Truncated Sequential CFAR Detectors Using Weighted Sign and Weighted Conditional Sign Tests

In this paper we propose truncated sequential constant false alarm rate ( CFAR) detectors which are approximations of the optimum sequential sign and sequential conditional sign detectors. The particular approximations used here allow exact evaluation of the performances. These detectors form a test statistic by taking the weighted sum of the outputs of a hard-limiter in one case and a dead-zone limiter in the other case. The test statistic, obtained after every observation, is compared with two constant thresholds. The tests terminate either when one of these thresholds is reached, or when the number of observations exceeds a predetermined truncation point. Design procedures which guarantee a CFAR are presented. Numerical comparisons with some reported detectors are given. The proposed truncated CFAR detectors gain improvement over their corresponding non-truncated CFA R detectors in reducing the average sample number under signal mismatch, at the expense of a more involved design process. The proposed truncated CFAR detector with conditional test is uniformly better than a previously reported CFAR detector. L Introduction We consider non-parametric detection of a constant signal corrupted by an additive noise with a symmetric probability density function (pdf). Let X~, X2, • • • be the observation random variables. We assume that they are independent and identically distributed (i.i.d.). Let the nominal signal strength be 01, 01 > 0. With these, the signal detection problem is that of testing the hypothesis H0 versus the alternative Hi, where © The Franklin Institute 00164)032/96 $9.50+0.00 0016-0032(95)00066-6 717 Pergamon A. Nasipuri and S. Tantaratana H0: X ~ f i x ) , f i x ) = f ( x ) i = 1,2 . . . . HI: X~"~f(x--O1), O, > 0 , i = 1 , 2 . . . . . (1) Here, X ~f(x) means that the pdf of X is f(x). Note that we use an upper-case letter to denote a random variable and a lower-case letter to denote its realization. By a constant false alarm rate (CFAR) detector we mean a detector whose false alarm probability is constant for any symmetric pdf f(x). There have been a large number of proposed fixed-sample-size (FSS) CFAR detectors, see (1-4) for examples. An FSS CFAR detector is generally designed by choosing a sample size large enough to give the desired probability of false-alarm (PEA) and acceptable probability of detection (POET) under some nominal pdff( . ) and nominal signalto-noise ratio. Since the number of observations is fixed for an FSS detector, its performance depends only on the test statistic. As opposed to FSS detectors, in sequential detectors the number of observations is not predetermined. It depends on the outcomes of the observations. The optimum (parametric) sequential test is the sequential probability ratio test (SPRT) (5), which is much more efficient than its corresponding FSS counterpart. A number of sequential non-parametric detectors have been proposed (6-9). In general, the average sample number (ASN) of a sequential test under H0 and Hi is lower than the sample size of the FSS test using the same test statistic and having the same PEA and PDET' However, the performance of a sequential detector suffers when the actual signal 0 is not the same as the nominal value used in the design process. When a sequential test has no upper bound on the number of observations, the test can occasionally result in a very long test, which is unacceptable in some situations. In (9), a sequential CFAR detector is proposed in which the sum of the outputs of a dead-zone limiter is compared with two thresholds as long as the number of non-zero outputs of the dead-zone limiter is below a pre-specified number, say M. Termination of the test occurs when the test statistic reaches either boundary or when the number of nonzero outputs of the dead-zone limiter reaches M. However, the test used in this detector is not a SPRT-Iike test, which suggests that improvement can be made. In addition, the test is not truly truncated, since the number of outputs of the deadzone limiter being bounded by M does not guarantee that there is an upper limit on the actual number of observations. Therefore, a particular realization can continue for a long time before it terminates, which