Diss / (Springer Series in Information Sciences 25) S. Haykin, J. Litva, T. J. Shepherd (auth.), Professor Simon Haykin, Dr. John Litva, Dr. Terence J. Shepherd (eds.)-Radar Array Processing-Springer-Verlag

XII Contributors

Stoica, Petre

Department of Control and Computers, Polytechnic Institute of Bucharest Splaiul Independentei 313, R-77 206 Bucharest, Romania

Viberg, Mats

Department of Electrical Engineering, Linkoping University

S-581 83 Linkoping, Sweden

Zhu,Zhaoda

Nanjing Aeronautical Institute, P.O. Box 1608, Nanjing 210016, China

2 S. Haykin, J. Litva, and TJ. Shepherd

for array calibration. Such problems will be discussed in detail in the following chapters, together with discussions on how they are being tackled.

The book is organized in three parts. Part I, consisting of Chaps. 2-4, deals with the related issues ofdetection and estimation. In Chap. 2, "Radar Detection Using Array Processing", Zhu and Haykin treat the processing as a statistical signal detection problem, and present optimal (in the sense of the NeymanPearson criterion) and sub-optimal methods for signal estimation in the cases of active or passive radar, sources of known or unknown bearing, and of known or unknown number. In Chap. 3, "Radar Target Parameter Estimation with Array Antennas", Nickel gives a broad and comprehensive overview of many array signal processing methods currently being researched and employed for estimating target range, angle, frequency, and received echo power. The merits or drawbacks ofeach method are pointed out in most instances, and demonstrated with selected associated experimental results. Chapter 4, "Exact and Large Sample Maximum Likelihood Techniques for Parameter Estimation and Detection in Array Processing", by Ottersten, Viberg, Stoica, and Nehorai, addresses the important subject of Maximum Likelihood (ML) methods of signal analysis. The aim of the chapter is to establish an analytical connection (in the asymptotic, large data length limit) with eigenstructure or subspace based methods. The performance of estimation and detection techniques described is tested in a set of numerical examples and simulations.

Part II, consisting of Chaps. 5 and 6, describes the use of systolic arrays for adaptive beamforming in the context of linear and planar antenna arrays, respectively. Advances in silicon device fabrication and very large scale integration promise the possibility of real-time implementation for many relatively sophisticated algorithms used in array processing. In Chap. 5, "Systolic Adaptive Beamforming", Shepherd and McWhirter describe in detail how least squares algorithms employed for side10be cancellation, target tracking, and other adaptive antenna applications can be cast in a parallel form, suitable for mapping onto parallel processor hardware architectures, and leading to muchenhanced data throughput rates. Most of the material described in this book concerns linear arrays; however, in Chap. 6, ''Two-Dimensional Adaptive Beamforming: Algorithms and Their Implementation", Ho and Litva extend the discussion of array processing to planar adaptive arrays, capable of scanning over both polar and azimuthal angles. Details are given of the problem eigenstructure, as well as a proposed parallel architecture for least squares processing.

Finally, in Part III, consisting of Chap. 7, "The Radio Camera", Steinberg outlines the problems of imaging at radio frequencies using large baseline arrays, and explains how the difficulties of instability and calibration can be tackled using adaptive beamforming techniques.

2. Radar Detection Using Array Processing

Z. Zhu and S. Haykin

With 2 Figures

In a radar system, the antenna part of the system is used not only for transmitting and receiving electromagnetic energy, but also as a spatial sampler of impinging wavefronts on the antenna aperture. Relevant information about the number, the directions, and the signal intensities of sources (i.e., targets in the case of an active radar or emitters in the case of a passive radar) may be extracted by properly processing the spatial samples of the impinging wavefronts incident on the receiving antenna aperture.

In a conventional radar system, the antenna has a continuous aperture and it is mechanically scanned. As such, it may realize optimum processing in a background of spatially white noise in the case of a single source. The system remains optimum for multiple sources as long as the angular spacing between them is larger than the antenna beamwidth. When the angular spacing between the sources is smaller than the antenna beamwidth, or the background noise is spatially coloured, a continuous aperture antenna can no longer provide optimum processing because it only integrates the impinging wavefronts on the aperture. In particular, the only way to improve the radar capabilities for detection, estimation and resolution in an environment of multiple sources for continuous aperture antennas is to narrow the beamwidth by increasing the physical aperture, a course of action that is undesirable in many practical situations.

