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Massive MIMO Radar for Target Detection

Stefano Fortunati, Luca Sanguinetti, Fulvio Gini, Maria S. Greco, Braham Himed

arXiv:1906.06191v4eess.SPcs.ITmath.IT

TL;DR

Target detection with unknown disturbance statistics is difficult to address without restrictive modeling assumptions. The paper derives a robust Wald-type detector for co-located MIMO radar from a single snapshot and studies its large-channel asymptotics. Its simulations report the specified false-alarm level at a finite channel count for two non-Gaussian disturbance models.

  • Problem

    Target-detection thresholds are difficult to derive without assumptions such as independent disturbance snapshots, a known disturbance distribution, constant target amplitude, and many temporal snapshots.

    Source: A. Motivation · p. 1

  • Method

    The paper develops a robust Wald-type detector for a single snapshot, with asymptotic distributions under both hypotheses that do not depend on the disturbance pdf.

    Source: B. Disturbance model · p. 4, C. The hypothesis test problem · p. 4, §III · p. 5

  • Results

    PF A = 10^-4 was achieved at N ≥ 10^4 for non-Gaussian stable autoregressive disturbance models of orders p = 3 and 6.

    Source: §V · p. 9

  • Takeaways & Limitations

    The results imply that sufficiently many virtual spatial channels can satisfy performance requirements without a priori knowledge of clutter statistics.

    Source: §V · p. 9

Abstract

from arXiv · show

Since the seminal paper by Marzetta from 2010, the Massive MIMO paradigm in communication systems has changed from being a theoretical scaled-up version of MIMO, with an infinite number of antennas, to a practical technology. Its key concepts have been adopted in the 5G new radio standard and base stations, where $64$ fully-digital transceivers have been commercially deployed. Motivated by these recent developments, this paper considers a co-located MIMO radar with $M_T$ transmitting and $M_R$ receiving antennas and explores the potential benefits of having a large number of virtual spatial antenna channels $N=M_TM_R$. Particularly, we focus on the target detection problem and develop a \textit{robust} Wald-type test that guarantees certain detection performance, regardless of the unknown statistical characterization of the clutter disturbance. Closed-form expressions for the probabilities of false alarm and detection are derived for the asymptotic regime $N\to \infty$. Numerical results are used to validate the asymptotic analysis in the finite system regime with different disturbance models. Our results imply that there always exists a sufficient number of antennas for which the performance requirements are satisfied, without any a-priori knowledge of the clutter statistics. This is referred to as the Massive MIMO regime of the radar system.

I. INTRODUCTION

The paper frames target detection as a test between target absence and presence in a resolution cell, where disturbance complicates the decision. Common solutions rely on simplifying assumptions that are often impractical, constraining how false-alarm thresholds can be set.

I. INTRODUCTION

  • The target test compares disturbance-only observations under H0 with observations containing the target signal under H1.The target contribution is modeled as an unknown deterministic amplitude multiplying a known signal that may vary over snapshots.Source: §I · p. 1
  • A radar detector compares a statistic of the collected snapshots with a threshold chosen to keep the false-alarm probability below its assigned value.The threshold is set using the statistic's probability density under H0.Source: §I · p. 1

A. Motivation

  • Deriving a detection solution is difficult because conventional GLR analysis relies on restrictive assumptions, and closed-form solutions are available only for a limited class of disturbance models.These assumptions include independent, identically distributed disturbance, a known disturbance pdf up to nuisance parameters, constant target amplitude, and K much larger than N.Source: A. Motivation · pp. 1–2
  • Secondary data can estimate disturbance nuisance parameters, but that approach assumes the disturbance statistics are shared across the tested and adjacent cells or times.The secondary dataset consists of signal-free snapshots from adjacent resolution cells sharing the tested cell's statistical characterization.Source: A. Motivation · p. 1

B. Contributions

The paper proposes robust target detection for co-located MIMO radar using a large number of virtual spatial channels, without prior knowledge of the disturbance model. It situates this approach against prior work and describes the radar signal and disturbance model used.

B. Contributions

  • The proposed detector uses a single snapshot and an asymptotic regime with N=MTMR growing unboundedly to avoid the usual assumptions and handle an unknown disturbance pdf.The paper reports PF A = 10^-4 achieved at N ≥ 10^4 for two non-Gaussian stable autoregressive disturbance models of orders 3 and 6.Source: B. Contributions · p. 2
  • The paper distinguishes its approach from its earlier work by requiring no a priori disturbance-model information, beyond the stated autocorrelation-decay condition.The earlier analysis assumed an autoregressive disturbance model of order 1.Source: B. Contributions · p. 2

C. Relevant literature

  • Prior MIMO radar work emphasized spatial resolution and related tasks, while robustness to generally unknown disturbance statistics in target detection remained underexplored.The cited prior asymptotic analyses commonly let both N and K grow, unlike this paper's single-snapshot setting.Source: C. Relevant literature · p. 2

D. Outline and Notation

  • The paper roadmap assigns the system and hypothesis-test model to Section II, the detector and asymptotic analysis to Section III, and numerical validation to Section IV.Concluding discussion follows in Section V.Source: D. Outline and Notation · pp. 2–3

A. Signal model

  • The signal model represents the received array output as a target return shaped by transmit and receive steering vectors, plus disturbance.The model leaves the disturbance covariance structure unrestricted apart from requiring its entries to decay at least polynomially with index separation.Source: A. Signal model · pp. 3–4

B. Disturbance model

The paper relaxes common white and Gaussian disturbance assumptions by restricting only how quickly autocorrelation decays, then builds a single-snapshot target test for that disturbance class.

