Evaluation of fractional, Iower-order statistics-based detection algorithms on real radar sea-clutter data

G .A. Tsi h ri ntzis C.L. Nikias

Indexing terms: Symmetric &ha-stable processes, Fractional lower-order stdstics, Radar signul processing, Radur target detection

Abstract: Alpha-stable distributions have recently been recognised in the signal processing community as simple, yet accurate, two- parameter statistical models for signals and noises that contain an impulsive component of various degrees of severity. On the basis of this finding, several signal processing problems have been addressed and solved within the framework of alpha-stable distributions and with the use of fractional, lower-order moments. The authors attempt to popularise these new signal processing tools within the radar community. In particular, they evaluate the goodness-of-fit of alpha-stable models in the radar environment and test the performance of new signal processing algorithms for signal detection on real radar sea-clutter data. They also include in the paper a brief review of the key ideas of signal processing with alpha- stable distributions, as well as a large number of references to the literature for further probing.

1 Introduction

The design of radar systems constitutes a highly challenging problem, characterised by very weak signal levels and strong interferences from either uninten- tional (clutter) or intentional (jammers) sources. In this unfavourable environment, complex tasks need to be performed by the radar sensors, such as detection and classification of very weak targets (embedded in clutter, noise and intentional jamming), robust beamforming to null interference (especially in constantly changing environments with minimal signal knowledge, moving arrays and uncertain element location and response), and automatic image analysis, segmentation and classification.

One main issue in this direction is that of developing proper statistical models for the interference that is present in radar returns and attains the form of spikes, due to clutter sources such as ocean waves, and glints,

0 IEE, 1997 IEE Proceedings online no. 19970933 Paper first received 18th June and in revised form 28th October 1996 G.A. Tsihrintzis is with the Communication Svstems Lab. Deuartment of Electrical Engineering, University of Virginia,dCharlottesville,LVA 22903- 2442, USA C.L. Nikias is with the Signal and Image Processing Institute, Department of Electrical Engineering ~ Systems, University of Southem California, Los Angeles, CA 90089-2564, USA

due to reflections from large, flat surfaces such as buildings or vehicles. The presence of these spikes obscures the target detection capability of the radar and degrades its performance. Usually, the K-distribu- tion is considered as the model for the amplitude statis- tics of sea-clutter [l-31:

In the above equation, K,(x) is a modified Bessel func- tion, c is a scale parameter and Y is a shape parameter. The K-distributed model for radar clutter arises from the assumption that the radar return consists of the sum of a large number of independent returns (‘speckle’) that vary in intensity with time. Other mod- els that have been proposed to statistically describe dutter fall within the class of spherically-invariant ran- dom processes [4, 51 and include the Weibull and the log-normal distributions as models for the clutter amplitude statistics. Target detection is usually done using either a bank of coherent detectors or a bank of quadratic energy detectors (e.g. [ 1-31 and references therein). However, in very spiky (impulsive) sea-clutter, the number of false alarms can be very high and cannot be reduced by varying the detection threshold [I]. Recently, the principles of higher-order statistics (HOS) have been used to propose tests to detect deterministic and non-Gaussian stochastic signals in Gaussian noise [6, 71. However the success of these methods in combat- ting the spiky nature of the interference has been very limited, especially for short observations [7].

In this paper, we attempt to raise research interest in the application of the recently proposed alpha-stable distributions to the modelling of radar clutter. Our motivation lies in the observation that [SI empirical impulsive noise probability density functions (PDFs) maintain a similarity to the Gaussian PDF, being bell- shaped, smooth and symmetric, but at the same time have significantly heavier tails. Additionally, these PDFs are often characterised by algebraic (inverse power) tails [SI. The above evidence, combined with a recent, increasing interest in the application of the theory of stable random variables and processes in statistical signal processing [9, lo], suggested that possi- ble quite accurate, yet simple models for large classes of impulsive noise may be the stable PDFs [I 11. Indeed, it can be shown theoretically that, under general assumptions, a broad class of impulsive noise follows a stable distribution [11, 121. The stable model has been tested with a variety of real data and was found to match the data with excellent fidelity [12], similar to that of the broadly accepted Middleton models [13-161.

