

The last equality holds because

k$,

qk(a)

=

1

and thus (14) where q@)(-) denotes the tth derivative of q,(a) with respect to a. Also, expanding the logarithm in a power series yields

1% [qkta)/pkl

=

1%

[I+

(ET/4)q~(O)/Pk +O(E’)]

= (ET~4)q~(O)~p~ +O(E2)

(15)

Now, from (9) (12) (13) and (15) we obtain

-4.4 = dE

= (nET2/16) c [q;2(0)/pJ+O(E2). k=

(16)

1

Substituting (15) into (10) and using ( 14) yields M

CJ:= (nE2T2/16) c [q;2(0)/&]+O(E3). k=

(17)

I

Consequently, by (6) we obtain 81 =

(T2/16):

[d?O)/Pkl.

(18)

.=I

We conclude that ARE(1,2) = el/sZ = c4 5 [q:2(0)/pk]

(19)

k=l

Vol. 322. No. 2, pp. 79-92, August Printed in Great Britain

1986

85

C. C. Lee and G. J. Jaw

and that ARE(1,3) = al/e3 = cr4

2 [q;2(0)/pk]. k= I

Table I gives some numerical values for ARE(I,2) and ARE(1,3). These ARES increase with b since a greater h implies less information loss by digital quantization. That ARE(1,2) is smaller than unity is predictable since the quadratic detector is the optimum in the limit as E -0. We conclude that, in detecting a very weak signal, the multinomial detector is slightly less efficient than the quadratic detector but is more efficient than the linear detector for b 2 3. b. The relative e~cienc~) (RE)

It should be noted that the ARE is simply a comparison of detector performances in the extreme case where E -+ 0. Therefore, despite the fact that ARE(l,2) is below unity, the multinomial detector can be superior to the quadratic detector when the signal is not too small. In this section, we use the relative efficiency (RE) to compare the performances of the multinomial detector, the quadratic detector, and the linear detector. The relative efficiency is defined as RE(i,j) = tij,‘ni where nj and Q are sample sizes required by Dj and Di, respectively, to achieve the same false alarm rate CIand probability of detection /?. The RE curves obtained are based on a Gaussian approximation for all three test statistics and IMSL library functions. The use of a Gaussian approximation is well justified because we used c = 1 -p = IO-i0 and relatively small signal-to-noise ratios and thus the sample sizes are very large. Let us order these detectors in the same way as we did in the previous section. For b = 3, Fig. 4 is a plot of RE(1,2) and RE(l,3) vs (E/02) which denotes the signal-to-noise ratio (SNR). We observe that RE( 1,2) exceeds unity when the SNR exceeds a threshold value at approximately -3 dB. For b = 4, a similar plot is given in Fig. 5 which shows that the threshold falls to approximately -6 dB. We also observe that, for b = 4, the multinomial detector is more efficient than the linear detector for any SNR smaller than 0 dB. The point of intersection of RE(1,2) and RE(1,3) (at approximately - 2.5 dB)

TABLE I

Numerical values for .4RE(1,2) and A RE( &th b = 2, 3, and 4 bits, respectively

b=2 b=3 b=4

86

1,3)

ARE(l,2)

ARE(1,3)

0.893 0.966 0.990

0.976 1.055 1.082

Journal of the Franklin Institute Pergamon Journals Ltd.

Radar Detector Using Quantized Samples 1.080

T

1.060

t

AM01 -1omo

-8.000

-6.000

-4.600

-2.OaO

O.o(M

SIGNAL TO NOISE RATIO(DB) FIG. 4. The relative efficiencies of the multino~a~

detector relative to the quadratic detector and the linear detector. b = 3, a = I -/I = 10-lo.

960

1,,,.,

-10.000

-8.000

-6.000

-4.000

-2.000

0.000

SIGNAL TO NOISE RATIO(D6) FIG. 5. The relative efficiencies of the multinomial

detector relative to the quadratic detector and the linear detector. b = 4, a = 1 -B = 10--‘“.

