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Multichannel adaptive signal detection: Basic theory and literature review

Weijian Liu, Jun Liu, Chengpeng Hao, Yongchan Gao, Yong-Liang Wang

arXiv:2102.03474v1stat.AP

TL;DR

Multichannel adaptive signal detection lacks broad overview coverage despite its long development and use across multichannel sensing problems. This paper provides a tutorial and literature review of its theory, detector design, relationships to filtering, representative performance, and future directions. Its central conclusion is that adaptive detection combines filtering and CFAR functions in one detector and can provide better detection performance than filtering-then-CFAR approaches.

  • Problem

    Multichannel adaptive signal detection has relatively few overview articles, while modern sensing produces multichannel data with unknown noise covariance requiring adaptive detection.

  • Method

    The paper develops a tutorial and state-of-the-art literature review covering detector criteria, filtering relationships, adaptive filtering, typical detectors, numerical examples, and research tracks.

  • Results

    Adaptive detection achieves filtering and CFAR processing simultaneously and can provide better detection performance than filtering-then-CFAR detection.

  • Takeaways & Limitations

    Adaptive detectors jointly use test and training data, usually possess CFAR properties, and embed filtering within the detection statistic.

  • Takeaways & Limitations

    Statistical performance remains to be studied for several detector settings, including Rao and Wald tests with subspace interference and 2S-GLRT under signal mismatch.

Abstract

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Multichannel adaptive signal detection jointly uses the test and training data to form an adaptive detector, and then make a decision on whether a target exists or not. Remarkably, the resulting adaptive detectors usually possess the constant false alarm rate (CFAR) properties, and hence no additional CFAR processing is needed. Filtering is not needed as a processing procedure either, since the function of filtering is embedded in the adaptive detector. Moreover, adaptive detection usually exhibits better detection performance than the filtering-then-CFAR detection technique. Multichannel adaptive signal detection has been more than 30 years since the first multichannel adaptive detector was proposed by Kelly in 1986. However, there are fewer overview articles on this topic. In this paper we give a tutorial overview of multichannel adaptive signal detection, with emphasis on Gaussian background. We present the main deign criteria for adaptive detectors, investigate the relationship between adaptive detection and filtering-then-CFAR detection, relationship between adaptive detectors and adaptive filters, summarize typical adaptive detectors, show numerical examples, give comprehensive literature review, and discuss some possible further research tracks.

1 Introduction

Multichannel adaptive detection addresses modern vector- or matrix-valued signal detection with unknown noise covariance by jointly using test and training data. The paper reviews its theory, detector criteria, relationships to filtering and adaptive filtering, representative detectors, performance, and research directions.

  • Modern radar and other sensing systems produce multichannel, vector-valued or matrix-valued data through multiple modules and diversity technologies.
  • Adaptive detection estimates unknown noise characteristics from training data while jointly using test data to construct a detector.The detector is compared with a threshold chosen to maintain a fixed probability of false alarm.
  • Its design targets adaptation to unknown or changing noise statistics, constant false alarm rate, and relatively simple processing.
  • Kelly’s 1986 generalized likelihood ratio test initiated multichannel adaptive detection for rank-one signals with known steering vectors in homogeneous Gaussian environments.
  • The review fills a literature gap by providing a tutorial and state-of-the-art survey covering Gaussian-background theory, detector relationships, typical detectors, numerical comparisons, and future research.

2 Basic theory

The basic theory formulates adaptive detection as testing signal absence against signal presence with unknown noise covariance estimated from training data. It develops major detector criteria and relates adaptive detection to filtering, CFAR processing, and adaptive filtering.

  • 2.1 Main detector design criteria: The binary test distinguishes H0, signal absence, from H1, signal presence, using test data and L training samples to estimate unknown noise covariance R.
  • 2.1 Main detector design criteria: GLRT, Rao, and Wald tests form the three main detector design criteria, with Fisher information supporting derivation of the latter two.
  • 2.1 Main detector design criteria: Relevant parameters describe quantities such as signal amplitude, whereas nuisance parameters include the noise covariance matrix; known parameters need not be estimated.
  • 2.1 Main detector design criteria: Two-step criteria first derive a detector assuming known noise covariance and then replace it with an estimate formed from training data.
  • 2.2 Relationship between adaptive detection and filtering-then-CFAR detection: Filtering-then-CFAR uses separate filtering and CFAR stages, while adaptive detection embeds both functions in its detection statistic.
  • 2.2 Relationship between adaptive detection and filtering-then-CFAR detection: Optimal detection and optimal filtering are equivalent in the analyzed formulation, but filtering methods such as STAP still require CFAR processing for final target detection.
  • 2.3 Relationship between adaptive detectors and adaptive filters: Adaptive filters maximize output SNR, whereas adaptive detectors maximize probability of detection at a fixed false-alarm probability, despite both estimating noise covariance from training data.

