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Subspace-based Second-Order Detection Architectures for Radar Signals in the Presence of Unstructured Interference

Nicola Acito, Danilo Orlando, Giuseppe Ricci

arXiv:2610.07185v1eess.SP

TL;DR

Second-order subspace detection has not incorporated intentional interference alongside the signal of interest, leaving noise-like jamming outside that formulation. This paper uses passive-mode training and an LRT-based estimation scheme for range-spread targets, and reports an advantage over a first-order competitor, including in sample-starved settings.

  • Problem

    Existing second-order subspace contributions do not incorporate intentional interference into the signal model, despite the need to address noise-like jammers.

    Source: §I · p. 2

  • Method

    The detector applies the likelihood ratio test with sequential maximum-likelihood estimates, using passive-mode training to estimate jammer parameters without target or clutter echoes.

    Source: §I · p. 2, §VI · p. 11

  • Results

    The SOJ-LRT outperforms the first-order competitor for the considered parameter values and can operate with low volumes of secondary data.

    Source: B. Detection Performance · p. 11, §VI · p. 12

  • Takeaways & Limitations

    Adaptive threshold selection approximately maintains nominal Pfa across the considered CNR and JNR ranges, with smaller deviations than a fixed threshold.

    Source: A. False-Alarm Analysis · p. 10

    Future work is proposed for heterogeneous clutter-power levels between primary and secondary data and for combining first- and second-order jamming components.

    Source: §VI · p. 12

Abstract

from arXiv · show

This paper proposes a subspace-based decision scheme to detect range-spread targets in presence of thermal noise, clutter, and intentional interference. The target and intentional interference (i.e., noise-like jammer) effects are confined to the covariance matrix of the received primary data vectors. The jammer is also present in the secondary data vectors necessary to estimate the clutter covariance matrix. Finally, an additional set of training data, acquired by the system in passive mode and, hence, not contaminated by target and clutter echoes, is used to avoid the jammer parameters are unidentifiable. The detection problem is addressed by deriving an architecture based upon the likelihood ratio test where the unknown second-order parameters are suitably estimated. To this end, a unified theoretical framework is defined also providing a geometrical interpretation of the estimation procedures. The performance assessment, conducted also in comparison to a natural competitor, highlights the effectiveness of the proposed solution.

I. INTRODUCTION

The introduction identifies a gap in second-order subspace detection: existing formulations do not include intentional interference alongside the signal of interest. It proposes an LRT-based detector for range-spread targets, using passive-mode training to address parameter identifiability.

I. INTRODUCTION

  • Subspace modeling accommodates target-signature uncertainty, including misalignment between nominal and actual signatures.The introduction attributes such uncertainty to hardware limitations, miscalibration, and unbalanced channels.Source: §I · p. 1
  • Earlier range-spread structured-interference models assumed interference did not affect secondary data, unlike the continuously transmitting jammer considered here.The earlier models are described in,.Source: §I · p. 2
  • Earlier second-order subspace formulations omit intentional interference, leaving the joint target-and-noise-like-jammer model unaddressed.The paper treats the target as representative of a fluctuating signal and the jammer as noise-like interference.Source: §I · p. 2
  • Passive-mode training, free of target and clutter echoes, helps resolve jammer-parameter identifiability while the jammer also affects secondary data.The passive samples represent thermal noise and a possible noise-like jammer.Source: §I · p. 2
  • The proposed LRT detector uses sequential maximum-likelihood estimation of unknown second-order parameters for range-spread target detection.The proposed detector is activated when a preliminary stage detects a noise-like jammer.Source: §I · p. 2

A. Notation

The notation section establishes conventions for complex matrices and vectors, common matrix operations, and the Gaussian random-matrix model used in the paper.

