Two-Dimensional Parametric Models for Signal Processing of Data Stationary in 1 D or 2 D: Maximum-Entropy Extensions and Time-Varying Relaxations

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1 Two-Dimensional Parametric Models for Signal Processing of Data Stationary in 1 D or 2 D: Maximum-Entropy Extensions and Time-Varying Relaxations YI Abramovich 1, BA Johnson 2, NK Spencer 3 1 Intelligence, Surveillance and Reconnaissance Division (ISRD), Defence Science and Technology Organisation (DSTO), PO Box 15, Edinburgh SA 5111, Australia YuriAbramovich@dstodefencegovau 2 RLM Management Pty Ltd and the University of South Australia, PO Box 22, Edinburgh RAAF Base SA 5111, Australia BenAJohnson@ieeeorg 3 Adelaide Research & Innovation Pty Ltd (ARI), c/- 2 Labs, ISRD, DSTO, PO Box 15, Edinburgh SA 5111, Australia NickSpencer@adelaideeduau 1 May 9, 28 1 Nick Spencer is supported by DSTO through the Centre of Expertise in Phased Array and Microwave Radar Systems at The University of Adelaide

2 1 Introduction and Background In a number of technical signal processing applications, the data stream to be processed is two-dimensional; for example, in airborne radar the 2 D data is collected using an antenna array comprising M identical sensors over a period of time known as the coherent processing interval (CPI), hence the two data dimensions are space and time Data is also recorded discretely in time: each CPI consists of N pulse repetitions The practical problem is that, given only a small number τ of training data vectors X j CN(, R) C MN 1, j = 1,,τ (1) we want to estimate the MN-variate covariance matrix R so that the adaptive filter (space-time adaptive processor) Ŵ(φ, Ω) = ˆR S(φ, Ω) (2) has an acceptable performance degradation compared with the optimal (clairvoyant) Wiener filter Ŵ opt (φ, Ω) = R S(φ, Ω) (3) Here S(φ, Ω) C MN 1 is the so-called steering vector that is a function of space and time (or Doppler frequency) In most cases, a moment of some probability density function (pdf), eg mean or median, for the random quantity [ ] 2 S H ˆR S η = S H ˆR [, 1] (4) RˆR S S H R S known as the signal-to-interference-plus-noise ratio (SINR) loss factor, serves as a performance measure for the given covariance matrix estimate ˆR ˆR(X 1,, X τ ) Usually the data is stationary in time, so that the MN-variate covariance matrix R is a positive-definite (pd) Toeplitz block matrix: an overall pd Toeplitz matrix composed of N N lots of unstructured M M blocks Moreover, in applications where the data stream is stationary in both time and space, such as the case with an accurately calibrated uniformly spaced linear antenna array (ULA), R has a Toeplitz-block-Toeplitz structure Stationarity (and ergodicity) in one or both data dimensions is exploited when multivariate autoregressive (parametric) models are used for adaptive covariance matrix estimation For data that is stationary in time, the autoregressive model AR M (n) (with n N) proposed for STAP applications in [12] provides a two-fold decrease in training support requirements First, the M N-variate Toeplitz block matrix estimate ˆT is uniquely specified for this model by its M(n + 1)-variate principal block (say, upper-left corner) Second, ergodicity in time means that we can generate (N n) different M(n + 1)-variate equally distributed training samples by applying sliding-window averaging over the entire CPI While not independent, these