On the other hand, array antennas or sampled aperture antennas acquire a set of spatial samples of the incoming wavefronts, and thereby permit more degrees of freedom in subsequent signal processing. We may thus improve radar capabilities of detection, estimation and resolution in an environment of multiple sources, or one that suffers from spatially coloured noise, by using sophisticated array signal processing [2.1-9]. Such an approach may be feasible by virtue ofcontinuing improvements in the cost, as well as capability, of digital signal processing hardware.

In this chapter, optimum and suboptimum array processing structures are investigated from the standpoint of the theory of statistical signal detection [2.10-13]. For the case of an active radar and single target with known direction, the averaged likelihood ratio test is known to be optimum in the sense of the Neyman-Pearson criterion, which appears to be of general usage in radar detection. The objective of the Neyman-Pearson criterion is to maximize the detection probability for a given false alarm probability. For the case of multiple sources of known number, with known or unknown directions, it is difficult to

4 Z. Zhu and S. Haykin

implement the averaged likelihood ratio test, and the generalized likelihood ratio test based on the maximum likelihood principle is the preferred method. The generalized likelihood ratio test is reasonable, even though not exactly optimum in the sense of the Neyman-Pearson criterion; it provides a combined detection-estimation solution. For the case of noncoherent radar and passive radar and sources of unknown direction, we introduce another generalized likelihood ratio test using a pseudo maximum likelihood estimate of signals based on the eigendecomposition method-[2.4] that decouples the subproblems of detection and estimation. For multiple sources of unknown number, we face an additional task of determining the number of sources. Two possible approaches are described. One approach is based on solving a multiple alternative hypothesis testing problem [2.4]. The other approach treats it as a problem of model-order selection [2.8].

The content of the chapter is organized as follows. The basic observation model in array processing is described first. Then, three different (and related) cases, i.e., active coherent radar, active noncoherent radar and passive radar, are considered in that order. The chapter concludes with a discussion of other aspects of radar array processing for detection.

2.1 Observation Model

To simplify the analysis, we will use complex signal notation for describing the narrowband process.

Consider, first, an active pulsed radar having an array antenna composed of M elements with arbitrary locations and arbitrary directional characteristics. In every pulse repetition period of the radar transmission, we can acquire a snapshot of the echo for every range gate. A snapshot for a specific range gate contains all the observation samples of the M antenna elements at the instant corresponding to the time sampling of that range gate. Assume that the radar has transmitted a total of N pulses. When there are K targets within the same range ring in the far field of the antenna, the observation sample of the mth element and the nth snapshot in the corresponding range gate may be written as

k=l

Here, Skn represents the signal sample of the kth target in the nth snapshot at a reference point of the array. We assume the use of a one-dimensional array; hence, only a single angular parameter per source is required. Extension to the case of a multi-dimensional array is straightforward. The wavefront received by the array from a target in the far-field may be modeled as a plane wave, so that the angle of arrival ofthe wavefront is the same for all elements of the array. The response of the mth element to a wavefront arriving at the angle Ok' due to the

Here, gm(Ok) is the amplitude response of the mth element to a wavefront impinging on it from direction Ok, and 'm(Ok) is the propagation delay between the reference point and the mth element for a wavefront arriving along direction Ok; Wo is the carrier angular frequency of echo signals. The sample of additive noise at the mth element in the nth snapshot is denoted by Wmn.

We may express the observation vector, namely, an M x 1 vector, from M samples of all elements in the nth snapshot for a certain range gate as follows:

is called the direction matrix.

During N pulse repetition periods, the totality of observation samples on the same range gate forms an M x N matrix

6 Z. Zhu and S. Haykin

where the K x N matrix

is called the noise matrix.

We have implicitly assumed that the direction vector a(Ok), and the direction matrix A are constant during the processing interval. The rationale here is that the processing interval, equal to N pulse repetition periods, is usually short enough to ensure that the directions of targets do not change significantly. On the other hand, the signal-in-space vector s" may vary with time; this accounts for the use of s" in the notation described above.

In the case of a passive radar, it may be not possible to discriminate a multiplicity of emitters in the range domain; nevertheless, they may be resolved in the direction domain by the use of array processing. The snapshot of observation samples is taken at the rate of time sampling. When the emitted signals are narrowband and have the same carrier frequency, the equations presented above remain valid.