29 IEE Proc -Radar, Sonar Navig., Vol. 144, No. 1, February 1997

The performance of optimum and suboptimum receivers in the presence of SUS impulsive interference was examined in [17], both theoretically and via Monte-Carlo simulation, and a method was presented for the real time implementation of the optimum nonlinearities. From this study, it was found that the corresponding optimum receivers perform in the presence of SaS impulsive interference quite well, while the performance of Gaussian and other suboptimum receivers is unacceptably low. It was also shown that a receiver designed on a Cauchy assumption for the first order distribution of the impulsive interference performed only slightly below the corresponding optimum receiver, provided that a reasonance estimate of the noise dispersion was available, which for real- time signal processing purposes could be obtained via the fast algorithms in [IS].

The study in [17] was, however, limited to coherent reception only, in which the amplitude and phase of the signals is assumed to be known exactly. The optimum demodulation algorithm for reception of signals with random phase in impulsive intereference and its corresponding performance was derived in [19] and tested against the traditional incoherent Gaussian receiver ([20], Chap. 4). Finally, the performance of asymptotically optimum multichannel structures for incoherent detection of amplitude-fluctuating bandpass signals in impulsive noise modelled as a bivariate, isotropic, symmetric, alpha-stable (BISaS) process was evaluated in [21]. In particular, our attention in [21] was directed to detector structures in which the different observation channels corresponded to spatially diverse receiving elements. However, our general findings hold for communication receivers of arbitrary diversity. We derived the proper test statistic by generalising the detector proposed by Izzo and Paura [22] to take into account the infinite variance in the noise model, and showed that exact knowledge of the noise distribution was not required for almost optimum performance. We also showed that receiver diversity did not improve the performance of the Gaussian receiver when operating in non-Gaussian impulsive noise and, therefore, a non-Gaussian detection algorithm could substitute for receiver diversity.

The present paper is devoted to an appraisal of the applicability of the alpha-stable model in the radar environment. More specifically, we evaluate the goodness-of-fit of the alpha-stable models in the radar environment and test the performance of the recently proposed new algorithms on real radar sea-clutter data. Our goal is twofold: (i) to popularise the concepts of alpha-stable distributions and fractional, lower-order statistics, as well as the basic signal processing algorithms that have been developed so far, among the radar community, and (ii) to test the proposed algorithms on real radar, sea-clutter data.

2 Symmetric, alpha-stable distributions

2.7 Univariate SaS distributions A univariate symmetric, a-stable ( S a S ) PDFfa(y, 6; .) is best defined via the inverse Fourier transform integral [9, 231

f a ( y , 6 ; x) = .i_, exp(i6w - ylwl")e-"Zdw (1)

and is completely characterised by the three parameters

1 "

30

a (characteristic exponent, 0 < a s 2), y (dispersion, y > 0) and 6 (location parameter, -CO < 6 < a).

The characteristic exponent a relates directly to the heaviness of the tails of the S a S PDF: the smaller its value, the heavier the tails. The value a = 2 corresponds to a Gaussian PDF, while the value a = 1 corresponds to a Cauchy PDF. For these two PDFs, closed-form expressions exist, namely:

(3)

For other values of the characteristic exponent, no closed-form expressions are known. All the SaS PDFs can be computed, however, at arbitrary argument with the real time method developed in [17]. The dispersion y is a measure of the spread of the S a S PDF, in many ways similar to the variance of a Gaussian PDF and equal to half the variance of the PDF in the Gaussian case (a = 2). Finally, the location parameter 6 is the point of symmetry of the S a S PDF.