3 samples significantly improve the quality of the M(n + 1)-variate covariance matrix estimate For data that is stationary in both dimensions, the corresponding 2 D autoregressive AR(m, n) model has been introduced for STAP applications [8] This model is uniquely specified by an (m + 1)(n + 1)-variate Toeplitz-block-Toeplitz matrix A combination of sliding-window averaging over the CPI (time) and spatial smoothing over the array aperture (space) then allows us to generate (M m)(n n) different (m + 1)(n + 1)-variate training samples from a single M N-variate one This substantial increase in the number of usable training data samples (naturally of smaller dimension), due to ergodicity in one or two dimensions, is the main reason for the superior stochastic accuracy of parametric model estimation compared with generic M N-variate Hermitian/Toeplitz block matrix estimation The drawback of any parametric model, and specifically AR M (n) and AR(m, n), is that the real-world covariance matrix does not exactly fit into any parametric model, and so for n < N and m < M, the reduced stochastic (finite-sample support driven) SINR losses are always offset by the SNR losses caused by the parametric model mismatch with the true covariance matrix Under these circumstances, meaningful evaluation of a particular parametric model may be performed for a credible phenomenological model of an MN-variate Toeplitz or Toeplitz-block-Toeplitz covariance matrix On the other hand, the SINR STAP performance that adopts a certain parametric model (parametric STAP) must be kept in focus of this development, rather than the strict properties of the (causal) AR models, that by itself may or may not contribute to the SINR performance improvement This is important because most studies have been concerned with causal AR M (n) or AR(m, n) model estimation, which on one hand is a significant problem for practical real-time implementation, while on the other hand may be not necessarily required in order to get an appropriate parametric STAP performance In particular, the necessary and sufficient conditions on a (m + 1)(n + 1)-variate pd Toeplitz-block-Toeplitz matrix to serve as covariance matrix of a causal AR(m, n) model, recently derived in [13], mean that not every such matrix may be directly used as an AR(m, n) covariance matrix model In this regard, we might consider extensions (completions) of an arbitrary (m + 1)(n + 1)-variate pd Toeplitz-block- Toeplitz matrix, other than the AR(m, n) one Specifically, in what follows, we consider the maximum-entropy (ME) completion of an M N-variate Toeplitz-block- Toeplitz covariance matrix given its (m + 1)(n + 1)-variate principal block, and ME completion of an M N-variate Toeplitz block matrix, without enforcing the Toeplitz properties of the completed M-variate blocks This point is important, due to the deliberate inconsistency of our estimation approach That is, on one hand we exploit the 2 D stationarity of the training data, and hence the Toeplitz-block-Toeplitz structure of the M N-variate covariance matrix when averaging temporally and spatially ( smoothing ); but on the other hand refuse to enforce these properties in our M N-variate covariance matrix estimate Such covariance matrix estimates and corresponding parametric models that do not enforce all the known properties of the underlying data, we call relaxations Note that a relaxation should always be an inferior covariance matrix estimate, in terms of SINR STAP performance, when the true covariance matrix of the

4 training data is indeed accurately described by an AR(m, n) model In practice, since the underlying Toeplitz-block-Toeplitz matrix does not exactly fit the parametric model, it is not obvious that (say) the best AR(m, n) model will always be better than its relaxation Again, only analysis using a phenomenological data model can provide a quantitative results on the comparative merits of the strict model and its relaxation In this regard, it is important that in most cases the initial direct data sample covariance matrix estimate is just a Hermitian and not a Toeplitz or Toeplitz-block-Toeplitz matrix, despite the above-mentioned fact that this estimate exploits these Toeplitz properties while averaging Therefore, instead of converting the (m + 1)(n + 1)-variate or M(n + 1)-variate sample covariance matrix into a Toeplitz-block-Toeplitz or Toeplitz matrix, respectively, for further extension (completion) to an M N-variate Toeplitz relaxation, we now propose a direct Hermitian (time-varying) relaxation 2 Stationary Relaxations for AR(m, n) Models Given an (m + 1)(n + 1)-variate pd matrix T t j k p q } for j k n, p q m (5) the restoration (reconstruction) of the entire M N-variate covariance matrix of a causal 2 D AR(m, n) model can be considered as a search for a particular matrix completion: Find T N (z) = t j k p q z j k p q for j k n p q m for j k > n p q > m (6) such that T N (z)} jk = for j k > n p q > m (7) pq Not surprisingly, this special completion exists only if the given matrix T satisfies certain requirements The necessary and sufficient conditions have recently been specified by the Geronimo Woerdeman Theorem 21 in [13] However, an arbitrary (m + 1)(n + 1)-variate pd Toeplitz-block-Toeplitz matrix may be completed in a number of suboptimal ways that could be treated as relaxations 21 Toeplitz-Block-Toeplitz Relaxation TbT(m, n) Here the ME completion problem is formulated as Find max log det T N (z) (8) where T N (z) is as in (6), but the to-be-completed elements z j k p q are also constrained by j k N 1 and p q M 1 The theory of Lundquist and Johnson [7]

