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Sequential Transmit Covariance Optimization for Wireless Tracking Exploiting Observation History

Kaiyue Hou, Shuowen Zhang

arXiv:2609.10124v1eess.SP

TL;DR

Nonlinearity in the observation model generally prevents closed-form posterior and predictive PDFs, motivating a Gaussian assumed-density approximation for tractable evaluation. The paper approximates the history-conditioned predictive PDF with a Gaussian assumed density, derives conditional PFIM and PCRB, and optimizes transmit covariance at each slot. The proposed design exploits historical observations, achieves lower conditional PCRB than benchmark schemes, and improves tracking accuracy over time.

  • Problem

    Nonlinearity in the observation model generally prevents closed-form posterior and predictive PDFs, motivating a Gaussian assumed-density approximation for tractable evaluation.

    Source: §III · p. 3

  • Method

    The paper approximates the history-conditioned predictive PDF with a Gaussian assumed density, derives conditional PFIM and PCRB, and optimizes transmit covariance at each slot.

    Source: §I · p. 1, B. Gaussian Approximation and Prior… · p. 4, §V · p. 5

  • Results

    The proposed design exploits historical observations, achieves lower conditional PCRB than benchmark schemes, and improves tracking accuracy over time.

    Source: §I · p. 2, §VII · p. 6

  • Takeaways & Limitations

    The conditional PCRB's explicit dependence on transmit covariance provides an objective for sequential transmit design before each new echo observation.

    Source: §I · pp. 1–2

Abstract

from arXiv · show

This paper studies a multiple-input multiple-output (MIMO) radar tracking system, where a multi-antenna base station (BS) aims to track the location of a moving target over multiple time slots based on the observed echo signals, initial prior probability density function (PDF) of the target state, and state evolution model. By exploiting the realized observation history in the past time slots, the BS sequentially updates the predictive state information and designs the transmit covariance matrix before collecting the current echo observation. Considering a Gaussian random-walk model for the target location and a Gauss-Markov model for the complex radar cross-section (RCS) coefficient, we propose an effective method to characterize the predictive PDF conditioned on the realized observation history via Gaussian approximation. Based on this, we derive the conditional posterior Fisher information matrix (PFIM) for the target state, and further characterize the conditional posterior Cram\'er-Rao bound (PCRB) for the mean-squared error (MSE) in estimating the target's location state as an explicit expression of the transmit covariance matrix. Next, we formulate the sequential transmit covariance matrix optimization problem to minimize the conditional PCRB for each time slot. Despite the non-convexity of the problem, we obtain the optimal solution via the Schur complement technique. Numerical results show that the proposed design effectively exploits the realized observation history, achieves a lower conditional PCRB than the benchmark schemes, and improves tracking accuracy over time.

I. INTRODUCTION

The paper addresses the challenge of characterizing tracking performance and designing transmit signals when decisions use both observation history and evolving target state. It develops sequential covariance design for a MIMO radar tracker under Gaussian random-walk location and Gauss-Markov RCS models.

I. INTRODUCTION

  • Transmit design matters because it determines echo quality, while tracking performance also depends jointly on state evolution and observation uncertainty.Over multiple time slots, the transmitter can adapt its probing strategy to observation history.Source: §I · p. 1
  • Prior numerical approaches to history-conditioned PCRB characterization can be computationally extensive with long observation histories and do not expose its explicit relationship to transmit signals.The paper identifies analytical characterization and sequential optimization based on observation history as open problems.Source: Figure 1 · p. 1
  • The proposed design derives a history-conditioned location PCRB as an explicit function of transmit covariance, then minimizes it at each time slot despite the original problem's non-convexity.The paper reports that the design exploits historical observations and improves tracking over time and against benchmark schemes.Source: §I · pp. 1–2

A. Target Location and RCS Coefficient Evolution Models

  • The tracker models the target's unknown horizontal location in each slot, with known fixed height, and assumes Gaussian random-walk motion with process noise capturing random horizontal displacement.The initial horizontal location has a Gaussian prior.Source: §II · p. 2, A. Target Location and RCS Coefficient… · p. 2
  • The complex RCS coefficient follows a first-order Gauss-Markov process, with temporal correlation determined by ρ and independent innovation noise.Its initial coefficient is zero-mean complex Gaussian, and the marginal distribution is complex Gaussian.Source: A. Target Location and RCS Coefficient… · p. 2

B. Signal Model for Tracking

The signal model describes a MIMO radar that tracks a target through reflected echoes received by the base station. It represents the slot-wise channel using target geometry, RCS, and transmit and receive array responses.

