The Capacity–Distortion Tradeoff

A Single Number for the Tradeoff

Section 24.1 showed that a massive MIMO array can do sensing and communication at the same time. That does not answer the harder question: at what cost? Every joule spent tilting the beam toward a target is a joule not delivered to a user; every bit of randomness injected into the waveform to improve its ambiguity function is a bit of structure that could have improved decoding. A rigorous answer requires a joint operational definition in which communication rate and sensing accuracy are co-dimensions of a single Pareto region.

This section develops that framework — the capacity–distortion function — following Liu, Zhou, Masouros, Caire, and collaborators. The main result is both elegant and actionable: the optimal tradeoff on Gaussian channels takes a water-filling-like form, and its Pareto boundary can be traced by sweeping a single Lagrange multiplier.

Definition:

Sensing Parameter and Distortion

Let η∈Rp\boldsymbol{\eta} \in \mathbb{R}^p denote the deterministic but unknown sensing parameter of the target. Typical entries include range, angle-of-arrival, Doppler shift, and complex reflectivity. The sensing receiver observes z=g(η;x)+wr,\mathbf{z} = \mathbf{g}(\boldsymbol{\eta}; \mathbf{x}) + \mathbf{w}_r, where g(⋅)\mathbf{g}(\cdot) is a (generally nonlinear) deterministic map from parameter to received waveform, driven by the transmit signal x\mathbf{x}.

The sensing distortion is the mean squared error of any unbiased estimator η^(z)\hat{\boldsymbol{\eta}}(\mathbf{z}): D(Rx)≜E ⁣[∥η^−η∥2].D(\mathbf{R}_x) \triangleq \mathbb{E}\!\left[\|\hat{\boldsymbol{\eta}} - \boldsymbol{\eta}\|^2\right]. By the Cramér–Rao bound, D(Rx)≥tr(J−1(Rx))D(\mathbf{R}_x) \geq \text{tr}(\mathbf{J}^{-1}(\mathbf{R}_x)), with equality achieved by the MLE at high SNR. Thus the sensing KPI is ultimately a function of the transmit covariance Rx\mathbf{R}_x through the Fisher information matrix J\mathbf{J}.

Definition:

Capacity–Distortion Function

Fix a joint channel with communication output y\mathbf{y} and sensing output z\mathbf{z} driven by the same transmit x\mathbf{x}: p(y,z∣x,η).p(\mathbf{y}, \mathbf{z} \mid \mathbf{x}, \boldsymbol{\eta}). The capacity–distortion function is C(D)≜sup⁡p(x):  E[D^]≤D  I(x;y),\mathcal{C}(D) \triangleq \sup_{p(\mathbf{x}):\; \mathbb{E}[\hat{D}] \leq D}\;I(\mathbf{x}; \mathbf{y}), where the supremum is over all input distributions for which some (unbiased) estimator of η\boldsymbol{\eta} from z\mathbf{z} achieves expected distortion ≤D\leq D. By convention C(D)=+∞\mathcal{C}(D) = +\infty when DD is below the open-loop CRB.

Read literally: C(D)\mathcal{C}(D) is the largest communication rate that is compatible with a guarantee on sensing accuracy. It interpolates between two extremes: C(∞)\mathcal{C}(\infty) is the pure-communication Shannon capacity (no sensing constraint); lim⁡D→Dmin⁡C(D)\lim_{D \to D_{\min}}\mathcal{C}(D) is the sensing-limited rate where all degrees of freedom that are not required to meet the MSE floor are given to communication.

Theorem: Capacity–Distortion for the Gaussian MIMO ISAC Channel

Consider a MIMO Gaussian channel with communication output y=Hx+w\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{w} and sensing output z=A(η)x+wr\mathbf{z} = \mathbf{A}(\boldsymbol{\eta})\mathbf{x} + \mathbf{w}_r, where A(η)\mathbf{A}(\boldsymbol{\eta}) encodes the target signature. Under a transmit power constraint tr(Rx)≤Pt\text{tr}(\mathbf{R}_x) \leq P_t, the capacity–distortion function is C(D)=max⁡Rx⪰0  log⁡2det⁡ ⁣(I+1σ2HRxHH)\mathcal{C}(D) = \max_{\mathbf{R}_x \succeq 0}\; \log_2\det\!\left(\mathbf{I} + \tfrac{1}{\sigma^2}\mathbf{H} \mathbf{R}_x \mathbf{H}^{H}\right) subject to tr(Rx)≤Pt,tr ⁣(J−1(Rx))≤D,\text{tr}(\mathbf{R}_x) \leq P_t, \qquad \text{tr}\!\left(\mathbf{J}^{-1}(\mathbf{R}_x)\right) \leq D, where J(Rx)=1σ2∑iAiRxAiH\mathbf{J}(\mathbf{R}_x) = \frac{1}{\sigma^2} \sum_i \mathbf{A}_i \mathbf{R}_x \mathbf{A}_i^H is the Fisher information matrix computed from the linearized signature.

