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A Unifying Framework for Adaptive Radar Detection in Homogeneous plus Structured Interference-Part II: Detectors Design
Domenico Ciuonzo, Antonio De Maio, Danilo Orlando
TL;DR
The paper studies adaptive multidimensional detection with unknown covariance and structured deterministic interference, extending GMANOVA. It derives several MIS-based detectors and establishes their CFARness, equivalences, and links to simpler models. Simulations compare their performance across operating conditions.
Problem
Adaptive detection lacks a comprehensive analysis for a general GMANOVA extension that includes structured deterministic interference and connects detectors to the MIS.
Method
The paper derives GLR, Rao, Wald, 2S-GLR, Durbin, Gradient, and LH detectors for I-GMANOVA and expresses them as functions of the MIS.
Results
All considered detectors are shown to be CFAR, while several pairs are statistically equivalent in the general model or important special cases.
Takeaways & Limitations
The resulting framework connects theoretically grounded detectors across I-GMANOVA and established simpler adaptive-detection scenarios.
Abstract
from arXiv · showhide
This paper deals with the problem of adaptive multidimensional/multichannel signal detection in homogeneous Gaussian disturbance with unknown covariance matrix and structured (unknown) deterministic interference. The aforementioned problem extends the well-known Generalized Multivariate Analysis of Variance (GMANOVA) tackled in the open literature. In a companion paper, we have obtained the Maximal Invariant Statistic (MIS) for the problem under consideration, as an enabling tool for the design of suitable detectors which possess the Constant False-Alarm Rate (CFAR) property. Herein, we focus on the development of several theoretically-founded detectors for the problem under consideration. First, all the considered detectors are shown to be function of the MIS, thus proving their CFARness property. Secondly, coincidence or statistical equivalence among some of them in such a general signal model is proved. Thirdly, strong connections to well-known simpler scenarios found in adaptive detection literature are established. Finally, simulation results are provided for a comparison of the proposed receivers.
I. INTRODUCTION
The paper addresses adaptive detection in the general I-GMANOVA model, which adds unknown structured deterministic interference to GMANOVA. It develops several detector families, expresses them through the MIS for CFARness, and analyzes special cases and equivalences.
- I. INTRODUCTION: I-GMANOVA generalizes GMANOVA by incorporating unknown double-subspace structured deterministic interference while encompassing several adaptive detection setups.These include point-like and extended targets or interference, single-steering and vector-subspace models, and standard GMANOVA.
- I. INTRODUCTION: The paper targets a gap in prior work, which lacked a comprehensive analysis connecting CFAR detectors to the MIS while accounting for structured deterministic interference.Earlier studies developed detectors for narrower models, including GLRT, Rao, and Wald tests, but did not establish the relevant MIS connection in the stated setting.
- I. INTRODUCTION: Closed-form GLRT, Rao, Wald, Gradient, Durbin, 2S-GLRT, and LH statistics are derived for the general model and specialized to important detection scenarios.The special cases include point-like targets, multidimensional signals, range-spread targets, and standard GMANOVA.
- I. INTRODUCTION: All considered detectors are expressed as functions of the MIS, establishing CFARness with respect to disturbance covariance and structured deterministic interference.The detector construction starts from the canonical form obtained in the companion paper.
- I. INTRODUCTION: Simulations compare the proposed detectors across relevant parameters and examine common performance trends.The paper is organized around problem formulation, detector derivation, special cases, simulations, and conclusions.
III. DETECTORS DESIGN
This section derives theoretically grounded detection statistics for the I-GMANOVA problem, beginning with GLR, Rao, and Wald criteria and adding Gradient, Durbin, and LH tests. The GLR is given in a Wilks’ Lambda form, and its expression as a function of the MIS establishes CFARness.
- III. DETECTORS DESIGN: The detector-design section derives GLRT, Rao, Wald, Gradient, Durbin, and LH statistics, including the two-step GLRT.Gradient and Durbin tests are treated as asymptotically distributed like the three principal criteria under mild conditions.
- III. DETECTORS DESIGN: The GLR is expressed in the well-known Wilks’ Lambda form, generalizing the earlier GLR to the interference scenario.The expression is obtained after substituting the closed-form maximum-likelihood estimates into the concentrated likelihood.
- III. DETECTORS DESIGN: The derivation uses maximum-likelihood covariance estimates under both hypotheses and closed-form estimates of the signal and interference parameters.The paper also states properties of the resulting covariance estimates for later use.
- III. DETECTORS DESIGN: The GLR is shown to be a function of the MIS, so its dependence on nuisance parameters is invariant and its CFAR property follows.The relevant nuisance parameters include the unknown disturbance covariance and structured interference in the underlying model.
B. Rao statistic
The Rao statistic is derived in closed form and rewritten solely in terms of the MIS, establishing its CFARness. The section also develops the Wald statistic and shows that it depends on the data through the MIS.
