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Adaptive Subspace Signal Detection and Performance Analysis in Nonzero-Mean Clutter

Weijian Liu, Zhenyu Xu, Jun Liu, Hui Chen, Yongxiang Liu

arXiv:2605.07434v1stat.OT

TL;DR

Subspace signal detection in nonzero-mean clutter had not been investigated, although ignoring the clutter mean can compromise CFAR or detection probability. This paper develops adaptive detectors and derives their PD/PFA expressions, identifying reduced test-statistic degrees of freedom and an SCR loss versus zero-mean clutter.

  • Problem

    Subspace signal detection in nonzero-mean clutter had not been investigated, although ignoring the clutter mean can compromise CFAR or detection probability.

    Source: §I · pp. 1–2

  • Method

    The paper develops adaptive subspace detectors using GLRT, Rao, Wald, gradient, and Durbin strategies and derives analytical PD and PFA expressions.

    Source: §I · p. 2

  • Results

    L/(L + 1) SCR loss from estimation bias and reduced test-statistic DOF distinguish nonzero-mean from zero-mean clutter; simulations and measured data validate the PD and PFA expressions.

    Source: §VI · p. 12

  • Takeaways & Limitations

    For practice, the authors recommend SGLRT-NMC without signal mismatch, SAMF-NMC for robust mismatched detection, and SRao-NMC when selectivity against mismatched signals is desired.

    Source: §VI · p. 12

Abstract

from arXiv · show

To solve the problem of detecting subspace signals in nonzero-mean clutter, we propose adaptive detectors, based on the strategies of generalized likelihood ratio test (GLRT), Rao test, Wald test, gradient test, and Durbin test. The results show that the detectors based on GLRT, Rao and Wald are structurally consistent with the subspace detectors in zero-means clutter. The analytic expressions for the probability of detection (PD) and probability of false alarm (PFA) of each detector are derived, and two major performance differences in the nonzero-mean clutter scenario are revealed. One is the loss of degree of freedom (DOF), which is reduced by 1 compared with the zero-mean clutter scenario. The second is the loss of signal-to-clutter (SCR) ratio. Simulation and measured data verify the effectiveness of the proposed detectors and demonstrate their practical value in real-world radar systems.

I. INTRODUCTION … A. GLRT-Based Detector

This paper develops adaptive detectors for subspace signals in nonzero-mean clutter and analyzes their statistical performance. It identifies structural relationships among the detectors and reductions in degrees of freedom and signal-to-clutter ratio relative to zero-mean clutter.

I. INTRODUCTION

  • Subspace signals occupy a signal space rather than one steering direction, with rank-one signals as a special case; examples include helicopter echoes and polarimetric radar returns.The paper notes that subspace variants of KGLRT, AMF, and DMRao had been developed in earlier work.Source: §I · p. 1
  • Nonzero-mean clutter can arise from background illumination and from outliers or interfering targets; ignoring it can undermine CFAR or severely degrade PD.The passages give hyperspectral imaging as an example and attribute outliers to discrete clutter or interference and interfering targets to radar-receiver sidelobes.Source: §I · p. 1
  • No prior work had addressed subspace detection in nonzero-mean clutter; unlike and, this study proposes detectors and analyzes them for complex-domain subspace signals and subspace mismatch.The cited comparison notes that focused primarily on GLRT-based detectors and PFA, while analyzed existing rank-one detectors in the real domain and considered clutter-mean or covariance-power mismatch.Source: §I · p. 2
  • The proposed GLRT-, Rao-, Wald-, gradient-, and Durbin-based detectors yield analytical PD and PFA expressions; nonzero-mean clutter reduces degrees of freedom by 1 and SCR by a factor of L/(L + 1).The GLRT-based, Rao-based, and Wald-based detectors are structurally consistent with zero-mean counterparts; the gradient test matches the GLRT statistic, and the Durbin test matches the Rao statistic.Source: §I · p. 2

II. DETECTION PROBLEM DESCRIPTION

  • The detection model uses nonzero-mean complex Gaussian clutter, a full-rank signal matrix A with p < N, and L IID signal-absent training samples; mean, covariance, and signal coordinates are unknown.Because the test and training data share clutter statistics and the training data are signal-absent under both hypotheses, the unknown parameters preclude an optimal detector.Source: §II · pp. 2–3

III. DETECTOR DESIGN

  • The detector-design section derives GLRT-, Rao-, and Wald-based detectors, while noting that gradient and Durbin tests are also used in detector design.The gradient-based detector is equivalent to its GLRT counterpart, and the Durbin-based detector is equivalent to the Rao-based detector.Source: §III · p. 3, A. GLRT-Based Detector · p. 3

A. GLRT-Based Detector

  • The GLRT-based subspace detector for nonzero-mean clutter is named SGLRT-NMC, and its mean-removed covariance and equivalent form facilitate statistical analysis.The paper identifies the mean-removed sample covariance under the null hypothesis and says the equivalent representation is more convenient for deriving statistical properties.Source: §I · p. 2, A. GLRT-Based Detector · p. 3

B. Rao Test-Based Detector · C. Wald Test-Based Detector

The section formulates a nonzero-mean-clutter Rao detector and Wald-based tests, including an adaptive Wald statistic. It also states the positive-definiteness condition required for the GLRT and Wald detectors.

