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Random ISAC Signals Deserve Dedicated Precoding

Shihang Lu, Fan Liu, Fuwang Dong, Yifeng Xiong, Jie Xu, Ya-Feng Liu, Shi Jin

arXiv:2311.01822v2cs.ITeess.SPmath.IT

TL;DR

Random ISAC signals can make conventional covariance-based sensing assumptions unreliable, particularly when only short frames are available. The paper defines ELMMSE and develops DDP and DIP precoding for sensing and ISAC. Its reported results show favorable sensing performance while meeting communication requirements, and it concludes that random signals warrant dedicated precoding.

  • Problem

    Random ISAC signals can vary sensing performance, while short frames may prevent the sample covariance from approximating its statistical counterpart reliably.

    Source: §I · p. 2, B. Sensing with Random Signals · p. 4

  • Method

    The paper defines ELMMSE and proposes DDP and DIP for random signals, extending the designs to ISAC with a penalty-based alternating-optimization method.

    Source: §I · p. 2, A. DDP for ISAC: A Penalty-Based AO… · p. 9

  • Results

    Both DDP and DIP achieve favorable sensing performance under Gaussian codebooks while satisfying the communication requirement, compared with conventional deterministic ISAC precoding.

    Source: §I · p. 2

  • Takeaways & Limitations

    The paper concludes that random ISAC signals deserve dedicated precoding designs, especially when the base station cannot accumulate sufficient raw sensory data.

    Source: §I · p. 2

    The work leaves open the setting where the ISAC base station has no prior knowledge of the sensing channel matrix.

    Source: §VI · p. 14

Abstract

from arXiv · show

Radar systems typically employ well-designed deterministic signals for target sensing, while integrated sensing and communications (ISAC) systems have to adopt random signals to convey useful information. This paper analyzes the sensing and ISAC performance relying on random signaling in a multi-antenna system. Towards this end, we define a new sensing performance metric, namely, ergodic linear minimum mean square error (ELMMSE), which characterizes the estimation error averaged over random ISAC signals. Then, we investigate a data-dependent precoding (DDP) scheme to minimize the ELMMSE in sensing-only scenarios, which attains the optimized performance at the cost of high implementation overhead. To reduce the cost, we present an alternative data-independent precoding (DIP) scheme by stochastic gradient projection (SGP). Moreover, we shed light on the optimal structures of both sensing-only DDP and DIP precoders. As a further step, we extend the proposed DDP and DIP approaches to ISAC scenarios, which are solved via a tailored penalty-based alternating optimization algorithm. Our numerical results demonstrate that the proposed DDP and DIP methods achieve substantial performance gains over conventional ISAC signaling schemes that treat the signal sample covariance matrix as deterministic, which proves that random ISAC signals deserve dedicated precoding designs.

I. INTRODUCTION

Random ISAC signals can undermine sensing, especially when short frames prevent their randomness from averaging out. The paper motivates dedicated precoding and frames the contrast with deterministic training-based sensing.

I. INTRODUCTION

  • Random communication signals support useful information-bearing transmission but can degrade sensing; deterministic sample-covariance assumptions may fail for short frames and massive arrays.The paper motivates dedicated precoding because covariance convergence requires frames much longer than the array size, while sensing receivers may lack caching capacity for massive raw-data accumulation.Source: §I · pp. 1–2
  • ELMMSE measures estimation error averaged over random signal realizations, and Jensen’s inequality gives a lower bound on the loss from random signaling.The paper also notes that the conclusion extends to convex sensing cost functions.Source: §I · p. 2
  • DDP and DIP minimize ELMMSE for Gaussian signals, respectively adapting to each data realization and keeping one precoder fixed across samples.The sensing-only DDP solution is closed form despite non-convexity, while DIP can use stochastic-gradient methods.Source: §I · p. 2
  • Both proposed schemes extend to ISAC under a communication-rate constraint, and numerical results report favorable sensing performance while satisfying that requirement.The ISAC problems use a penalty-based alternating-optimization approach.Source: §I · p. 2

II. SYSTEM MODEL

  • The considered ISAC base station estimates the target impulse response matrix from sensing echoes while serving a communication user.The model includes transmit and receive antennas, communication and sensing channels, and additive noise.Source: §II · pp. 2–3

A. Sensing with Deterministic Training Signals

  • Orthogonal deterministic training yields an LMMSE-optimal water-filling precoder, but the stated training assumption is unsuitable when L < NT.The limitation follows from the rank-deficient signal in that regime.Source: A. Sensing with Deterministic Training… · p. 3

B. Sensing with Random Signals

For random Gaussian signals, covariance-based approximations can require long frames, motivating an ergodic sensing metric and precoders that account for signal randomness. The section distinguishes adaptive DDP from fixed, offline-optimized DIP and formulates the rate-constrained ISAC setting.

