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From Capacity–Distortion to a Practical Design

The capacity–distortion function of Section 24.2 gives the theoretical Pareto boundary. It does not tell an engineer how to synthesize a waveform that actually lives on that boundary. This section fills the gap: we derive a semidefinite program (SDP) that designs the transmit signal covariance $\mathbf{R}_x$ to match a desired sensing beampattern while guaranteeing each communication user a minimum SINR. The SDP is the standard workbench for all practical ISAC precoder designs, and generalizes cleanly to cell-free architectures in Section 24.4.

Definition:
Transmit Beampattern

For a transmit signal $\mathbf{x}(t)$ with covariance $\mathbf{R}_x = \mathbb{E}[\mathbf{x}(t)\mathbf{x}^H(t)]$ , the transmit beampattern in the direction $\theta$ is $P(\theta; \mathbf{R}_x) \triangleq \mathbf{a}^H(\theta) \mathbf{R}_x \mathbf{a}(\theta),$ where $\mathbf{a}(\theta) \in \mathbb{C}^{N_t}$ is the transmit array steering vector at angle $\theta$ . The total radiated power is $\int_{-\pi/2}^{\pi/2} P(\theta; \mathbf{R}_x)\,d\theta \propto \text{tr}(\mathbf{R}_x)$ .

The beampattern is a quadratic form in $\mathbf{a}(\theta)$ . Because $\mathbf{R}_x$ enters linearly, any design objective or constraint that is a linear functional of $\mathbf{R}_x$ is compatible with semidefinite programming.

Semidefinite Relaxation (SDR)

A convex relaxation technique for nonconvex quadratic problems in a vector $\mathbf{v}$ : lift the optimization variable to the rank-one matrix $\mathbf{V} = \mathbf{v}\mathbf{v}^H \succeq 0$ and drop the rank-one constraint. The resulting SDP is convex. If the optimal $\mathbf{V}^\star$ has rank one, SDR is tight and $\mathbf{v}^\star$ is recovered by eigendecomposition; otherwise a randomization step extracts an approximate solution.

Definition:
Beampattern-Matching Problem

Given a desired beampattern $P_d(\theta)$ (e.g., a flat mainlobe over the target region and low sidelobes elsewhere), the beampattern matching problem is $\min_{\mathbf{R}_x \succeq 0, \eta > 0}\; \sum_{i=1}^{M} |\eta P_d(\theta_i) - \mathbf{a}^H(\theta_i) \mathbf{R}_x \mathbf{a}(\theta_i)|^2$ subject to per-antenna or sum power constraints on $\mathbf{R}_x$ and user SINR constraints discussed below.

The scale factor $\eta$ is necessary because absolute beampattern levels depend on the total transmitted power; only the shape is intrinsic to the problem.

Theorem: Joint ISAC Precoder Design via SDR

Let the downlink channel to user $k$ be $\mathbf{H}_{k}$ with target SINR $\gamma_k$ , and let $P_d(\theta_i)$ be the desired sensing beampattern sampled at $M$ angles $\theta_1, \ldots, \theta_M$ . The joint ISAC precoder design problem $\min_{\mathbf{R}_x \succeq 0, \{\mathbf{R}_k\}, \eta}\; \sum_{i=1}^{M} w_i |\eta P_d(\theta_i) - \mathbf{a}^H(\theta_i)\mathbf{R}_x\mathbf{a}(\theta_i)|^2$ subject to $\text{(i) SINR:}\quad \frac{\mathbf{H}_{k}^{H} \mathbf{R}_k \mathbf{H}_{k}}{\sum_{j \neq k} \mathbf{H}_{k}^{H} \mathbf{R}_j \mathbf{H}_{k} + \sigma^2} \geq \gamma_k,\quad \forall k,$ $\text{(ii) Consistency:}\quad \mathbf{R}_x = \sum_{k=1}^{K} \mathbf{R}_k,$ $\text{(iii) Power:}\quad \text{tr}(\mathbf{R}_x) \leq P_t,$ is a convex SDP in the per-user covariances $\mathbf{R}_k \succeq 0$ . If each $\mathbf{R}_k^\star$ has rank 1, its principal eigenvector is the optimal per-user beamformer.

The SINR constraints are linear in $\mathbf{R}_k$ : rearranging gives $\mathbf{H}_{k}^{H} \mathbf{R}_k \mathbf{H}_{k} \geq \gamma_k(\sum_{j\neq k}\mathbf{H}_{k}^{H} \mathbf{R}_j \mathbf{H}_{k} + \sigma^2)$ . Every term is linear in the matrix variables — no rank-one constraint is imposed on $\mathbf{R}_k$ , so the lifted formulation is convex by construction. The objective is a convex quadratic in $\mathbf{R}_x$ . The miracle is that, as shown by Liu et al., the optimal solution often has rank-one $\mathbf{R}_k^\star$ automatically, making SDR tight.

Show Hint

Expand the SINR ratio and multiply through by the denominator to linearize it in $\mathbf{R}_k$ .