Vol. 322, No. 2, p&k lSL-92, August 1986 Printed in Great Britain

87

C. C. Lee and G. J. Jaw serves as a threshold between the quadratic detector and the linear detector ; either detector performs better on one side but worse on the other. Note that one important advantage of the multinomial detector is that its usefulness is not restricted by the magnitude of the SNR. In addition, in case the noise process begins to deviate from Gaussian, the linear detector and the quadratic detector quickly become inappropriate (13-14) while the multinomial detector is still applicable, provided that those multinomial parameters pk and qk(a) are measured at the envelope detector’s output instead of being computed from Rayleigh and Rician distributions prior to the design of detector. c. The probability of detection With the false alarm rate, the signal-to-noise ratio and the sample size (the number of pulses) fixed, it is clear that the probability of detection is a performance measure. For the radar detection problem considered, Figs 6-8 show comparisons among the multinomial detector (with b = 4), the quadratic detector, and the linear detector in terms of the probability of detection (plotted as a function of the false alarm rate). Figures 6 and 7 were obtained using the Gaussian approximation for all three test statistics (since large sample sizes are involved) while Fig. 8 is a simulation result. Note that these figures are basically ROC (receiver operating characteristics) curves of those three detectors under consideration. When the

1.200

T

FIG. 6. The curves of the probability of detection vs the false alarm rate for the quadratic detector, the linear detector and the multinomial detector with b = 4.2048 samples averaged, signal-to-noise ratio = - 10 dI3. 88

Journalofthe

Franklin Institute Pergamon Journals Ltd.

Radar Detector Using Quantized Sbnples

I.OcW

&lo

g

a

.mo

Jioo

‘200

FIG. 7. The curves of the probability of detection vs the false alarm rate for the quadratic detector, the linear detector, and the multinomial detector. 128 samples averaged, signalto-noise ratio = -2 dB.

signal-to-noise ratio is small (Fig. 6), we see that the quadratic detector is the best and that the multinomial detector is not far from the best. For a medium signalto-noise ratio (Fig. 71, the multinomial detector is more powerful than the other two detectors, as expected. When the signal-to-noise ratio is large (Fig. 8), the linear detector and the multinomial detector are close in performance, and the quadratic detector is less powerful. Note in particular that Fig. 7 also gives an indication of how much the quadratic detector and the linear detector deviate from optimum in the middle range of signal-to-noise ratio.

For multiple-puIse detection of a radar signal with sure amplitude and random phase, it was shown that the multinomial detector may be advantageous in comparison with the quadratic detector and the linear detector since its usefulness is not restricted by the magnitude of the signal-to-noise ratio. Also, the multino~al detector performs postdetection by digital counting which avoids the difficulty of video storage in automatic detection systems. However, the multinomial detector is not appropriate when the signal amplitude is neither known nor pre-estimated. In this case, some means of simultaneously estimating qk(a) and detecting is required. One possibility is the use of the Pearson’s chi-square test (9) for which only P,+are required and the test statistic is independent of the signal strength. Vol. 322, No. 2, pp. 79-92, August 1986 Printed in Great Britain

89

C. C. Lee and G. J. Jaw

0.000

10-6

8

’

;

L

10-5

a ;

8

10-4

n ;

’

10-3

f’(FA)

c

;

*

10-2

3

4

10-l

FIG. 8. The curves of the probability of detection vs the false alarm rate for the quadratic detector, the linear detector, and the multinomial detector with b = 4.4 samples averaged, SNR = 7 dB.

In stating the detection problem in Section I, it was assumed that the noise process is Gaussian. In case this assumption is invalid, the quadratic detector and the linear detector may become inappropriate. Note that, under a non-Gaussian situation, the Rayleigh and Rician are not the correct distributions for the envelope samples ; the correct ones are in general extremely difficult to determine, For the multinomial detector, all the input statistics are summarized by the set of probability parameters pk and &(ff). If these parameters are measured directly at the envelope detector’s output (instead of being computed from Rayleigh and Rician distributions in channel modeling), the resulting multinomial detector is still applicable no matter how inaccurate the Gaussian assumption is. The measurement of the required multinomial probabilities is easy because the maximum likelihood estimates for them are the relative frequencies of occurrence. This advantage makes it possible to generalize the concept of the multinomial detector to adaptive systems where the input statistics must be updated periodically. The integers m,, VZ~,. . . , mj collected by the binary counters of Fig. 3b can be stored and used to update channel statistics from time to time.