is undesirable. In this paper we propose truncated sequential CFAR detectors using a weighted sign and a weighted conditional sign test statistics. After each observation before the truncation point, (an approximation of) the likelihood ratio of the quantized observations is compared with two constant thresholds. Truncation occurs when the total number of observations at the input of the quantizer reaches a prespecified number. A randomized threshold test must be employed to maintain CFAR operation. The quantization is performed by a hard limiter in one case and by a dead-zone limiter in the other. These detectors differ from the one described in (9) in that they are approximations to the optimal SPRT test. Also, the actual number of observations is upper bounded. In Section II, we consider near-optimum sequential sign and dead-zone limiter detectors. A Markov chain formulation of the non-truncated sequential CFAR detectors for computing the exact ASN and 7 1 8 Journal of the Franklin Institute Elsevier Science Ltd Truncated Sequential CFAR Detectors power functions is presented in this section. In Section III we consider truncation of these sequential tests. Design of the truncation point and the two thresholds to maintain CFAR performance is presented. Some numerical examples are given in Section IV. Note that the performances we obtain in this paper are exact, not approximations as is the case in (6). IL Sequential CFAR Detectors In this section we describe CFAR detectors using sign, conditional sign, weighted sign and weighted conditional sign tests. For these detectors, the observation samples are first passed through a memoryless quantizer. Since the input samples to the quantizer, X1, )(2 . . . . . are i.i.d., it follows that the output random variables are also i.i.d. Consequently, an SPRT based on the quantized observations can be designed. The pdf of the quantizer output will depend on the type of quantization employed. 2.1. Sequential conditional sign detector Consider quantizing each observation X/using a dead-zone limiter, which gives an output U~ defined by + i if X i > c U~= if -c<~X~<<,c i f X , < c (2) where c is some constant chosen by the designer. Let po(c) =Po(Ui = +1) = 1 F ( c O ) qo(c) =Po(Ui = --1) = F ( c O ) ro(c) =Po(Ui=O) = l-po(c)--qo(c) (3) where F(x) = S ~_ ~ f(t) dt and Po(') is the probability when the signal strength is 0. Note that under H0, 0 = 0 and po(c) = qo(c) for symmetricf(x). CFAR operation can be achieved using a conditional test. Define po(c) p~(c) =Po(U~= +llU~-¢0) -po(c)+qo(c) qo(c) q~(c) =Po(U~= I IU ,=~0) -po(c)+qo(c)" (4) Note that q'o(c) = 1 p ' o ( c ) . Since p~ (c) = q~ (c) = 0.5 for all symmetric pdfs, a test based on Ui conditioned on Ui 4:0 will be CFAR (4). The sequential conditional sign detector is VoL 332B, No. 6, pp. 717 734, 1995 Printed in Great Brilain. All rights reserved 7 1 9 A. Nasipuri and S. Tantaratana f >~ log Pp~A w ~ accept Hi ~. e(u, Inl ,U/ ~ O) l°gP(UilHo,Ui # 0) ~< log 1 --PDET1 --PEA (5) i=/-"al ¢" accept H0 otherwise ~ continue. Define the random variable Z/as follows: {log [2p~, (c)] if U /= +1 / ZSi=/log[2q~ l(c)] if V / = l . (6) L0 if Ui = 0 For a signal strength 0, 2/assumes the value log 2p~, (c) with probability (w.p.) po(c), the value log 2(1 p ~ (c)) w.p. qo(c), and the value 0 w.p. ro(c). Let Np be the number of observations (out of n observations) greater than c and N, be the number of observations (out of n observations) whose absolute values are greater than c. Then the test statistic in Eq. (5) becomes Y/= np log 2p01 (C ) 71-( n c np) log 2(1 --p~ (e)) /=1 (7) and the detector in Eq. (5) can be written as l ip "~ 1 P D E T og ~ -nc log 2(1 p ~ (c)) t> ' ¢ log [p~, (c)/(1 -Po,( ))] ~ a c c e p t H l 1 PDET _ _ l C log 1 --PFA --n~ log2(1 --Po,()) ~< --* accept H0 log [p~, (c)/(1 -p~, (c))] otherwise ~ continue (8) In the special case when the parameter c is set to zero, the quantizer (2) becomes a hard-limiter, with the output U/being + 1 w.p. po(O) (when X/> 0) and 1 w.p. qo(O) (when X/< 0). For brevity we will denote po(O) and qo(O) by Po and qo, respectively. An SPRT based on the outputs of the hard-limiter becomes the sequential sign detector: (" P D E T ] log ~zi '~ I P D E T i=l ] ~< 1 o g ~ [. otherwise where --*acceptH~ --* accept H0 ---, continue (9) Journal of