- Email: [email protected]
A CFAR processor for the detection of unknown random signals in nonstationary correlated noise
SIGNAL
PROCESSING ELSEVIER
Signal Processing 48 (1996) 17-26
A CFAR processor for the detection of unknown random signals in nonstationary correlated noise Q.T. Zhang* Department
of Electrical Engineering, Ryerson Polytechnic
University. 350 Victoria Street, Toronto, Ontario, Canada A45B 2K3
Received 3 June 1993; revised 1.5December 1994 and 18 September 1995
.4bstract Signal detection in a nonstationary environment has been widely studied in the literature, either in the context of known deterministic signals in unknown correlated noise or in the context of unknown random signals in white noise. However, the detection of unknown random signals in unknown correlated noise has not been adequately described although it is of practical importance. Application of the conventional normalization-plus-integration technique to such situations results in performance degradation and failure to achieve a constant false-alarm rate (CFAR). An efficient and computationally simple CFAR detection procedure is proposed in this paper, based on the comparison of the covariance structure of the data vector under test with that of the interference. The theoretical detection performance of the new test, in a closed form, is presented. It is shown that for interference with moderate and high autocorrelation, the new test provides about 6 to 12 dB improvement over the conventional normalization technique. Zusammenfassung Die Signaldetektion in einer instationlren Umgebung ist in der Literatur ausfiihrlich untersucht worden, und zwar entweder im Kontext eines bekannten determinierten Signals in unbekanntem korreliertem Rauschen oder in dem eines unbekannten Zufallssignals in we@‘em Rauschen. Die Detektion eines unbekannten Zufallssignals in unbekanntem korreliertem Rauschen ist jedoch bislang nicht angemessen beschrieben worden trotz ihrer praktischen Bedeutung. Die Anwendung konventioneller Techniken mit Normalisierung und Integration in solchen Situationen fiihrt zu verringerter Leistungsftihigkeit und dem Verfehlen einer konstanten Fehlalarmrate (CFAR). In diesem Beitrag wird nun eine einfache CFAR-Detektionsprozedur vorgeschlagen; sie beruht auf dem Vergleich der Strukturen der Datenvektorund der Interferenz-Kovarianzen. Die theoretische Detektionsfihigkeit des neuen Tests wird in geschlossener Form vorgestellt. Es wird gezeigt, da13 der neue Tests fi.ir Interferenzen mit mll3iger bis starker Autokorrelation eine Verbesserung von ca. 6 bis 12 dB gegeniiber der iiblichen Normalisierungstechnik bringt. Rksumir La dCtection d‘un signal dans un environnement non stationnaire a Ctt largement &udiCe dans la littkrature, soit dans le contexte de signaux dkterministes connus dans un bruit corrtlt: inconnu ou dans le contexte de signaux alCatoires inconnus dans un bruit blanc. Toutefois, la dCtection de signaux alhatoires inconnus dans un bruit corr&lC inconnu n’a pas encore CtC d&rite de faGon adirquate bien qu’elle soit d’importance pratique. L’application de la technique
* E-mail: kzhang(@ee.ryerson.ca; fax: (416) 979-5280 0165-l 684/96/$15.00 c 1996 Elsevier Science B.V. All rights reserved SSDI 0165-1684(95)00120-4
18
Q. T. Zhang / Signal Processing
48 (1996) 17-26
conventionnelle de normalisation-plus-inkgration & de telles situations a pour risultat une d&gradation des performances et Cchoue $ maintenir un taux de fausse alarme constant (CFAR). Nous proposons dans cet article une proctdure de dktection CFAR efficace et simple au niveau calcul, baste sur la comparaison des structures de covariance du vecteur de donnkes test& et de l’interfkrence. Les performances de dttection thkoriques du nouveau test sont prksenttres sous forme analytique. 11est monk que pour une interfirence ayant une autocorr&lation mod&&eou klevte, le nouveau test fournit une amklioration de l’ordre de 6 a 12 dB par rapport k une mkthode de normalisation conventionnelle. Keywords: Adaptive detection; CFAR processor; Generalized normalization field
1. Introduction
Signal detection in a nonstationary noise environment has been described extensively in the literature. In the derivation of a desired test procedure, one usually assumes either the model of a known deterministic signal in unknown correlated noise [ll, 71, or the model of an unknown random signal in white noise [3,9,2]. The solution proposed for the former case is an adaptive implementation of the matched filter [l l] or a generalized likelihood ratio test [7], while that for the latter is the normalization technique [3,9]. However, detection of UIZknown random signals in unknown correlated noise has not been adequately described. Deterministic signal models and the corresponding test procedures are applied only to nonfluctuating targets. On the other hand, most of targets encountered in radar and sonar are fluctuating, and are therefore more suitably described by stochastic models [3,12]. Very often, the stochastic nature of signals is also attributed to the multipath propagation, especially in an underwater environment. The general idea of the normalization technique is to compare the squared amplitudes of the received data with the noise power to determine the presence of a target signal. When processing with multiple pulses, the normalization technique performs two functions: normalize the power of the received data to achieve a constant false-alarm rate (CFAR) and integrate the useful information in the data to enhance the detection performance. These goals can be achieved, indeed, provided that the noise is a spatially uncorrelated process. Unfortunately, the white noise assumption may not be valid in practice. In fact, radar clutter and
technique; Nonstationary
correlated noise
sonar reverberation are usually spatially correlated. Application of the conventional normalization technique to these situations will cause two problems: performance degradation [3] and loss of the CFAR property. The second problem is a severe obstacle to the system implementation since the threshold of the test depends on the noise covariance matrix, which is typically unknown or time varying. The primary objective of this paper is to develop a CFAR processor for detecting unknown complex random signals in unknown correlated interference. Here the term interference is used to include clutter or reverberation plus the receiver thermal noise. For convenience, hereafter the terms interference and noise will be used interchangeably. Correlated data convey both amplitude and Doppler information, which in turn are contained in the covariance matrix of the data. In general, the covariance structure of the interference differs from that of a target since they have different geometries and velocities. Thus, when a target is present, the covariance structure of the data under test will change relative to its neighborhood. Such statistical variations in covariance structure will be exploited to construct a new procedure for detecting unknown random signals in unknown correlated noise. An important feature of the detection problem under consideration is that the unknown signal covariance matrix has more parameters than the length of the data under test, making it difficult to use the maximum likelihood principle for the derivation of the test statistic. A heuristic argument is therefore used instead. The new technique is applicable to both stationary and nonstationary environments. For the latter,
Q. T. Zhang 1 Signal Processing 48 (I 996) 17-26
it needs an additional weak assumption of local stationarity, which is satisfied in most practical situations. The remainder of the paper is organized as follows. The problem formulation and the test statistic construction are described in Section 2, followed by a performance analysis in Section 3. Section 4 compares the detection performance of the new scheme with the conventional normalization technique, and Section 5 concludes the paper with a summary.