(1) J. V. DiFranco and W. L. Rubin, “Radar Detection”, Prentice-Hall, EngIewood Cliffs, 1968. Journal

90

of the Franklin Institute Pergamon Journals Ltd.

Radar Detector

Using Quantized

Samples

(2) C. W. Helstrom “Statistical Theory of Signal Detection”, 2nd Edn., Pergamon Press, New York, 1975. (3) A. D. Whalen, “Detection of Signals in Noise”, Academic Press, New York, 1971. (4) M. Abramowitz and I. A. Stegun, “Handbook of mathematical Functions”, Dover, New York, 1970. (5) B. Widrow, “Statistical analysis of amplitude-quantized sampled-data systems”, AIEE Trans. Appi. Indust., Vol. Xl, pp. 555-568, Jan. 1961. (6) S. A. Kassam and J. B. Thomas, “Generalizations of the sign detector based on conditional tests”, IEEE Trans. Commun., Vol. COM-24, pp. 481-487, May 1976. (7) C. Lee and J. B. Thomas, ‘Detectors for multinomial input”, ZEEE Trans. Aero. Elec. Sys., Vol. AES-19, pp. 288-297, March 1983. (8) P. Papantoni-Kazakos and D. Kazakos (eds), “Nonparametric Methodr in Communications”, Dekker, New York, 1977. (9) P. J. Bickel and K. A. Doksum, ‘~~athemat~~al Statistics : Basic Ideas and Selected Topics”, Holden-Day, San Francisco, 1977. (10) J. D. Gibson and J. L. Melsa, “Introduction to Nonparametric Detection with Applications”, Academic Press, New York, 1975. (11) G. E. Nother, “On a theorem of Pitman”, Ann. Math. Statist., Vol. 26, pp. 64-68, 1955. (12) J. Capon, “On the asymptotic efficiency of locally optimum detectors”, IRE Trans. Information Theory, Vol. IT-7, pp. 67-71, Jan. 1961. (13) J. W. Modestino and A. Y. Ningo, “Detection of weak signals in narrowband nonGaussian noise”, IEEE Trans. Information Theory, Vol. IT-25 pp. 592-600, Sept. 1979. (14) N. 11. Lu and B. A. Eisenstein, “Detection of weak signals in non-Gaussian noise”, IEEE Trans. Information Theory, Vol. IT-27, pp. 75.5-771, Nov. 1981. Appendix.

First and ~~eu~~ advents

The test statistic of the multinomial

ofT,, detector is

T, = f nk log

k=1

For the multinomial

kddhl.

vector (n,, n2, . , n,J, we have (9) E(Q) = nh,,

E(n,2) = n(n-

l)h: +nhk

and E(n,n,) = n(n - l)h,h,

for

j # k,

where hk = pk when H0 is true and h* = y&) when H, is true. It follows that

and that

Vol. 322, No. 2, pp. 79-92, August Printed in Great Rritam

1986

91

C. C. Lee and G. J, Jaw Also, we have

E[TZlff~l =E

f j=

f

njnk

I k=

log

[qk(a)/pkl

l‘%

[%(a)/Pjl

1

,=I

k=l

k=, j#k

m =

kT,

b@-

m

+

[qk@)/pkl)

lh?+n~kl{log

m

1 1 [n(nj=l

l)pjpkl

log

[q,(a)/pjl

log

[qk (“)/pkl

k=l

/+k

=

f k=

I

npk

{log [qk(u)/Pk]} *+ f

,=,

2 incn-

l)p]Pkl

log

[qj(“)/pjl

log

[qk(“)/pkl

k=l

and thus 0: = var [T,IH,] = E[TiIH,,]-[E(T,IH,,)]*

= kE,

npk{log

[qk(“)/Pk1)2--n

kc,pk

log

[qk@)/pkl

‘.

1

I

In the same way, we obtain 0;’ = var[T,IH,]

= 5 k=l

92

nqk(a){log

[qk(a)/pkl}Z-

n

t k=

qk@)

log

[qk@)/pkl

I

Journal of the Franklin Institute Pergamon Journals Ltd.