The non-Gaussian ( a f: 2) S a S distributions maintain many similarities to the Gaussian distribution, but at the same time differ from it in some significant ways. For example, a non-Gaussian SaS PDF maintains the usual bell shape and, more importantly, non-Gaussian S a S random variables satisfy the linear stability prop- erty [23]. However, non-Gaussian S a S PDFs have much sharper peaks and much heavier tails than the Gaussian PDF. As a result, only their moments of order p < a are finite, in contrast with the Gaussian PDF, which has finite moments of arbitrary order. These and other similarities and differences between Gaussian and non-Gaussian SUS PDFs and their impli- cations on the design of signal processing algorithms are presented in detail in the tutorial paper [9], to which the interested reader is referred. For illustration purposes, we show in Fig. 1 plots of the S a S PDFs for location parameter S = 0, dispersion y = 1, and for characteristic exponents a = 0.5, 1, 1.5, 1.99 and 2. The curves in Fig. 1 have been produced by calculation of the inverse Fourier transform integral in eqn. 1.

0.7

0.6 i

argument of PDF Fig. 1 dispersion and various c aracteristic exponents ~ a = 2

Symmetric, athastable ,PDFs of zero location parameter, unit

a = 1.99 a = 1.5 a = 1 .... a = 0 . 5

_ _ _

... .

IEE Proc.-Radar, Sonar Navis, Vol. 144, No. I , February 1997

2.2 Bivariate isotropic symmetric, alpha- stable (BISaS) distributions Multivariate stable distributions are defined as the class of distributions that satisfy the linear stability property. In particularg an n-dimensional distribution function F(x), x E Rn is called stable if for any independent, identically distributed random vectors XI, X2 with joint distribution F(x) and arbitrary constants al, a2, there exist a E R, b E R”, and a random vector X with distribution F(x), such that alXl + a2X2 has the same distribution as aX + b. Unfortunately, the class of multivariate stable distributions cannot be parameterised [Note I]. Fortunately, however, the subclass of multivariate stable distributions that arise in impulsive noise modelling fall within the family of isotropic multivariate stable distributions which can be given a parametric form.

More specifically, the bivariate isotropic symmetric alpha-stable (BISaS) probability density function (PDF) Jh,y,61,62(x1, x2) is given by the inverse Fourier transform

fLY,Y,S1,S* (21, z2)

w1 dw2 x e - 2 ( x 1 W 1 t x 2 W Z ) d

(4) where the parameters a and y are termed the character- istic exponent and the dispersion, respectively, and 6, and S2 are location parameters. The characteristic exponent generally in the interval 0 < a I 2 and relates to the heaviness of the tails, with a smaller exponent indicating heavier tails. The dispersion y is a positive constant relating to the spread of the PDF. The two marginal distributions obtained from the bivariate dis- tribution in eqn. 4 are univariate S a S with characteris- tic exponent a, dispersion y and location parameters 6, and S2, respectively [9, 121. In the rest of the paper, we are going to assume d1 = 6, = 0, without loss of gener- ality, and drop the corresponding subscripts from all our expressions.

Unfortunately, no closed-form expressions exist for the general BISaS, PDF except for the special cases of a = 1 (Cauchy) and a = 2 (Gaussian):

where p” = xf + x,”. For the remaining (non-Gaussian, non-Cauchy) BISaS distributions, power series exist [9, 121, but are not of interest to this paper and therefore are not given here.

2.3 BlSaS models for impulsive noise Consider a narrowband receiver operating in S a S impulsive interference n(t) of dispersion y. Let

n(t) = n, cos(2nfOt) - n, sin(2nfot)

= R{ (n, + in,) exp(i2nfot)} (6) where n, and n, are the ‘in-phase’ and ‘quadrature’ components of the interference and nc + in, is its corre- sponding complex amplitude. The joint PDF of the two components can be shown [I21 to be the BISaS PDF of Note 1: The characteristic function of any multivariate stable distribution can be shown to attain a certain nonparametric form. The details can he found in [9, 121 and references therein.

dispersion y, i.e.