2. The test statistic Suppose we are given m x 1 complex data vectors y and {xk, k = 1, . . , n; n > m}, where y is the data set to be processed and x, are the reference data, which contain interference only. The reference data (xk} are usually obtained from the nearby cells at different range rings or from the previous scans, depending on the statistical characteristics of the given environment. The cells immediately adjacent to those under test are excluded from the reference data to avoid the effect of target signal splitting over the range. The purpose of signal processing is to choose one of the following hypotheses: H,: H,:
y contains interference alone, y contains signal plus interference.
Given the above data, we can estimate the covariance matrix of the interference, as shown by
iix=i E x,x:,
19
The condition II > m is required in order for the inverse of the sample covariance matrix to be valid. The corresponding decision rule can be written as
where A, is the preset threshold. The new statistic differs from the conventional normalization technique in three respects. First, it processes the data on a vector basis and is therefore capable of extracting the useful information in the data effectively. Second, in the new scheme the received vector is normalized by the estimated interference covariance matrix. This implies that the signal detection is based on the comparison of the full covariance structure of the data. Such a comparison becomes obvious if we rewrite (2) as (4)
Indeed, the summation inside the trace operation represents the sample estimate of the reference covariance matrix and (yyt) represents the sample estimate of the covariance matrix of the data under test. Finally, the new scheme allows the normalization and integration to be accomplished in a single step, rendering itself computationally efficient. The assumptions made in this paper are in order: The random vectors xk are independent, identically m-dimensional complex Gaussian distributed with mean zero and covariance matrix R,. Symbolically, we write
nk=l
with the dagger denoting the Hermitian transpose. This covariance matrix provides a statistical structure against which the data under test can be compared. Recall that in the conventional technique, received and reference data are compared on the basis of power level, which, however, is only partial information about the covariance matrix. Thus, as a natural extension of the conventional normalization technique, we suggest using a new test statistic defined by -1
y,
n > m.
(2)
xk -
Wr,(O,R,).
(5)
The random vector y is independent of _%k,and distributed as Y w Wn(W,).
(6)
Under hypothesis Ho, y has the same distribution as xk. That is, under this condition, R, = R,. The Gaussian assumption on the data is made here to facilitate the performance analysis, but is not necessary for the use of the proposed scheme. Observing Eq. (2), we can find two simple but interesting properties of the new test.
Q. T. Zhang / Signal Processing 48 (I 996) 17-26
20
Property 1. The new test includes the conventional normalization as a special case. SpeciJically, when m = 1, all vectors in (2) become scalars and lo reduces to (7) where 8: is the sample estimate of the interference variance, given by
(8)
vectory, then average this PDF overy to obtain the desired result. In the case of real data with given y, it has been shown ([S, pp. 95-961) that 1 is Wishart distributed. By following the procedure used by Giri in proving his Theorem 3.1 of [4], it is quite straightforward to show that conditional on y, 1 has a one-dimensional complex Wishart distribution [6, 51, with (n - m + 1) degrees of freedom and covariance matrix (ytR; ‘y)-‘. More explicitly we can write the conditional PDF as fi(z,y)
=
(y+R;
‘y)“-,+ ‘znemexp( -
Eq. (7) has exactly the same form as the conventional normalization test except for a constant factor. Property 2. The new test IO is invariant under an arbitrary linear transformation of the data.
This property can easily be examined by replacing y and & in (2) with Ay and AXE, where A is a nonsingular m x m matrix. It implies that under a zero-mean Gaussian assumption on the received data, the new test has the property of constant false alarm rate since changes in interference have no effect on the test statistic. For ease of theoretical analysis, we will use the reciprocal of lo as our test statistic such that
ytR; ‘yz)
r(n - m + 1)
(10) We need to average fi(zly) over y to obtain the unconditional probability density function. Let us denote a=y+R;‘y,
p=n-mm,
D=$(.)
and rewrite (10) as aP+
lzPe-az
fi(ZlY) =
P!
(12) Averaging with respect to the distribution m-vector y, we find
1 =y+(~;=,xkx,t)-ly
HI 6 A,
n > m,
EY[e-“‘1
Ho
with n denoting the corresponding
threshold.
(11)
= (n”‘det R,)-’
s
of the
...
s
exp [ - zytR; ‘y - ytR; ‘y] d”y
3. Performance analysis
= (?‘det R,)-’ To determine the detection performance of the test 1 given in (9), we need to determine its probability density function (PDF) under both hypotheses Ho and H1. Since 1 is a complicated sample function of the observed data, it is difficult to obtain the PDF through direct calculation. We will instead take an indirect, two-step procedure. We first evaluate the conditional PDF of 1 given random
s
s
...
exp [ - yt (R; ’ f zR; ’ )y] d”y
1 = det R, det(R; ’ + zR; ’ ) = [det(Z + zR,R; ‘)I-‘,
(13)
Q,T Zhang 1 Signal Processing 48 (1996) 17.-26
which, when combined J(z) = E,Cfi(z
with (12), gives
Iv)1
2(-Lq~+l
E,.[e-“‘I
2$-L))“’
[det(l+
(14)
zR$R;‘)]-‘.
