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The Fundamental Question: How Much Can One Signal Do?

Every communication signal carries randomness (the data). Every sensing waveform benefits from determinism (known transmit signal improves matched-filter SNR). The information-theoretic question at the heart of ISAC is: how should a single signal balance randomness (for communication) and structure (for sensing)?

The Liu/Caire framework answers this by decomposing the transmit signal into a deterministic component (optimal for sensing) and a random component (carrying information). The CRB-rate region quantifies the achievable tradeoff.

Definition:
ISAC Capacity-Distortion Region

The ISAC capacity-distortion region is the set of achievable pairs $(R_c, D_s)$ where $R_c$ is the communication rate and $D_s$ is the sensing distortion (e.g., MSE of the scene estimate):

$\mathcal{C}_{\mathrm{ISAC}} = \left\{(R_c, D_s) \;\middle|\; \exists \, \mathbf{R}_x \succeq \mathbf{0}, \; \operatorname{tr}(\mathbf{R}_x) \leq P_t : R_c \leq C(\mathbf{R}_x), \; D_s \geq D(\mathbf{R}_x)\right\}$

where $C(\mathbf{R}_x) = \log\det(\mathbf{I} + \mathbf{H}\mathbf{R}_x\mathbf{H}^H / \sigma^2_{c})$ is the communication capacity and $D(\mathbf{R}_x) = \operatorname{tr}\!\left[(\mathbf{I} + \mathbf{A}\mathbf{R}_x\mathbf{A}^{H} / \sigma^2_{s})^{-1}\right]$ is the MMSE sensing distortion.

The boundary of $\mathcal{C}_{\mathrm{ISAC}}$ is the Pareto frontier: no system can simultaneously improve both metrics beyond this boundary. The shape depends on the alignment between $\mathbf{H}$ (communication channel) and $\mathbf{A}$ (sensing matrix).

🎓CommIT Contribution(2023)

ISAC Capacity-Distortion Tradeoff

Y. Xiong, F. Liu, Y. Cui, J. Yuan, G. Caire — IEEE Trans. Inform. Theory

Xiong, Liu, Cui, Yuan, and Caire established the rigorous information-theoretic framework for the ISAC tradeoff under Gaussian channels. Their key contributions:

Deterministic-random decomposition: The optimal ISAC signal has a deterministic component (known to the sensing receiver, maximising Fisher information) and a random component (carrying communication data, maximising mutual information).
CRB-rate region: They characterised the complete Pareto frontier between communication rate and sensing CRB, showing that the tradeoff is governed by the eigenvalue overlap between the communication channel and sensing steering matrices.
Gaussian optimality: Gaussian signalling achieves the boundary of the capacity-distortion region --- a fortunate result since Gaussian is also the standard model for communication (Shannon) and radar (Van Trees).

This framework connects ITA Chapter 18 (information-theoretic limits) to the imaging model $\mathbf{y} = \mathbf{A}\mathbf{c} + \mathbf{w}$ of this book: the "sensing side" of ISAC IS the forward model, and $D_s$ is the MMSE of the scene estimate.

ISACcapacity-distortioninformation-theoryCommITView Paper →

Theorem: ISAC Rate-Distortion Pareto Frontier

For an ISAC system with $N_t$ transmit antennas, communication channel $\mathbf{H} \in \mathbb{C}^{K_u \times N_t}$ , and sensing matrix $\mathbf{A} \in \mathbb{C}^{M_s \times N_t}$ , the Pareto-optimal transmit covariance $\mathbf{R}_x^*$ solves:

$\max_{\mathbf{R}_x \succeq 0} \; C(\mathbf{R}_x) + \mu \, \mathrm{MI}_{\mathrm{sens}}(\mathbf{R}_x) \quad \text{s.t.} \quad \operatorname{tr}(\mathbf{R}_x) \leq P_t$

where $\mu \geq 0$ is a Lagrange multiplier that traces the Pareto frontier. At $\mu = 0$ , the solution reduces to the communication-optimal waterfilling. As $\mu \to \infty$ , the solution maximises sensing MI.

When $\mathbf{H}$ and $\mathbf{A}$ share the same column space, the tradeoff is mild. When they are orthogonal, the tradeoff is severe.

The ISAC tradeoff is fundamentally about spatial resource allocation: the limited degrees of freedom ( $N_t$ antennas) must serve communication users and sensing targets. When users and targets are in similar directions, both functions benefit from the same beams --- the tradeoff nearly vanishes.

Proof

Lagrangian formulation

The Pareto frontier of the bi-objective problem $\max(C, \mathrm{MI}_s)$ is obtained by scalarising: $\mathcal{L} = C(\mathbf{R}_x) + \mu \, \mathrm{MI}_s(\mathbf{R}_x) - \nu(\operatorname{tr}(\mathbf{R}_x) - P_t)$ .