A. Notation

  • Vectors and matrices use boldface lowercase and uppercase letters, respectively; det, Tr, transpose, and conjugate transpose denote standard matrix operations.The section also defines the Frobenius norm and diagonal-matrix notation.Source: A. Notation · p. 3
  • CN_{N,K}(m, Σ, I) denotes K IID N-dimensional circularly invariant Gaussian vectors with mean m and covariance Σ.The covariance is specified as positive definite.Source: A. Notation · p. 3
  • For Hermitian matrices, ≻0 and ⪰0 denote positive definiteness and positive semidefiniteness, while I and 0 denote identity and null matrices or vectors.Their dimensions depend on context.Source: A. Notation · p. 3

II. PROBLEM FORMULATION

The problem formulation models primary, secondary, and passive-mode data in a multichannel radar setting with clutter, thermal noise, and a noise-like jammer. It frames detection as deciding whether primary data also contain target components.

II. PROBLEM FORMULATION

  • The jammer continuously illuminates the radar, so its contribution affects all collected data, including the secondary data used to estimate interference parameters.The primary data matrix has statistically independent columns, and the model assumes independence across the combined data sets.Source: §II · p. 3
  • The binary test distinguishes primary data containing clutter, jammer, and thermal noise from data that also contain target components.The target and jammer steering vectors are assumed to lie in known subspaces spanned by H and J.Source: §II · p. 3
  • An additional passive-mode data set represents thermal noise and jammer without target or clutter echoes and can estimate jammer covariance and thermal-noise power under either hypothesis.The passive set is collected by opening the receiver before transmission and beyond instrumental range.Source: §II · p. 3
  • The section specifies probability-density expressions for primary data under both hypotheses and for the secondary and passive-mode data used in later derivations.The primary-data density includes a covariance contribution from target components under the alternative hypothesis.Source: §II · pp. 3–4

III. DETECTOR DESIGNS

The detector design uses maximum likelihood estimation to derive a decision scheme, addressing mathematically involved joint likelihood maximization through tractable subproblems.

III. DETECTOR DESIGNS

  • The decision scheme is designed using a maximum-likelihood approach to the detection problem.The section first introduces technical results needed for the derivation.Source: §III · p. 4
  • When joint maximization over unknown parameters is mathematically involved, the design uses subproblems to obtain closed-form estimates that are suboptimal in the maximum-likelihood sense.These subproblems make the parameter-estimation step tractable.Source: §III · p. 4
  • Technical results are introduced as preliminary steps toward deriving the decision scheme.Source: §III · p. 4

A. Preliminary results

The preliminary results give eigenvalue-based solutions for positive-semidefinite matrix estimates and a scalar noise-variance estimate. They also interpret one matrix solution as a projection onto the positive-semidefinite cone.

A. Preliminary results

  • The maximizer clips eigenvalues of R below one to zero, yielding a positive-semidefinite estimate that shares R’s eigenvectors and is its Frobenius-norm projection onto the positive-semidefinite cone.The estimate is obtained by setting negative entries of Λ − I to zero.Source: A. Preliminary results · pp. 4–5
  • The matrix optimization’s geometrical interpretation is projection of an unconstrained solution onto the cone of positive-semidefinite Hermitian matrices.The Frobenius-norm projection preserves the unconstrained solution’s eigenvectors and clips negative eigenvalues to zero.Source: A. Preliminary results · pp. 4–5
  • For objectives generalized by a positive-definite matrix A or a constrained matrix F, the maximizers likewise use eigenvectors and eigenvalues of corresponding whitened matrices.The stated solution for B uses A^−1/2RA^−1/2; the related result uses (C + γI)^−1/2R(C + γI)^−1/2.Source: A. Preliminary results · pp. 4–5
  • The scalar noise-variance equation has a unique positive solution, found by locating the interval containing its candidate solution.Its left-hand side is piecewise linear and increasing, while h(σ_n^2) is concave.Source: A. Preliminary results · p. 6

B. LRT-based Detector

The detector starts from a likelihood ratio test and substitutes estimates of unknown parameters derived from the available data. Estimation proceeds from passive training data to secondary data and then primary data.