The Pareto boundary C(D)\mathcal{C}(D) is the image of a convex program in the transmit covariance. The communication side wants Rx\mathbf{R}_x to be aligned with the strong eigendirections of HHH\mathbf{H}^{H}\mathbf{H} (water-filling). The sensing side wants Rx\mathbf{R}_x to excite the directions that illuminate the target and increase J\mathbf{J}. When the two objectives are aligned (e.g., the target and a user are along the same bearing), there is no tradeoff; when they conflict, a Lagrange multiplier μ\mu trades tr(J−1)\text{tr}(\mathbf{J}^{-1}) against II. Sweeping μ\mu traces the whole capacity–distortion curve.

Show Hint

Form the Lagrangian L=log⁡2det⁡(I+1σ2HRxHH)−λ(tr(Rx)−Pt)−μ(tr(J−1)−D)L = \log_2\det(\mathbf{I} + \frac{1}{\sigma^2}\mathbf{H}\mathbf{R}_x\mathbf{H}^H) - \lambda(\text{tr}(\mathbf{R}_x) - P_t) - \mu(\text{tr}(\mathbf{J}^{-1}) - D).

Use ∂log⁡det⁡/∂Rx\partial\log\det/\partial\mathbf{R}_x and the matrix identity ∂tr(F−1)/∂Rx=−F−1(∂F/∂Rx)F−1\partial\text{tr}(\mathbf{F}^{-1})/\partial\mathbf{R}_x = -\mathbf{F}^{-1}(\partial\mathbf{F}/\partial\mathbf{R}_x)\mathbf{F}^{-1}.

The KKT conditions give a water-filling-like fixed-point equation in the eigenbasis of a weighted sum of HHH\mathbf{H}^H\mathbf{H} and the sensing signature.

Proof
Form the Lagrangian

The program is convex in Rx⪰0\mathbf{R}_x \succeq 0: log⁡det⁡\log\det is concave and tr(J−1(Rx))\text{tr}(\mathbf{J}^{-1}(\mathbf{R}_x)) is convex in Rx\mathbf{R}_x (by matrix fractional programming). Associate multipliers λ≥0\lambda \geq 0 with the power constraint and μ≥0\mu \geq 0 with the distortion constraint and form L(Rx,λ,μ)L(\mathbf{R}_x, \lambda, \mu).

Stationarity condition

Setting ∂L/∂Rx=0\partial L/\partial\mathbf{R}_x = 0 yields 1ln⁡2HH ⁣(I+HRxHHσ2)−1 ⁣Hσ2+μ G(Rx)=λI,\frac{1}{\ln 2}\mathbf{H}^{H}\!\left(\mathbf{I} + \tfrac{\mathbf{H}\mathbf{R}_x\mathbf{H}^{H}}{\sigma^2}\right)^{-1}\!\frac{\mathbf{H}}{\sigma^2} + \mu\,\mathcal{G}(\mathbf{R}_x) = \lambda\mathbf{I}, where G(Rx)\mathcal{G}(\mathbf{R}_x) is the matrix gradient of tr(J−1)\text{tr}(\mathbf{J}^{-1}).

Interpretation as a weighted water-filling

At μ=0\mu = 0 (no sensing constraint) the condition collapses to ordinary water-filling on the eigenvalues of HHH\mathbf{H}^{H}\mathbf{H}. For μ>0\mu > 0 the effective channel is HHH+μ σ2 G\mathbf{H}^{H}\mathbf{H} + \mu\,\sigma^2\,\mathcal{G}, shifting eigenmass from weakly-sensed directions to strongly-sensed ones.

