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On the Fundamental Limits of Bistatic ISAC Over Vector Channels
Dongsheng Peng, Lei Yu, Ping Chen, Jun Chen
TL;DR
Bistatic ISAC raises the question of how communication rate trades off against sensing accuracy when the sensing receiver does not know the transmitted message, while many exact characterizations require a global degradation order. The paper extends the analysis to parallel and general vector channels, obtaining single-letter results and an exact capacity-distortion-power characterization for vector Gaussian channels with additive Gaussian states.
Problem
Many exact single-letter characterizations for bistatic ISAC rely on a global degradation order, which may not hold across subcarriers or in multi-antenna channels.
Source: §I · p. 2Method
The paper analyzes mixed-degradation parallel channels and general vector channels, using channel enhancement and a Fisher-information-based matrix extremal inequality for vector Gaussian channels.
Source: §I · p. 2Results
Matching achievability and converse characterize C(D) for parallel channels, and Gaussian superposition coding exactly characterizes C(D, P) for vector Gaussian channels with additive Gaussian states.
Source: §I · p. 2Takeaways & Limitations
Joint common-layer coding across parallel components exploits complementary degradation directions and yields a rate gain over independent coding.
Source: §I · p. 2, §V · p. 11
Abstract
from arXiv · showhide
This paper studies the fundamental limits of bistatic integrated sensing and communications (ISAC) systems over vector channels. We first consider a class of parallel channels in which, on each component channel, the channel from the transmitter (Tx) to the communication receiver (Rx) is degraded with respect to the channel from the Tx to the sensing Rx, or vice versa, where the direction of degradation may vary across component channels. For this class of channels, we establish matching achievability and converse results, which yield a single-letter characterization of the capacity-distortion function $C(D)$. For parallel Gaussian channels with additive Gaussian states, the evaluation of the capacity-distortion-power function $C(D,P)$ under the squared error distortion measure reduces to a joint power and distortion allocation problem across the component channels. We then extend the analysis to general vector channels and derive single-letter lower and upper bounds on $C(D)$. In particular, for vector Gaussian channels with additive Gaussian states, we prove the optimality of Gaussian superposition coding without requiring a global degradation order between the communication and sensing channels, by combining the channel enhancement technique with a matrix extremal inequality based on Fisher information. This leads to an exact characterization of $C(D,P)$.
I. INTRODUCTION
Bistatic ISAC asks how communication rate trades off against sensing accuracy when the sensing receiver does not know the transmitted message. This paper characterizes that tradeoff for parallel channels with mixed degradation directions and extends the analysis to general vector channels, including an exact result for vector Gaussian channels with additive Gaussian states.
II. SYSTEM MODEL
- The capacity-distortion function C(D) is the maximum communication rate achievable subject to a constraint on expected state-estimation distortion.The framework models the sensing target as a random channel state.Source: §I · p. 1
I. INTRODUCTION
- Existing exact single-letter characterizations mainly require a global degradation order, which can fail across OFDM subcarriers or in multi-antenna channels.In bistatic sensing, the information-bearing signal is unknown to the sensing receiver and acts as interference for state estimation.Source: §I · pp. 1–2
III. PARALLEL CHANNELS
- For parallel channels with potentially different degradation directions, matching achievability and converse results yield a single-letter characterization of C(D).The distortion is additive across component channels under the stated parallel-channel setup.Source: §I · p. 2, §III · p. 3
- Jointly encoding the common layer across components exploits complementary degradation directions and achieves a rate gain over independent coding.The sensing receiver’s surplus decoding capability on one component set can compensate for the cloud-layer rate loss on the other.Source: §I · p. 2, §III · p. 4
I. INTRODUCTION
- For vector Gaussian channels with additive Gaussian states, Gaussian superposition coding is optimal without a global degradation order, giving an exact C(D, P) characterization.The proof combines channel enhancement with a matrix extremal inequality based on Fisher information.Source: §I · p. 2
A. Achievability
Achievability uses superposition coding with a common layer jointly encoded across component channels and private messages on selected channels. The sensing receiver decodes the common layer, while the communication receiver successively decodes both layers.
A. Achievability
- The common layer is jointly encoded across all component channels, exploiting their complementary degradation directions.This is the central feature of the superposition scheme.Source: A. Achievability · p. 4
- The transmitter sends the common codeword on channels in A and private codewords on channels in B; the sensing receiver decodes the common message.The communication receiver decodes the common message first, then the private message.Source: A. Achievability · p. 4
- The sensing receiver estimates each state symbol using the decoded common-layer codeword, and the expected distortion is bounded using a genie-aided estimate.The proof selects a codebook with decoding error tending to zero and asymptotic distortion no greater than D.Source: A. Achievability · pp. 4–5
B. Converse
The converse single-letterizes the rate and distortion bounds for the parallel channels, using degradedness and time sharing. It also identifies the extra condition needed for the stated extension to sequence-wise logarithmic loss.