B. Sensing with Random Signals

  • For NT = 32, approximately 4096 samples are needed to keep approximation error below −20 dB, so short-frame systems may not reliably use the sample-covariance approximation.The approximation assumes L ≫ NT and the required accumulation can be impractical when the base station cannot collect long frames.Source: B. Sensing with Random Signals · pp. 3–4
  • ELMMSE is the sensing estimation error averaged over random signal realizations, addressing performance variation that an instantaneous LMMSE does not capture.The paper formulates the resulting dedicated precoder design as a stochastic optimization problem.Source: B. Sensing with Random Signals · p. 4
  • DDP adapts the precoder to each known signal realization and can minimize ELMMSE, whereas DIP fixes the precoder and can be optimized offline using Gaussian training samples.DDP has high real-time implementation complexity; DIP offers a performance-complexity tradeoff.Source: B. Sensing with Random Signals · p. 4

C. ISAC Precoding with Gaussian Signals

  • The ISAC formulation uses capacity-achieving Gaussian signals and imposes a required communication-rate threshold on dedicated precoding.The paper assumes perfect communication-channel state information and identifies imperfect CSI as future work.Source: C. ISAC Precoding with Gaussian Signals · p. 4

III. SENSING-ONLY PRECODING WITH GAUSSIAN SIGNALS

Gaussian signaling raises the sensing error relative to deterministic orthogonal training, so the section develops ELMMSE-oriented precoding for sensing-only systems. Its data-dependent design attains the minimum ELMMSE through a closed-form solution, while requiring per-realization computation.

III. SENSING-ONLY PRECODING WITH GAUSSIAN SIGNALS

  • The deterministic LMMSE is a lower bound on the ELMMSE for Gaussian signals, and the Jensen bound becomes tight as frame length L grows.For small L, treating the sample covariance as deterministic may be unreliable, motivating ELMMSE rather than lower-bound-only optimization.Source: §III · pp. 4–5
  • The sensing-only section motivates a data-independent alternative and an SGP approach to reduce DDP’s computational burden.The DIP precoder remains fixed across data realizations and can be trained offline with locally generated signal samples.Source: §III · p. 5, A. DDP: A Closed-Form Solution · p. 6

A. DDP: A Closed-Form Solution

  • The DDP design decomposes into parallel deterministic subproblems because the transmitter knows each signal realization in the monostatic setup.Each realization receives its own precoding matrix.Source: A. DDP: A Closed-Form Solution · p. 5
  • DDP minimizes ELMMSE by adapting the precoder to each signal realization, attaining the minimum at the cost of substantial per-realization computational complexity.The non-convex per-realization problem has a closed-form modified water-filling solution.Source: A. DDP: A Closed-Form Solution · pp. 5–6

B. DIP: An SGP-Based Precoding Method

The DIP design minimizes ELMMSE with a precoder shared across data realizations, using stochastic gradients and projection to enforce the transmit-power budget. A momentum-based variant adapts the update for complex-valued optimization.

B. DIP: An SGP-Based Precoding Method

  • SGP estimates gradients from one or mini-batch signal samples and projects each update onto the feasible region to satisfy the transmit-power budget.The learning rate may decay over iterations; larger batches may improve performance while requiring more gradient computations.Source: B. DIP: An SGP-Based Precoding Method · p. 6
  • A large constant step size can converge quickly but only to a large neighborhood of the optimum, while improved accuracy requires decreasing the step size.The convergence of SGD critically depends on a positive step size.Source: B. DIP: An SGP-Based Precoding Method · p. 6
  • MB-SGP adapts Adam-style momentum to complex gradients by updating the second moment with the squared Frobenius norm of the gradient.The classical Adam method cannot be applied directly to the paper’s complex-variable optimization problem.Source: B. DIP: An SGP-Based Precoding Method · pp. 6–7
  • The SGP and MB-SGP gradient calculations scale with batch size and the cube of the transmit-antenna count, with MB-SGP adding a second-moment update.The supplied complexity passage gives these scaling factors but not a complete expression.Source: B. DIP: An SGP-Based Precoding Method · p. 7

C. DIP in the High-SNR Regime

For the DIP design, the paper derives a high-SNR approximation to the ELMMSE objective and uses its structure to simplify precoder optimization. The approximation fits the theoretical objective well in the high-SNR regime.

