Ferkans — Interactive Telecom Tutor

One Precoder, Two Jobs

A MIMO-OTFS-ISAC transmitter has one precoder $\mathbf{F}$ and two jobs: deliver information to $K$ communication users with high rate, and illuminate the sensing scene so that targets can be estimated with low variance. These are not the same objective. A precoder matched perfectly to the comms channels may leave the sensing directions in a null; a precoder that illuminates all directions equally may waste spatial diversity for the users. The joint beamforming problem picks a single $\mathbf{F}$ that negotiates the two — and this section shows how.

Definition:
Joint ISAC Beamformer

Consider a BS with $N_t$ transmit antennas serving $K$ communication users and sensing $T_{\text{tgt}}$ targets. The transmit signal on the DD grid is $\mathbf{x}[\ell, k] \;=\; \mathbf{F}_c\, \mathbf{s}_c[\ell, k] \,+\, \mathbf{F}_s\, \mathbf{s}_s[\ell, k] \;\in\; \mathbb{C}^{N_t},$ where $\mathbf{F}_c \in \mathbb{C}^{N_t \times K}$ is the communication precoder (one column per user), $\mathbf{F}_s \in \mathbb{C}^{N_t \times N_s^{\text{sens}}}$ is the sensing precoder (one column per sensing stream), and $\mathbf{s}_c, \mathbf{s}_s$ are the DD-domain data and sensing waveforms respectively. The joint beamformer is the stacked matrix $\mathbf{F} = [\mathbf{F}_c, \mathbf{F}_s]$ .

The transmit covariance is $\mathbf{R}_x \;=\; \mathbf{F}\mathbf{F}^H \;=\; \mathbf{F}_c \mathbf{F}_c^H + \mathbf{F}_s \mathbf{F}_s^H.$ The comms objective depends on $\mathbf{F}$ directly (through user channel products); the sensing objective depends only on $\mathbf{R}_x$ .

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Theorem: Sensing Depends Only on the Covariance

Let $\mathbf{H}_{i} = a_i \mathbf{a}_r(\theta_i) \mathbf{a}_t(\phi_i)^H$ be the per-path MIMO-OTFS channel. The Fisher information for target parameters $\Theta = \{(\tau_i, \nu_i, \phi_i, \theta_i, a_i)\}_{i=1}^{P}$ depends on the transmit signal only through the covariance $\mathbf{R}_x = \mathbb{E}[\mathbf{x}\mathbf{x}^H]$ : $[J(\Theta)]_{pq} \;=\; \frac{2}{\sigma_w^2} \, \Re\left\{\mathrm{tr}\!\left(\frac{\partial \mathbf{H}_{p}}{\partial \Theta_p} \mathbf{R}_x \frac{\partial \mathbf{H}_{q}^{H}}{\partial \Theta_q}\right)\right\}.$

Consequence. Two beamformers $\mathbf{F}, \mathbf{F}'$ with the same $\mathbf{F}\mathbf{F}^H$ produce the same sensing performance.

Radar cares only about what energy is transmitted in each direction, not about which stream it came from or how it was coded. Comms, on the other hand, cares about the full precoder structure because different columns serve different users and the columns' inner products determine inter-user interference. This asymmetry — scalar vs. matrix-valued objective — is the source of the design flexibility in joint ISAC beamforming.

Proof

Expected received signal

Per path, the received signal (after averaging over data) is $\mathbf{y}_i = \mathbf{H}_{i} \mathbf{x} + \mathbf{w}$ with covariance $\mathbf{H}_{i} \mathbf{R}_x \mathbf{H}_{i}^{H} + \sigma_w^2 \mathbf{I}$ .

Log-likelihood

The log-likelihood of $\Theta$ given the observations is, for Gaussian noise and the sparse-path model, a quadratic form in $(\partial \mathbf{H}_{i} / \partial \Theta_i) \mathbf{x}$ .

