Sum S&C Mutual Information in ISAC

• Sum S&C MI is a metric that unifies sensing and communication mutual information in ISAC systems, enabling performance optimization for both deterministic and random signaling scenarios.
• It employs gradient-based waveform and precoder designs, explicitly linking information-theoretic measures with classical metrics like MMSE and Bayesian CRB.
• Weighted-sum designs reveal convex trade-offs, effectively balancing energy allocation across eigen-modes in multi-user and multi-target ISAC environments.

Sum Sensing & Communication Mutual Information (S&C MI) refers to the unified metric formed by adding the mutual information (MI) from the sensing function and the communication function within an integrated sensing and communication (ISAC) system. This sum provides a rigorous, information-theoretic criterion for evaluating and optimizing joint system performance, and can be precisely formulated for a range of channel models with both deterministic and random signaling. The S&C MI framework formalizes the shared operational units and mathematical structure of sensing MI (SMI) and communication MI (CMI), enables gradient-based waveform or precoder design for joint tasks, and connects to classical performance bounds (MMSE, Bayesian CRB) via explicit large-system limits.

1. Formal Definition and Additivity

Let $I_s$ denote the sensing MI—the mutual information between a signal reflecting environment parameters and the observed receiver output, often conditioned on a known transmit waveform; let $I_c$ denote the communication MI—the mutual information between transmitted and received symbols over the communication channel. In practical ISAC settings, both are measured in the same units (bits or nats) and possess closed-form expressions under linear-Gaussian models.

The sum S&C MI is then

$J_{\mathrm{sum}} = I_s + I_c$

and optionally, in weighted form,

$$J_{\mathrm{sum}}(\alpha) = \alpha I_c + (1 - \alpha) I_s$$

with $\alpha \in [0, 1]$ a user-selected tradeoff parameter (Ouyang et al., 2022, Peng et al., 2023, Xie et al., 2024, Piao et al., 2023). This metric is well-defined for deterministic (fixed waveform) and random (Gaussian codebook) signaling scenarios.

2. Sensing and Communication MI: Channel Models

In prototypical MIMO Gaussian models:

• Sensing Channel (random linear model):

$$\mathbf{Y}_s = \mathbf{H}_s \mathbf{X} + \mathbf{N}_s$$

where $\mathbf{H}_s$ is a random target response with prior covariance $\mathbf{R}$; $\mathbf{X}$ is the (known or random) transmit waveform; $\mathbf{N}_s$ is Gaussian noise.

The SMI is:

$$I_s = \mathbb{E}_{\mathbf{S}}\left[ \log \left| \mathbf{I} + \sigma_s^{-2} (\mathbf{I}_{N_R} \otimes \mathbf{X})^H \mathbf{R} (\mathbf{I}_{N_R} \otimes \mathbf{X}) \right| \right]$$

or, for deterministic $\mathbf{X}$,

$$I_s = \log_2 \det\left[ \mathbf{I} + \frac{1}{\sigma_s^2} \mathbf{R}_G^{1/2} \mathbf{X}_s \mathbf{X}_s^H \mathbf{R}_G^{1/2} \right]$$

(Ouyang et al., 2022, Xie et al., 2024).

• Communication Channel (standard MIMO):

$$\mathbf{Y}_c = \mathbf{H}_c \mathbf{X} + \mathbf{N}_c$$

with $\mathbf{H}_c$ as the channel matrix, $\mathbf{N}_c$ as noise.

The CMI is:

$$I_c = \log_2 \det\left[ \mathbf{I} + \frac{1}{\sigma_c^2} \mathbf{H}_c \mathbf{Q}_c \mathbf{H}_c^H \right]$$

or for random codebooks,

$$I_c(\Phi) = \log \lvert \mathbf{I} + \sigma_c^{-2} \mathbf{H}_c \Phi \mathbf{H}_c^H \rvert$$

where $\Phi = \mathbf{F}\mathbf{F}^H$ (Ouyang et al., 2022, Xie et al., 2024).

Sum S&C MI thus evaluates to a sum of (potentially log-det) terms that share algebraic structure and unit consistency.

3. Optimization: Weighted-Sum Design

The central problem is to maximize (or constrain) the sum S&C MI over waveform or precoder variables, subject to total power constraints:

$$\max_{\Phi \succeq 0,\ \operatorname{tr}(\Phi) \le P} J_{\mathrm{sum}}(\Phi) = I_s(\Phi) + I_c(\Phi)$$

where

• $I_s(\Phi)$ admits tractable large-system approximations via random matrix theory, e.g.,

$$I_s(\Phi) = \sum_{j=1}^{K} \bar\varrho_j(\Phi) + \mathcal{O}\left(1/N_S\right)$$

with each $\bar\varrho_j(\Phi)$ given in closed form involving the eigenstructure of channel covariances and a fixed-point equation for $\delta(\cdot)$ (Xie et al., 2024).