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Adaptive detectors combine filtering and CFAR functions within detection statistics, using test and training data to form specific structures. Their relationship to adaptive filters depends on the detector: some are filter outputs, whereas others are not.

  • The SMI and AMF can be interpreted as outputs of adaptive filters, whereas KGLRT and DMRao cannot.
  • The SMI whitening operation uses the inverse sample covariance matrix, while projection implements signal integration.Together, these operations provide adaptive-filtering functionality.
  • AMF and SMI have the same filtering performance because they have the same output SNR.
  • AMF has the CFAR property, whereas SMI does not, despite their related construction from covariance-based quantities.
  • Adaptive detection embeds filtering and CFAR processing simultaneously in its detection statistic, avoiding independent filtering.Filtering-then-CFAR detection requires two independent procedures.

3 Literature Review

Adaptive detection is organized by target extension, signal mismatch, and the statistical properties of the background noise. These criteria define the major problem classes reviewed in the paper.

  • Adaptive detection is categorized by point versus distributed targets, signal mismatch presence, and background statistical properties.

3.1 Adaptive detection for point targets in the absence of signal mismatch

The literature review covers point-target detectors under homogeneous and partially homogeneous environments, beginning with Kelly’s Gaussian rank-one formulation. It also includes extensions to subspace signals and notes a CFAR limitation of SMI.

  • Kelly’s KGLRT detects a known-steering-vector point target in Gaussian homogeneous noise with unknown covariance estimated from IID signal-free training data.
  • The modified SMI’s detection threshold depends on the noise covariance matrix, so SMI does not have the CFAR property.
  • KGLRT, AMF, and DMRao were devised for homogeneous environments, while ACE extends point-target detection to partially homogeneous environments.
  • ACE was shown to be uniformly most powerful invariant in a partially homogeneous environment.
  • Subspace-signal detectors extend point-target methods to signals with unknown coordinates lying in a known signal subspace.

3.2 Adaptive detection for distributed targets in the absence of signal mismatch

Distributed-target detection has produced GLRT, Rao, Wald, and related detectors for homogeneous and partially homogeneous environments, including unknown steering and subspace-constrained cases. Its statistical performance is harder to derive than for point-target detectors.

  • High-resolution radar can improve distributed-target detection by reducing clutter energy per range bin and reducing target fluctuation.
  • For same-direction distributed targets, GLRT and 2S-GLRT were derived in homogeneous and partially homogeneous environments, with corresponding Rao and Wald tests also developed.
  • When the steering vector is unknown, proposed distributed-target detectors include GLRT, 2S-GLRT, M2S-GLRT, and SNT.
  • For steering vectors confined to a known subspace, GADD was proposed for the direction-detection problem.
  • DOS models represent matrix-valued distributed signals whose row and column elements lie in known subspaces with unknown coordinates.
  • Statistical performance is more difficult to derive for distributed-target detectors than for point-target detectors.

3.3 Adaptive detection in the presence of signal mismatch

Signal mismatch occurs when actual and nominal steering vectors differ, motivating detectors that balance robustness against selectivity. The review covers fixed, tunable, cascaded, and weighted designs for controlling this directivity.

  • Mismatch sources and detector goals: Signal mismatch can arise from array errors, target maneuvering, or sidelobe jamming, requiring detector designs suited to different mismatch sources.Robust detectors favor performance under mismatch, whereas selective detectors favor rejecting signals away from the nominal direction.
  • Robust and selective designs: Robust-detector design includes subspace modeling, signal-subspace enlargement, and angle or Doppler constraints formulated as semidefinite programs.These approaches address mismatch by broadening or constraining the assumed signal model.
  • Robust and selective designs: Selective detectors modify hypothesis tests with constrained fictitious signals, including signals orthogonal to the nominal signal in whitened space.Other designs add random unknown fictitious signals under both hypotheses, as in DN-AMF.
  • Directivity control: Fixed detectors cannot adjust between robustness and selectivity, limiting flexibility when detecting mismatched signals.The review identifies tunable, cascaded, weighted, and combined detectors as responses to this limitation.
  • Directivity control: Tunable detectors smoothly change directivity through one or two parameters and commonly include conventional detectors as special cases.Examples include KGLRT–AMF, KRAO, and KMABORT; cascaded and weighted detectors provide additional directivity-control mechanisms.