Average

Taking the expectation over the data streams $\mathbf{s}_c, \mathbf{s}_s$ converts $\mathbf{x}\mathbf{x}^H \to \mathbf{R}_x$ . The Fisher information depends on the data covariance only.

Covariance equivalence

Any two precoders with $\mathbf{F}\mathbf{F}^H = \mathbf{F}'\mathbf{F}'^H$ produce identical $J(\Theta)$ and hence identical CRLBs. $\blacksquare$

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Key Takeaway

Sensing cares about $\mathbf{R}_x$ ; comms cares about $\mathbf{F}$ . This structural asymmetry is the lever for joint beamforming: among all precoders yielding a target $\mathbf{R}_x$ (same sensing performance), pick the one that maximizes comms rate. This immediately reduces the joint problem to a well-posed constrained optimization.

Definition:
Joint Beamforming Optimization

The joint ISAC beamforming problem for MIMO-OTFS is $\begin{aligned} \max_{\mathbf{F}} \;&\; R_{\text{sum}}(\mathbf{F}) \\ \text{s.t.} \;&\; \mathrm{CRB}_{\text{sens}}(\mathbf{F}\mathbf{F}^H) \leq \gamma, \\ \;&\; \mathrm{tr}(\mathbf{F}\mathbf{F}^H) \leq P_t. \end{aligned}$ Here $R_{\text{sum}}$ is the sum rate across $K$ users, $\mathrm{CRB}_{\text{sens}}$ is a scalar aggregate of the sensing CRLB matrix (e.g., trace of the CRB matrix for target positions), $\gamma$ is the sensing tolerance, and $P_t$ is the total transmit power.

Equivalent formulation using the covariance: $\begin{aligned} \max_{\mathbf{R}_x \succeq 0} \;&\; R_{\text{sum}}(\mathbf{R}_x) \\ \text{s.t.} \;&\; \mathrm{CRB}_{\text{sens}}(\mathbf{R}_x) \leq \gamma, \quad \mathrm{tr}(\mathbf{R}_x) \leq P_t. \end{aligned}$ The optimal $\mathbf{F}^\star$ is obtained by Cholesky factorization of $\mathbf{R}_x^\star$ . Semidefinite programming (SDP) solves the covariance formulation globally when $R_{\text{sum}}$ is SDP-representable (as it is for MMSE-equalized ZF users).

The Convexity Check

The covariance formulation above is a convex problem when $R_{\text{sum}}(\mathbf{R}_x)$ is concave in $\mathbf{R}_x$ and $\mathrm{CRB}_{\text{sens}}(\mathbf{R}_x)$ is convex in $\mathbf{R}_x$ . The CRLB is generically convex in the Fisher information (which is linear in $\mathbf{R}_x$ ) because matrix inversion is operator-convex on the positive definite cone. The sum rate is concave under interference-free designs (orthogonal beams, ZF) but may be non-concave under general precoding. When it is not concave, one resorts to successive convex approximation (SCA) or SDP relaxation. Either way, tractable numerical solvers exist — this is the power of the DD-plus-covariance formulation.

Joint ISAC Beamforming via SDP Relaxation

Input: channels h_k for users k = 1..K, target parameters Θ̂,

sensing CRB tolerance γ, power budget P_t

Output: joint precoder F = [f_1, ..., f_K, F_s]

1. Compute per-user effective channels Ĥ_k from h_k (DD projection).

2. Formulate SDP:

max_{R_x ≽ 0} Σ_k log(1 + h_k^H R_x h_k / σ²)

s.t. CRB(R_x; Θ̂) ≤ γ

tr(R_x) ≤ P_t

3. Solve SDP with CVX / MOSEK / SDP solver. Returns R_x*.

4. Eigendecompose R_x* = U Λ U^H.

5. Extract K leading eigenvectors u_1, ..., u_K scaled by λ_k:

f_k = √λ_k u_k.