f ( % ns) = f a , y ( n c , n s ) (7) From eqns. 5 and 7 we can see that the in-phase and quadrature components of narrowband S a S interfer- ence are not independent, except in the Gaussian case (a = 2) [12]. Moreover, the joint distribution of the two components is heavy-tailed when compared to the Gaussian case. Finally transformation of the Fourier integral in eqn. 4 in radial co-ordinates ( p = d(n:+ n;), 0 = tan-’(nJn,)) gives the following joint PDF of the envelope p and the phase 0 of narrowband S a S inter- ference:

f ( P , Q ) = & { ~ ~ ~ U e x l ’ ( Y U ~ ) ~ ~ ( U ~ ) ’ l z U 1 1

2n = - f e , a , y ( P ) P > 0, Q E [0,2.ir) ( 8 )

where Jo(.) is the zero-order Bessel function of the first kind. Eqn. 8 clearly shows that the phase 6 of narrow- band S a S interference is uniformly distributed in the interval [0, 2 4 and independent of the corresponding interference envelope. These facts are in agreement with the corresponding results for the Gaussian case [ 121; however, the envelope distribution is heavy-tailed when compared with the Gaussian model. Indeed, it can be shown [12] that the envelope distribution

has algebraic tails

(9)

where B(a, y) is a positive constant, independent of p. In Fig. 2, we show the amplitude probability distribu- tion (APD), i.e. Pr{A > a } of the amplitude A of BIScrS random variables of unit dispersion and various characteristic exponents and compare them to the Gaussian distribution (dashed line).

i -20 -4

0 5 20 LO 60 80 95 99 Pr A >a], %

Fig. 2 noise unit dispersion and various c furacteristic exponents

APD of bivariate, isotro ic, symmetric, alpha-stable narrowband

3 Fractional, lower-order statistics

We consider a real or complex random variable 5 of zero location parameter, such that its fractional lower- order pth moment is finite,

E{lSl”> < 0 (10) where 0 < p < 00 and E{.} indicates statistical expectation. We will call < a pth-order random variable. Next we consider two real or complex pth- order random variables of zero location parameter <

31 IEE Proc.-Radar, Sonar Nuvig., Vol. 144, No. I , February 1997

and q. We define their pth-order fractional correlation as [24]

(5, r l ) P = E{5(rl)'P-1') (11) where

for real random variables and

for complex random variables. In eqns. 12 and 13, sgn(.) denotes the signum function, while the overbar denotes complex conjugation, respectively.

The above definitions are clearly seen to reduce to the usual second-order (SOS) and higher-order (HOS) statistics in the cases where those exist and can be easily extended to include random processes and their corresponding fractional correlation sequences. For example, if { X k } , k = 1, 2, 3, ..., is a discrete-time random process, we can define its fractional, pth-order correlation sequence as

which, for p = 2, gives the usual autocorrelation sequence.

The fractional, lower-order statistics (FLOS) of a random process have been found useful in designing algorithms that exhibit resistance to outliers and allow for robust processing of impulsive, as well as Gaussian data.

A pth-order random process {X,}, k = 1, 2, 3, ..., will be called pth-order stationary if its corresponding pth- order correlation sequence pJn, m) in eqn. 14 depends only on the difference 1 = m - n of its arguments. Sam- ple averages can be used to define the FLOS of an ergodic stationary observed time series { X k ] , k = I , 2, 3, ..., similarly to ensemble averages:

N

The basic properties of the FLOS of S a S processes with zero location parameter are summarised as: P1: For any cl and c2, we have

(a151 + a 2 C 2 , r l ) p = m K 1 , v ) p Jr a2(Cz,v)?, (16)

( 5 , r l ) P = 0 (17)

(alvl +a2v2, alrl + a 2 v 2 ) p = lbllp(vl,vl)p+ I 4 P ( v 2 , v a ) p

(18)

P2: If and q are independent, then

while the converse is not true. P3: If y1 and q2 are independent, then

P4: For a stationary p-th order random process {X,}, k = 1, 2, 3, ..., its p-th order correlation and the corre- sponding sample average satisfy

p, ( l ) 5 pp(0) I = 0 , z t l , *2, . . . (19)

T P ( Z ) 5 T P ( 0 ) I = 0, hl, *2, , I . (20)

4 Estimation of the model parameters

4.1 Estimation problem formulation In radar applications, the following two problems arise in practice [Note 21.

4. I . I Real signals: Let XI, X,, ..., X , be observed

32

independent realisations of an SaS random variable X of unknown characteristic exponent a and dispersion y and zero location parameter 6 = 0. We attempt to estimate the exact parameters of the SaS distribution of X from the observed realisations. The proposed estimation procedure has a hierarchical rather than a simultaneous structure, as explained in the subsequent paragraphs. More details on the rationale behind these estimators and on their performance can be found in [18, 251.