The value of F, can be found from a look-up table [lo], or by using Formula (26.6.5) of [ 1, p. 9461. Before concluding this subsection, we should point out that the new detector is CFAR only if the data are Gaussian; it is not a CFAR in the nonparametric sense.
(15)
qf detection
3.2. Probability Eq. (14) provides a basis to study the false alarm rate and detection probability of the new test.
Let i,, denote the eigenvalues hypothesis H1, we can write det(l+
3.1. The false alarm rate
zR,R;‘)-’
= fi
of RYR; ‘. Under
(1 + &z)-l
k=l
Under hypothesis reduces to
Ho, R,R;
1 = I and, thus, (14)
(21)
~;(zIH~,=~(-D)~+~(~ P!
(m + p)!zp
=_ p!(m
with the residue coefficients
+z)-”
-
l)!(l
+ z)m+p+l’
(16)
This equation can be used to determine the threshold for a preset false alarm rate. In fact, the evaluation of threshold can considerably be simplilied if we realize that the new variable defined by the linear transform
‘k =
j;fi+k[l -%I-’
m n-m+1
1
=
(17)
follows the F-distribution with 2(n - m + 1) and 2m degrees of freedom. As a convention, we write
Suppose to be
the probability
Pf, = 3
ep+1
n-m+1 FE,
,;,
akhk (1
+
~kz)P+2’
(23)
< I,(Hi)
of false alarm (Pr,) is preset
(19)
m
l)ZP
Hi
(18)
and F, denotes the lower a-point of the F-distribution. The required threshold for the test statistic I is then equal to I, =
+
under
which, when integrated from 0 to the threshold level [,, yields the probability of detection Pd = Pr(z
t - F(2(n - m + 1),2m).
(p
(22)
(21) and (14). the PDF
By combining becomes
m
[=
given by
(20)
(24) where p = n - m. Clearly, the detection probability of the new test depends on the statistics of the data only through the eigenvalues of RL’R,. It is also clear that Pd is a simple function of the threshold value.
Q. T. Zhang J Signal Processing 48 (1996) 17-26
22
where p in (29) is a parameter used to control the signal-to-noise ratio (SNR), defined by
4. Performance comparison In this section, we examine the new test by comparing its detection performance with that technique. of the conventional normalization The test statistic of the conventional scheme is defined by
t=’
(25)
U'
SNR = 10 log,,
trace {pR,} trace {R,} ’
As an illustration, we consider the situation in which r&i, k) and rx(i, k) have the following form: Ik-il
r&k)
= rs
r,(i,k)
=
(e
jO.3rr
k-i
1
,
r(ck-il(e-jO.lx)k-i,
where lJ =y+y,
(26)
”
u =
1 x& k=l
We need the probability density function of t to evaluate its detection performance. This PDF can be obtained by using the formula for the PDF of the quotient of two random variables whose PDFs, in turn, can be obtained from their moment generating functions. We denote the eigenvalues of the matrices R, and R, by {ak} and {ak}, respectively. As shown in Appendix A, the detection probability is given by P,(T) = Pr{t > T) =
j$lcj [det (I+
pi ’ TR,)] - “.
In this expression, T > 0 is a prespecified shold, and cj are the coefficients given by
(27) thre-
(28) When hypothesis H, is true, R, = R, and thus pj = Dlj. Accordingly, in determining the required threshold, Eq. (27) is still applicable except that all fij should be replaced by ctj. Let r,(i, k) and rx(i, k) denote the (i, k)th entries of the signal and inteference (noise) covariance matrices, respectively. Then we can write the covariance matrix of the data under test as
4 = Cr&,k)l + drdi, k)l,
(29)
where r, > 0 and rc > 0 represents the magnitudes of the autocorrelation of the signal and interference. We note that R, and R,are Toeplitz matrices. Fig. 1 shows the probability of detection versus SNR for various probability of false alarm (Pr,), in which the following parameters are used: n = 40,
m = 10,
r, = 0.72, rs = 0.80.
From Fig. 1, we see that the new scheme is superior to the conventional technique by about 6 dB for various values used for the CFAR. Fig. 2 illustrates the effect of the data length m on the detection performance, in which n = 40, Pf,= 10m5, rc = 0.72 and r, = 0.80. As expected, the detection performance improves with increased data length. A longer data length implies more information is available. For all three values of m, again, 5 to 6 dB improvement of the new scheme over the conventional method is observed. Finally, we show the influence of the autocorrelation of the interference on the detection performance in Fig. 3, where the parameters are n = 40,
m = 10,
Pf, = 10-5, r, = 0.8.
The curves are plotted for r, equal to 0.5,0.72 and 0.9, respectively. The new test obviously outperforms the conventional method. It is interesting to observe that the detection performance of the new
Q.T. Zhang / Signal Processing
48 (I 996) 17-26
23
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 04
SNR (dB) Fig.
1. Performance
comparison
of the new and conventional
schemes.