5 can be modified to include this case, whereby the optimum completion must satisfy where N M jk=1 pq=1 j k=v p q=s τ jk pq = for v > n s > m (9) T N (z) τpq jk } jk=1,, N (1) pq=1,, M As in every ME Toeplitz completion, this optimality condition means that the elements in the inverse matrix at positions that in the direct matrix correspond to the same covariance lag t v s sum to zero For an arbitrary (m + 1)(n + 1)-variate pd Toeplitz-block-Toeplitz matrix, its M N-variate Toeplitz-block-Toeplitz ME completion is not an autoregressive matrix, since those elements in the inverse sum to zero, but are not necessarily zero individually For this reason, we call this model TbT M (m, n) The problem (8) is a linear matrix inequality (LMI) analytic center problem that is solved by the interior-point Nesterov Nemirovskii routine [1] Expressions for the gradient and Hessian for this complex-valued LMI problem are derived in [1] 22 Autoregressive AR M (n m) or AR N (m n) Relaxations In (8) we enforced the Toeplitz-block-Toeplitz structure of the ME completion, but ended up with a nonar relaxation Another option is to enforce the multivariate AR structure of the completion, although it will end up being only a Toeplitz block matrix For example, we may consider the ME completion of the M(n + 1)-variate Toeplitz-block-Toeplitz matrix that is the solution to Find max log det T n+1 (z) (11) where T n+1 (z) = The unique optimal solution t j p q for j n; p q m z j p q for j n; m < p q < M (12) T n+1 (m) Toep [T (m),,t n (m)] (13) can now be completed as an MN-variate pd Toeplitz block matrix in an AR M (n)- type completion: Find max Z log dettoep [T (m),, T n (m), Z n+1,, Z N ] (14) This ME Burg completion T N (n m) will then obey the AR M (n) property: T N (n m)} = for j k > n (15) jk but the completed matrices Z n+1,,z N are generally not Toeplitz We call this model the AR M (n m) relaxation

6 Note that the complete symmetry of the (m+ 1)(n+ 1)-variate Toeplitz-block- Toeplitz matrices in both space and time dimensions permits us to also propose a AR N (m n) relaxation Indeed, we may first ME-complete the given T (m+1)(n+1) matrix to the N(m + 1)-variate one in Find max log det T m+1 (z) (16) where T m+1 (z) = t j k p for j k n; p m z j k p for j k > n; p m (17) and then build an AR N (m) model using the optimal completion T m+1 (n) Toep [T (n),, T m (n)] (18) where T j (n) denotes the j th N-variate temporal Toeplitz block, rather than the M- variate spatial Toeplitz block as before This AR N (m)-based completion of a matrix T N (m n) that gives the AR N (m n) relaxation then possesses the band-inverse property common to AR models: T N (m n)} = for p q > m (19) pq Note that these models, while different, are all uniquely specified (for any particular ME completion type) by the same given (m+1)(n+1)-variate pd Toeplitzblock-Toeplitz matrix, and only if this matrix satisfies the Geronimo Woerdeman conditions do all of them collapse to a single causal 2 D AR(m, n) model It is important that the relative merits of the TbT(n, m), AR M (n m) and AR N (m n) models, in terms of the SINR loss factor, depend on the properties of the underlying true Toeplitz-block-Toeplitz covariance matrix 3 Nonstationary Relaxations for AR M (n) and AR M (n m) Models 31 TVAR M (n) Relaxation for the AR M (n) Model Recall that the MN-variate Toeplitz block matrix for the model AR M (n) can be uniquely reconstructed from a given (arbitrary) M(n + 1)-variate pd Toeplitz block matrix using the Dym Gohberg band method, a triangular factorization of the inverse matrix [4]: where V is a triangular matrix whose elements are T (n) = VV H (2) V jk Yjk Y kk for k j minn, k + m} otherwise (21)

