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Joint State Sensing and Communication: Optimal Tradeoff for a Memoryless Case
Mari Kobayashi, Giuseppe Caire, Gerhard Kramer
TL;DR
Separated sensing and communication can struggle with mobility or large state dimension and may reduce data rate, motivating joint design. The paper characterizes the capacity-distortion tradeoff for memoryless channels with i.i.d. states and proposes an input-distribution optimization method. Its examples show a gain for joint sensing and communication over a separation-based approach.
Problem
Separated sensing and communication can perform poorly in high-mobility or large-state-dimension settings, and allocating resources to sensing can reduce data rate.
Source: §I · p. 1Method
For memoryless channels with i.i.d. states, the paper characterizes the capacity-distortion tradeoff and formulates its maximization as a convex input-distribution optimization with an iterative algorithm.
Source: §I · p. 1, §IV · p. 3Results
The binary-channel example shows a significant gain for the joint approach over separation-based time-sharing between (D, C) = (0, 0) and (q, q).
Source: A. Binary Channel with Multiplicative… · p. 5Takeaways & Limitations
The examples demonstrate benefits of joint sensing and communication compared with a separation-based design.
Source: §I · p. 1, A. Binary Channel with Multiplicative… · p. 5The characterization is restricted to memoryless channels with i.i.d. state sequences.
Source: §I · p. 1
Abstract
from arXiv · showhide
A communication setup is considered where a transmitter wishes to simultaneously sense its channel state and convey a message to a receiver. The state is estimated at the transmitter by means of generalized feedback, i.e. a strictly causal channel output that is observed at the transmitter. The scenario is motivated by a joint radar and communication system where the radar and data applications share the same frequency band. For the case of a memoryless channel with i.i.d. state sequences, we characterize the capacity-distortion tradeoff, defined as the best achievable rate below which a message can be conveyed reliably while satisfying some distortion constraint on state sensing. An iterative algorithm is proposed to optimize the input probability distribution. Examples demonstrate the benefits of joint sensing and communication as compared to a separation-based approach.
I. INTRODUCTION
The paper studies joint state sensing and communication because separated sensing and data transmission can perform poorly under mobility or large state dimension and can reduce data rate. It models a memoryless channel with i.i.d. states, strictly causal output feedback for transmitter estimation, and receiver state knowledge, then defines the achievable rate under a distortion constraint.
I. INTRODUCTION
- Prior joint radar-communication work lacks a fundamental framework for the optimal sensing–communication tradeoff; this paper addresses that gap for the memoryless case.Separated designs can perform poorly with high mobility or large state dimension, and dedicating resources to sensing reduces data rate because no data symbols are sent during sensing.Source: §I · p. 1
II. SYSTEM MODEL
- The transmitter estimates the channel state from a strictly causal channel output, while the receiver has perfect state knowledge.The channel is memoryless with an i.i.d. state sequence.Source: §I · p. 1, §II · p. 1
I. INTRODUCTION
- The contributions are a capacity-distortion characterization for memoryless channels with i.i.d. states, an input-distribution optimization algorithm, and examples comparing joint and separated designs.Source: §I · p. 1
II. SYSTEM MODEL
- A rate-distortion pair is achievable when message error tends to zero and expected state distortion is at most D; C(D) is the supremum achievable rate.Source: §II · p. 2
III. CAPACITY-DISTORTION TRADEOFF
The section characterizes C(D) for a state-dependent memoryless channel with i.i.d. states. Its expression optimizes over input distributions meeting the distortion constraint and uses a deterministic state estimator without loss of generality.
III. CAPACITY-DISTORTION TRADEOFF
- The section establishes the characterization through useful lemmas followed by converse and achievability proofs.Source: §III · p. 2
- Theorem 1 characterizes C(D) by maximizing over input distributions satisfying the distortion constraint under the specified joint distribution of state, input, outputs, and estimate.Source: §III · p. 2
- C(D) is nondecreasing and concave for D at least Dmin.Source: §III · p. 2
- Without loss of generality, the state estimator can be chosen deterministic and based on the input and feedback output.Source: §III · p. 2
A. Converse
The converse uses error-probability and channel-dependence arguments to bound achievable communication rates by the capacity-distortion characterization.
A. Converse
- The converse invokes Fano’s inequality and the Markov relation between message history, state and input, and channel output to bound the rate.Source: A. Converse · p. 2
- The rate bound uses the concavity and nondecreasing properties of C(D) to relate the converse terms to the distortion constraint.Source: A. Converse · p. 3
B. Achievability
Achievability uses random coding with input distribution and estimator chosen for a slightly relaxed distortion target. Typicality decoding and feedback-based estimation yield vanishing error and the desired distortion as the relaxation vanishes.
B. Achievability
- For bounded distortion, the achievability proof chooses an input distribution and estimator for D/(1+ε), then randomly generates codewords known to encoder and decoder.The proof also extends to distortion functions with a letter s* whose expected distortion is bounded by dmax.Source: B. Achievability · p. 3
- The encoder transmits the selected codeword, the decoder uses joint typicality with the state sequence, and the encoder estimates state from its input and feedback output.Source: B. Achievability · p. 3
- Expected distortion is bounded by separating decoding-error and successful-decoding events; continuity of C(D) gives the desired tradeoff as ε tends to zero.Source: B. Achievability · p. 3
- The decoding error probability tends to zero when R < I(X; Y |S), by the law of large numbers and packing lemma.Source: B. Achievability · p. 3
IV. NUMERICAL METHOD FOR OPTIMIZATION
With an input cost constraint in addition to the distortion constraint, the paper formulates the optimization over input distributions and proposes an iterative algorithm.
