Adaptive OFDM waveform design for spatio-temporal-sparsity exploited STAP radar
2.1 Introduction
The problem of detecting slowly moving targets using airborne radars, particularly in the presence of background clutter and hostile electronic countermeasures or jamming, has led to the development of the space–time adaptive processing (STAP) algorithms in the radar community. Since the publication of a seminal paper by Brennan and Reed [1], the STAP techniques have been extensively researched and well documented in the literature over the last few decades [2–6];
interested readers may refer to [7–9] and references therein for a comprehensive survey of different STAP methodologies.
Conventionally, STAP involves the design of a two-dimensional spatio- temporal filter in order to cancel out the interference effects from the measurements of the primary range gate [also commonly known as the cell-under-test (CUT)] that contains a target. Assuming that the secondary range gates adjacent to the CUT are target free and have the same statistical characteristics of the interference returns as in the CUT [2], they are used to estimate the interference covariance matrix required for the computation of the STAP filter weights. The estimation accuracy of the interference covariance matrix, and consequently the effectiveness of the STAP filter, depends on a number of homogeneous secondary measurements used (given by the Reed-Mallett-Brennan (RMB) rule) [10]. Unfortunately, in a fully adaptive STAP, the number of required secondary measurements is so large that they do not satisfy the essential homogeneity property due to the intrinsic non- stationarity of the interference statistics.
To overcome such practical limitations of a fully adaptive STAP, several partially adaptive STAP algorithms are proposed. Popular among them are various rank-reduction techniques that assume the dominant interferences to be confined within a low-dimensional subspace. In general, these methods transform the STAP filtering process from its original high-dimensional space to a lower dimensional subspace, for example, by applying eigen-decomposition on the interference cov- ariance matrix [11–13]. The data-dependent rank-reduction algorithms, such as the principal components [14,15], relative importance of eigenbeam [6, Ch. 5], cross- spectral metric [16], parametric adaptive matched filter [17,18] and multistage Wiener filter [19], are shown in [20] to provide improved STAP performance at the expense of higher computational cost when compared with the data-independent methods, such as the joint-domain localization [9,21].
While the radar community continued the development of various partially adaptive STAP methods (and also recently knowledge-aided or knowledge-based STAP techniques [22–27]), the research efforts have been primarily biased toward the improvement of receiver signal processing algorithms, which can intelligently utilize the secondary measurements to estimate the interference covariance matrix required for the computation of the STAP filter weights. However, the measure- ments collected at the receiver are mere representations of the interactions of the transmitted signal with the operational scenarios. Therefore, if the transmit signal is kept fixed (non-adaptive), then depending on the variabilities in the environmental
conditions, the STAP performance may heavily deteriorate. This is why we propose to operate the transmitter and receiver of a STAP radar in a closed-loop fashion via an adaptive waveform design module.
With the recent technological advancements in the fields of flexible waveform generators and high-speed signal processing hardware, it is now possible to gen- erate and transmit sophisticated radar waveforms that are dynamically adapted to the sensing environments on a periodic basis (potentially on a pulse-by-pulse basis) [28–32]. This dynamic transmit-signal adaptation becomes possible due to the proper utilizations of relevant information fed back by the receiver regarding the target and interference characteristics. Consequently, the radar systems can poten- tially achieve a significant performance gain by appropriately reciprocating to any changes in the target and interference responses, particularly in the defence appli- cations involving fast-changing scenarios.
Now, to obtain the full benefits of the adaptive waveform design techniques, it is necessary for us to employ such a waveform whose parameters can be easily modified in accordance to the system objectives. To fulfil this goal, in this work, we use an orthogonal frequency division multiplexing (OFDM) radar signal [33–36] to detect a target using a STAP technique; see also [37,38]. The motivation of employing an OFDM signal is that it offers a very efficient way to optimally tailor its spectrum as each of the designed OFDM coefficients determines the transmit energy at a particular subchannel. Therefore, depending on the frequency vari- abilities of the target and interference responses, we adaptively synthesize the OFDM spectral parameters in order to improve the system performance. In addition, the OFDM signalling technique increases the frequency diversity of the system by providing additional information about the target, as different scattering centres resonate at different frequencies [39,40], and thus improves the target-detectability from the interfering signals.
However, due to the addition of one extra dimension in terms of frequency, the use of an OFDM signal for the STAP applications also causes one disadvantage by increasing the adaptive degrees of freedom fromMN toLMN, whereM denotes the number of antenna-array elements,N is the pulse repetition periods and Lis the number of OFDM subcarriers. Thereby, the construction of a fully adaptive OFDM-STAP becomes practically impossible because of the requirement of a larger number of homogeneous secondary data and the computational burden of a bigger matrix inversion [1,10]. To circumvent these challenges, we propose a sparsity-based STAP algorithm. We observe that the target and interference spectra are inherently sparse in the spatio-temporal domain, as the clutter responses occupy only a diagonal ridge on the spatio-temporal plane and the jammer signals interfere only from a few spatial directions. Hence, we exploit that sparsity to develop an efficient OFDM-STAP technique requires considerably lesser number of secondary data compared to the other existing STAP techniques and produces a near-optimum STAP performance. Similar spatio-temporal sparsity-exploited STAP techniques, but without the OFDM signalling scheme, are lately proposed in [41–54]. It is also to be noted here that an OFDM-STAP filter operates on the received data as a
whole, in contrast to the subband STAP approach which partitions the data among different subchannels by assuming frequency independence [55,56].