C. DIP in the High-SNR Regime

  • At high SNR, the asymptotic ELMMSE formulation fits its theoretical value very well, as shown against transmit SNR.The formulation is presented for L > NT and sufficiently large P/(σ^2_s NR).Source: C. DIP in the High-SNR Regime · p. 7, Figure 3 · p. 7
  • In the high-SNR regime, the optimal DIP precoder aligns with the eigenspace of RH, reducing the design to transmit-power allocation within that eigenspace.The resulting optimization computes the eigenvalues in Ψ.Source: C. DIP in the High-SNR Regime · pp. 7–8
  • The reduced power-allocation problem remains non-convex, so the paper applies SCA and reports that its objective is non-increasing and the algorithm converges.The feasible region is convex, and the objective is bounded below by the limited power budget.Source: C. DIP in the High-SNR Regime · p. 8

IV. ISAC PRECODING WITH GAUSSIAN SIGNALS

The ISAC design minimizes ELMMSE subject to communication-performance requirements. The paper proposes DDP and DIP solutions, using penalty-based alternating optimization and a high-SNR simplification for DIP.

IV. ISAC PRECODING WITH GAUSSIAN SIGNALS

  • The ISAC precoding problem minimizes ELMMSE while satisfying communication performance requirements.Source: §IV · p. 8
  • The paper proposes a tailored penalty-based alternating optimization algorithm for the DDP design and an SGP-AO algorithm for DIP.Source: §IV · p. 8
  • For DIP, the paper uses a high-SNR approximation of the objective to further simplify the design.Source: §IV · p. 8

A. DDP for ISAC: A Penalty-Based AO Method

The ISAC DDP design handles communication-rate constraints while minimizing sensing error for precoders dependent on each signal realization. A penalty-based alternating-optimization method separates covariance and precoder updates, but its outcome depends on the penalty parameter.

A. DDP for ISAC: A Penalty-Based AO Method

  • The rate in (12) remains achievable when the precoder depends on the signal realization, although the resulting non-Gaussian transmitted signal leaves that rate without a capacity interpretation.The paper states that the closed-form channel capacity is unavailable in this case.Source: A. DDP for ISAC: A Penalty-Based AO… · p. 8
  • Requiring each realization-specific precoder to meet the communication-rate threshold decomposes the ISAC DDP design into N parallel subproblems.Source: A. DDP for ISAC: A Penalty-Based AO… · p. 9
  • Introducing an auxiliary covariance variable and penalizing its equality with the precoder covariance enables alternating updates of the covariance and precoder.The covariance subproblem is convex, while the precoder subproblem is non-convex; the covariance update can use a numerical toolbox such as CVX.Source: A. DDP for ISAC: A Penalty-Based AO… · p. 9
  • A gradient-projection update for the precoder uses the convex feasible region to compute a projection that maintains feasibility.Source: A. DDP for ISAC: A Penalty-Based AO… · p. 9
  • The penalty-based AO objective is non-increasing and the algorithm attains a stationary point, but the converged solution may fail the communication requirement depending on the penalty parameter.Increasing the penalty can improve performance but generally requires more iterations.Source: A. DDP for ISAC: A Penalty-Based AO… · p. 9

B. DIP for ISAC: A Penalty-Based SGP-AO Method

For ISAC DIP, the precoder is independent of the random signal, so the design can use a covariance auxiliary variable and stochastic-gradient alternating updates. The resulting SGP-AO method adapts the penalty-based AO framework to a stochastic objective.

B. DIP for ISAC: A Penalty-Based SGP-AO Method

C. DIP for ISAC in the High-SNR Regime

The high-SNR DIP design approximates its stochastic objective with a deterministic covariance-based problem and solves it using successive convex approximation. Simulations report rapid convergence for the compared sensing and ISAC algorithms, with MB-SGP requiring fewer iterations than SGP while achieving comparable estimation performance.

C. DIP for ISAC in the High-SNR Regime

  • In the high-SNR regime, the DIP design approximates the stochastic objective by a deterministic function of the signal covariance, then applies successive convex approximation.The feasible set is recast as convex in the covariance variable.Source: C. DIP for ISAC in the High-SNR Regime · p. 10

V. SIMULATION RESULTS

  • The simulations use 10 mini-batch samples for SGP and penalty-based SGP-AO, with RH eigenvalues drawn uniformly from [1, 10].Source: §V · p. 10