- B. Rao statistic: The Rao statistic is derived in closed form through Fisher-information-based expressions and compact matrix equalities.
- B. Rao statistic: The Rao statistic is rewritten as a function only of the MIS, proving its CFARness.
- C. Wald statistic: The Wald statistic is shown to depend on the data matrix uniquely through the MIS, specifically its first component.
D. Gradient statistic
The Gradient test is obtained in closed form and shown to be CFAR because its statistic is a function of the MIS. The section also relates alternative detectors through statistical equivalence and motivates a computationally simpler Lawley–Hotelling approximation.
- D. Gradient statistic: The Gradient statistic avoids Fisher-information inversion and compressed likelihood evaluation, making it formally simpler to compute than GLR, Wald, and Rao statistics.
- D. Gradient statistic: Under mild technical conditions, the Gradient test is asymptotically equivalent to the GLR, Rao, and Wald statistics.
- D. Gradient statistic: The Gradient test satisfies the CFAR property after its statistic is expressed through MIS-based equalities.
- E. Durbin statistic: The Durbin statistic is statistically equivalent to the Rao statistic for the considered hypothesis-testing model, and therefore is also CFAR.
- F. Two-step GLR: The two-step GLR is statistically equivalent to the Wald statistic and therefore also has the CFAR property.
- G. Lawley-Hotelling statistic: The Lawley–Hotelling statistic uses a first-order determinant approximation, det[I_M + Υ] ≈ 1 + Tr[Υ], for a small perturbation matrix.
IV. DETECTORS IN SPECIAL CASES
Special-case reductions connect the proposed detectors to established adaptive detection statistics. In point-like and structured-interference settings, several tests become statistically equivalent or recover familiar GLR, Rao, and AMF forms.
- A. Adaptive (Vector Subspace) Detection of a Point-like Target: In the single-steering case without interference, the Rao statistic reduces to the well-known Rao statistic for that scenario.
- A. Adaptive (Vector Subspace) Detection of a Point-like Target: In the single-steering case without interference, the Wald and two-step GLR statistics reduce to the Adaptive Matched Filter.
- A. Adaptive (Vector Subspace) Detection of a Point-like Target: For point-like targets without interference, the Gradient statistic is statistically equivalent to Kelly’s GLR.
- A. Adaptive (Vector Subspace) Detection of a Point-like Target: For point-like targets without interference, the LH statistic is statistically equivalent to the GLRT through a monotone transformation.
- B. Adaptive Vector Subspace Detection with Structured Interference: With structured interference and M = 1, the Gradient statistic is statistically equivalent to the GLR, and the LH statistic is also statistically equivalent to the GLR.
C. Multidimensional Signals
For multidimensional signals, the paper specializes the detector expressions under a no-interference model with full signal subspace. The resulting formulas recover earlier results and reveal equivalences among the proposed statistics.
- C. Multidimensional Signals: The Rao/Durbin statistic coincides with the specific result obtained in prior multidimensional-signal work, originally derived as a modified two-step GLRT procedure.
- C. Multidimensional Signals: The Wald and two-step GLR specialization coincides with the specific result obtained in prior multidimensional-signal work.
- C. Multidimensional Signals: In the multidimensional signal setup, the Gradient statistic coincides with the Rao statistic.
- C. Multidimensional Signals: In the multidimensional signal setup, the LH statistic coincides with the Wald and two-step GLR statistics.
D. Range-spread Targets
Under the range-spread specialization, the paper reduces the general detectors to closed forms and establishes equivalence among several of them.
- D. Range-spread Targets: The specialized model assumes no interference and reduces the signal and interference subspaces to the range-spread canonical form.
- D. Range-spread Targets: The GLR specialization follows from the general statistic using the model equalities and Sylvester’s determinant theorem.
- D. Range-spread Targets: The Rao, Wald, 2S-GLR, Gradient, and LH statistics are each specialized from their general formulations for this setting.
- D. Range-spread Targets: Gradient and LH statistics are statistically equivalent to the GLR for range-spread targets with rank-one signal subspace.
E. Standard GMANOVA
The standard GMANOVA specialization removes structured interference and recovers established adaptive detection formulations, while simulations compare detector performance across sample regimes.
- E. Standard GMANOVA: The simulations use M = 3, N = 8, vector-subspace cases with r,t equal to (2,4) or (4,2), and K equal to 12 or 19.
- E. Standard GMANOVA: As K grows, all detectors converge to non-adaptive performance, whereas sample-starved conditions produce substantial differences among them.
- E. Standard GMANOVA: For r = 4 and t = 2 with K = 12, Rao and Gradient outperform Wald and LH; for r = 2 and t = 4, Wald and LH prevail above approximately 18 dB.
- E. Standard GMANOVA: The numerical conclusions are illustrative rather than general because performance depends on N, K, M, r, and t.