B. LRT-based Detector

  • The decision statistic is an LRT with estimated parameters, and its threshold η controls the probability of false alarm.The estimates include noise variance and jammer covariance from X, clutter covariance from ZS, and target covariance from ZP.Source: B. LRT-based Detector · p. 6
  • Under H1, the procedure first estimates noise variance and Q from X, then estimates clutter covariance from ZS using those estimates.The final step estimates P from ZP with the preceding parameter estimates held fixed.Source: B. LRT-based Detector · pp. 6–7
  • The likelihood for X is recast using a Cholesky factorization and a change of variables to support estimation of Q and noise variance.The derivation defines Γ = J†J and S_X = XX†/K_J.Source: B. LRT-based Detector · p. 7

I + QLL†

This section derives the parameter-estimation steps used in the detector under the alternative and null hypotheses. It uses the additional data set for noise and jammer parameters, secondary data for clutter, and primary data for target covariance under H1.

I + QLL†

  • The additional data X yields noise-variance and Q estimates by maximizing its likelihood, using a transformed variable Y and the eigenvalues of a matrix derived from X.The nonlinear equation for the noise estimate has a unique solution; the resulting estimate is then used to obtain Q.Source: I + QLL† · p. 7
  • The clutter covariance estimate is obtained by maximizing the ZS likelihood after substituting the noise and jammer estimates.The maximization uses eigenvectors of the whitened secondary-data covariance SS,T.Source: I + QLL† · pp. 7–8
  • Under H1, target covariance P is estimated from ZP after fixing the noise, jammer, and clutter estimates obtained in earlier steps.The primary-data likelihood is recast using G = σ̂_n^2I + M̂_c + JQ̂J† and an eigenvalue-based maximization.Source: I + QLL† · p. 8
  • Under H0, the procedure repeats the X-based estimates of noise variance and Q and the ZS-based clutter estimate before evaluating the primary-data likelihood.The paper refers to the resulting detector as the second-order jamming-affected LRT (SOJ-LRT).Source: I + QLL† · p. 8

IV. AN IDENTIFIABILITY ISSUE

The identifiability analysis shows that the secondary-data likelihood cannot uniquely determine the jammer and background covariance parameters. A coordinate transformation into the jammer-signature subspace exposes the ambiguity.

IV. AN IDENTIFIABILITY ISSUE

  • A QR decomposition writes J = U_jR, with U_j spanning the jammer-signature subspace and R upper triangular and invertible.A unitary change of coordinates maps U_j to E = [I 0]^T before partitioning the transformed matrices.Source: §IV · p. 8
  • After the coordinate change, the likelihood depends on the partitioned covariance blocks and transformed jammer covariance, making their interchangeable choices explicit.The text identifies the non-unique choices as M_V,11 and Q_R.Source: §IV · p. 8
  • Infinitely many choices of the partitioned background covariance block and transformed jammer covariance yield the same likelihood value.Thus, estimating Q and M from ZS alone has an identifiability issue.Source: §IV · p. 8

V. PERFORMANCE ASSESSMENT

The assessment uses Monte Carlo experiments to evaluate SOJ-LRT false-alarm and detection performance against a benchmark under varying training-data volumes and interference conditions.

V. PERFORMANCE ASSESSMENT

  • Performance is evaluated by Monte Carlo counting, with detection probability measured at nominal Pfa = 10^-4 across specified primary, secondary, and additional training-data sizes.The simulations use N = 16 channels, r = 3, q = 4, and KP = 16; KS and KJ each range over {12, 16, 24}.Source: §V · pp. 8–9
  • The simulated target and jammer occupy subspaces formed from the most significant eigenvectors of prolate matrices, while clutter power is set to CNR = 30 dB.The jammer components in secondary and additional data are generated using the same procedure as in primary data.Source: §V · p. 9
  • The benchmark statistic can be implemented only when the secondary-data covariance SS is invertible with probability 1, namely when KS ≥ N.Thus, the competitor cannot be evaluated in the sample-starved setting KS = 12 < N.Source: §V · p. 9

A. False-Alarm Analysis

With a fixed threshold, SOJ-LRT false-alarm probability varies substantially with jammer power; an adaptive threshold strategy approximately maintains the nominal rate across tested conditions.