SNR (dB) Fig. 2. The effect of the data length on the detection
scheme improves as the interference becomes more correlated. This is in contrast to the conventional scheme whose detection performance degrades with increased interference correlation. As a result, the new scheme is considerably superior to the conventional test as rc increases. With the above parameter setting, the performance improvement
performance
for I, = 0.5 is only about 1.2 dB. As r, increases to 0.72 and 0.90, however, the performance improvement becomes 6 and 12.5 dB, respectively. These results are not difficult to understand since for a small rcr the noise tends to be uncorrelated and under this situation, as shown in Section 2, the new test is nearly identical to the conventional test. As r,
24
Q.T. Zhang / Signal Processing
------E ‘S 4g
0.7-
z ,x ‘3
0.5 -
g k
0.6 -
48 (1996) 17-26
c”n”enu”na,
Pfa= lE-5 n=40, m=lO Curve
1 2 3
0.4 -
rc 0.50 0.72 0.90
1
0.3 0.2 -
,’ ,’ ,’ ,’ : ,’ ,’ ,’ ,’
/l/J
0.1 0 -1.5
-10
-5
,’ ,’ ,’ ‘,, ‘I ,I ,,* ,_ _-I’ __,’ __,*’ --0 5
10
15
20
1
25
SNR (dB) Fig. 3. The effect of the interference autocorrelation
increases, the estimation of the noise variance used in the conventional test is based on highly correlated data and thus has a very poor accuracy, resulting in degradation in the probability of detection. On the contrary, the new scheme is capable of exploiting the covariance structure of the noise data for signal detection. Interference with a high correlation provides more statistical structures for the new scheme to distinguish signal from noise and, hence, a better detection performance. Besides its superior detection performance, the new test has the property of constant false alarm, which is not shared by the conventional normalization technique. As shown in (27), the threshold of t for a given Pr, is a function of R,. In other words, the threshold of the conventional test depends on the covariance structure of the noise environment, which is usually unknown or time varying, making the implementation of the test difficult in practice.
5. Conclusion In this paper, we have considered the problem of detecting random signals in an unknown twodimensional correlated noise field. By comparing the covariance structures of the reference data and
on the detection performance.
the data under test, we obtain a generalized normalization test. The new test provides a powerful means of extracting the signal information in the second-order statistics. Its statistical performance is therefore superior to the conventional normalization technique. For a noise field with correlation coefficient greater than 0.7, the performance improvement ranges from 6 to 12 dB. This result is important in practice since many radar data, such as ground clutter, sea clutter and weather clutter, have a correlation coefficient of the same order or even higher. The new test retains the property of constant false alarm rate. For a given CFAR, the threshold can easily be determined by using the F-distribution. We have derived a simple formula, in a closed form, for the performance evaluation of the new test. The implementation of the new test is simple. The new scheme can operate in a changing environment adaptively if we update the sample interference covariance matrix by discarding the old reference data. With its CFAR property, ease of implementation and good detection performance achieved with little prior information taken into account, the new scheme is an attractive candidate for the detection of unknown random signals in an unknown correlated noise environment.
Q.T Zhang J Signal Processing
48 (1996) I7- 26
Acknowledgements
= 9-l
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada, to which the author wishes to express his sincere appreciation. The author is also most grateful to the anonymous reviewer for his useful comments and suggestions which improved the formula for the detection performance and the presentation of the paper.
fi j=
25
(1 +
Spj)-’
1
(A.4) where ,Oj are the eigenvalues of R, and cj are the residue coefficients defined in (28). Thus the PDF of v can be written as
(A.5) Appendix A. Derivation
of Eq. (27) By using the formula for the density function of the quotient of two independent random variables, we find for the probability density function of t = vIu:
Since xk - C'N,,,(O,R,), for each k, we have E[e-“‘“1
=(n’“det
Rx)-' ..
s
fxt) =
exp [ - sx+x - x+R; 'x]d”x
s‘x 0
ufl.(Wfu(u)du:
= [det (1+ sR,)]- '
= fi:
(1 + SXJl,
(A.11
x exp
k=l
where %kare the eigenvalues of R,.The PDF of u is the inverse Laplace transform
with hk* denoting the partial-fraction coefficients that we shall not need to evaluate. Carrying out the transform. we find which after integrated u 3 0,
64.3) for the density function of the denominator U. Similarly the PDF of u is the inverse Laplace transform
.f;:(v)= Y- ’ [det(I + sR,)] -1
P,(T)
= Prft
from T to x
s T IH1 3
gives
Q.T. Zhang 1 Signal Processing 48 (1996) 17-26
26
Comparing (A.7) with (A.l) and (A.2), and putting s = T/cck there, we obtain the probability of detection as P,(T) = f j=
cj[det(Z + PjlTRx)]-“,
T > 0.(A.8)
1
This formula can also be used to determine the false alarm rate by setting flj = Clj.
References Cl] M. Abramowitz and LA. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. [2] J.V. DiFranco and W.L. Rubin, Radar Detection, Artech House, Dedham, MA, 1980, pp. 291-294. [3] J.L. Eaves and E.K. Reedy, Principles of Modern Radar, Van Nostrand Reinhold Co., New York, 1987, pp. 255-262 and 343-367. [4] N. Giri, “On the complex TZ- and Sz-tests”, Ann. Math. Statist., Vol. 36, 1965, pp. 665-670.
[S] N.G. Goodman, “Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction)“, Ann. Math. Statist., Vol. 34, 1964, pp. 152-177. [6] A.T. James, “Distributions of matrix variates and latent roots derived from normal samples”, Ann. Math. Statist., Vol. 35, 1964, pp. 475-501. [7] E.J. Kelly, “An adaptive detection algorithm”, IEEE Trans. Aerospace Electronic Systems, Vol. AES-22, No. 1, 1986, pp. 115-127. [S] R.J. Muirhead, Aspects of Multivariate Statistical Theory, Wiley, New York, 1982. [9] R. Nitzberg, “Constant-false-alarm-rate processor for locally non-stationary clutter”, IEEE Trans. Aerospace Electronic Systems, Vol. AES-9, No. 3, May 1973, pp. 399-405. [lo] E.S. Pearson and H.O. Hartley, eds., Biometrika Tablesfor Statisticians, Vol. II, Cambridge University Press, Cambridge, England, 1976, pp. 170-183. [11] I. Reed, J.D. Mallet and L.E. Brennan, “Rapid convergence rate in adaptie array”, IEEE Trans. Aerospace Electronic Systems, Vol. AES-10, No. 6, 1974, pp. 853-863. [12] P. Swerling, “Detection of fluctuating pulsed signals in presence of noise”, IRE Trans. Inform. Theory, Vol. IT-3, September 1957, pp. 175-178.