B. Disturbance model

  • The disturbance class consists of stationary, discrete, circular complex processes whose autocorrelation decays polynomially at a rate meeting Assumption 1.Uniform and strong mixing characterize dependence between observations separated by a specified number of lags.Source: B. Disturbance model · p. 4
  • Stable second-order stationary ARMA processes, including stable AR processes, satisfy the stated autocorrelation-decay assumption, which also accommodates non-Gaussian and potentially heavy-tailed disturbances.The paper notes that ARMA models can approximate second-order statistics of processes with continuous power spectral density.Source: B. Disturbance model · p. 4
  • The results require array volume to increase with antenna count, so they do not apply to space-constrained array topologies.The paper describes the assumption as otherwise broad, covering most practical disturbance models.Source: B. Disturbance model · p. 4

C. The hypothesis test problem

  • The test considers H0: x = c versus H1: x = ᾱv + c for a single snapshot and must be applied separately to each resolution-cell steering vector.The disturbance vector entries are sampled from a process satisfying Assumption 1.Source: C. The hypothesis test problem · p. 4
  • Because the disturbance density is unknown, the proposed Wald-type test uses an estimator's asymptotic statistics and needs no explicit data-density form.It requires an asymptotically normal, N-consistent signal-parameter estimator and a consistent estimate of its error covariance.Source: C. The hypothesis test problem · pp. 4–5

III. MAIN RESULTS

The paper derives a robust Wald-type test and establishes the needed properties of a least-squares signal-amplitude estimator under dependent disturbance. Its covariance estimate is consistent when the truncation lag grows more slowly than N^1/3.

A. Estimation of ᾱ under dependent data √

  • The least-squares objective yields an estimator that coincides with mismatched maximum likelihood under a white Gaussian disturbance assumption.The paper presents least squares as the natural objective for its measurement model.Source: A. Estimation of ¯α under dependent… · p. 5
  • Under Assumption 1, the least-squares estimator of ᾱ is N-consistent and asymptotically normal.These properties provide the estimator characterization used to construct the Wald-type test.Source: A. Estimation of ¯α under dependent… · p. 5
  • A consistent estimate of the estimator's error covariance can be formed under uniform or strong mixing, although its expression requires an expectation under the unknown disturbance density.The estimate uses a truncation lag l.Source: A. Estimation of ¯α under dependent… · p. 5
  • For the covariance estimate to be consistent, the truncation lag must grow with N while remaining o(N^1/3).The theorem states this rate condition under Assumption 1.Source: A. Estimation of ¯α under dependent… · p. 6

B. A robust Wald-type test

The robust Wald statistic has asymptotic null and alternative distributions that support detection without prior knowledge of the disturbance density. Its null calibration is CFAR across the stated disturbance class, and its detection probability tends to one as virtual channel count grows.

B. A robust Wald-type test

  • Unlike the adaptive matched filter, the robust Wald statistic needs no secondary data and applies to disturbance distributions satisfying Assumption 1, with asymptotic CFAR over that class.The cited comparison says the adaptive matched filter assumes Gaussian disturbance and needs homogeneous secondary data for CFAR.Source: B. A robust Wald-type test · p. 6
  • As N grows, both test-statistic distributions converge to χ-squared distributions with 2 degrees of freedom, yielding the asymptotic threshold λ̄ = −2 ln P_FA for any preassigned P_FA and disturbance satisfying Assumption 1.This gives the statistic an asymptotic CFAR property over that disturbance class.Source: B. A robust Wald-type test · p. 6
  • In the special case of orthonormal waveforms and spatially and temporally white Gaussian disturbance, the robust test matches the asymptotic detection performance of the perfectly matched likelihood-ratio test.Unlike that likelihood-ratio test, the robust test does not require a priori knowledge of Gaussianity; in more realistic settings, the likelihood-ratio test may be infeasible or lack a closed form.Source: B. A robust Wald-type test · pp. 6–7
  • The robust Wald test's detection probability tends to 1 as N grows and depends on disturbance covariance, radar geometry, and the waveform matrix through v.The probability expression uses the true, generally unknown covariance matrix Γ.Source: B. A robust Wald-type test · p. 7

IV. NUMERICAL ANALYSIS

The numerical study sets up a co-located array experiment with orthogonal waveforms and a single far-field target. It represents the virtual channels as an equivalent phased array and uses Monte Carlo averaging to assess the asymptotic analysis.