the Franklin Institute 720 Elsevier Science Ltd Truncated Sequential CFAR Detectors (log2p0~ if U~ = +1 (lO) Z i = l o g t ~1 0) ( l o g 2 ( 1 p 0 , ) if U i = I If Np is the number of positive observations out of a total of n observations, then the test in Eq. (9) can also be described by np P D E T l o g -n log 2(1 --POl) P F A ~" lo t [P0l/(7 -Po,)] ~ accept HI 1 P D E T log 1 P F A n 10g 2 (1 --Po,) ~< ~ accept H0 log [p0,/(1 --po,)] otherwise ---, continue (11) Note that as opposed to the sequential sign detector, the sequential conditional sign detector ignores the observations lying between c and c. It was shown in (6) that the efficiency of the sequential conditional sign detector is maximized when c/a = 0.6 under a Gaussian noise of variance tr 2. Analytical expressions for the ASN and power functions of the SPRT are difficult to obtain. Wald's results can be used for the design of the sequential tests, but only approximate expressions are available for the ASN and power functions. 2.2. Sequential weighted sign and weighted conditional sign detectors In this section we describe CFAR detectors using weighted sign and weighted conditional sign statistics. These detectors are approximations of the SPRTs in the previous sections. However, exact evaluations of the ASN and power functions of these proposed detectors can be obtained using Markov chain formulation (10, 11). Consider the sequential sign detector described in Eq. (9) and define A + = log 2p0, and A= log2(1 --Po,). Let us approximate A + and Aby A + ~ J A , A~ K A (12) where J, K are integers and A is an appropriate step size. This is a generalization of the detector in (12), which corresponds to the case of J = 1 and K = 2. In addition, we choose two integers A and B such that 1Og~Ax ~ AA, log 1 ~ P D E T 1 -PVA ~ -BA. (13) Therefore, the test statistic in Eq. (9) becomes ~ ze = JAnp--KA(n--np), where ~ , i = l zi = JA if Xi >~ 0 and KA if Xi < 0. zi can be viewed as the weighted sum of i=1 the outputs of a hard limiter whre JA and KA are the weights corresponding to the outputs + 1 and 1 , respectively. The sequential sign detector becomes the sequential weighted sign detector: Vol. 332B, No. 6, pp. 717 734, 1995 Printed in Great Britain. All rights reserved 721 A. Nasipuri and S. Tantaratana (/> A ~ accept H0 1 n | Si~_, zi = JnpK(n--%) I, <~ -B ~ accept HI. (14) (otherwise ~ continue This detector can be described by a Markov chain with a finite number of states and absorbing boundaries at A and B . A positive transition (when X~ >~ 0) from state s leads to state (s + J), and a negative transition (when X~ < 0) to state ( s K). The states of the Markov chain represent the possible values of the sum ~ z~ after i--1 n observations. As an example, the Markov chain for J = 2 and K = 3 is described in Fig. 1. In this figure the additional overflow states ( B 1 ) and ( B 2 ) are introduced below the lower boundary ( B), and the overflow state (A + 1) is introduced above the upper boundary (A) to take care of the possible overflow at the termination of the test. Let ej denote the expected number of times that state j is occupied before absorption. Then these expected values satisfy the set of equations (10) A 1 ej 6j + ~ 0 = ekpkj, j = B + I , B + 2 . . . . . A--1 (15) k= --B+ I where 6j = 1 i f j = 0 and it is zero otherwise, and p0j is the probability of transition from state k to s ta te j when the signal strength is 0. The ASN function ASN(O) of the sequential test is equal to the average number of transitions of the system before absorption starting from state 0 when the signal strength is 0. It is given by A--1 ASN(O) = ~ ej. (16) j= -B+ 1 The set of linear equations in Eq. (15) can be expressed as e = P0e+V (17) where e is the vector consisting of the unknown expected values, P0 is the state transition matrix and V is a vector of state probabilities at the start of the test (it has a value of one at the position corresponding to state 0, and a value of zero everywhere else). For the example in Fig. 1, we have 1 1 ~ ~ +P I ~ ~ q ~ 0 ~ . P0 PO P0 P0 ...P0 P0 __t LOWER ABSORBING STARTING STATE BOUNDARY 1 UPPER ABSORBING BOUNDARY FIG. 1. State diagram of the sequential weighted sign test, J = 2, K = 3. Journal of the Franklin Institute 722 Elsevier Science Ltd Truncated Sequential C F A R Detectors