C. Signal Model for Tracking

  • The system estimates and tracks the target's time-varying horizontal location from target-reflected echo signals received at the base station.The target location, RCS, and channel remain constant within each time slot.Source: B. Signal Model for Tracking · p. 2
  • The slot-wise round-trip channel depends on target distance, azimuth and elevation, RCS, and the transmit and receive array responses.The transmit and receive arrays are uniform planar arrays placed on the x-z plane.Source: B. Signal Model for Tracking · p. 2
  • Each received sample is modeled as the channel acting on the transmitted signal plus circularly symmetric complex Gaussian receiver noise.The transmit sample covariance is defined for the signals sent in each time slot, under a transmit power constraint.Source: B. Signal Model for Tracking · p. 2

C. Overall Tracking Procedure

At each slot, the base station designs signals using the previous observation history, sends them, and collects new echoes. It updates the history and uses it for the next slot's design.

C. Overall Tracking Procedure

  • At the start of each slot, the base station designs transmit signals from the previous observation history, initial priors, and state evolution models before collecting current echoes.The designed signals are transmitted over the slot's sample intervals.Source: C. Overall Tracking Procedure · p. 3
  • After collecting current echoes, the base station estimates the target location from the updated history, which informs transmit design in the next slot.The procedure repeats across time slots.Source: C. Overall Tracking Procedure · p. 3

III. TRACKING PERFORMANCE CHARACTERIZATION VIA CONDITIONAL PCRB

The paper characterizes location-estimation error with a conditional PCRB based on the previous observation history. Because the nonlinear observation model generally prevents a closed-form predictive PDF, it uses a Gaussian assumed-density approximation for tractable evaluation.

III. TRACKING PERFORMANCE CHARACTERIZATION VIA CONDITIONAL PCRB

  • The tracking state combines the two-dimensional target location with the real and imaginary parts of the complex RCS coefficient, and evolves through a linear state transition with process noise.The transition matrix is block diagonal, with identity for location and ρ-scaled identity for the RCS components.Source: §III · p. 3
  • The conditional PFIM for the joint target state is decomposed into information from the current echo observation and prior information in the predictive PDF.The target location and complex RCS coefficient are both unknown state parameters.Source: §III · p. 3
  • The nonlinear observation model generally leaves the posterior and predictive PDFs unavailable in closed form, motivating Gaussian assumed-density approximation for tractable PFIM evaluation.The predictive distribution is formed by propagating the posterior through the state transition model.Source: §III · p. 3

A. Conditional Data PFIM

The section derives the conditional data PFIM from the current echo observation given the realized observation history. It expresses the information blocks in terms of the transmit covariance, while noting that the exact predictive PDF is generally intractable to characterize.

A. Conditional Data PFIM

  • The conditional data FIM is derived from the current echo likelihood given past observations, with the transmitted signals known at the receiver.The transmit signals are determined before the current observation is collected.Source: A. Conditional Data PFIM · p. 3
  • The instantaneous data FIM is partitioned into target-location, location–RCS cross-information, and RCS blocks.The location and cross-information blocks are expressed in terms of transmit-covariance-dependent coefficients.Source: A. Conditional Data PFIM · pp. 3–4
  • Averaging the instantaneous data FIM over the predictive PDF gives the conditional data PFIM, which is affine in W_s.The exact predictive PDF generally has no closed-form expression, making this characterization difficult.Source: A. Conditional Data PFIM · p. 4

B. Gaussian Approximation and Prior Information Matrix

The method approximates the posterior and predictive state densities with Gaussian information states that summarize observation history through means and covariances. Propagation under the linear Gaussian transition model yields the assumed predictive PDF used to form prior information.