7 where V qq V q+n,q = V 11 V n+1,1 = T M(n+1) I M M M (22) for q N n, while for q N n + s; s = 1,, n V qq V Nq = T M(n+1 s) I M M M (23) The TVAR M (n) relaxation is constructed by adopting the same reconstruction formula (2), except that we use the pd Hermitian (sample) block matrix ˆR M(n+1) instead of the Toeplitz block matrix T M(n+1) in (22) and (23) The resulting block matrix ˆR(n) is pd and band inverse by construction, ie ˆR (n)} pq = for p q > n (24) and has the same multivariate Burg spectrum (specified by the first block-column of its inverse) as the M(n + 1)-variate sample matrix ˆR M(n+1) The analogous 2 D (space-time) TVAR M (n m) relaxation is constructed using the same restoration principle (2) (24), but acting in both dimensions 4 SINR Performance of Parametric STAP: AFRL Data Results As discussed in the Introduction, we choose to evaluate different parametric models for STAP applications by analyzing the SINR loss factor with respect to the optimum (clairvoyant) Wiener filter (4) for phenomenological clutter covariance matrices, as provided by the AFRL dataset The AFRL covariance matrix model is based on the spatial aperture M = 11 and temporal aperture N = 32 DARPA KASSPER radar scenario [2, 5], but with simplified electromagnetic calculations and an ideal ULA that gives rise to a Toeplitz-block-Toeplitz covariance matrix First, Fig 1 presents the distribution of optimum STAP filter gains over the conventional matched filter at the fixed elevation angle φ = 465 that is appropriate for range bin 2 These gains are calculated on a discrete 2 D grid that cover the entire range of azimuthal angle θ and Doppler frequency ω; however, radar scientists sometimes prefer to work in terms of sin θ and normalized Doppler frequency ω/2π The optimum STAP gain q is calculated similarly to (4): q(θ, ω) = SH T SS H TS [S H S] 2 (25)

8 AFRL rb2 optimal STAP gains, max normalised Doppler 5 Figure 1 Azimuth-Doppler distribution of optimal STAP gains with respect to the conventional matched filter for range bin 2 98 AFRL clairvoyant fwd AR 11 (8) SINR losses normalised (a) Doppler [8] AFRL clairvoyant fwd TVAR (8 5) SINR losses normalised (b) Doppler 5 5 [9] Figure 2 Clairvoyant (forward-reconstructed) (a) AR 11 (8) and (b) TVAR 11 (8 5) SINR losses relative to optimal STAP filter processing Fig 2(a) illustrates the SINR losses (4) for the traditional [3,6,9,11] AR M (n) model of (temporal) order n = 8, calculated for the clairvoyant matrices for range bin 2 in each dataset, with almost the same distribution of losses The maximum SINR loss factor for the AFRL Toeplitz-block-Toeplitz matrix is 8 As expected for a low-order AR M (n) model, maximum losses occur close to the clutter ridge, whereas away from the ridge the losses drop to only a fraction of a These results confirm the very high potential efficiency of the AR M (n) models

9 AFRL, dim M(n+1), clairv NN SINR losses, m=5, n=8 98 AFRL, clairv NN SINR losses, m=5, n= normalised (a) Doppler 5 [9] normalised (b) Doppler 5 [7] Figure 3 Comparison of clairvoyant SINR losses for the rigorous Nesterov Nemirovskii TbT 11 (5, 8) model for the (a) partial M(n + 1)-variate and (b) full MN-variate AFRL data with n N for STAP applications, as demonstrated in [3,9] At Fig 3 we show the loss distributions for the TbT 11 (5, 8) and AR 11 (8 5) models calculated for the clairvoyant matrix T (m+1)(n+1) that, as expected, is hardly distinguishable from Fig 2(a) This means that an exact restoration of the Toeplitzblock-Toeplitz properties of the reconstructed M N-variate covariance matrix is unnecessary (from the SINR viewpoint), and that the large spatial-order reduction from m max = M 1 = 1 to m = 5 costs only 11 in maximum SINR degradation Finally, at Fig 4 we present an example stochastic realization of SINR losses for τ = 5, 4, 2, 1 iid training samples calculated for the TVAR 11 (8 5) relaxation model Of course, for a single training snapshot we observe significant SINR losses, up to 16, but already for five samples the SINR losses do not exceed 39 Clearly, stationarity and ergodicity in the spatial domain permit a noticeable STAP performance improvement, though for the AFRL radar model with M N, this improvement is less profound than the one due to temporal-order reduction (n N) We may expect different results for applications other than the AFRL scenario, however, the proposed set of 2 D parametric stationary models and their time-varying autoregressive relaxations provide a good tool-box for parametric STAP design 5 Summary and Conclusions For a multivariate process that is stationary in its temporal dimension ( slow time in the radar application), we have introduced various 2 D parametric models that rely on autoregressive and maximum-entropy extension principles, motivated by their application in STAP radar space-time adaptive processing These parametric models have fewer free parameters than the model for an arbitrary MN-variate