The objective is $\\|\\mathbf{p}_d - \\mathbf{p}(\\mathbf{R}_x)\\|^2$ with $\\mathbf{p}(\\mathbf{R}_x)$ linear in $\\mathbf{R}_x$ — a convex quadratic.

Feasibility of each user's rank-one reduction follows because the dual KKT conditions yield complementary slackness with a dual matrix of rank $N_t - 1$ .

Proof

Linearize SINR

Rewrite the SINR constraint as $\mathbf{H}_{k}^{H} \mathbf{R}_k \mathbf{H}_{k} - \gamma_k \sum_{j\neq k} \mathbf{H}_{k}^{H} \mathbf{R}_j \mathbf{H}_{k} \geq \gamma_k \sigma^2.$ Both sides are linear in the matrix variables. This is a Linear Matrix Inequality (LMI) in $\{\mathbf{R}_k\}$ .

Convexify the beampattern objective

The objective is $\sum_i w_i (\eta P_d(\theta_i) - \mathbf{a}_i^H \mathbf{R}_x \mathbf{a}_i)^2$ , which is a convex quadratic in $(\eta, \mathbf{R}_x)$ . It is typically implemented by introducing slack variables $s_i$ with $s_i \geq |\eta P_d(\theta_i) - \mathbf{a}_i^H \mathbf{R}_x \mathbf{a}_i|$ and minimizing $\sum_i w_i s_i^2$ — a second-order cone program embedded inside the SDP.

Solve as an SDP

The resulting problem has the form $\min\; f_0(\mathbf{R}_x, \{\mathbf{R}_k\}, \eta, \mathbf{s})\quad\text{s.t.}\quad \{\mathbf{R}_k\} \succeq 0,\; \text{LMIs, SOCs, equalities}.$ Any interior-point SDP solver (SeDuMi, SDPT3, MOSEK) returns an optimal solution in polynomial time. For $N_t = 64$ and $K = 8$ , modern solvers handle this in under a second.

Extract the precoders

From $\mathbf{R}_k^\star$ , the per-user beamformer $\mathbf{v}_k$ is obtained by principal eigendecomposition: $\mathbf{v}_k = \sqrt{\lambda_1^{(k)}}\,\mathbf{u}_1^{(k)}$ . If $\mathbf{R}_k^\star$ has rank $> 1$ , Gaussian randomization (Luo–Ma–So–Yang) recovers a near-optimal rank-one solution with provable approximation ratio. $\blacksquare$

Joint ISAC Beampattern Design via SDR

Complexity: Interior-point SDP:

\mathcal{O}(N_t^{6.5})

worst case;

\mathcal{O}(N_t^{3.5})

with structured solvers. Rank-one typical on LOS channels.

Input: Channels

\{\mathbf{h}_k\}_{k=1}^{K}

, target SINRs

\{\gamma_k\}

,

desired beampattern samples

\{(\theta_i, P_d(\theta_i))\}_{i=1}^{M}

,

power budget

P_t

, noise variance

\sigma^2

.

Output: Per-user beamformers

\{\mathbf{v}_k\}_{k=1}^{K}

and sensing covariance

\mathbf{R}_s

.

1. Build steering matrix

\mathbf{A} \in \mathbb{C}^{N_t\times M}

with columns

\mathbf{a}(\theta_i)

.

2. Solve the SDP

3.

\quad\min_{\{\mathbf{R}_k\}\succeq 0,\,\mathbf{R}_s \succeq 0,\,\eta} \;\sum_i w_i (\eta P_d(\theta_i) - \mathbf{a}_i^H \mathbf{R}_x \mathbf{a}_i)^2

4.

\quad\text{s.t.}\quad \mathbf{R}_x = \sum_k \mathbf{R}_k + \mathbf{R}_s,\; \text{tr}(\mathbf{R}_x) \leq P_t,

5.

\quad\quad \mathbf{H}_{k}^{H}\mathbf{R}_k\mathbf{H}_{k} \geq \gamma_k\!\left(\textstyle\sum_{j\neq k}\mathbf{H}_{k}^{H}\mathbf{R}_j\mathbf{H}_{k} + \mathbf{h}_k^H\mathbf{R}_s\mathbf{h}_k + \sigma^2\right)

.

6. for

k = 1, \ldots, K

do

7.

\quad \lambda_1^{(k)}, \mathbf{u}_1^{(k)} \leftarrow

principal eigenpair of

\mathbf{R}_k^\star

8.

\quad \mathbf{v}_k \leftarrow \sqrt{\lambda_1^{(k)}}\,\mathbf{u}_1^{(k)}

9. end for

10. return

\{\mathbf{v}_k\}

and

\mathbf{R}_s^\star

.

The extra sensing-only covariance $\mathbf{R}_s$ (not tied to any user) absorbs the "dither" term of Liu–Caire: the deterministic sensing-optimal component added on top of the random communication beams. Without $\mathbf{R}_s$ , the design is restricted to per-user beamforming and cannot realize the full Pareto boundary.