4.1.2 Complex signals: Let A,, AZ, ..., A N be observed independent realisations of the envelope of a BISaS random variable X of unknown characteristic exponent a, and dispersion y and zero location param- eters 61 = 6, = 0. We attempt to estimate the exact parameters of the BISaS distribution of X from obser- vations of the realisations of its envelope. The pro- posed procedure is a variation of the estimators in [l8, 251.

4.2 Estimator of the characteristic exponent

4.2.1 Real signals: For the estimation of the characteristic exponent a of the PDF, we propose the following algorithm. Consider a segmentation of the data into L nonoverlapping segments, each of length K = NIL: {X1,X2,...,XN} = { X ( 1 ) , X ( 2 ) , - , X ( L ) } (21) where X(4 = { X ( ~ - I / N / L + ~ , X(I-l)N/L+Z, ...) XINIL}, 2 = 1, 2, ..., L. This segmentation is done arbitrarily for the time being and the reason for considering it will become apparent momentarily. Optimisation of the segmenta- tion is a topic of present and future research.

and _X, be the maximum and the minimum of the data segment X(4. We then define

Let

Zl = logXL (22)

Zl = - log(-&) (23) and the corresponding standard deviations

I . L < L

With these definitions in mind, the estimate for the characteristic exponent a of the SaS PDF takes the form

4.2.2 Complex signals: Again we consider a seg- mentation of the data into L nonoverlapping segments, each of length K = NIL: {Ai ,A2, . . . , A N } = { A ( I ) , A ( 2 ) , . . . , A ( L ) } (27)

..., L. Let 3 be the maximum of the data segment A(4. We define

where A(2) = {A(l- l )N/I ,+l~ AJ1-1/N/L+2, ...> AIN/L)$ = 1, 2,

(28) - al = log21

Note 2: The problem and the algorithms can be directly extended to include estimation of location parameters.

IEE Proc.-Radar. Sonar Navig., Vol. 144, No. 1, February I997

and

L-

I _ L - L

Then

4.3 Estimator of the dispersion

4.3.1 Real signals: For the estimation of the dispersion y of an S a S PDF, we propose the following estimator which is based on the theory of fractional lower-order moments of the PDF:

L

where r(.) indicates the Gamma function, C(p, a) has been defined as

and the choice of the order p (0 < p < &/2) of the fractional moment is arbitrary.

As we can see, the dispersion estimator requires knowledge of the characteristic exponent and the location parameter of the S a S PDF. Thus, the dispersion estimate must be computed after estimates for the characteristic exponent and the location parameter have been obtained.

4.3.2 Complex signals: For the estimation of the dispersion y of a BISaS PDF, we propose the following estimator, which is based on the theory of fractional lower-order moments of the PDF:

(33)

where D(p, $) has been defined as

and the choice of the order p (0 < p < 612) of the frac- tional moment is arbitrary.

4.4 Application on real sea-clutter data The proposed estimators were applied on real radar sea-clutter data, provided by R.D. Pierce, Naval Surface Warfare Center, Carderock Division, Bethesda, Maryland, USA. A typical set of 1000 samples of the I and Q components of measured radar clutter data are shown in Fig. 3. A total of N = 320000 samples of clutter were processed. We considered L = 320 segments, each of length K = 1000. The- estimation algorithm returned the parameter values 6 = 0, a = 1.85 and 7 = 0.19 [Note 31. For these parameters we can compute the theoretical amplitude probability distribution (APD) [12] and compare it to the empirical one. A comparison of the theoretical and empirical Note 3: We examined the three estimation cases: estimation from the I component only, estimation from the Q component only and estimation from the real envelope. The retumed estimates were basically identical, which is consistent with the fact that the marginals of a B I S d distribu- tion are S a .