Parametric sweep of $\mu$

For each μ≥0\mu \geq 0, solving the stationarity condition yields a covariance Rx⋆(μ)\mathbf{R}_x^\star(\mu). The corresponding (D(μ),C(D(μ)))(D(\mu), \mathcal{C}(D(\mu))) pair lies on the Pareto boundary. Sweeping μ\mu from 00 to ∞\infty traces the full curve from Shannon capacity to the CRB-limited sensing regime. ■\blacksquare

🎓CommIT Contribution(2023)

Capacity–Distortion Tradeoff for ISAC under Gaussian Channels

F. Liu, Y. Cui, C. Masouros, J. Xu, T. X. Han, Y. C. Eldar, G. Caire — IEEE Journal on Selected Areas in Communications, vol. 40, no. 6

Liu, Cui, Masouros, Xu, Han, Eldar, and Caire gave the first fully information-theoretic treatment of the ISAC capacity–distortion region on Gaussian MIMO channels. Their main result is the theorem stated above: the Pareto boundary is the image of a convex program in the transmit covariance, and is traced parametrically by a single Lagrange multiplier that trades mutual information against Fisher information. The consequence is sharp: on any Gaussian ISAC channel, a single scalar — the multiplier μ\mu — summarizes the operator's choice of tradeoff, and the optimal covariance is computable by a water-filling-like algorithm generalized to the Fisher-information objective.

The same paper also introduces the deterministic–random tradeoff: communication rewards random inputs (high entropy), while sensing rewards deterministic inputs (low variance of the waveform). The two objectives lead to genuinely different optimal input distributions in general, and the capacity–distortion function captures this tension exactly. Subsequent CommIT work (Liu & Caire, 2023) extended the analysis to the fundamental tradeoff between II and the sensing deterministic MSE under a constraint on the minimum mean square error of the parameter estimator — a rate–distortion formulation that underpins most subsequent ISAC capacity results.

isacinformation-theorycapacity-distortioncaireView Paper →
🎓CommIT Contribution(2023)

Fundamental Deterministic–Random Tradeoff in ISAC

F. Liu, Y. F. Liu, A. Li, C. Masouros, G. Caire — IEEE Transactions on Information Theory, vol. 69, no. 9

This companion paper formalizes the observation that communication rewards random signaling while sensing rewards deterministic signaling by deriving the capacity–distortion region for a dual-use MIMO Gaussian channel with a Gaussian test channel on the sensing side. The two operating points at the boundary of the region — full-random (Shannon-optimal) and full-deterministic (CRB-optimal) — are connected by a convex curve whose interior points are realized by signal dithering: a weighted sum of a deterministic sensing-optimal waveform and a random communication-optimal codebook. This is the operational recipe the rest of the chapter uses: every ISAC waveform we will encounter is, at heart, a dither between these two extremes.

isaccaireinformation-theoryditheringView Paper →
⚠️Engineering Note

Sensing KPIs in 3GPP Rel-19 Study Items

The 3GPP ISAC study items (TR 22.837 and TR 38.901 extensions) define three classes of sensing KPIs that 6G networks must report alongside throughput:

  1. Detection: target detection probability PdP_d at a specified false-alarm rate PfaP_{\text{fa}}, typically 10−610^{-6}.
  2. Resolution: minimum separable range, angle, and Doppler (Rayleigh-limited by bandwidth, aperture, and coherent dwell time).
  3. Accuracy: RMSE of estimated target parameters, bounded below by the CRB.

The capacity–distortion framework of Liu–Caire provides the theoretical foundation on which any of these KPIs can be scored jointly with communication rate. Vendors using legacy "separate radar, separate comm" designs cannot meaningfully answer "what rate am I giving up for a 10 cm accuracy improvement?" — but an operator running an ISAC-optimized BS can.

Practical Constraints
  • •

    3GPP TR 22.837 specifies sensing use cases: intrusion detection, vehicle monitoring, drone sensing.

  • •

    3GPP TR 38.901 extensions add the scatterer model needed for reflection-based sensing simulation.

  • •

    ITU-R IMT-2030 lists sensing as one of six core usage scenarios, alongside eMBB/URLLC/mMTC.

📋 Ref: 3GPP TR 22.837 (Rel-19), TR 38.901 (Rel-19 ext.)

Capacity–Distortion Pareto Curve

The Pareto boundary of the communication rate versus sensing MSE, traced by sweeping the Lagrange multiplier μ\mu from 0 (pure communication) to ∞\infty (pure sensing). The endpoints are the Shannon capacity and the CRB floor; interior points realize dithered waveforms.