B. Rao Test-Based Detector

  • The GLRT and Wald detectors depend on S2 being positive definite; when L ≥ N + 1, S2 is positive definite with probability 1.Source: B. Rao Test-Based Detector · p. 3
  • The Rao statistic in (15) is named the subspace-based Rao test statistic for nonzero mean clutter (SRao-NMC).Appendix B-2 shows an alternative expression for (15).Source: B. Rao Test-Based Detector · pp. 3–4

C. Wald Test-Based Detector

IV. STATISTICAL PERFORMANCE ANALYSIS · A. The PD and PFA of the SGLRT-NMC in (10)

The section derives the SGLRT-NMC’s statistical behavior and its PD and PFA under signal mismatch, where the true steering vector may lie partly outside the assumed subspace. It characterizes the distributions governing the detector and mismatch loss.

IV. STATISTICAL PERFORMANCE ANALYSIS

  • The analysis allows signal mismatch: the true steering vector p0 need not lie entirely in the assumed subspace, while matching is the zero-mismatch-angle case.Source: §IV · p. 4

A. The PD and PFA of the SGLRT-NMC in (10)

B. The PD and PFA of the SRao-NMC in (16) · C. The PD and PFA of the SAMF-NMC in (20)

The sections derive analytical detection and false-alarm probability expressions for SRao-NMC and SAMF-NMC, enabling threshold calculation and performance prediction.

B. The PD and PFA of the SRao-NMC in (16)

C. The PD and PFA of the SAMF-NMC in (20)

  • For SAMF-NMC, PD is expressed analytically, with P1(βηW) obtained from (37) and ηW as its detection threshold.Source: C. The PD and PFA of the SAMF-NMC in… · p. 5
  • SAMF-NMC PFA is derived using a treatment similar to that for (49).Its expression is analytical.Source: C. The PD and PFA of the SAMF-NMC in… · p. 6
  • PD expressions for all three proposed detectors and PFA expressions for SRao-NMC and SAMF-NMC use numerical integration; SGLRT-NMC PFA is closed-form.The analytical PFA expressions yield accurate thresholds without time-consuming Monte Carlo simulations, aiding system parameter design and optimization.Source: C. The PD and PFA of the SAMF-NMC in… · p. 6
  • Detection performance depends on SCR, signal mismatch cos^2 θ, subspace dimension p, training-data count L, and clutter covariance characteristics.Analytical PD expressions support prediction across SCR conditions and help guide detection-performance improvements and radar-system design.Source: C. The PD and PFA of the SAMF-NMC in… · p. 6

V. NUMERICAL EXAMPLES · A. Practical Radar Sensing Application Context

The numerical examples connect the proposed adaptive detectors to an X-band maritime radar application and set up performance evaluation with simulated and measured data. The application context maps the subspace-signal, nonzero-mean-clutter model to temporal Doppler processing and distinguishes it from conventional detector models.

V. NUMERICAL EXAMPLES

A. Practical Radar Sensing Application Context

  • The detectors generalize to spatial, temporal, and space-time multichannel scenarios; the case study applies temporal Doppler processing to X-band maritime surveillance radar.The section frames this case study as a link between analytical work and real-world implementation.Source: A. Practical Radar Sensing Application… · p. 6
  • The example sets 9.4 GHz center frequency and 6 MHz bandwidth, yielding approximately 0.032 m wavelength and 25 m range resolution; it uses 24 nearby range cells for training.Those training cells span 600 m; possible nonzero-mean clutter sources include interferers, coastal sidelobes, and uneven sea-surface illumination.Source: A. Practical Radar Sensing Application… · p. 6
  • For the radar example, the observation dimension is N = 12 pulses per CPI, forming a 12 × 1 test vector that captures target and clutter Doppler characteristics.The example uses a 1 kHz PRF and 12 ms CPI.Source: A. Practical Radar Sensing Application… · p. 6
  • Three ships in one 25 m range cell with distinct radial velocities motivate a p = 3 signal subspace, as does target smearing across three Doppler bins during a long CPI.The smearing example uses a maneuvering ship whose velocity varies from 3 m/s to 7 m/s during the 12 ms CPI.Source: A. Practical Radar Sensing Application… · p. 6
  • Detection performance is compared with SGLRT, SAMF, and SRao; separately, SGLRT-NMC is compared with GLRT-NMC for rank-1 signals in nonzero-mean clutter.The other comparator is the SGLRT for subspace signals in zero-mean clutter.Source: A. Practical Radar Sensing Application… · pp. 6–7
  • The proposed SGLRT-NMC combines a subspace-signal model with nonzero-mean clutter, unlike GLRT-NMC for rank-1 signals or SGLRT for subspace signals in zero-mean clutter.Table I compares their test statistics and characteristics, including their statistical distributions under H1 and output SCR.Source: A. Practical Radar Sensing Application… · p. 7