Equation (2.11) is the basic observation model used throughout this chapter. The matrix X is observable. The matrix A is a known function of (}i' ... , OK for a given array antenna. The matrices S and Ware unknown except for some a priori information about them. Our task is, based on the observation data acquired by the array antenna, to decide whether sources are present or not, and to estimate the number of the sources when it is unknown.

2.2 Coherent Radar Detection

In the case of an active radar, the two cases of coherent and noncoherent detection require separate attention. In this section the coherent radar case is investigated. The noncoherent radar case is considered in the next section.

2.2.1 Signal and Noise Model

For a coherent radar, the signal sample of the kth target and the nth snapshot may be expressed as

(2.14)

where the amplitude bk, and the phase cPk' are unknown constants independent of n. The signal is non-fluctuating dl,lring the N snapshots. The constant phase implies a zero Doppler frequency of the signal. Otherwise, Doppler filtering of

the time series of observation samples is needed, but the signal processing in the spatial domain discussed below will not be influenced.

The noise model is as follows: We assume that the noise vector w" is a complex, stationary and ergodic Gaussian vector process with zero mean and a positive definite covariance matrix 0'2CO' where 0'2 is the power of complex noise averaged over all elements of the array, and Co is the normalized covariance matrix. Thus, we assume

(2.15)

where the notation tr signifies the trace of a square matrix.

In an active radar, noise includes system noise, clutter and interference. It is only when the system noise is dominant, or the clutter and interference are spatially white, that we may assume Co = I, where I is the identity matrix. In general, Co is not equal to 1.

We assume that 0'2 and Co are known in the active radar case. In practice, 0'2 and Co are unknown, and we have to estimate them by the use of reference noise samples. In active pulsed radar, observation samples collected in adjacent range gates in the same pulse repetition period, or collected in preceding pulse repetition periods in the same range gate under test, may be used as reference noise samples. This assumes: (1) the absence of target signals in these samples, and (2) local stationarity of the noise in range or time domain. The rationale for making these two assumptions is similar to that in implementing a Constant False Alarm Rate (CFAR) processor in conventional active pulsed radar [2.14J, where the detector outputs of nearby range cells are averaged to obtain a background estimate, or the detector output of each resolution cell is averaged over several scans to obtain the background estimate. Needless to say, the quality of the estimate of noise covariance significantly influences the performance of radar detection, estimation and resolution. However, we will not discuss this issue further.

In general, if there is clutter in the background of the received signal, N noise samples of an element of array in different snapshots are correlated, and some form of decorrelation or pre-whitening processing is required for the time series of observation samples; however, it will not influence the spatial signal processing discussed below. Consequently, we assume that the noise vectors w1 , ••• , WN are independent and identically distributed.

2.2.2 Detection of Targets with Known Directions

The detection of targets with known directions implies surveillance in steered directions of the array antenna.

Two cases, single target and multiple targets, will be considered. First, we discuss the detection of a single target with known direction.

8 Z. Zhu and S. Haykin

Assuming K = 1, and omitting the subscript k in (2.4) and (2.14), we may express the observation vector in a coherent radar for the case of·a single target as follows:

Here, 8 and a(8) are known, and the phase I/> is assumed to be uniformly distributed on the interval [0, 2n]. Temporarily, we assume that b is known.

In order to decide whether a target is present or not, we introduce the following binary hypothesis testing scenario:

HI: x" = sa(8) + w" ,

(2.18)

for n = 1, ... ,N. The conditional likelihood function for hypothesis HI is

(2.19)

where the superscript H signifies Hermitian transposition. Using (2.17), we obtain

The averaged likelihood ratio is therefore

Using the definition of the modified Bessel function of the first kind of zero order, namely,

Io(z) = Yezcos(~-fl)de,

o 2n

we may rewrite the expression for averaged likelihood ratio as

(2.22)

where the parameter q is defined by

The modified Bessel function Io(2bqju2 ) is a monotonically increasing function of q. Therefore, the decision rule may be based on the test statistic q. According to (2.23), the optimum detection for a single target with known direction, assuming coherent detection, has the structure shown in Fig. 2.1a. It has the form of a spatial correlation-coherent integration-linear detection.

The inverse matrix Co1 is a positive definite Hermitian matrix. By the Cholesky decomposition [2.15], we may write Co1 = DHD, where D is a positive definite lower triangular matrix. We will show immediately that D may serve as a spatially whitening operator for the noise vector Wn • Consider the vector

(2.24)

The covariance matrix of w~ is, by definition,

Hence, D represents a spatially whitening transformation. Using D, we may thus express the test statistic as

(2.25)