Performance analysis of order-statistic CFAR detectors in time diversity systems for partially correlated chi-square targets and multiple target situations: A comparison

Abstract

In radar systems, detection performance is always related to target models and background environments. In “time diversity systems”, the assumed Swerling complete correlation (slow fluctuation) and complete decorrelation (fast fluctuation) target models may not predict the actual system performance when the target returns are partially correlated. The probability of detection is shown to be sensitive to the degree of correlation among the received pulses. In this paper, we derive exact expressions for the probability of false alarm ($P_{\mathrm{fa}}$) and the probability of detection ($P_{\mathrm{d}}$) of a pulse-to-pulse partially correlated target with 2K degrees of freedom in a pulse-to-pulse completely decorrelated thermal noise for the order statistics constant false alarm rate (OS-CFAR) detector and the censored mean level CFAR (CMLD-CFAR). The complete analysis is carried out for the “non-conventional time diversity system” (NCTDS) and multiple target situations. The obtained results are compared with the detection performance of the “conventional time diversity system” (CTDS).

Introduction

In automatic detection, received signals might fade due to target fluctuations. According to Swerling's models, if only one pulse per scan hits a target, we cannot distinguish between cases I and II and cases III and IV. However, if multiple pulses are transmitted per antenna scan, the problem of detecting slow fluctuating targets (complete correlation) and fast fluctuating targets (complete decorrelation) can be easily overcome. Nevertheless, we should take into consideration the partial correlation of the target signal; otherwise the processor fails to predict the actual system performance. In other words, the more we know about the statistics of the target signal, the better the detection is.

In the literature of constant false alarm rate (CFAR) detection, the echoed signals of the transmitted pulses are processed non-coherently within the same receiver. The non-coherent integration accumulates M pulses and processes them as an entity to form the noise level estimate. Dealing with either uncorrelated or partially correlated data samples, one often seeks to improve detection while maintaining a CFAR. Several authors have considered different applications of the non-coherent integration. Here, we only list a few of them [1], [2], [3], [4], [5], [6]. In [1], Kanter has studied the detection performance of a non-coherent integration detector accumulating M-correlated pulses from a Rayleigh target with two degrees of freedom. Complete correlation and complete decorrelation of the target returns yielding Swerling models I and II, respectively, were treated as extreme cases of the target correlation coefficient. The noise was assumed to be uncorrelated. Wiener [2] extended the work in [1] by deriving exact expressions for the probabilities of detection for partially correlated chi-square targets with four degrees of freedom. In this case, the limiting bounds of complete correlation and complete decorrelation of the target returns yield the Swerling models III and IV. The work done in [1], [2] used fixed threshold detection. It is known that radar detectors with fixed threshold cannot maintain a CFAR, and thus adaptive threshold detection is considered. Kim and Lee [3] and Han [4] exploited the results in [1], [2] and used the method of residues to develop the generalized order statistics (GOS) CFAR detector and the OS CFAR detector, respectively. Hou [5] also used the method of residues to evaluate exact formulas for the detection performance for the chi-square family with 2K degrees of freedom. Ritcey [6] analyzed the detection performance of the maximum ML detector (MX-MLD) for both non-fluctuating and chi-square fluctuating target models with 2K degrees of freedom.

The idea of processing independently the received target pulses to yield preliminary decisions in distributed CFAR detection, was first suggested by Himonas and Barkat [7], [8]. They studied the case of partially correlated target returns with different architectures of time diversity and distributed CFAR detectors to minimize the effect of the correlation factor among the received target pulses. They called it “time diversity systems” referring to multiple-pulse systems. In [7], they considered the detection performance of the generalized censored mean level detectors (GCMLD), which is known to assume no prior knowledge of the number of interfering targets, for time diversity and multiple target situations. Then, they proposed in [8] a distributed architecture for the generalized two-level GTL-CMLD CFAR when a clutter power transition is present in the range resolution cells. They showed that an appropriate choice of the fusion rule yields a detection performance to be less sensitive to the degree of correlation among target returns. El Mashade [9], [10] has thoroughly developed this idea by considering the integration of all the individual noise level estimates. More precisely, as shown in Fig. 1, the reference samples of the individual pulse returns are ranked in an ascending order. Then, each ordered window is processed by the suited one-pulse order statistic algorithm. Finally, the obtained noise level estimates are added to get the overall background level. We shall call it the “non-conventional technique” with respect to the conventional non-coherent integration technique. In [9] he analyzed the detection performance of the trimmed-mean (TM) CFAR processor for a homogeneous background, in which the reference cells of each sweep (each processed pulse) are first ranked in an ascending order, trimmed, and then added up to obtain the final noise level estimates. Using the same concept, he analyzed in [10], the linearly combined order-statistic (LCOS) CFAR detectors in non-homogeneous background environments for the Swerling II model. Consequently, for the sake of comparison, we shall adopt in this paper the terminology “conventional time diversity system” (CTDS) to refer to the non-coherent integration and “non-conventional time diversity system” (NCTDS) to refer to the technique used in [9], [10].

Of particular interest to radar applications is the presence of interfering targets in the reference cells, which is known to degrade the performance of CFAR detectors. To alleviate this problem, a lot of detectors have been proposed in the literature. The aim of these algorithms is to show how robust a CFAR detector can be in multiple target situations by taking into account the non-homogeneity of the background either analytically or by simulation.

In summary, we observe that the previous work using the NCTDS did not show a comparison of any of the analyzed order statistics detectors with their corresponding detectors for the CTDS in neither single nor multiple target situations. Furthermore, the two time diversity systems did not consider the general case of a pulse-to-pulse partially correlated chi-square target with 2K degrees of freedom. To complete the study, we introduce, in this paper, a detailed detection analysis for a mathematical model representing the case of detecting a pulse-to-pulse chi-square partially correlated target with 2K degrees of freedom embedded in a pulse-to-pulse Rayleigh and uncorrelated thermal noise for the OS-CFAR detector and the CMLD-CFAR. The complete analysis is carried out for the NCTDS. The paper is organized as follows. In Section 2, we formulate the statistical model and set the assumptions. In Section 3, we derive exact false alarm probabilities for the two CFAR detectors. Then, in Section 4, we give the moment generating function (MGF) of the test cell under hypothesis H₁ in terms of K and use it to derive exact detection probabilities for the two CFAR detectors. Next, in Section 5, by deriving sets of detection curves, we show the effect of the degree of correlation of the target returns, the number of degrees of freedom, the number of processed pulses and the number of interfering targets present in the range resolution cells on the detection performance of the detectors. We also compare the detection performance of the CTD and the NCTDS. Swerling's cases I–IV are handled as extreme limits of the proposed model. A summary along with a conclusion are given in Section 6.