PROCESSING ELSEVIER
Signal Processing 48 (1996) 17-26
A CFAR processor for the detection of unknown random signals in nonstationary correlated noise Q.T. Zhang* Department
of Electrical Engineering, Ryerson Polytechnic
University. 350 Victoria Street, Toronto, Ontario, Canada A45B 2K3
Received 3 June 1993; revised 1.5December 1994 and 18 September 1995
.4bstract Signal detection in a nonstationary environment has been widely studied in the literature, either in the context of known deterministic signals in unknown correlated noise or in the context of unknown random signals in white noise. However, the detection of unknown random signals in unknown correlated noise has not been adequately described although it is of practical importance. Application of the conventional normalization-plus-integration technique to such situations results in performance degradation and failure to achieve a constant false-alarm rate (CFAR). An efficient and computationally simple CFAR detection procedure is proposed in this paper, based on the comparison of the covariance structure of the data vector under test with that of the interference. The theoretical detection performance of the new test, in a closed form, is presented. It is shown that for interference with moderate and high autocorrelation, the new test provides about 6 to 12 dB improvement over the conventional normalization technique. Zusammenfassung Die Signaldetektion in einer instationlren Umgebung ist in der Literatur ausfiihrlich untersucht worden, und zwar entweder im Kontext eines bekannten determinierten Signals in unbekanntem korreliertem Rauschen oder in dem eines unbekannten Zufallssignals in we@‘em Rauschen. Die Detektion eines unbekannten Zufallssignals in unbekanntem korreliertem Rauschen ist jedoch bislang nicht angemessen beschrieben worden trotz ihrer praktischen Bedeutung. Die Anwendung konventioneller Techniken mit Normalisierung und Integration in solchen Situationen fiihrt zu verringerter Leistungsftihigkeit und dem Verfehlen einer konstanten Fehlalarmrate (CFAR). In diesem Beitrag wird nun eine einfache CFAR-Detektionsprozedur vorgeschlagen; sie beruht auf dem Vergleich der Strukturen der Datenvektorund der Interferenz-Kovarianzen. Die theoretische Detektionsfihigkeit des neuen Tests wird in geschlossener Form vorgestellt. Es wird gezeigt, da13 der neue Tests fi.ir Interferenzen mit mll3iger bis starker Autokorrelation eine Verbesserung von ca. 6 bis 12 dB gegeniiber der iiblichen Normalisierungstechnik bringt. Rksumir La dCtection d‘un signal dans un environnement non stationnaire a Ctt largement &udiCe dans la littkrature, soit dans le contexte de signaux dkterministes connus dans un bruit corrtlt: inconnu ou dans le contexte de signaux alCatoires inconnus dans un bruit blanc. Toutefois, la dCtection de signaux alhatoires inconnus dans un bruit corr&lC inconnu n’a pas encore CtC d&rite de faGon adirquate bien qu’elle soit d’importance pratique. L’application de la technique
* E-mail: kzhang(@ee.ryerson.ca; fax: (416) 979-5280 0165-l 684/96/$15.00 c 1996 Elsevier Science B.V. All rights reserved SSDI 0165-1684(95)00120-4
18
Q. T. Zhang / Signal Processing
48 (1996) 17-26
conventionnelle de normalisation-plus-inkgration & de telles situations a pour risultat une d&gradation des performances et Cchoue $ maintenir un taux de fausse alarme constant (CFAR). Nous proposons dans cet article une proctdure de dktection CFAR efficace et simple au niveau calcul, baste sur la comparaison des structures de covariance du vecteur de donnkes test& et de l’interfkrence. Les performances de dttection thkoriques du nouveau test sont prksenttres sous forme analytique. 11est monk que pour une interfirence ayant une autocorr&lation mod&&eou klevte, le nouveau test fournit une amklioration de l’ordre de 6 a 12 dB par rapport k une mkthode de normalisation conventionnelle. Keywords: Adaptive detection; CFAR processor; Generalized normalization field
1. Introduction
Signal detection in a nonstationary noise environment has been described extensively in the literature. In the derivation of a desired test procedure, one usually assumes either the model of a known deterministic signal in unknown correlated noise [ll, 71, or the model of an unknown random signal in white noise [3,9,2]. The solution proposed for the former case is an adaptive implementation of the matched filter [l l] or a generalized likelihood ratio test [7], while that for the latter is the normalization technique [3,9]. However, detection of UIZknown random signals in unknown correlated noise has not been adequately described. Deterministic signal models and the corresponding test procedures are applied only to nonfluctuating targets. On the other hand, most of targets encountered in radar and sonar are fluctuating, and are therefore more suitably described by stochastic models [3,12]. Very often, the stochastic nature of signals is also attributed to the multipath propagation, especially in an underwater environment. The general idea of the normalization technique is to compare the squared amplitudes of the received data with the noise power to determine the presence of a target signal. When processing with multiple pulses, the normalization technique performs two functions: normalize the power of the received data to achieve a constant false-alarm rate (CFAR) and integrate the useful information in the data to enhance the detection performance. These goals can be achieved, indeed, provided that the noise is a spatially uncorrelated process. Unfortunately, the white noise assumption may not be valid in practice. In fact, radar clutter and
technique; Nonstationary
correlated noise
sonar reverberation are usually spatially correlated. Application of the conventional normalization technique to these situations will cause two problems: performance degradation [3] and loss of the CFAR property. The second problem is a severe obstacle to the system implementation since the threshold of the test depends on the noise covariance matrix, which is typically unknown or time varying. The primary objective of this paper is to develop a CFAR processor for detecting unknown complex random signals in unknown correlated interference. Here the term interference is used to include clutter or reverberation plus the receiver thermal noise. For convenience, hereafter the terms interference and noise will be used interchangeably. Correlated data convey both amplitude and Doppler information, which in turn are contained in the covariance matrix of the data. In general, the covariance structure of the interference differs from that of a target since they have different geometries and velocities. Thus, when a target is present, the covariance structure of the data under test will change relative to its neighborhood. Such statistical variations in covariance structure will be exploited to construct a new procedure for detecting unknown random signals in unknown correlated noise. An important feature of the detection problem under consideration is that the unknown signal covariance matrix has more parameters than the length of the data under test, making it difficult to use the maximum likelihood principle for the derivation of the test statistic. A heuristic argument is therefore used instead. The new technique is applicable to both stationary and nonstationary environments. For the latter,
Q. T. Zhang 1 Signal Processing 48 (I 996) 17-26
it needs an additional weak assumption of local stationarity, which is satisfied in most practical situations. The remainder of the paper is organized as follows. The problem formulation and the test statistic construction are described in Section 2, followed by a performance analysis in Section 3. Section 4 compares the detection performance of the new scheme with the conventional normalization technique, and Section 5 concludes the paper with a summary.