KKT conditions

Setting $\partial\mathcal{L}/\partial\mathbf{R}_x = \mathbf{0}$ : $\mathbf{H}^H(\sigma^2_{c}\mathbf{I} + \mathbf{H}\mathbf{R}_x\mathbf{H}^H)^{-1}\mathbf{H} + \mu\mathbf{A}^{H}(\sigma^2_{s}\mathbf{I} + \mathbf{A}\mathbf{R}_x\mathbf{A}^{H})^{-1}\mathbf{A} = \nu\mathbf{I}.$ This is a generalised waterfilling across the joint communication-sensing eigenspace.

Interpretation

The solution allocates power to eigenvectors that serve both functions. At $\mu = 0$ , all power goes to the communication eigenmodes (standard waterfilling). As $\mu$ increases, power shifts toward sensing eigenmodes. The rate at which the frontier curves reflects the alignment between $\mathbf{H}$ and $\mathbf{A}$ . $\blacksquare$

Definition:
CRB for ISAC Target Estimation

For an ISAC system estimating target angle $\theta$ with transmit covariance $\mathbf{R}_x$ , the CRB is:

$\mathrm{CRB}(\theta) = \frac{\sigma^2}{2 L \, \operatorname{Re}\!\left[\alpha^2 \dot{\mathbf{a}}^H(\theta)\mathbf{R}_x \dot{\mathbf{a}}(\theta) - \frac{|\alpha^2 \dot{\mathbf{a}}^H(\theta)\mathbf{R}_x\mathbf{a}(\theta)|^2}{\alpha^2 \mathbf{a}^{H}(\theta)\mathbf{R}_x\mathbf{a}(\theta) + \sigma^2}\right]}$

where $\dot{\mathbf{a}}(\theta) = \partial\mathbf{a}/\partial\theta$ is the steering vector derivative and $L$ is the number of snapshots.

The CRB depends on the transmit covariance through the beampattern energy $\mathbf{a}^{H}\mathbf{R}_x\mathbf{a}$ and its angular gradient.

Minimising CRB requires concentrating energy toward the target AND having a steep beampattern gradient (sharp beam). This creates tension with communication, which may require broad beams toward users in different directions.

Theorem: Optimality of Gaussian Signalling for ISAC

For the ISAC system with Gaussian noise, the optimal transmit signal distribution that maximises communication capacity while satisfying a sensing MI constraint is Gaussian:

$\mathbf{x} \sim \mathcal{CN}(\mathbf{0}, \mathbf{R}_x^*)$

where $\mathbf{R}_x^*$ solves the capacity-distortion tradeoff SDP. Gaussian signalling simultaneously achieves the capacity of the communication channel and maximises Fisher information for Gaussian-noise sensing.

Gaussian signals are capacity-achieving for Gaussian channels (Shannon) and maximise Fisher information for Gaussian-noise parameter estimation (Van Trees). This fortunate coincidence means there is no loss in using the same Gaussian signal for both functions --- the dual use is information-theoretically optimal.

Proof

Communication optimality

By Shannon's capacity theorem, for a Gaussian channel $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{w}$ with $\mathbf{w} \sim \mathcal{CN}(\mathbf{0}, \sigma^2\mathbf{I})$ , the capacity-achieving input is $\mathbf{x} \sim \mathcal{CN}(\mathbf{0}, \mathbf{R}_x)$ .

Sensing optimality

For sensing, the MI between the scene $\mathbf{c}$ and observation $\mathbf{y}$ under Gaussian prior and noise is maximised by Gaussian input (maximum entropy for given covariance). The Fisher information for deterministic parameter estimation depends only on $\mathbf{R}_x$ , not on the specific distribution.

Conclusion

Since Gaussian input is optimal for both objectives and the tradeoff is parameterised by $\mathbf{R}_x$ alone, the capacity-distortion boundary is achieved by Gaussian signalling. $\blacksquare$

Example: ISAC with Orthogonal Communication and Sensing Channels

An ISAC system has $N_t = 4$ antennas. The communication user is at broadside ( $\theta_u = 0^\circ$ ) and the target is at $\theta_t = 60^\circ$ . The channel is $\mathbf{h} = \mathbf{a}(0^\circ)$ and the sensing steering vector is $\mathbf{a}(60^\circ)$ . Compute $\mathbf{h}^H\mathbf{a}(60^\circ)$ and discuss the tradeoff severity.

Solution

Inner product

$\mathbf{h}^H\mathbf{a}(60^\circ) = \sum_{m=0}^{3} e^{j\pi m \sin(60^\circ)} = \sum_{m=0}^{3} e^{j\pi m \cdot 0.866}$ . This geometric sum gives $|S|^2/N_t^{2} \approx 0.048$ .