A. Convergence Examples

  • MB-SGP requires fewer iterations than SGP across SNR settings while achieving comparable estimation performance.The simulations use L = 32 for the sensing-only comparison.Source: A. Convergence Examples · p. 10, Figure 5 · p. 11
  • At 32 dB transmit SNR and L = 40, SCA converges within 10 iterations to an objective value close to its equal-power initialization.The paper says this indicates equal-power allocation over the eigenspace of RH for high-SNR SGP-based DIP.Source: A. Convergence Examples · p. 10, Figure 6 · p. 11
  • The penalty-based SGP-AO algorithm converges within tens of iterations and satisfies penalty and communication-rate constraints under different parameter settings.Source: A. Convergence Examples · pp. 10–11

B. Sensing-Only Precoding

In sensing-only scenarios, DDP and DIP outperform water-filling, while DDP performs best and DIP offers a lower-complexity alternative. In ISAC scenarios, dedicated precoding improves the sensing–communication tradeoff over a deterministic benchmark, with the DDP–DIP gap narrowing as frame length grows.

B. Sensing-Only Precoding

  • DDP and DIP outperform classical water-filling for both L < NT and L = NT, and DDP consistently performs best.Water-filling has an error floor when L < NT because the data samples are rank-deficient, and it overlooks signal randomness in both frame-length cases.Source: B. Sensing-Only Precoding · p. 11, Figure 8 · p. 12
  • As frame length increases at fixed total power, random-signal average sensing performance approaches deterministic-signal LMMSE, but a performance gap remains.The paper highlights accounting for signal randomness when delay requirements or limited computation, storage, and signal processing prevent sufficiently long frames.Source: B. Sensing-Only Precoding · p. 12, Figure 9 · p. 12

C. ISAC Precoding

  • DDP improves sensing by nearly 1.5 dB over DetOpt while meeting the required communication rate; DIP gains 3 bps/Hz at L = 32 = NT and 4 bps/Hz at L = 24 < NT at the same sensing performance.At the maximum rate requirement, DIP and DetOpt have the same worst-case sensing performance.Source: Figure 10 · p. 12, C. ISAC Precoding · p. 13
  • In the high-SNR regime, the approximation-based SCA algorithm closely aligns with penalty-based SGP-AO as cumulative frame count increases.The paper concludes that high SNR can compensate for sensing loss from signal randomness, allowing deterministic-signal precoding with negligible performance loss in such cases.Source: C. ISAC Precoding · p. 13, Figure 11 · p. 13
  • As L increases, DDP and DIP asymptotically attain consistent sensing performance, so DDP's data-specific advantage may become small.Higher rate thresholds degrade sensing performance; at a fixed communication requirement, accumulated frames improve it.Source: C. ISAC Precoding · p. 13, Figure 12 · p. 13

VI. CONCLUSION

  • The proposed dedicated precoders address random ISAC signals in multi-antenna systems; future work includes other radar metrics and sensing without prior channel knowledge.The paper identifies mobile perceptive networks and massive-MIMO ISAC as potential application settings.Source: §VI · pp. 13–14

APPENDIX

The appendix supports a high-SNR approximation under stated validity conditions and derives a structured optimal solution. It also recasts the optimization and establishes equivalences used in the proof.

A. Proof of Lemma 2

  • The high-SNR asymptotic expression follows after substituting the moment approximation, which is valid when (L − NT)^4 is sufficiently large.The proof also ties the approximation to small γ, attainable with sufficiently high transmit power for fixed NR and σ²_s.Source: QHW SSHW HQ · p. 14

B. Proof of Theorem 2

  • The proof rewrites the objective using R = QΩQH and establishes that the resulting problem is equivalent to problem (29).It uses the unitary property of Q and the stated forms of the transformed objective functions.Source: B. Proof of Theorem 2 · p. 14
  • The optimal solution has a diagonal structure with entries ordered decreasingly, yielding Ωopt = QHΨ↑Q, where Ψ↑ has ascending diagonal entries.The proof recasts problem (29) and uses the sorted inverse eigenvalues of Λ to obtain the structure.Source: B. Proof of Theorem 2 · p. 14

C. Proof of Proposition 2

The proof considers a Gaussian linear channel and lower-bounds mutual information for X = WS conditioned on Hc. It then maximizes the bound over input distributions using independence and transmitter channel knowledge.

C. Proof of Proposition 2

  • The proof starts from a Gaussian linear channel and lower-bounds the mutual information between Yc and X = WS conditioned on Hc.The stated steps use the mutual-information chain rule and nonnegativity of I(Yc; W | Hc).Source: C. Proof of Proposition 2 · p. 14
  • Maximizing both sides over input distributions yields a bound involving channel capacity C.The proof denotes the channel capacity as C before taking the maximum.Source: C. Proof of Proposition 2 · p. 14
  • The proof invokes independence of W and S and perfect transmitter knowledge of Hc in its final step.Source: C. Proof of Proposition 2 · p. 15

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