3.4 Adaptive detection in interference

Interference complicates adaptive detection through masking and deception, motivating detectors for coherent, noise, subspace, and incompletely known interference. The literature develops hypothesis-test-based methods while examining how interference rejection affects detection.

  • Interference types and effects: Interference may be intentional or unintentional and can accompany both noise and the signal of interest.The review distinguishes coherent interference from noise interference in subsequent detector designs.
  • Interference types and effects: Noise interference masks radar targets by raising the noise level and detection threshold, whereas coherent interference can deceive radar by imitating a target.The masking effect reduces target-detection sensitivity when CFAR is maintained.
  • Subspace interference: Subspace-interference studies derive GLRT, Rao, and Wald detectors for settings with known or estimated noise covariance matrices.In homogeneous environments, related detectors whiten noise similarly but reject subspace interference differently.
  • Subspace interference: When interference-subspace information is incomplete, studies consider unknown subspaces, uncertainty in signal and interference, and completely unknown coherent interference.The AORD was reported to outperform other detectors for completely unknown interference.
  • Additional interference problems: Adaptive interference detection also includes GLRT-based tests for identifying noise interference and detecting signals when interference occupies only part of the training data.These works extend the review beyond standard coherent-interference models.

3.5 Adaptive detection with limited training data

Adaptive detectors require sufficient training data to estimate unknown covariance matrices, but practical environments may provide too few IID samples. The review surveys prior-information, structural, and dimension-reduction strategies to mitigate this requirement.

  • Training-data requirement: The RMB rule states that adaptive filtering needs at least 2N − 3 IID training data to maintain 3 dB SNR loss relative to the optimum filter.Adaptive detection requires more than 2N − 3 IID training data to maintain 3 dB SNR loss relative to the optimum detector.
  • Training-data requirement: In airborne STAP with 30 antennas, 40 pulses, and 10 MHz bandwidth, meeting the RMB requirement can require data from roughly 36 km.The IID assumption may not hold across such a wide range.
  • Mitigation strategies: Two main approaches alleviate limited-IID-training requirements: incorporating a priori information and reducing data dimension.A priori methods include Bayesian, parametric, and special-structure-based approaches.
  • Mitigation strategies: Bayesian, parametric, and covariance-structure methods reduce estimation burden by modeling distributions, approximating spectra with low-order AR models, or exploiting matrix structure.Parametric methods use few parameters and can reduce both training-data requirements and computational complexity.
  • Mitigation strategies: Low-rank covariance modeling can make principal-component SCM approximations preferable to the SCM itself with limited training data.Other structural approaches exploit persymmetry, spectral symmetry, or combinations of special structures.
  • Dimension reduction and alternatives: A priori methods may lose performance when prior information departs substantially from reality, while dimension reduction projects data into a lower-dimensional subspace.Reduced-dimension detectors apply this transformation before adaptive processing, including JDL-GLR and subspace-based designs.
  • Dimension reduction and alternatives: Some approaches seek to detect multichannel signals in unknown noise without training data by estimating covariance from echo signals in the test data.This is presented as an additional direction beyond prior-information and dimension-reduction methods.

3.6 Adaptive detection for MIMO radar

MIMO radar expands adaptive detection across distributed, colocated, and hybrid architectures. The reviewed work studies detector derivations and performance under waveform, Doppler, covariance, clutter, and target-motion variations.