Null-Forming Toward Communication Users

A dual viewpoint on the SDP above is that the sensing beam $\mathbf{v}_s$ should form nulls in the directions of communication users, so that the sensing waveform does not appear as interference to the comm receivers. In the large-array regime this nulling is essentially free: with $N_t \gg K$ , the sensing beam lives in the $(N_t - K)$ -dimensional orthogonal complement of the channel subspace, and the sidelobe penalty for the $K$ nulls is a small loss of $\sim K/N_t$ in the sensing array gain. This is another concrete statement of why massive MIMO is the natural ISAC platform.

SDR-Designed ISAC Beampattern

Compare the ideal target beampattern (flat-top over the target region), the unconstrained sensing-only SDR solution, and the joint ISAC solution that also satisfies per-user SINR constraints. The sidelobe level and mainlobe width trade off against the spatial room left for communication.

Parameters

N_t

32

Target center angle (deg)30

Target mainlobe width (deg)10

K

2

Per-user SINR target (dB)10

Example: Two-User ISAC with a 32-Element ULA

A BS with $N_t = 32$ antennas serves two users at angles $\theta_1 = -20^\circ$ and $\theta_2 = 20^\circ$ with target SINR 10 dB each. The radar must illuminate a $\pm 5^\circ$ mainlobe centered at $\theta_t = 60^\circ$ . Qualitatively describe the solution of the SDR problem.

Solution

Decompose the degrees of freedom

Two communication users consume 2 spatial DoF via (soft) ZF. The remaining $32 - 2 = 30$ DoF form the null space in which the sensing beam can live without interfering with comm. The sensing beam's effective aperture in its orthogonal subspace has array gain close to $32 - 2 = 30$ , i.e., only 0.28 dB below the unconstrained 32-element gain.

SDR solution structure

Solving the SDP yields two rank-one user covariances $\mathbf{R}_1 = p_1 \mathbf{v}_1\mathbf{v}_1^H$ and $\mathbf{R}_2 = p_2 \mathbf{v}_2\mathbf{v}_2^H$ where $\mathbf{v}_k$ is approximately the regularized zero-forcer for user $k$ . The sensing covariance $\mathbf{R}_s$ is a low-rank matrix with energy concentrated along $\mathbf{a}(60^\circ)$ and its neighbors, projected onto the orthogonal complement of $\text{span}\{\mathbf{h}_1, \mathbf{h}_2\}$ .

Power split

With the target SINR of 10 dB, communication consumes $\approx\gamma/(1+\gamma)$ times the per-user power budget; the rest goes to $\mathbf{R}_s$ . For $\gamma = 10$ , comm gets $\approx 10/11 \approx 91\%$ of the power per user; sensing takes whatever is left of the total budget after subtracting the two user contributions. Reducing the target SINR frees more power for sensing — this is the Lagrange multiplier from Theorem TCapacity–Distortion for the Gaussian MIMO ISAC Channel in its operational form. $\blacksquare$

🚨Critical Engineering Note

Monostatic Self-Interference and the Full-Duplex Problem

The monostatic ISAC model in Section 24.1 assumes the base station can receive echoes while transmitting. In practice, colocated transmit and receive chains suffer from enormous self-interference: the direct coupling between the Tx array and the Rx array is typically 80–100 dB above the weakest detectable target echo. Three mitigation strategies are used in practice, each with tradeoffs:

Spatial isolation: physically separate Tx and Rx subarrays (10–20 dB suppression, at the cost of doubling antenna count).
Analog cancellation: subtract a scaled copy of the Tx signal before the LNA (30–40 dB suppression, requires per-path calibration).
Digital cancellation: subtract a model of the leakage in baseband (20–30 dB, nonlinearities limit further gain).

Cumulative 70–90 dB is achievable with state-of-the-art full-duplex designs. Bistatic ISAC — Section 24.4 — sidesteps the problem entirely by using a spatially separated sensing receiver, which is the main operational argument for cell-free ISAC architectures.

Practical Constraints

•
Typical full-duplex self-interference floor: 70–90 dB suppression end-to-end.
•
Each 10 dB of residual self-interference costs ~5 dB of maximum target range.
•
Tx/Rx array isolation of $> 40$ dB requires physical separation $> 0.5\lambda$ for bulk coupling.

📋 Ref: 3GPP TR 38.858 (full-duplex NR study)

Common Mistake: SDR Is Not Always Tight — Randomization May Be Needed

Mistake:

A student implements the SDR and finds that the optimal $\mathbf{R}_k^\star$ has rank 2 or 3; they conclude the SDP is broken.

Correction:

SDR tightness is not guaranteed a priori. On line-of-sight-dominated channels with few users, the per-user covariances $\mathbf{R}_k^\star$ are often rank one by the KKT conditions; on scattered channels with many users, they can be higher rank. When rank $> 1$ , use Gaussian randomization: draw random vectors from $\mathcal{CN}(\mathbf{0}, \mathbf{R}_k^\star)$ , scale each to satisfy the SINR constraint, and pick the best. Luo–Ma–So–Yang give a provable constant-factor approximation ratio in this setting.

Quick Check

A 64-element BS must serve 8 users at specified SINRs and also illuminate a single target beam. Roughly how many spatial DoF are available to the sensing beam after ZF-style user constraints?

1

8

56

64