B. Converse
- A time-sharing variable Q selects a channel use, yielding single-letter variables from the block observations and inputs.The construction defines W := (Q, WQ), X := XQ, and Yi := Yi,Q.Source: B. Converse · p. 5
- Degradedness lets the converse combine stronger and weaker component outputs, then bound the B-channel rate terms by I(Xk;Y1k|Uk) + I(Uk;Y2k).The componentwise bound follows from the product structure and Csiszár sum identity.Source: B. Converse · pp. 5–6
- The converse bounds the rate by the single-letter expression of Theorem 1 under distortion constraint D, with the blocklength and slack tending to their limits.Correlated component inputs can be replaced by products of their marginals without changing the rates or distortion.Source: B. Converse · p. 6
- The stated converse theorem uses symbol-wise distortion; extending it to sequence-wise logarithmic loss requires Xk ↔ Y1k ↔ (Y2k, Sk) for k in B.The passage states that no additional condition is needed for k in A.Source: B. Converse · p. 6
C. A Special Gaussian Case
For parallel Gaussian channels with independent additive Gaussian states and squared-error sensing, the capacity-distortion-power evaluation becomes a joint allocation problem. Gaussian superposition inputs suffice for the optimization.
C. A Special Gaussian Case
- For Gaussian component channels, the noises and states are mutually independent across channels and independent of the input, and sensing uses squared-error distortion.The channel model includes an input power constraint, defining C(D, P).Source: C. A Special Gaussian Case · p. 6
- The degradation direction is determined by comparing the communication receiver noise variance with the sensing receiver noise plus state variance.The passage identifies B as the channels satisfying N1k < N2k + NSk.Source: C. A Special Gaussian Case · p. 6
- The input-distribution optimization can be restricted to Gaussian superposition inputs.For channels in A, the stated distortion is N2kNSk/(N2k + NSk).Source: C. A Special Gaussian Case · p. 7
- Computing C(D, P) reduces to jointly allocating power and distortion across the component channels.The allocation problem is described for the parallel Gaussian case.Source: C. A Special Gaussian Case · p. 7
1) Optimal Power Allocation:
The power allocation is derived from the Lagrangian stationarity condition and has water-filling form, with classical water-filling recovered at the two endpoint weights.
1) Optimal Power Allocation:
- Setting the derivative of L with respect to Pk to zero, subject to Pk ≥ 0, yields the water-filling solution.The power water level is νP.Source: 1) Optimal Power Allocation · p. 7
- At θ = 1, the allocation reduces to classical water-filling over N1k.Source: 1) Optimal Power Allocation · p. 7
- At θ = 0, the allocation reduces to classical water-filling over N2k + NSk.Source: 1) Optimal Power Allocation · p. 7
2) Optimal Distortion Allocation:
The section treats distortion allocation in parallel Gaussian channels and develops single-letter bounds for general vector channels. In the additive-state vector Gaussian case, the lower bound is tight even without a global degradation order.
2) Optimal Distortion Allocation:
- The distortion allocation uses a distortion water level and projects component allocations onto their feasible intervals.When the upper bound is active, that component carries only private information, and its weighted rate is independent of θ.Source: 2) Optimal Distortion Allocation · p. 7
- Joint coding offers a noticeable capacity-distortion gain over separate coding, and the gap shrinks as the distortion constraint is relaxed.The comparison is for two parallel Gaussian component channels under squared error distortion.Source: 2) Optimal Distortion Allocation · p. 7, Figure 2 · p. 8
IV. GENERAL VECTOR CHANNELS
- General vector channels need not have product structure or a degradation order, as can occur when MIMO channel components are coupled.For this setting, the paper derives single-letter lower and upper bounds on C(D).Source: §IV · pp. 7–8
B. A Special Gaussian Case
- For additive-state vector Gaussian channels, the lower bound is tight without requiring a positive-semidefinite ordering between the communication and sensing noise covariances.The model assumes positive-definite noise and state covariance matrices and a trace power constraint.Source: §IV · p. 8, B. A Special Gaussian Case · p. 8
A. Single-Letter Bounds
- The general-vector lower and upper bounds use an auxiliary variable with cardinality at most |X| + 1.The respective bounds include I(X; Y1|U) + I(U; Y2) and I(U; Y2) + I(X; Y1, Y2|U).Source: A. Single-Letter Bounds · p. 8
1) Proof of the Lower Bound:
The lower-bound construction uses independent Gaussian layers for the channel input. Conditioning on the common layer turns the sensing observation into a Gaussian estimation problem.
1) Proof of the Lower Bound:
- The achievable construction sets X = U + V, with independent Gaussian layers U and V whose covariance matrices sum to the input covariance.Both layers are independent of the communication noise, sensing noise, and state.Source: 1) Proof of the Lower Bound · p. 9
- Given U, the sensing observation reduces to Y2 − U = V + G, where G combines sensing noise and the Gaussian state.The combined noise has covariance Σg = Σ2 + Σs.Source: 1) Proof of the Lower Bound · p. 9
- The conditional estimation error is expressed through Σe(A) := Σs − Σs(A + Σg)^−1Σs.This covariance formula supplies the distortion calculation for the Gaussian superposition input.Source: 1) Proof of the Lower Bound · p. 9
2) Proof of the Upper Bound:
The upper-bound proof translates the sensing distortion constraint into an equivalent covariance constraint, then combines an extremal inequality with channel enhancement. Letting the enhancement slack vanish yields equality between the capacity-distortion function and its Gaussian characterization.