10 AFRL 8 sample fwd TVAR (8 5) SINR losses normalised (a) Doppler [ 39] AFRL 4 sample fwd TVAR (8 5) SINR losses normalised (b) Doppler [ 49] AFRL 2 sample fwd TVAR (8 5) SINR losses AFRL 1 sample fwd TVAR (8 5) SINR losses normalised (c) Doppler 7 [ 77] normalised (d) Doppler 2 [26] Figure 4 Sample TVAR 11 (8 5) SINR losses for (a) τ = 8, (b) τ = 4, (c) τ = 2, and (d) τ = 1 snapshots Hermitian covariance matrix, which results in our goal of a reduced requirement for training-sample support, but unfortunately also leads to an extra STAP performance degradation caused by the mismatch between any true clutter covariance matrix and its parametric model To quantify the performance of the introduced models, we therefore focused on the STAP filter performance degradation (SINR loss) due to replacing the true covariance matrix in the optimum Wiener filter by the estimate from the parametric model This criterion enables any invertible covariance matrix model to be considered for STAP design On the other hand, meaningful recommendations regarding these parametric models can only be justified for a particular application and having a trustworthy high-fidelity phenomenological clutter covariance matrix provided For this reason, our performance analysis has been conducted for the radar clutter covariance matrix model of an airborne side-looking radar, with a nominally uniformly spaced linear antenna array (M = 11 sensors) with no antenna errors, and

11 a periodic waveform (N = 32 repetitions) This clutter is stationary in slow time over the coherent processing interval, and is described by a range-dependent MNvariate Toeplitz-block-Toeplitz matrix, whose structure is N N lots of M M ( inner ) spatial blocks As an alternative to the 2 D AR(m, n) model, that may not exist for an arbitrary pd Toeplitz-block-Toeplitz matrix, we suggested a new family of parametric models that are uniquely specified by an arbitrary (m + 1)(n + 1)-variate pd Toeplitz-block-Toeplitz matrix, and introduced them as ME completions In particular, we demonstrated that an ME completion that is also constrained to be Toeplitz-block-Toeplitz is not necessarily an AR-type matrix (but this does not preclude this matrix for STAP filter design) In our two other completions AR M (n m) and AR N (m n), we enforced the AR properties in the completion, but only demanded Toeplitz-block (rather than Toeplitz-block-Toeplitz) properties, so that some completed blocks are non-toeplitz All the introduced ME completions of an arbitrary (m + 1)(n + 1)-variate pd Toeplitz-block-Toeplitz matrix collapse to the same AR(m, n) model, if the necessary and sufficient conditions given by Geronimo and Woerdeman are satisfied Strictly speaking, the two models AR M (n m) and AR N (m n) must be treated as relaxations (relaxed models), since the fundamental Toeplitz-block-Toeplitz property relied on for sliding-window averaging over both dimensions is not retained in the final covariance matrix model Yet, we demonstrated that for AFRL data this deliberate inconsistency does not lead to a noticeable STAP performance degradation Our sole parametric model performance criterion has been radar-relevant STAP SINR, and this encouraged us to expand the (pre-existing but vaguely defined) relaxation idea much further Most studies on parametric STAP consider estimation of a causal AR model that is consistent with the adopted ergodicity properties In adaptive settings, this means that when constructing an accurate Toeplitz-block- Toeplitz matrix model, the sample Hermitian (m + 1)(n + 1)-variate matrix is first transformed into a pd Toeplitz-block-Toeplitz matrix, then completed using the convex programming technique, for example While possible, this rigorous approach is impractical due to significant computational requirements Rather than Toeplitz-block-Toeplitz solutions, we therefore proposed a new class of Hermitian time-varying autoregressive (TVAR) relaxations, that can be directly calculated from the (m + 1)(n + 1)-variate Hermitian sample matrices These relaxations were derived for adaptive applications with (m + 1)(n + 1)-variate sample Hermitian-block matrices, and when applied to the true (m + 1)(n + 1)- variate Toeplitz-block-Toeplitz matrices, respectively, they generally lead to models different from AR M (n m), AR M (n m) or AR N (m n) Yet, for the AFRL covariance matrix for range bin 2, the rigorous AR M (n m) model calculated for the 2th AFRL covariance matrix was practically the same as that of its relaxation TVAR 11 (8 5) Overall, for a high-fidelity trusted phenomenological clutter covariance matrix model, there is a modest performance degradation associated with our 2 D parametric models which offer significant reductions in training-sample support requirements and calculations for STAP filter design

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