Tradeoff analysis

The communication and sensing directions have correlation $\approx 5\%$ . A beam toward the user provides very little sensing energy at $60^\circ$ and vice versa. This is a severe tradeoff: dedicating power to one function provides almost no benefit to the other.

Joint beamforming improvement

With $N_t = 4$ , we have 4 spatial DOF. Two are consumed by the communication and sensing beams; two remain for sidelobe control. Joint optimisation improves over naive splitting, but the fundamental orthogonality limits the gain.

Example: ISAC with Aligned Channels — No Tradeoff

Consider a $2 \times 2$ MIMO ISAC system with $\mathbf{H} = \mathbf{I}_2$ and sensing direction $\mathbf{a} = [1, 1]^T/\sqrt{2}$ . Show that the Pareto frontier degenerates to a single point.

Solution

Communication rate

$R_c = \log_2(1 + p_1/\sigma^2) + \log_2(1 + p_2/\sigma^2)$ with $p_1 + p_2 = P_t$ . Maximised by waterfilling: $p_1 = p_2 = P_t/2$ .

Sensing MI

$\mathrm{MI}_s = \log_2(1 + (p_1 + p_2)/(2\sigma^2_{s})) = \log_2(1 + P_t/(2\sigma^2_{s}))$ . This is constant for all power allocations satisfying $p_1 + p_2 = P_t$ .

Conclusion

Since the sensing direction $[1,1]^T/\sqrt{2}$ receives equal energy regardless of the power split, there is no tradeoff. The Pareto frontier is a single point. This occurs whenever $\mathbf{a}$ lies in the column space of $\mathbf{H}$ with equal projections on all eigenmodes.

Common Mistake: Confusing CRB and MI as Sensing Metrics

Mistake:

Using the CRB as the sensing metric when the goal is scene reconstruction (imaging), or using MI when the goal is single- target tracking.

Correction:

CRB is appropriate for estimating a small number of deterministic parameters (angle, range, velocity of individual targets). MI is appropriate when the scene is a random field to be reconstructed --- exactly the imaging scenario of this book. For ISAC imaging, use MI or MMSE distortion. For ISAC tracking, use CRB. The optimal waveform design differs significantly between the two metrics.

Quick Check

When the communication channel $\mathbf{H}$ and the sensing matrix $\mathbf{A}$ have orthogonal column spaces, the ISAC Pareto frontier is:

A straight line connecting the two extreme points

A convex curve above the line

A single point

Correction:

A straight line connecting the two extreme points

With orthogonal subspaces, power allocated to communication provides zero sensing benefit and vice versa. The tradeoff is linear --- no "free sensing" is possible.

⚠️Engineering Note

Practical CRB Gaps in ISAC Systems

The information-theoretic CRB assumes perfect knowledge of the target model (single point target, known noise variance, no clutter). In practice, ISAC sensing performance is degraded by:

Clutter: Environmental reflections compete with target echoes. The effective $\sigma^2$ includes clutter power.
Finite alphabet: Communication uses QAM, not continuous Gaussian. The gap is typically 1--3 dB.
Channel estimation errors: The communication channel $\mathbf{H}$ is estimated, not perfectly known.
Waveform constraints: PAPR limits, constant-modulus constraints, and guard intervals reduce effective sensing energy.

Practical ISAC systems typically operate 5--10 dB above the CRB, depending on clutter severity.

Key Takeaway

The ISAC capacity-distortion region defines the fundamental limits of joint sensing and communication. The Pareto frontier is traced by varying the Lagrange multiplier $\mu$ weighting sensing vs. communication. Gaussian signalling is optimal. The tradeoff severity depends on the alignment between $\mathbf{H}$ and $\mathbf{A}$ --- and connecting this to the imaging forward model is the unique contribution of the RFI perspective.

ISAC Information-Theoretic Limits

The Fundamental Question: How Much Can One Signal Do?

Definition: ISAC Capacity-Distortion Region

ISAC Capacity-Distortion Tradeoff

Theorem: ISAC Rate-Distortion Pareto Frontier

Lagrangian formulation

KKT conditions

Interpretation

Definition: CRB for ISAC Target Estimation

Theorem: Optimality of Gaussian Signalling for ISAC

Communication optimality

Sensing optimality

Conclusion

Example: ISAC with Orthogonal Communication and Sensing Channels

Inner product

Tradeoff analysis

Joint beamforming improvement

Example: ISAC with Aligned Channels — No Tradeoff

Communication rate

Sensing MI

Conclusion

Common Mistake: Confusing CRB and MI as Sensing Metrics

Quick Check

Practical CRB Gaps in ISAC Systems

Key Takeaway

Definition:
ISAC Capacity-Distortion Region

Definition:
CRB for ISAC Target Estimation