2) Proof of the Upper Bound:
- The converse combines the distortion characterization, an extremal inequality, and an enhanced-channel construction.The enhancement uses an optimization over covariance matrices and a slack parameter δ.Source: 2) Proof of the Upper Bound · p. 9
- The sensing distortion constraint is represented through an equivalent covariance matrix obtained from conditional Fisher information.The construction averages the conditional Fisher information matrices and defines the equivalent covariance as their inverse minus Σg.Source: 2) Proof of the Upper Bound · p. 9
- The enhanced-channel argument bounds C(D, P) by RG(D + κδ, P + Kδ), then takes δ to zero to obtain C(D, P) ≤ RG(D, P).The passage establishes feasibility under the slackened constraints before taking the limit.Source: 2) Proof of the Upper Bound · p. 10
- The lower and upper bounds combine to give the exact characterization C(D, P) = RG(D, P).For the illustrated channel, the capacity-distortion curves are tangent at D* = 0.8.Source: 2) Proof of the Upper Bound · p. 10
V. CONCLUSION
For parallel channels with varying degradation directions, joint coding across components exploits their complementarity. For general vector channels, the paper establishes single-letter bounds and obtains an exact characterization for additive-state vector Gaussian channels.
V. CONCLUSION
- For parallel channels with differing degradation directions, jointly encoding the common layer exploits complementarity and gains rate over independent component coding.The paper establishes a single-letter characterization of C(D) for this class.Source: §V · p. 11
- For general vector channels without a global degradation order, the paper derives single-letter lower and upper bounds on C(D).Source: §V · p. 11
- For additive-state vector Gaussian channels, Fisher-information-based distortion characterization, channel enhancement, and a matrix extremal inequality yield an exact C(D, P) characterization.Gaussian superposition coding is optimal, with the sensing receiver partially decoding to assist state estimation.Source: §V · p. 11
APPENDIX A PROOF OF THEOREM 3
Theorem 3’s converse uses time-sharing to reduce block variables to a single-letter Markov structure and bound rate under the distortion constraint. Lemma 1’s proof uses Fisher-information inequalities to establish its covariance and distortion properties.
APPENDIX A PROOF OF THEOREM 3
- The converse introduces a uniform time-sharing variable to obtain U ↔ X ↔ (Y1, Y2, S) and bound the block rate and distortion.The proof then takes the blocklength to infinity and lets the slack tend to zero to obtain its stated capacity bound.Source: Appendix A · p. 11
APPENDIX B PROOF OF LEMMA 1
- Fisher-information, Cramér–Rao-type, matrix-convexity, and Jensen inequalities are used to establish property (i) of Lemma 1.The proof also uses covariance ordering and the order-reversing property of matrix inversion.Source: Appendix B · p. 11
- Lemma 1’s property (ii) identifies the conditional distortion as tr Σe(ˆΣV).The proof conditions on U and uses the independence of the Gaussian state S from U.Source: Appendix B · p. 12
APPENDIX C PROOF OF LEMMA 2
The proof of Lemma 2 interpolates between two Gaussian noise covariances and uses Fisher-information inequalities and matrix convexity to establish (52).
APPENDIX C PROOF OF LEMMA 2
- The proof interpolates the noise covariance from ˜Σ1 to Σg while preserving the corresponding endpoint distributions.It defines Y(t) = X + N(t), with covariance Ωt = ˜Σ1 + t∆.Source: Appendix C · p. 12
- A Fisher-information inequality and operator Jensen inequality bound the interpolated covariance quantities.The proof defines a mapping φt and uses its operator convexity, including the singular case by continuity.Source: Appendix C · p. 12
- Integrating the derivative bound over t ∈ [0, 1] and rearranging yields (52).The derivative step invokes the multivariate de Bruijn identity and Jacobi’s formula.Source: Appendix C · p. 12
APPENDIX D PROOF OF LEMMA 3
The proof of Lemma 3 establishes an optimizer through KKT conditions and shows that a transformed objective is concave. It also proves a covariance ordering used in the argument.
APPENDIX D PROOF OF LEMMA 3
- The feasible set is compact, and Slater’s condition makes the KKT conditions necessary for the stated optimization.The proof establishes existence of an optimal Σ∗V before applying the KKT conditions.Source: Appendix D · p. 12
- The covariance ordering ˜Σ1 ⪯ Σg follows from the order-reversing property of matrix inversion.This ordering supports the proof’s construction of ∆ as a positive semidefinite matrix.Source: Appendix D · pp. 12–13
- The transformed objective ˜f is concave because the proof establishes convexity of g using the worst additive noise theorem.The proof also uses the chain rule, nonnegative mutual information, independence, and conditional Gaussianity.Source: Appendix D · p. 13
- The optimal Σ∗V also maximizes ˜f over Vδ, establishing property (iii).The argument uses the concavity of ˜f and the stated matrix order constraints.Source: Appendix D · p. 13