• $I_c(\Phi)$ is again log-det in $\Phi$.

Gradient-projection methods are effective for such problems, based on closed-form gradients for both $I_s$ and $I_c$. The design can also be posed as a constrained maximization (maximizing $I_s$ subject to $I_c \ge R_0$), which is tackled using ADMM with projections onto the feasible set (Xie et al., 2024).

In weighted-sum settings, a single parameter $\alpha$ traces out the trade-off surface between SMI and CMI (Peng et al., 2023, Piao et al., 2023). Closed-form solutions can sometimes be obtained via generalized water-filling in the presence of simultaneous log-det objectives.

4. Connection to Classical Sensing Metrics

In the large-system limit, SMI is tightly linked with classical estimation-theoretic performance measures:

Metric	Mathematical Connection	Reference Section
Sensing MI ($I_s$)	Information gain over $\mathbf{h}_s$	(Xie et al., 2024)
MMSE, LMMSE	Minimizing error in $\mathbf{h}_s$ estimation	$J_{\mathrm{ELMMSE}}$
Bayesian CRB (EBCRB)	Lower bound on $\mathrm{var}(\hat{\mathbf{h}}_s)$	$$I_s = \log|\mathbf{R}| - \mathbb{E}_{\mathbf{S}} \log|\Psi_{\mathrm{BCRB}}|$$
Information–estimation duality	Maximizing $I_s$ $\Longleftrightarrow$ minimizing EBCRB	(Xie et al., 2024)

This connection explains the operational significance of SMI: maximizing mutual information yields lower parameter uncertainty as measured by (E)CRB, and vice versa.

5. S&C MI in Uplink and Downlink ISAC Systems

For both downlink and uplink ISAC, the sum S&C MI metric unifies waveform/precoder design and enables rate-region analysis:

• Downlink: Joint maximization over sensing and communication waveforms/precoders under total power, with SMI and CMI as joint objectives. Iterative methods (e.g., alternating water-filling, superposition, WMMSE algorithms) yield practical solutions (Ouyang et al., 2022, Peng et al., 2023).
• Uplink: S&C signals overlap; MI-based metrics characterize rate–rate tradeoff surfaces, and joint optimization strategies include sequential interference cancellation and joint matrix power allocation (Piao et al., 2023).

In all cases, the sum S&C MI captures the achievable region for joint communication and sensing performance, generalizing scalar rate trade-offs to vector-valued policy spaces.

6. Numerical Results and Trade-off Curves

Monte Carlo and asymptotic analyses consistently show:

• The large-system closed-form SMI is highly accurate (errors $<1$ nat for typical $N_S$, $N_T$).
• Joint MI-optimizing designs achieve strictly larger SMI for fixed $I_c$, and vice versa, than designs optimizing only one objective (Xie et al., 2024, Piao et al., 2023).
• Weighted-sum MI curves exhibit a convex trade-off ("rate–rate" frontier) as the weighting parameter $\alpha$ is varied, with maximal $J_{\mathrm{sum}}$ at an interior value rather than at the endpoints.
• Practical allocation solutions (water-filling or KKT-based) efficiently balance energy across eigen-modes according to their joint sensing and communication value (Piao et al., 2023).

7. Extensions, Practical Considerations, and Open Problems

• Computational Efficiency: Log-det and trace formulations admit efficient evaluation and gradient computation. For moderate antenna/channel sizes, second-order methods are practical; for large systems, random-matrix and blockwise computation are critical (Xie et al., 2024).
• Multi-user and multi-target ISAC: The sum S&C MI framework generalizes to systems with multiple users/targets by summing per-user/target MIs and applying block-diagonal optimization (Peng et al., 2023).
• Robust/Adaptive Design: Extensions to imperfect CSI, non-Gaussian noise, or robust design are open research problems (Ouyang et al., 2022).
• Physical Layer Benchmarking: S&C MI provides a rigorous, unit-consistent benchmark for comparing ISAC architectures, as shown by dominance of ISAC rate regions over FDSAC in simulations (Ouyang et al., 2022).
• Connection to Information–Estimation Dualities: The duality between MI maximization and MMSE/CRB minimization positions S&C MI as a unifying metric for both information-theoretic and estimation-driven ISAC system analysis (Xie et al., 2024).

References: (Ouyang et al., 2022, Peng et al., 2023, Piao et al., 2023, Xie et al., 2024)