APDs is given in Figs. 4 and 5, respectively, from which the good fit between the model and the real data is clear.

2

0 -2

-61 I

0 200 400 600 BOO 1000 b

hg.3 a I component of measured radar clutter b Q component of measured radar clutter

Real radar sea clutter data

0 0 1 2 3 4 5 6 7

Theoretical APD for a = 1.85 and y = 0.19 X

Fig.4

E 9 10

1.0

0 0 1 2 3 L 5 6 7 8 9 10

Empirical APD from real radar sea clutter datu X

Fig.5

5 Signal detection

5.7 Likelihood ratios In this Section, we compare the performance of coher- ent and incoherent, data-adaptive detectors on real radar sea-clutter.

IEE Proc-Radar, Sonar Mavig., Vol. 144, No. 1, February 1997 33

5. I. I Coherent deterministic signal detection: First, we considered the detection of completely known signals in radar clutter. Theoretically, the problem is formulated as a need to decide between the two hypotheses

X ( k ) = n ( k ) under hypothesis HO (35)

~ ( k ) = A + n ( k ) under hypothesis HI (36) where k = 1, 2, ..., M and {x(k)} and {n(k)} are the observation sequence and a sequence of independent, identically distributed radar clutter samples, respec- tively. A is the known signal amplitude. The detection problem, therefore, consists of deciding whether the observed data sequence {x(k)} contains noise only or if a constant signal is also present.

The Gaussian, the data-adaptive Cauchy, and the data-adaptive optimum receiver compute the test statistics

M

tG[z(1),2(2),...,Z(M)] c z ( k ! (37) k=l

respectively, and compare them to an appropriately set threshold. In the above expressions, we have indicated with y and .y the estimates of the dispersion of the noise returned from eqn. 31 when applied to a data block of length A4 under the hypotheses (noise only, 6 = A = 0) and H I (signal present, 6 = A # 0), respectively. On the other hand, the computation of the S(a = 1.85)s PDF at arbitrary argument needs to be done with the method of [17]. We find that the following expressions are appropriate for real-time computations:

f l . S 5 ( Y , A ; z) = (?'-1/1'85fi.S5(1, 0; (7)-1/1'85(x - A ) ) (40)

with

f1.85(1,0; Z) = 4.8291 x 1 0 - 6 ~ 8 - 1.5905 x 10-4z7 +0.002z6- 0.01Zz5+ 0 , 0 3 4 2 ~ ~ - 0 . 0 2 4 3 ~ ~ - 0 . 0 6 4 4 ~ ~ - 0.00015~ + 0.2829 0.13/ ( z ~ . * ~ ) + 1.115/ ( z ~ . ~ )

if 1x1 5 5 1 + 1 0 . 8 5 1 4 / ( ~ ~ . ~ ~ ) + 122.3833/(~' .~) if 1x1 > 5 The performance of the data-adaptive Cauchy and

the Gaussian receivers was tested on the provided real radar clutter data. In particular, we divided the data into blocks of length A4 = 100, considered a signal amplitude A = 0.1, and derived the corresponding receiver operating characteristics (ROCs) via averaging over 3200 blosks. To compute the dispersion estimates and y, we considered the assumption a = 1.85. In Figs. 6 and 7, we show the ROC of the data- adaptive Cauchy (solid line) and the Gaussian (dotted line) receivers for the I and Q components, respectively, of the clutter. Clearly, the data-adaptive Cauchy receiver outperforms the Gaussian, which is an

34

additional indication of the high accuracy of the SaS model. These results are in full agreement with the theoretical results in [ 171.

r

t

1 oo 10-4

IO4 1 0-* 18 probability of false alarm

Fig. 7 component of clutter)

~ Cauchy ..... Gaussian

ROC for data-adaptive Cauchy against Gaussian receivers ( Q

5.7.2 Incoherent deterministic signal detec- tion: We consider the hypothesis testing problem:

X(k) = n ( k ) ~ ( k ) = ez4s(k) + n(k) under hypothesis HI (41)

where k = 1, 2, ..., A4 and {s(k)} is a known, generally complex signal, is a random phase uniformly distributed in [0, 2 4 , and {n(k)} is a sequence of bivariate isotropic symmetric, alpha-stable (BISaS) random variables of dispersion y Moreover we assume that the two hypotheses are a priori equiprobable and attempt to decide which hypothesis is true on the basis of the observed sequence {x(k)}.