B. Gaussian Approximation and Prior Information Matrix

  • A Gaussian information state summarizes the posterior observation history through its mean and covariance, providing a tractable approximation to the posterior PDF.The representation admits closed-form propagation under the linear Gaussian state-evolution model.Source: B. Gaussian Approximation and Prior… · p. 4
  • Propagating the posterior Gaussian information state under the linear Gaussian transition model yields the assumed predictive PDF for the next slot.Its mean and covariance recursively incorporate the tracking observation history.Source: B. Gaussian Approximation and Prior… · p. 4
  • The assumed predictive PDF approximates the intractable history-conditioned predictive PDF and is used to construct prior information for transmit design.Because the assumed predictive PDF is Gaussian, the negative Hessian of its log density is constant.Source: B. Gaussian Approximation and Prior… · p. 4
  • The information update can introduce dependence between target location and RCS, which is retained during state prediction.The initial location and RCS are independent.Source: B. Gaussian Approximation and Prior… · p. 4

C. Assumed Density Conditional PFIM and Posterior Update

The assumed conditional PFIM combines prior and data information, with data-information blocks characterized through coefficient matrices. After the current observation, the method updates the posterior Gaussian information state and propagates it forward.

C. Assumed Density Conditional PFIM and Posterior Update

IV. PROBLEM FORMULATION

At each time slot, the transmit covariance is designed from the assumed predictive PDF to minimize the conditional PCRB under a transmit-power constraint. The resulting problem is non-convex but can be transformed into a convex SDP using the Schur complement.

IV. PROBLEM FORMULATION

  • The per-slot design chooses W_s from the assumed predictive PDF to minimize the conditional PCRB, subject to tr(W_s) ≤ P.Source: §IV · p. 5
  • The problem is non-convex because of the matrix inverse in its objective and the structure of the conditional information matrix.Source: §IV · p. 5
  • The Schur complement equivalently transforms the problem into a convex semidefinite program.Source: §IV · p. 5

V. OPTIMAL SOLUTION TO (P1-s)

The transmit-covariance problem is reformulated as a convex SDP using auxiliary matrices and Schur-complement constraints. The SDP has the same optimal value as the original problem, so its optimal covariance is also optimal for the original problem.

V. OPTIMAL SOLUTION TO (P1-s)

  • An auxiliary matrix Ψs is constrained below the conditional information matrix and used to express the matrix inequality through a Schur complement.The passage states that the relevant conditional information block is positive definite.Source: §V · p. 5
  • The trace-inverse objective is converted to a convex form by introducing a positive semidefinite auxiliary matrix Zs that upper-bounds Ψs^-1.The inverse inequality is represented with a Schur-complement constraint.Source: §V · p. 5
  • The constraints become affine in the decision and auxiliary matrices, making the reformulated problem a convex SDP.The two formulations have the same optimal value, and an optimal covariance from the SDP is also optimal for the original problem.Source: §V · p. 5

VI. NUMERICAL RESULTS

The proposed design's conditional PCRB decreases over time and remains below the benchmark schemes as tracking observations accumulate. The posterior becomes more concentrated around the true target location, while the MAP estimate follows its trajectory.

VI. NUMERICAL RESULTS

  • The evaluation uses 10 time slots, 20 transmit antennas, 24 receive antennas, and a 20 dBm transmit-power setting.The target location follows a Gaussian random walk with Qs = 5×10^-3I2 m2 for s ≥ 1; the RCS coefficient follows a Gauss-Markov model with ρ = 0.99.Source: §VI · pp. 5–6
  • As time slots increase, the proposed design's conditional PCRB decreases and remains below the benchmark schemes; at s = 0, it matches Benchmark II.Benchmark I uses isotropic covariance, while Benchmark II optimizes covariance without exploiting observation history.Source: §VI · p. 6
  • As tracking observations accumulate, the posterior location distribution concentrates around the true target, and the MAP estimate gradually follows the target trajectory.The location marginal is obtained by integrating out the RCS coefficient.Source: §VI · p. 6

VII. CONCLUSIONS

The paper derives a Gaussian-assumed density conditional PCRB for target horizontal-location MSE and optimizes transmit covariance at each time slot. Numerical results report reduced posterior uncertainty and improved tracking accuracy over time.

VII. CONCLUSIONS

  • The paper derives a Gaussian-assumed density conditional PCRB for estimating the target's horizontal location as an explicit function.Source: §VII · p. 6
  • At each time slot, transmit covariance is optimized to minimize the conditional PCRB through an equivalent convex SDP using the Schur complement.Source: §VII · p. 6
  • Numerical results show that the design exploits historical observations, reduces posterior uncertainty, and improves tracking accuracy over time.Source: §VII · p. 6

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