  • MIMO radar categories: MIMO radar uses multiple transmit and receive elements with linearly independent or orthogonal waveforms, forming distributed and colocated categories.Distributed antennas are widely separated, whereas colocated antennas are closely spaced.
  • MIMO radar categories: The review treats distributed, colocated, and other MIMO radar types in separate adaptive-detection discussions.This organization reflects the emerging research activity around MIMO radar target detection.
  • Distributed MIMO radar: Distributed MIMO radar can outperform traditional phased-array radar at high SNR because spatial diversity alleviates target scintillation.The cited work attributes the improvement to spatial diversity gain exceeding coherent processing gain.
  • Distributed MIMO radar: Distributed-MIMO studies derive GLRT and two-step detectors for moving targets, persymmetric covariance, arbitrary waveforms, and arbitrary noise time-correlation.One analysis reports an inherent diversity–integration trade-off and no uniformly optimum waveform design strategy.
  • Waveforms and noise: Most cited distributed-MIMO studies adopt orthogonal waveforms, while nonorthogonal waveforms can cause detection loss under white Gaussian noise.The loss result may not apply directly to colored-noise settings.
  • Colocated and other MIMO radar: Colocated-MIMO research includes GLRTs, two-step GLRTs, training-free detection, jammer suppression, robust Wald tests, and frequency-diverse-array detectors.Some results report that sufficient virtual antennas can meet required performance without prior knowledge of noise statistics.
  • Colocated and other MIMO radar: Other architectures include phased-MIMO, hybrid MIMO phased arrays, transmit-subaperturing MIMO, and multisite radar-system MIMO.Distributed-colocated and distributed-phased MIMO are distinguished by waveform organization and sub-array configuration.

4 Typical adaptive detectors for different detection problems

The paper organizes adaptive detectors by detection problem and signal-mismatch strategy, then relates their statistical properties to detection performance under representative conditions.

  • Subspace signal models: Subspace signal models generalize rank-one models, making the matched subspace detector a building block that includes the rank-one matched filter as a special case.The point-target detectors discussed in this subsection are therefore based on subspace signal models.
  • Detector families: For known subspace signals, SGLRT, SRao, and SAMF are subspace generalizations of KGLRT, DMRao, and AMF, while ASD generalizes ACE under power mismatch.In the partially homogeneous environment, the test and training covariance matrices differ by an unknown positive scaling factor.
  • Signal mismatch: Signal mismatch motivates selective or robust detectors because it can arise from antenna error, mutual coupling, target maneuvering, strong targets, or sidelobe jamming.The paper introduces factitious-jammer constructions under H0 or both hypotheses to design selective detectors, including SABORT, W-SABORT, and DN-SAMF.
  • Selective detectors: The resulting detectors include SABORT and W-SABORT, which are linked to ADD-MSR2 and ADD-MSR1, plus DN-SAMF, linked to a Rao test and a subspace DN-AMF.SABORT uses a jammer orthogonal to the signal subspace in quasi-whitened space, whereas W-SABORT uses true whitened-space orthogonality.
  • Statistical distributions: Analytical PD and PFA expressions follow from conditional detector distributions, the loss-factor distribution, and their statistical dependences; the AED distribution also permits a direct derivation.The SMF with known covariance has a complex noncentral Chi-square distribution under H1 with p degrees of freedom and noncentrality parameter ρ.
  • Numerical comparisons: In the reported example, SGLRT has the highest adaptive-detector PD, about 4 dB SNR loss at PD = 0.9 versus SMF, while AED is most robust to mismatch and SABORT is preferable when selectivity is needed.SABORT remains below PD 0.5 when cos2 φ < 0.55 regardless of SNR, whereas SAMF can reach PD 0.9 at sufficiently high SNR even at cos2 φ = 0.
  • CFAR properties: All discussed adaptive detectors are CFAR with respect to R, but only ASD has CFAR in the partially homogeneous environment.The paper also notes that DN-SAMF can perform well when the system dimension N is sufficiently large.

5 Conclusions

The paper surveys multichannel adaptive detection, emphasizing its unified filtering and CFAR role, while identifying open challenges in non-Gaussian environments and under broader noise assumptions.

  • Adaptive detection jointly uses test and training data to form detectors that combine filtering and CFAR processing in a simple procedure.The paper contrasts this with filtering-then-CFAR detection and reports better detection performance for adaptive detection.
  • The review covers detector design criteria, relationships with filtering-based approaches and adaptive filters, and comparisons of typical adaptive detectors.
  • Future work includes analyzing statistical performance under signal mismatch or interference and developing detectors for colored noise with unknown covariance matrices.The paper notes that many compressive-sensing-based detectors currently assume known noise or white Gaussian noise with unknown variance.
  • The review mainly focuses on Gaussian backgrounds, although practical environments may exhibit non-Gaussian characteristics such as compound-Gaussian clutter.The paper directs readers to existing work for compound-Gaussian clutter when relevant clutter properties are known in advance.
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