We propose [19] the use of a receiver that computes the test statistic

under hypothesis HO

IEE Proc.-Radar, Sonar Navig., Vol. 144, No. I , February I997

and compares it to a threshold r. If lc > q, then HI is decided, otherwise Ho. In the above, E2 = ix(k)12 + ls(k)I2+e, El = -2%{x(F)s*(k)} and E2 = -23{x(k)s (k ) } , where the superscript denotes the complex conjugate. The proposed receiver is the optimum test for the prac- tical problem of detection of an incoming signal the phase of which varies randomly from sample to sample within the same block. However, it still outperforms the traditional Gaussian envelope detector in real radar sea-clutter, as seen in the following tests, and has been shown to be robust in the entire class of BISaS noises [19]. The integral in eqn. 42 has been computed in [19] in a form that is amenable to real time computations.

In our tests, we chose a chirp signal as the test signal {s (k) } . A chirp signal belongs to the class of phase modulated (PM) waves that have the form

s ( t ) = exp[i(w,t + b t2 ) ]

w ( t ) = w, + 2bt

(43) where mc is the carrier frequency and b is the chirp rate. The instantaneous frequency of a chirp signal is

(44) and increases linearly with time. Chirp signals are widely used in radar applications because they possess long duration in both the time and frequency domains. The large frequency bandwidth provides better range resolution while the long duration in time enables the radar to transmit more energy, thus improving the out- put signalhoise ratio [26].

We assumed the hypothesis testing problem of eqn. 41 with the signal considered as unknown. In place of the unknown signal s(k), we used its least squares estimate x(k) [26] and derived the ROCs of Fig. 8 for the Gaussian (dotted line) and the Cauchy (solid line) receivers.

r. m u , . . .---.--

probability of false alarm Fig. 8 receiver

ROC for data-adaptive Cauchy against Gaussian incoherent

~ Cauchy ..... Gaussian

5.1.3 Fluctuating deterministic signal detec- tion: The detection problem that we examine in this Section of the paper can be formulated as the situation where we need to decide between the two hypotheses

Ho ? k j = f i k ,

k = 1 , 2 , . . * , K ; j = 1 , 2 , . . . , J (45) In this problem, k = 1, 2, ..., K indexes K independent

channels, along which a fading target is observed, and j - 1, 2, ..., J indexes J time samples along observation channels. We assume that the noise samples f ikj and the signal amplitudes Ak B 0 are independent across and along channels, while the fading is slow enough to allow for A, to be constant over the observation inter- val. The random phases 0, are independent across paths and uniformly distributed in [0, 2 4 . Finally, 5 is descriptive of the transmitted signal strength. In gen- eral, we use a tilde 'I' to denote a complex envelope.

Assuming the fading variables Ak to be independent, Rayleigh-distributed, we obtain the asymptotically optimum test statistic [21]

-

K t=C-- wk CIA2 +QF) 2 + w k

k=l

where

w k = E{A;} t2pT u 3 g p ( u ) f k ( u ) d u (47)

In eqns. 46 and 47, we have defined

F k = 7r U 3 9 ; ( U ) f k ( U ) d U (51) 1- with

J

(52) 1 P = lim -E 1Sj12

J-03 J j=1

andfk(.) denoting the PDF of the envelope of the noise in the kth channel, i.e. &(.) = j&J.), as in eqn. 8.

The Gaussian detector can be written in the form [21]

while the Cauchy detector attains the form [21]

We have tested the above Gaussian and Cauchy detectors for the case of K = 3 channels, with E{Af} = 0.5, E{A?} = 0.1 and E{A;> = 0.01. We assumed that J = 16 samples were taken along each channel and set P = 1 and 5 = 1. The performance of the Gaussian and the Cauchy receivers is shown in Fig. 9, from which it is clear that the Cauchy receiver outperforms the Gaussian one without additional computational complexity.