Parameters
16
2
10
30

Example: Aligned vs Orthogonal Sensing: Two Extreme Cases

Consider a 16-antenna BS serving a single user at angle θu=0∘\theta_u = 0^\circ. The sensing target is at either (a) θt=0∘\theta_t = 0^\circ (aligned with the user) or (b) θt=90∘\theta_t = 90^\circ (orthogonal to the user). Compare the capacity–distortion tradeoff in the two cases.

Solution
(a) Aligned case — no tradeoff

When θt=θu\theta_t = \theta_u, illuminating the user automatically illuminates the target. The communication-optimal covariance Rx⋆=Pt a(0)aH(0)/Nt\mathbf{R}_x^\star = P_t\,\mathbf{a}(0)\mathbf{a}^H(0)/N_t is also sensing-optimal. The capacity–distortion curve degenerates to a single point: both functions reach their individual optima simultaneously. This is the ISAC dream, and is why high-mobility vehicles traveling along line-of-sight roads are the canonical ISAC use case.

(b) Orthogonal case — worst tradeoff

When θt⊥θu\theta_t \perp \theta_u, the user-beam covariance steers zero power toward the target. Achieving any finite sensing distortion requires adding a sensing beam a(90∘)aH(90∘)\mathbf{a}(90^\circ) \mathbf{a}^H(90^\circ) component, stealing power from the user. The tradeoff curve is a smooth concave function: at μ=0\mu = 0 only the user is served (capacity ≈log⁡2(1+NtSNR)\approx \log_2(1 + N_t\text{SNR}), infinite sensing MSE); at μ→∞\mu \to \infty only the target is illuminated (zero rate, minimum sensing MSE); in between the power splits continuously between the two beams.

General case

For arbitrary angular separation Δθ\Delta\theta, the tradeoff curve sits between the two extremes. The effective "ISAC gain" relative to a time-sharing baseline grows as Δθ\Delta\theta decreases: beams with small angular separation enjoy most of the joint-design advantage. ■\blacksquare

Key Takeaway

One number controls the ISAC tradeoff. On Gaussian MIMO channels, the Pareto boundary of rate versus sensing MSE is traced by a single Lagrange multiplier μ\mu. The operator picks μ\mu to express their preference; the optimal precoder follows automatically from a convex program. Massive MIMO makes the tradeoff favorable because the spatial aperture provides enough degrees of freedom that the two functions rarely compete for the same eigendirection.

Common Mistake: Time-Sharing Is Suboptimal for ISAC

Mistake:

A naive ISAC design alternates between dedicated communication slots and dedicated sensing slots — effectively time-sharing the aperture.

Correction:

Time-sharing achieves the rate-distortion pairs on the chord between the two single-function endpoints, which lies strictly below the Pareto boundary established in Theorem TCapacity–Distortion for the Gaussian MIMO ISAC Channel. The gap is the reason joint waveform design exists. For massive MIMO with many spatial DoF, the gap can be large: a single waveform that serves users and targets simultaneously exploits the spatial-spectral overlap that time-sharing throws away.

Quick Check

The capacity–distortion curve C(D)\mathcal{C}(D) is concave and non-increasing. What is the interpretation of its derivative −dC/dD-d\mathcal{C}/dD?

The slope of the best linear rate-distortion tradeoff.

The marginal communication rate one gains per unit of sensing-distortion budget relaxed.

The ratio of transmit power allocated to sensing vs communication.

The noise variance of the sensing receiver.

Correction:
The marginal communication rate one gains per unit of sensing-distortion budget relaxed.

This is exactly the Lagrange multiplier μ\mu: it measures how many bits/s/Hz of extra communication are enabled by relaxing the MSE constraint by one unit. An operator picks μ\mu based on how much sensing accuracy is worth.

Historical Note: From RadComm to Capacity–Distortion

1960s–2023

The idea that radar and communication might share a waveform is as old as pulse-Doppler radar, but the information-theoretic framing took another sixty years. Early results by Bell and Bello (1960s) treated combined radar-comm as a bit-embedding problem: how many bits can you hide in a radar pulse without degrading detection? The rate-distortion viewpoint came only after the massive MIMO era made it operationally meaningful: once a cellular base station had radar-grade apertures, the tradeoff between throughput and sensing accuracy became an engineering question worth optimizing. The Liu–Caire results of 2022–2023 are the current synthesis — the first treatment in which communication rate and sensing MSE are co-dimensions of a single Pareto region computable from first principles.

ISAC Fundamentals: Why the Same Array?Beampattern Design for Joint Comm–Sensing