IV. NUMERICAL ANALYSIS

  • The numerical setup uses uniform linear transmit and receive arrays, orthogonal waveforms, and one far-field target.It sets W = I_M_T and S = I_M_T.Source: §IV · p. 7
  • The receive elements are spaced by d and transmit elements by M_R d, producing a virtual steering vector equivalent to an N-element phased array.Here N = M_R M_T.Source: §IV · p. 7
  • The reported numerical results average 10^6 Monte Carlo simulations and choose truncation lag l = floor(N^1/4).The lag choice is used for the theorem's estimator procedure.Source: §IV · p. 7

A. Disturbance models

The experiments use two circular, second-order stationary autoregressive disturbance models driven by i.i.d. t-distributed innovations. Scenario 1 concentrates disturbance power near spatial frequency ν = 0, while Scenario 2 spreads it across the frequency range with multiple peaks.

A. Disturbance models

  • Scenario 1 uses an AR(3) disturbance with i.i.d. t-distributed innovations; λ = 2 sets the innovation tails and σ²_w = 1.The disturbance power is normalized to σ² = r_C[0] = 1.Source: A. Disturbance models · pp. 7–8
  • Scenario 2 uses a circular, second-order stationary AR(6) disturbance with t-distributed innovations and power spread across spatial frequency ν with multiple peaks.In contrast, Scenario 1 concentrates power mostly around ν = 0.Source: A. Disturbance models · p. 8
  • The signal-to-noise ratio under H1 is defined as SNR ≜ 10 log10(|ᾱ|²/σ²).Source: A. Disturbance models · p. 8

B. Performance Analysis

In both disturbance scenarios, the robust Wald test reaches the nominal false-alarm probability at sufficiently large virtual-channel counts, and detection probability approaches one in Scenario 1. The conclusion identifies N ≥ 10⁴ as the Massive MIMO regime for the tested non-Gaussian autoregressive disturbances.

B. Performance Analysis

  • For both scenarios, estimated P_FA reaches the nominal 10⁻⁴ for N ≥ 10⁴, with similar curve progress across scenarios and spatial frequencies.The figures plot false-alarm probability against N for Scenarios 1 and 2.Source: Figure 5 · p. 8, B. Performance Analysis · p. 8
  • For SNR ≥ −20 dB, estimated P_D approaches 1 at N ≥ 10⁴ in Scenario 1, matching the closed-form prediction.Fig. 6 compares estimated and nominal P_D at SNR values −20, −10, and −5 dB, with ν = 0 and nominal P_FA = 10⁻⁴.Source: Figure 6 · p. 8, B. Performance Analysis · p. 8

V. CONCLUDING REMARKS AND DISCUSSIONS

  • The tested non-Gaussian, stable AR(3) and AR(6) disturbances reach the preassigned P_FA = 10⁻⁴ at N = M_T M_R ≥ 10⁴, defining the radar’s Massive MIMO regime.The conclusion says sufficiently many virtual antennas can satisfy performance requirements without a priori knowledge of disturbance statistics.Source: §V · p. 9
  • The authors propose extending the Massive MIMO concept to other radar problems and describe it as a shift in MIMO radar modeling, operation, theory, and implementation.Source: §V · p. 9

APPENDIX A PROOF OF THEOREM 1

Appendix A outlines the proof of the LS estimator’s √N-consistency and asymptotic normality under Assumption 1. It establishes consistency by identifying the true amplitude as the expected objective’s minimizer and invokes a mixing-process central limit theorem for asymptotic normality.

A. Consistency

  • Under Assumption 1, the expected objective is minimized at the true amplitude because the residual has zero mean, establishing consistency of the LS estimator.Source: A. Consistency · p. 9

B. Asymptotic normality

  • The asymptotic-normality argument expands the real-valued LS objective around the true amplitude and applies the mean value theorem.The proof then uses a central limit theorem established for strong mixing processes.Source: B. Asymptotic normality · pp. 9–10
  • Under Assumption 1, the mixing-process central limit theorem and circularity of the residual process yield the stated asymptotic-normality result.Source: B. Asymptotic normality · p. 10

APPENDIX B ASYMPTOTIC DISTRIBUTION OF ΛRW

The appendix derives the asymptotic distribution of Λ_RW using the standard procedure discussed in [53, Ch. 9].

APPENDIX B ASYMPTOTIC DISTRIBUTION OF ΛRW

  • The appendix follows the standard procedure in [53, Ch. 9] to establish the asymptotic distribution of Λ_RW.Source: Appendix B · p. 10

A. Asymptotic distribution of ΛRW under H0

Under the null, the test is rewritten and its asymptotic distribution is obtained by applying continuity and Slutsky arguments. The limiting statistic is central chi-squared with two degrees of freedom.

A. Asymptotic distribution of ΛRW under H0

B. Asymptotic distribution of ΛRW under local alternatives

The paper analyzes Λ_RW under local alternatives, which are introduced for mathematical purposes to approximate detection probability near the null. It derives an asymptotic detection-probability expression and relates B_N(0) to the power calculation.

B. Asymptotic distribution of ΛRW under local alternatives

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