2. The test statistic Suppose we are given m x 1 complex data vectors y and {xk, k = 1, . . , n; n > m}, where y is the data set to be processed and x, are the reference data, which contain interference only. The reference data (xk} are usually obtained from the nearby cells at different range rings or from the previous scans, depending on the statistical characteristics of the given environment. The cells immediately adjacent to those under test are excluded from the reference data to avoid the effect of target signal splitting over the range. The purpose of signal processing is to choose one of the following hypotheses: H,: H,:
y contains interference alone, y contains signal plus interference.
Given the above data, we can estimate the covariance matrix of the interference, as shown by
iix=i E x,x:,
19
The condition II > m is required in order for the inverse of the sample covariance matrix to be valid. The corresponding decision rule can be written as
where A, is the preset threshold. The new statistic differs from the conventional normalization technique in three respects. First, it processes the data on a vector basis and is therefore capable of extracting the useful information in the data effectively. Second, in the new scheme the received vector is normalized by the estimated interference covariance matrix. This implies that the signal detection is based on the comparison of the full covariance structure of the data. Such a comparison becomes obvious if we rewrite (2) as (4)
Indeed, the summation inside the trace operation represents the sample estimate of the reference covariance matrix and (yyt) represents the sample estimate of the covariance matrix of the data under test. Finally, the new scheme allows the normalization and integration to be accomplished in a single step, rendering itself computationally efficient. The assumptions made in this paper are in order: The random vectors xk are independent, identically m-dimensional complex Gaussian distributed with mean zero and covariance matrix R,. Symbolically, we write
nk=l
with the dagger denoting the Hermitian transpose. This covariance matrix provides a statistical structure against which the data under test can be compared. Recall that in the conventional technique, received and reference data are compared on the basis of power level, which, however, is only partial information about the covariance matrix. Thus, as a natural extension of the conventional normalization technique, we suggest using a new test statistic defined by -1
y,
n > m.
(2)
xk -
Wr,(O,R,).
(5)
The random vector y is independent of _%k,and distributed as Y w Wn(W,).
(6)
Under hypothesis Ho, y has the same distribution as xk. That is, under this condition, R, = R,. The Gaussian assumption on the data is made here to facilitate the performance analysis, but is not necessary for the use of the proposed scheme. Observing Eq. (2), we can find two simple but interesting properties of the new test.
Q. T. Zhang / Signal Processing 48 (I 996) 17-26
20
Property 1. The new test includes the conventional normalization as a special case. SpeciJically, when m = 1, all vectors in (2) become scalars and lo reduces to (7) where 8: is the sample estimate of the interference variance, given by
(8)
vectory, then average this PDF overy to obtain the desired result. In the case of real data with given y, it has been shown ([S, pp. 95-961) that 1 is Wishart distributed. By following the procedure used by Giri in proving his Theorem 3.1 of [4], it is quite straightforward to show that conditional on y, 1 has a one-dimensional complex Wishart distribution [6, 51, with (n - m + 1) degrees of freedom and covariance matrix (ytR; ‘y)-‘. More explicitly we can write the conditional PDF as fi(z,y)
=
(y+R;
‘y)“-,+ ‘znemexp( -
Eq. (7) has exactly the same form as the conventional normalization test except for a constant factor. Property 2. The new test IO is invariant under an arbitrary linear transformation of the data.
This property can easily be examined by replacing y and & in (2) with Ay and AXE, where A is a nonsingular m x m matrix. It implies that under a zero-mean Gaussian assumption on the received data, the new test has the property of constant false alarm rate since changes in interference have no effect on the test statistic. For ease of theoretical analysis, we will use the reciprocal of lo as our test statistic such that
ytR; ‘yz)
r(n - m + 1)
(10) We need to average fi(zly) over y to obtain the unconditional probability density function. Let us denote a=y+R;‘y,
p=n-mm,
D=$(.)
and rewrite (10) as aP+
lzPe-az
fi(ZlY) =
P!
(12) Averaging with respect to the distribution m-vector y, we find
1 =y+(~;=,xkx,t)-ly
HI 6 A,
n > m,
EY[e-“‘1
Ho
with n denoting the corresponding
threshold.
(11)
= (n”‘det R,)-’
s
of the
...
s
exp [ - zytR; ‘y - ytR; ‘y] d”y
3. Performance analysis
= (?‘det R,)-’ To determine the detection performance of the test 1 given in (9), we need to determine its probability density function (PDF) under both hypotheses Ho and H1. Since 1 is a complicated sample function of the observed data, it is difficult to obtain the PDF through direct calculation. We will instead take an indirect, two-step procedure. We first evaluate the conditional PDF of 1 given random
s
s
...
exp [ - yt (R; ’ f zR; ’ )y] d”y
1 = det R, det(R; ’ + zR; ’ ) = [det(Z + zR,R; ‘)I-‘,
(13)
Q,T Zhang 1 Signal Processing 48 (1996) 17.-26
which, when combined J(z) = E,Cfi(z
with (12), gives
Iv)1
2(-Lq~+l
E,.[e-“‘I
2$-L))“’
[det(l+
(14)
zR$R;‘)]-‘.