We first develop a sparsity-based STAP technique by formulating a realistic sparse-measurement model for OFDM radar that accounts for measurements over multiple frequencies. Though both the spatial and temporal frequencies are con- tinuously varying parameters, the superiority of sparse signal processing can still be availed byjust discretizingthe spatio-temporal plane into a finite number of spatio- temporal frequency grids [57]. Then, the sparse nature of the target and interference spectra on this spatio-temporal grid structure is exploited by the standard sparse- recovery techniques to produce an estimate of the clutter covariance matrix. In particular, to design the STAP filter, we use the secondary range gate data and estimate the interference-only covariance matrix in two steps: (i) apply the LASSO estimator [58] as a sparse-recovery technique to obtain the clutter response from the secondary measurements and (ii) premultiply the sparsely recovered clutter responses with the known sparse-measurement matrix to compute the clutter cov- ariance matrix. One additional benefit of this approach is that the estimated clutter support provides the necessary information to form a masking matrix [44], which is used in the sparse-recovery of target response from the primary gate measurements, and thus, we can avoid the explicit estimation and inversion of the interference covariance matrix.
In addition to designing the STAP filter, we propose to optimally design the transmit OFDM signals by maximizing the output signal-to-interference-plus-noise ratio (SINR) in order to improve the STAP performance. However, in practical scenarios, the computation of output SINR depends on the estimated value of the interference covariance matrix, which we obtain by applying the sparse recovery algorithm. Obtaining good estimates of the interference covariance matrix heavily influence the achievable improvement of output SINR via adaptive waveform design. Therefore, we investigate the effects of different choices of the transmit signal parameters on the accuracy of sparse recovery technique by computing the coherence measure [59–61] of the sparse-measurement matrix. Subsequently, we provide a closed-form expression of the optimal OFDM coefficients by reasonably considering that the estimation accuracy of the interference covariance matrix insignificantly depends on the signal parameters.
We present several numerical examples to demonstrate the OFDM-STAP performance obtained by employing spatio-temporal sparsity framework, and the achieved performance gain due to adaptive OFDM waveform design. We analyze the performance in terms of the SINR-loss measure and the receiver operating characteristics (ROCs) and make comparative characterizations with respect to the ideal STAP technique which has the full knowledge about the target and inter- ference characteristics. Our analyses include both the idealistic and non-idealistic [including the presence of temporal decorrelation effects caused by the intrinsic clutter motion (ICM) [6, Ch. 4, 62]] STAP scenarios with a Doppler-unambiguous clutter characteristic. The scattering coefficients of the target are also varied to construct two scenarios having different target-energy distributions across different subchannels. We show that the spatio-temporal sparsity-based OFDM-STAP
approach yields near-optimum performances by utilizing a substantially small number of secondary measurements. For example, in the ideal scenario, only two to five secondary data are found to be enough to produce a near-optimum perfor- mance. In the presence of temporal decorrelation, although we get a wider main- beam clutter notch, the near-optimum SINR-loss performance is still attained by using only two to five secondary measurements. A significant amount of improvement in performance due to the use of adaptive OFDM waveform is demonstrated by computing the ROCs for two different target responses. For example, in the presence and absence of temporal decorrelation effects, we respectively observe approximately 3 and 6 dB of improvement in detection performance.
The rest of the chapter is organized as follows. In Section 2.2, we first develop a parametric sparse-measurement model for OFDM-STAP radar. Then, in Section 2.3, we describe a sparse recovery technique to estimate the interference covariance matrix and to design the STAP filter weights. An adaptive OFDM waveform design algorithm is proposed in Section 2.4. We discuss the numerical results in Section 2.5.
Conclusions and possible future work are given in Section 2.6.
2.1.1 Notations
We present here some notational conventions that are used throughout this chapter.
We use math italic (a) for scalars, lowercase bold (a) for vectors and uppercase bold (A) for matrices. For a matrix A2Ckm, AT, AH and trf gA denote the transpose, conjugate transpose and trace ofA, respectively.Ik represents an iden- tity matrix of dimensionk. blkdiagð Þ forms a block-diagonal matrix with non- zero submatrices only on the main diagonal. Ref g is the real part and j jis the magnitude of a complex quantity. Among different types of vector norms, we consider the‘1and‘2norms, expressed ask k1andk k2, respectively. In addition, h;i,andare the inner product, Kronecker product and point-wise Hadamard product operators, respectively. For a random variablea,EðaÞand varðaÞ, respec- tively, denote the mean and variance ofa.