B. Performance Evaluations Using Simulated Data · C. Performance Evaluations Using Measured Data

Simulations show that the proposed nonzero-mean-clutter detectors maintain stable false-alarm and detection performance where conventional detectors can degrade. Measured-data evaluations confirm dataset-dependent detection advantages and mismatch behavior, after preprocessing to support the assumed Gaussian model.

B. Performance Evaluations Using Simulated Data

C. Performance Evaluations Using Measured Data

VI. CONCLUSION

The paper develops adaptive subspace detectors for nonzero-mean clutter and identifies their performance differences from zero-mean clutter. Simulations and real-world experiments validate the analytical PD/PFA expressions and demonstrate robust detection across clutter conditions.

VI. CONCLUSION

  • The proposed SGLRT-NMC, SRao-NMC, and SAMF-NMC extend adaptive detection from rank-one signals to general subspace signals in nonzero-mean clutter.Analysis finds one fewer degree of freedom due to joint clutter-mean and covariance estimation, and an SCR loss by a factor of L/(L + 1) from estimation bias.Source: §VI · p. 12
  • Simulations and real-world experiments validate the PD/PFA analyses and show the proposed detectors maintain robust performance across clutter mean power and signal-angle relationships, unlike conventional detectors.SGLRT-NMC is recommended without signal mismatch, SAMF-NMC for robust detection under mismatch, and SRao-NMC for high selectivity against mismatched signals; no detector universally outperforms the others.Source: §VI · p. 12
  • Future work includes non-Gaussian clutter, reduced training-data needs using historical data or covariance structure, and robust designs for other mismatch types.The proposed directions include nonzero-mean clutter mismatch and time-varying clutter statistics.Source: §VI · p. 12

1) Derivations of the SGLRT-NMC:

The SGLRT-NMC is derived by maximizing the likelihood under the null and alternative hypotheses, then combining the resulting likelihoods. Because directly solving for the alternative-hypothesis signal parameter is analytically difficult, the derivation reformulates the covariance estimate to obtain it.

1) Derivations of the SGLRT-NMC:

  • Under H0, Wirtinger differentiation yields the MLE of R for a given µ; setting the derivative with respect to µ to zero then gives its MLE and the substituted likelihood expression.The derivation applies the matrix determinant lemma when substituting the mean estimate.Source: 1) Derivations of the SGLRT-NMC · p. 12
  • Under H1, the derivation first obtains the MLE of R and substitutes it into the likelihood, then obtains the MLE of µ given α.Source: 1) Derivations of the SGLRT-NMC · p. 12
  • The alternative-hypothesis derivation reformulates the covariance estimate and reorganizes S1 to obtain the MLE of α, then takes the likelihood ratio’s (L + 1)-th root to yield the final GLRT.Directly differentiating with respect to α is challenging because of multiple summation terms; the reformulated covariance can be viewed as the sample covariance matrix with the nonzero mean removed.Source: 1) Derivations of the SGLRT-NMC · pp. 12–13

2) Derivations of the SRao-NMC:

The derivation partitions the Fisher information matrix by α, µ, and R, then uses its block structure to obtain the Rao statistic. Substituting (70) yields the final statistic in (15).