The value of F, can be found from a look-up table [lo], or by using Formula (26.6.5) of [ 1, p. 9461. Before concluding this subsection, we should point out that the new detector is CFAR only if the data are Gaussian; it is not a CFAR in the nonparametric sense.
(15)
qf detection
3.2. Probability Eq. (14) provides a basis to study the false alarm rate and detection probability of the new test.
Let i,, denote the eigenvalues hypothesis H1, we can write det(l+
3.1. The false alarm rate
zR,R;‘)-’
= fi
of RYR; ‘. Under
(1 + &z)-l
k=l
Under hypothesis reduces to
Ho, R,R;
1 = I and, thus, (14)
(21)
~;(zIH~,=~(-D)~+~(~ P!
(m + p)!zp
=_ p!(m
with the residue coefficients
+z)-”
-
l)!(l
+ z)m+p+l’
(16)
This equation can be used to determine the threshold for a preset false alarm rate. In fact, the evaluation of threshold can considerably be simplilied if we realize that the new variable defined by the linear transform
‘k =
j;fi+k[l -%I-’
m n-m+1
1
=
(17)
follows the F-distribution with 2(n - m + 1) and 2m degrees of freedom. As a convention, we write
Suppose to be
the probability
Pf, = 3
ep+1
n-m+1 FE,
,;,
akhk (1
+
~kz)P+2’
(23)
< I,(Hi)
of false alarm (Pr,) is preset
(19)
m
l)ZP
Hi
(18)
and F, denotes the lower a-point of the F-distribution. The required threshold for the test statistic I is then equal to I, =
+
under
which, when integrated from 0 to the threshold level [,, yields the probability of detection Pd = Pr(z
t - F(2(n - m + 1),2m).
(p
(22)
(21) and (14). the PDF
By combining becomes
m
[=
given by
(20)
(24) where p = n - m. Clearly, the detection probability of the new test depends on the statistics of the data only through the eigenvalues of RL’R,. It is also clear that Pd is a simple function of the threshold value.
Q. T. Zhang J Signal Processing 48 (1996) 17-26
22
where p in (29) is a parameter used to control the signal-to-noise ratio (SNR), defined by
4. Performance comparison In this section, we examine the new test by comparing its detection performance with that technique. of the conventional normalization The test statistic of the conventional scheme is defined by
t=’
(25)
U'
SNR = 10 log,,
trace {pR,} trace {R,} ’
As an illustration, we consider the situation in which r&i, k) and rx(i, k) have the following form: Ik-il
r&k)
= rs
r,(i,k)
=
(e
jO.3rr
k-i
1
,
r(ck-il(e-jO.lx)k-i,
where lJ =y+y,
(26)
”
u =
1 x& k=l
We need the probability density function of t to evaluate its detection performance. This PDF can be obtained by using the formula for the PDF of the quotient of two random variables whose PDFs, in turn, can be obtained from their moment generating functions. We denote the eigenvalues of the matrices R, and R, by {ak} and {ak}, respectively. As shown in Appendix A, the detection probability is given by P,(T) = Pr{t > T) =
j$lcj [det (I+
pi ’ TR,)] - “.
In this expression, T > 0 is a prespecified shold, and cj are the coefficients given by
(27) thre-
(28) When hypothesis H, is true, R, = R, and thus pj = Dlj. Accordingly, in determining the required threshold, Eq. (27) is still applicable except that all fij should be replaced by ctj. Let r,(i, k) and rx(i, k) denote the (i, k)th entries of the signal and inteference (noise) covariance matrices, respectively. Then we can write the covariance matrix of the data under test as
4 = Cr&,k)l + drdi, k)l,
(29)
where r, > 0 and rc > 0 represents the magnitudes of the autocorrelation of the signal and interference. We note that R, and R,are Toeplitz matrices. Fig. 1 shows the probability of detection versus SNR for various probability of false alarm (Pr,), in which the following parameters are used: n = 40,
m = 10,
r, = 0.72, rs = 0.80.
From Fig. 1, we see that the new scheme is superior to the conventional technique by about 6 dB for various values used for the CFAR. Fig. 2 illustrates the effect of the data length m on the detection performance, in which n = 40, Pf,= 10m5, rc = 0.72 and r, = 0.80. As expected, the detection performance improves with increased data length. A longer data length implies more information is available. For all three values of m, again, 5 to 6 dB improvement of the new scheme over the conventional method is observed. Finally, we show the influence of the autocorrelation of the interference on the detection performance in Fig. 3, where the parameters are n = 40,
m = 10,
Pf, = 10-5, r, = 0.8.
The curves are plotted for r, equal to 0.5,0.72 and 0.9, respectively. The new test obviously outperforms the conventional method. It is interesting to observe that the detection performance of the new
Q.T. Zhang / Signal Processing
48 (I 996) 17-26
23
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 04
SNR (dB) Fig.
1. Performance
comparison
of the new and conventional
schemes.