A. False-Alarm Analysis

  • With a fixed threshold, Pfa depends pronouncedly on JNR, with deviations exceeding one order of magnitude at high jammer powers.CNR dependence is generally weak, becoming noticeable mainly when KS = 12 < N and CNR is low.Source: A. False-Alarm Analysis · p. 9
  • The adaptive strategy builds a threshold lookup table for each data-size triplet at reference CNR = 40 dB, with entries calibrated to nominal Pfa = 10^-4 over JNR values from 20 to 59 dB.At detection, the threshold is linearly interpolated at the estimated JNR; the reported assessment also includes JNR values off the LUT grid.Source: A. False-Alarm Analysis · pp. 9–10
  • Adaptive threshold selection approximately maintains Pfa at its nominal value across the considered CNR and JNR ranges and all tested data-size triplets.Deviations are larger for KS = 12 < N, consistent with less accurate clutter covariance estimation and CNR dependence not accounted for by the one-dimensional LUT, but remain substantially smaller than with a fixed threshold.Source: A. False-Alarm Analysis · p. 10

B. Detection Performance

In the tested scenarios, SOJ-LRT outperforms the benchmark detector, while both detectors benefit from more secondary data. SOJ-LRT performance declines as JNR rises but improves with additional training data.

B. Detection Performance

  • SOJ-LRT significantly outperforms the competitor across the considered secondary-data settings and retains a detection advantage even at KS = 24.The competitor's statistic cannot be evaluated for the sample-starved case KS = 12 < N, whereas SOJ-LRT can operate with low secondary-data volumes.Source: Figure 3 · p. 11, B. Detection Performance · p. 11
  • Both detectors improve as KS increases, with a more pronounced improvement for the competitor and a narrowing performance gap.The paper links this trend to more accurate estimation of the clutter covariance matrix as secondary-data volume increases.Source: B. Detection Performance · p. 11
  • SOJ-LRT generally outperforms the competitor, but its performance decreases as JNR increases while the competitor appears insensitive to JNR.The paper attributes the competitor's JNR robustness to projecting primary data onto the orthogonal complement of the jammer subspace; increasing KJ improves SOJ-LRT, especially at JNR = 40 and 50 dB.Source: B. Detection Performance · p. 11

VI. CONCLUSIONS

The proposed detector uses second-order models for target and jammer effects, with passive-mode training data to resolve an identifiability issue. Its likelihood-ratio architecture uses sequential maximum-likelihood estimates and an adaptive threshold; the assessment reports performance above a first-order competitor, including in sample-starved scenarios.

VI. CONCLUSIONS

  • Passive-mode training data, free of target and clutter echoes, estimate jammer parameters and resolve the identifiability issue from jointly estimating jammer and clutter covariance components using secondary data alone.The jammer is present in the secondary data alongside clutter.Source: §VI · p. 11
  • A likelihood-ratio test with sequential maximum-likelihood estimation provides closed-form second-order parameter estimates and a geometrical interpretation of the estimation procedures.Source: §VI · p. 11
  • The proposed solution outperforms its first-order competitor and also works under sample-starved scenarios.Figure 4 compares detector performance across JNR values of 30, 40, and 50 dB, with KJ varied across panels.Source: §VI · p. 12, Figure 4 · p. 12
  • An adaptive threshold interpolated from a lookup table using estimated JNR is introduced to keep Pfa approximately constant as JNR varies.Source: §VI · pp. 11–12
  • Future work includes extending the approach to heterogeneous environments with different clutter power levels in primary and secondary data, and combining first- and second-order jamming components.Source: §VI · p. 12
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