5.2 Moment methods for random signal detection In this Section, we look at moment-based methods for signal detection and classification. As an illustrative example, we consider the detection of a stochastic, FIR

35 IEE Proc-Radar, Sonar Navig.. Vol. 144, No. 1, February 1997

signal in clutter. More specifically, we consider the hypothesis testing problem: Ho : 21 = Wl

HI : 2l = 1 = 0 , 1 , 2 ) . ” ) M (55) 4

SkU1-k + Wl k=O

where {U/,} is a sequence of iid S a S random variables, i s k } , k = 0, 1, 2, ..., q, is a known signal sequence, and {wk} is a sequence of S a S random noise variables inde- pendent of the FIR signal. Finally, we are going to assume that M > q. For the dependence structure of the signal and the noise, we are not making any assumptions beyond those stated above.

1 oo

t 0 .- c

1 : lo-’ QI U

0 x c

c d .- ._ n n g

I

10-3 I 0

I 0” IO-* 10’ 10 probability of false alarm

Fig. 9 ~ Cauchy .... Gaussian

ROC for Cauchy against Gaussian receiver under Rayleigh fading

We propose a detection rule that consists of comput- ing the test statistic

- M 1

T P = - 1 I1JTLJP, p < a/2 n = O

M

where yn = Ef=O~q- l~n- l . If the test statistics exceeds a threshold, hypothesis HI is declared otherwise hypothesis Ho is declared. This proposed test statistic is a direct generalisation of the second-order statistics (SOS)-based energy detector [27] and HOS-based detectors [28].

probability of false alarm Fig. 10

~ FLOS detector _ _ _ HOS detector

SOS detector

ROC of FLOS-, SOS- and HOS-based detector

The performance of our proposed FLOS-based detector relative to its SOS- and HOS-based counterparts is illustrated with the following example. The test signal is the stochastic FIR. signal xi = 0 . 3 ~ ~ + 0.2ul_, - O.lul-, + O . ~ U ~ - ~ , where the variables {al} are iid, Laplace-distributed random variables of unit variance and the sequence { wl} are sea-clutter samples. We chose A4 = 100 samples per block, a FLOS of order p = 1, and a HOS statistic based on fourth-order cumulants [28]. The ROCs of the three detectors are shown in Fig. 10. Clearly, the performance of the fourth-order cumulant-based detector is the lowest of the three. The proposed FLOS-based detector gives the highest performance.

6 Summary, conclusions and future research

In this paper, we tested the validity of the recently pro- posed alpha-stable distributions as models for radar clutter. We fitted an alpha-stable model to 320000 sam- ples of real sea-clutter and found that the best fit gives rise to an estimated characteristic exponent a = 1.85 for the available data set. We proceeded to evaluate the performance of algorithms that we recently proposed on the real data and exanimed the cases of detection of both deterministic (completely known or with unknown parameters) and random signals using generalised like- lihood ratios and fractional, lower-order moment-based methods. In all cases examined, we found that the pro- posed algorithms outperformed existing ones, especially at the low probabilities of false alarm that any realistic radar would operate.

Thus, it seems that further validation of the alpha- stable model is due for characterisation of interference in modern radar systems. This validation should include extensive comparison with other statistical models currently in use. Further research seems to be due also in the design of new radar systems, in which the signal processors are based on alpha-stable models. Specific radar signal processing issues that need to be addressed within the framework of alpha-stable distri- butions and fractional, lower-order statistics include beamforming and bearing estimation, time-frequency distributions, application of alpha-stable fractals in radar, and radar image processing. This and similar research is currently under way and its results will be announced shortly.

7 Acknowledgments

This work was supported by the Office of Naval Research under contract N00014-92-J-1034. The authors wish to thank Dr. P. Tsakalides of the Signal and Image Processing Institute, Department of Electrical Engineering - Systems, University of Southern California for his help with some of the computer programming in Section 5.

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