SNR (dB) Fig. 2. The effect of the data length on the detection
scheme improves as the interference becomes more correlated. This is in contrast to the conventional scheme whose detection performance degrades with increased interference correlation. As a result, the new scheme is considerably superior to the conventional test as rc increases. With the above parameter setting, the performance improvement
performance
for I, = 0.5 is only about 1.2 dB. As r, increases to 0.72 and 0.90, however, the performance improvement becomes 6 and 12.5 dB, respectively. These results are not difficult to understand since for a small rcr the noise tends to be uncorrelated and under this situation, as shown in Section 2, the new test is nearly identical to the conventional test. As r,
24
Q.T. Zhang / Signal Processing
------E ‘S 4g
0.7-
z ,x ‘3
0.5 -
g k
0.6 -
48 (1996) 17-26
c”n”enu”na,
Pfa= lE-5 n=40, m=lO Curve
1 2 3
0.4 -
rc 0.50 0.72 0.90
1
0.3 0.2 -
,’ ,’ ,’ ,’ : ,’ ,’ ,’ ,’
/l/J
0.1 0 -1.5
-10
-5
,’ ,’ ,’ ‘,, ‘I ,I ,,* ,_ _-I’ __,’ __,*’ --0 5
10
15
20
1
25
SNR (dB) Fig. 3. The effect of the interference autocorrelation
increases, the estimation of the noise variance used in the conventional test is based on highly correlated data and thus has a very poor accuracy, resulting in degradation in the probability of detection. On the contrary, the new scheme is capable of exploiting the covariance structure of the noise data for signal detection. Interference with a high correlation provides more statistical structures for the new scheme to distinguish signal from noise and, hence, a better detection performance. Besides its superior detection performance, the new test has the property of constant false alarm, which is not shared by the conventional normalization technique. As shown in (27), the threshold of t for a given Pr, is a function of R,. In other words, the threshold of the conventional test depends on the covariance structure of the noise environment, which is usually unknown or time varying, making the implementation of the test difficult in practice.
5. Conclusion In this paper, we have considered the problem of detecting random signals in an unknown twodimensional correlated noise field. By comparing the covariance structures of the reference data and
on the detection performance.
the data under test, we obtain a generalized normalization test. The new test provides a powerful means of extracting the signal information in the second-order statistics. Its statistical performance is therefore superior to the conventional normalization technique. For a noise field with correlation coefficient greater than 0.7, the performance improvement ranges from 6 to 12 dB. This result is important in practice since many radar data, such as ground clutter, sea clutter and weather clutter, have a correlation coefficient of the same order or even higher. The new test retains the property of constant false alarm rate. For a given CFAR, the threshold can easily be determined by using the F-distribution. We have derived a simple formula, in a closed form, for the performance evaluation of the new test. The implementation of the new test is simple. The new scheme can operate in a changing environment adaptively if we update the sample interference covariance matrix by discarding the old reference data. With its CFAR property, ease of implementation and good detection performance achieved with little prior information taken into account, the new scheme is an attractive candidate for the detection of unknown random signals in an unknown correlated noise environment.
Q.T Zhang J Signal Processing
48 (1996) I7- 26
Acknowledgements
= 9-l
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada, to which the author wishes to express his sincere appreciation. The author is also most grateful to the anonymous reviewer for his useful comments and suggestions which improved the formula for the detection performance and the presentation of the paper.
fi j=
25
(1 +
Spj)-’
1
(A.4) where ,Oj are the eigenvalues of R, and cj are the residue coefficients defined in (28). Thus the PDF of v can be written as
(A.5) Appendix A. Derivation
of Eq. (27) By using the formula for the density function of the quotient of two independent random variables, we find for the probability density function of t = vIu:
Since xk - C'N,,,(O,R,), for each k, we have E[e-“‘“1
=(n’“det
Rx)-' ..
s
fxt) =
exp [ - sx+x - x+R; 'x]d”x
s‘x 0
ufl.(Wfu(u)du:
= [det (1+ sR,)]- '
= fi:
(1 + SXJl,
(A.11
x exp
k=l
where %kare the eigenvalues of R,.The PDF of u is the inverse Laplace transform
with hk* denoting the partial-fraction coefficients that we shall not need to evaluate. Carrying out the transform. we find which after integrated u 3 0,
64.3) for the density function of the denominator U. Similarly the PDF of u is the inverse Laplace transform
.f;:(v)= Y- ’ [det(I + sR,)] -1
P,(T)
= Prft
from T to x
s T IH1 3
gives
Q.T. Zhang 1 Signal Processing 48 (1996) 17-26
26
Comparing (A.7) with (A.l) and (A.2), and putting s = T/cck there, we obtain the probability of detection as P,(T) = f j=
cj[det(Z + PjlTRx)]-“,
T > 0.(A.8)
1
This formula can also be used to determine the false alarm rate by setting flj = Clj.
References Cl] M. Abramowitz and LA. Stegun, Handbook of Mathematical Functions, Dover, New York, 1972. [2] J.V. DiFranco and W.L. Rubin, Radar Detection, Artech House, Dedham, MA, 1980, pp. 291-294. [3] J.L. Eaves and E.K. Reedy, Principles of Modern Radar, Van Nostrand Reinhold Co., New York, 1987, pp. 255-262 and 343-367. [4] N. Giri, “On the complex TZ- and Sz-tests”, Ann. Math. Statist., Vol. 36, 1965, pp. 665-670.
[S] N.G. Goodman, “Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction)“, Ann. Math. Statist., Vol. 34, 1964, pp. 152-177. [6] A.T. James, “Distributions of matrix variates and latent roots derived from normal samples”, Ann. Math. Statist., Vol. 35, 1964, pp. 475-501. [7] E.J. Kelly, “An adaptive detection algorithm”, IEEE Trans. Aerospace Electronic Systems, Vol. AES-22, No. 1, 1986, pp. 115-127. [S] R.J. Muirhead, Aspects of Multivariate Statistical Theory, Wiley, New York, 1982. [9] R. Nitzberg, “Constant-false-alarm-rate processor for locally non-stationary clutter”, IEEE Trans. Aerospace Electronic Systems, Vol. AES-9, No. 3, May 1973, pp. 399-405. [lo] E.S. Pearson and H.O. Hartley, eds., Biometrika Tablesfor Statisticians, Vol. II, Cambridge University Press, Cambridge, England, 1976, pp. 170-183. [11] I. Reed, J.D. Mallet and L.E. Brennan, “Rapid convergence rate in adaptie array”, IEEE Trans. Aerospace Electronic Systems, Vol. AES-10, No. 6, 1974, pp. 853-863. [12] P. Swerling, “Detection of fluctuating pulsed signals in presence of noise”, IRE Trans. Inform. Theory, Vol. IT-3, September 1957, pp. 175-178.