IV. NUMERICAL METHOD FOR OPTIMIZATION
- The setup adds an average input-cost constraint to the distortion-constrained problem.The cost function is defined as an average over the input sequence, with an expected-cost bound.Source: §IV · p. 3
- The maximization is convex with respect to the input distribution, and the authors propose a modified Blahut-Arimoto algorithm.The input must satisfy both the distortion and cost constraints.Source: §IV · p. 3
A. Problem Formulation
The formulation reduces the sensing-and-communication problem to optimizing the input distribution under simultaneous distortion and cost constraints. It assumes the feasible set is nonempty and uses an alternating-optimization approach to characterize the solution.
A. Problem Formulation
- The stated objective is to maximize I(X; Y |S).Source: A. Problem Formulation · p. 3
- The estimator can be computed a priori, allowing the original problem to be rewritten explicitly in terms of PX.The formulation uses the mutual-information functional over PX and PY|XS|PS.Source: A. Problem Formulation · p. 3
- The feasible set F(B, D) may be empty for some values of (B, D), but the paper assumes it is nonempty and notes that it is convex and compact.Both constraints are linear in PX.Source: A. Problem Formulation · p. 3
- Because the solution need not meet both constraints with equality, the paper incorporates one cost as a penalty and focuses on the other constraint.The resulting optimization is parameterized by the penalty.Source: A. Problem Formulation · p. 4
- The derivation follows standard Blahut-Arimoto alternating optimization, using a conditional pmf QX|YS; the proof of the stated result is omitted.The theorem includes a nonnegative multiplier chosen so the input-cost constraint holds with equality.Source: A. Problem Formulation · p. 4
B. Modified Blahut-Arimoto Algorithm
The modified algorithm iteratively optimizes the input distribution and yields a capacity-distortion pair for each fixed input cost. Varying its penalty parameter traces the tradeoff, from the ordinary capacity-cost function to distortion minimization.
B. Modified Blahut-Arimoto Algorithm
- The method is based on a general result on alternating optimization.The procedure is proposed for the penalized optimization problem.Source: B. Modified Blahut-Arimoto Algorithm · p. 4
- The algorithm starts from a uniform input distribution and iterates primal and dual variable updates to obtain a convergent input distribution.The resulting distribution yields a capacity-distortion pair for each fixed input cost B.Source: B. Modified Blahut-Arimoto Algorithm · p. 4
- Varying µ produces the capacity-distortion tradeoff for fixed input cost B, ranging from the standard capacity-cost function at µ = 0 to distortion minimization as µ →∞.The latter is for a given input cost B.Source: B. Modified Blahut-Arimoto Algorithm · p. 4
V. EXAMPLES
The examples compare the proposed joint scheme with a separation-based approach, defined as dividing resources between sensing and communication.
V. EXAMPLES
- The examples illustrate gains of the proposed joint scheme over the separation-based approach.The comparison is the stated purpose of the two examples; no numerical outcome is specified in these passages.Source: §V · p. 4
- Separation divides resources between state sensing with generalized feedback and data communication without feedback.For simplicity, generalized feedback is assumed perfect, Z = Y.Source: §V · p. 4
A. Binary Channel with Multiplicative Bernoulli State
The binary multiplicative-state example characterizes a capacity–distortion tradeoff and compares joint design with separation. Its reported endpoints show the rate–distortion extremes, while the q = 0.4 plot reports a joint-design gain.
A. Binary Channel with Multiplicative Bernoulli State
- The example uses a binary multiplicative channel with q in [0, 1/2] and Hamming distortion; its input distribution is parameterized by p to maximize C(D).The stated cost function is c(1) = 0 and c(0) = q.Source: A. Binary Channel with Multiplicative… · pp. 4–5
- At p = 0, sending X = 1 gives Dmin = 0 but C(D) = 0; at p = 1/2, C(Dmax) = q for Dmax = q/2.These are the two stated extreme points of the tradeoff.Source: A. Binary Channel with Multiplicative… · p. 4
- At q = 0.4, the joint approach significantly outperforms separation’s time-sharing between (D, C) = (0, 0) and (q, q).The distortion q is achieved by the fixed estimator ŝ = 0, independent of feedback.Source: A. Binary Channel with Multiplicative… · p. 5
B. Real Gaussian Channel with Rayleigh Fading
The fading AWGN example evaluates quadratic-distortion tradeoffs at P = 10 dB, using Gaussian input for unconstrained capacity and 2-PAM for minimum distortion. The separation baseline is described by two corner points.
B. Real Gaussian Channel with Rayleigh Fading
- The example sets P = 10 dB and assumes i.i.d. unit-variance Gaussian noise and fading.The input satisfies the stated average-power constraint.Source: B. Real Gaussian Channel with Rayleigh… · p. 5
- 1.213 bit/channel use is the unconstrained capacity with Gaussian input; minimum distortion is 0.091 with 2-PAM, whose capacity is 0.733 bit/channel use.The capacity–distortion tradeoff is shown in Fig. 3.Source: B. Real Gaussian Channel with Rayleigh… · p. 5, Figure 3 · p. 5
- The plotted tradeoff is calculated by applying a modified Blahut–Arimoto algorithm to a quantized real AWGN channel and M-ary PAM.The example focuses on quadratic distortion.Source: B. Real Gaussian Channel with Rayleigh… · p. 5
- The separation scheme has corners (Dmin, 0) and (Dmax, Cmax): it devotes full resources to estimation at the first and ignores feedback, sending Gaussian data, at the second.Here Dmax = var[S] = 1.Source: B. Real Gaussian Channel with Rayleigh… · p. 5