2) Derivations of the SRao-NMC:

  • The FIM is partitioned into parameter blocks for α, µ, and R.The partition includes the cross-information blocks between these parameters.Source: 2) Derivations of the SRao-NMC · p. 13
  • The block-diagonal FIM shows that IR,R(Θ) has no influence on Θr, supporting the subsequent block-matrix inversion.The inversion uses the block matrix inversion lemma and the definition in (86).Source: 2) Derivations of the SRao-NMC · p. 13
  • Substituting (70) into the Rao statistic derived for given µ and R yields the final statistic in (15).The statistic is obtained after setting α = 0 and ignoring the constant.Source: 2) Derivations of the SRao-NMC · p. 13

APPENDIX B EQUIVALENT FORMS OF THE PROPOSED DETECTORS · 1) Equivalent Forms of the SGLRT-NMC: · 2) Equivalent Forms of the SRao-NMC:

Appendix B derives equivalent forms of the proposed SGLRT-NMC and SRao-NMC detectors using determinant and inversion lemmas, connecting the derivations to the detector expressions in (10) and (16).

1) Equivalent Forms of the SGLRT-NMC:

  • For the SGLRT-NMC, the matrix determinant lemma is applied to the numerator of (6) to obtain an initial equivalent form.The derivation then expresses S0 in terms related to S2 to facilitate statistical performance analysis.Source: 1) Equivalent Forms of the SGLRT-NMC · p. 14
  • Using (8), the derivation expresses S0 in terms related to S2, then substitutes that relation into (99) and applies the matrix determinant lemma again.Source: 1) Equivalent Forms of the SGLRT-NMC · p. 14
  • The resulting SGLRT-NMC expression in (101) is shown equivalent to (102), with t′ SGLRT-NMC = tSGLRT-NMC − 1.Using (11), the derivation rewrites (102) as (10).Source: 1) Equivalent Forms of the SGLRT-NMC · p. 14

2) Equivalent Forms of the SRao-NMC:

3) Equivalent Forms of the SAMF-NMC: · APPENDIX C CONDITIONS FOR S2 POSITIVE DEFINITION

The section derives equivalent SAMF-NMC expressions through matrix inversions and substitutions, then gives sample-size conditions under which the SCM estimates are positive definite with probability 1.

3) Equivalent Forms of the SAMF-NMC:

APPENDIX C CONDITIONS FOR S2 POSITIVE DEFINITION

  • Substituting (9) into (7) yields the stated intermediate expression.Source: Appendix C · p. 14
  • Substituting (9) into (8) leads to another stated intermediate expression.Source: Appendix C · p. 14
  • Substituting (111) into (112) gives the subsequent result in the positive-definition derivation.Source: Appendix C · p. 14
  • L ≥ N + 1 ensures that the SCM denoted by R-hat is positive definite with probability 1, according to Theorem 3.1.4 in.Source: Appendix C · p. 14
  • L ≥ N ensures that the other stated SCM is positive definite with probability 1.Source: Appendix C · p. 14

APPENDIX D GRADIENT TEST-BASED DETECTOR · APPENDIX E DURBIN TEST-BASED DETECTOR · APPENDIX F GENERATION METHODS FOR SIGNAL MATRIX A, TRUE SIGNAL COMPONENT p0, AND CLUTTER’S NONZERO MEAN

The gradient-test statistic is equivalent to the GLRT-based detector, and the Durbin test statistic is mathematically equivalent to the Rao test statistic. The appendices also describe procedures for generating the signal subspace, target signal, and nonzero-mean clutter.

APPENDIX D GRADIENT TEST-BASED DETECTOR

  • The gradient-test statistic is equivalent to the GLRT-based detector by monotonic-statistic equivalence.The appendix states that the two statistics are mutually monotonically increasing functions.Source: Appendix D · p. 15

APPENDIX E DURBIN TEST-BASED DETECTOR

  • The Durbin test statistic is mathematically equivalent to the Rao test statistic.With µ and R given, the two statistics share the same structure and yield identical results after substitution.Source: Appendix E · p. 15

APPENDIX F GENERATION METHODS FOR SIGNAL MATRIX A, TRUE SIGNAL COMPONENT p0, AND CLUTTER’S NONZERO MEAN

  • Signal-matrix columns follow an array-manifold structure with normalized frequencies randomly drawn from (−0.5, 0.5), ensuring a full-rank signal subspace.The matrix A is formed from the steering vectors a_m.Source: Appendix F · p. 15
  • A normalized target component is formed from Aα0, while the last left singular vector of R^-1A supplies an orthogonal component.The method uses singular value decomposition of R^-1A.Source: Appendix F · p. 15
  • Among 300 blended target candidates, the method selects the one closest to the preset cos^2 θ and scales it to satisfy the specified SCR.The blend combines the normalized target component and the orthogonal component.Source: Appendix F · p. 15
  • Among 500 clutter steering-vector candidates, the method selects the frequency closest to the preset cos^2 ϕ and scales the resulting mean to the preset amplitude power ξ.Candidate normalized frequencies are drawn from [−0.5, 0.5).Source: Appendix F · pp. 15–16
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