- E. Standard GMANOVA: For point-like targets with possible point-like interference, Gradient and LH tests are statistically equivalent to Kelly’s GLRT.
- E. Standard GMANOVA: For multidimensional signals, Rao is statistically equivalent to Gradient, while Wald or 2S-GLR is statistically equivalent to LH.
IX. DERIVATION OF RAO STATISTIC
The Rao derivation constructs the statistic from the Fisher information and covariance estimates, using block-matrix structure and projection identities to obtain a closed form.
- IX. DERIVATION OF RAO STATISTIC: The likelihood gradient required by the Rao test is derived using complex differentiation and composition of intermediate gradients.
- IX. DERIVATION OF RAO STATISTIC: Permutation and block-matrix identities simplify the Fisher-information inversion and preserve the required matrix structure.
- IX. DERIVATION OF RAO STATISTIC: The derivation defines the covariance-weighted matrix Ω and partitions it to evaluate the relevant Fisher-information block.
- IX. DERIVATION OF RAO STATISTIC: The Rao statistic is obtained in closed form after substituting the covariance estimate and data-dependent quantities into the derived block expressions.
X. DERIVATION OF WALD STATISTIC
The Wald derivation evaluates the estimated signal parameters and the inverse Fisher-information block, then combines them into the closed-form statistic.
- X. DERIVATION OF WALD STATISTIC: The derivation evaluates the inverse Fisher-information block at the alternative covariance estimate using a 2 × 2 block-matrix inversion.
- X. DERIVATION OF WALD STATISTIC: The null and alternative signal estimates are specified, with the alternative estimate represented by the maximum-likelihood signal matrix.
- X. DERIVATION OF WALD STATISTIC: Substitution of these estimates and information-block expressions yields the closed-form Wald statistic.
- X. DERIVATION OF WALD STATISTIC: The matrix equalities used in the derivation reduce the required block inverse and establish the preceding statistic expression.
XI. DERIVATION OF GRADIENT STATISTIC
The Gradient statistic is derived from intermediate results for the Rao and Wald statistics, using real-valued equivalents and matrix identities. The resulting expression is shown to coincide with the Rao test for the I-GMANOVA model.
- XI. DERIVATION OF GRADIENT STATISTIC: The Gradient statistic is obtained by exploiting intermediate results from the derivations of the Rao and Wald statistics.Equations (121) and (145) provide the starting results for the derivation.
- XI. DERIVATION OF GRADIENT STATISTIC: The derivation converts complex inner products and Hermitian quadratic forms into equivalent real-valued block-symmetric expressions.The derivation uses the real/imaginary decomposition of complex vectors and properties of Hermitian matrices.
- XI. DERIVATION OF GRADIENT STATISTIC: Standard vec(·) properties, matrix identities, and closed-form optimization steps complete the Gradient-statistic derivation.The proof uses vec(·), block inverses, and substitutions into the defining statistic.
- XI. DERIVATION OF GRADIENT STATISTIC: The derived Gradient statistic coincides with the Rao test for the I-GMANOVA model.The equivalence follows by comparing the final expression with the Rao statistic reported for the same model.
XIII. PROOFS OF USEFUL EQUALITIES
The paper proves matrix equalities needed to establish CFARness of the proposed detectors and then confirms detector equivalences numerically in representative scenarios. The simulations show different equivalence groupings for point-like, multidimensional, and range-spread signal models.
- XIII. PROOFS OF USEFUL EQUALITIES: The proofs establish equalities fundamental to proving CFARness for GLR, Wald, LH, Rao, Durbin, and Gradient statistics.Separate proof steps address the equalities used for Rao/Durbin and for GLR, Wald, LH, and Gradient statistics.
- XIII. PROOFS OF USEFUL EQUALITIES: The matrix proofs rely on block partitioning, the structure of E_t, Woodbury identity, and block-inverse formulas.These identities simplify the matrices appearing in the detector expressions.
- XIV. SIMULATION RESULTS SHOWING SPECIFIC: In the point-like signal and interference setup, GLR, Gradient, and LH tests are statistically equivalent in Pd-versus-ρ simulations.The setup uses Pfa = 10^-4, M = 1, r = 2, t = 4, K = 13, and N = 8.
- XIV. SIMULATION RESULTS SHOWING SPECIFIC: For multidimensional signals, Rao and Gradient tests are equivalent, as are Wald and LH tests, in the reported simulations.The setup uses Pfa = 10^-4, N = r = 8, M = 8, and K = 24.
- XIV. SIMULATION RESULTS SHOWING SPECIFIC: For range-spread targets with a rank-one signal subspace and no interference, GLR, Gradient, and LH tests are statistically equivalent.The setup uses Pfa = 10^-4, M = 8, r = 1, t = 0, K = 24, and N = 8.