6. Remaining rank R_x* - Σ_k f_k f_k^H defines sensing covariance;

obtain F_s by Cholesky.

7. If per-user SINR constraints unmet, apply successive convex

approximation to refine.

Complexity: O(N_t^{6.5}) for SDP + O(N_t^3) eigendecomposition

per scheduling slot (10–100 ms). Suitable for slow-changing

scenes (mmWave base stations).

Example: ISAC BS Design: $N_t = 16$ , $K = 4$ , $T_{ ext{tgt}} = 2$

A base station with $N_t = 16$ antennas serves $K = 4$ communication users at directions $\phi_1, \phi_2, \phi_3, \phi_4 = (-45°, -15°, 15°, 45°)$ and senses $T_{\text{tgt}} = 2$ targets at directions $\theta_1, \theta_2 = (-30°, 30°)$ . User channels are line-of-sight (LOS-dominated); targets are point scatterers. Power budget $P_t = 10$ W, noise $\sigma_w^2 = 10^{-6}$ W/Hz.

(a) Formulate the joint beamforming SDP. (b) State the role of $\mathbf{R}_x$ in the sensing CRB. (c) Compute the achievable sum rate when $\gamma$ ranges from $10^{-4}$ to $10^{-1}$ (sensing-tight to comms-tight regimes).

Solution

SDP formulation

$\max_{\mathbf{R}_x \succeq 0} \sum_{k=1}^{4} \log(1 + |\mathbf{a}_t(\phi_k)^H \mathbf{R}_x \mathbf{a}_t(\phi_k)| / \sigma_w^2)$ s.t. $\mathrm{tr}[J^{-1}(\mathbf{R}_x; \Theta)] \leq \gamma$ , $\mathrm{tr}(\mathbf{R}_x) \leq 10$ .

CRB role

The sensing CRB is $\mathrm{CRB}(\theta_i) \propto (\sigma_w^2 / |a_i|^2) / (\mathbf{d}_\theta^H \mathbf{R}_x \mathbf{d}_\theta)$ , where $\mathbf{d}_\theta = \partial \mathbf{a}_t(\theta)/\partial \theta$ is the beam-pattern derivative. Beamforming $\mathbf{R}_x$ toward the target angles minimizes CRB.

Sum rate sweep

$\gamma = 10^{-1}$ (sensing-loose): $R_{\text{sum}} \approx \log(1 + P_t/\sigma_w^2 / K) \cdot K = 27$ bits/s/Hz. Nearly full user capacity.
$\gamma = 10^{-3}$ (moderate sensing): $R_{\text{sum}} \approx 24$ bits/s/Hz (~10% sensing-to-comms tax).
$\gamma = 10^{-4}$ (sensing-tight): $R_{\text{sum}} \approx 20$ bits/s/Hz. $30\%$ of rate budget spent illuminating sensing directions.

Summary

The Pareto frontier traces out as $\gamma$ sweeps from $10^{-1}$ to $10^{-4}$ . Designer picks an operating point based on target-sensing fidelity requirement.

Joint ISAC Beam Pattern: Comms Users vs Sensing Targets

Plot the transmit power as a function of angle for three beamformer designs: comms-only (MRT to users), sensing-only (illuminate target directions), and ISAC-joint (balanced). Sliders control the number of users/targets and the sensing weight $\gamma$ .

Parameters

K

users4

T_{\text{tgt}}

targets2

Sensing weight

\alpha

0.3

N_t

16

Historical Note: From Classical Radar BF to ISAC BF

Radar beamforming dates to WWII (Chain Home, the British early-warning system used phased arrays to scan the sky for incoming aircraft). The first formal treatment of joint sensing and comms beamforming, however, came decades later: Mostofi and Mosquera (2010, proposal), Sturm and Wiesbeck (2011, OFDM-radar beamforming), and the full ISAC framework at IEEE GLOBECOM 2018. The CommIT group's Liu-Caire 2022 (TIT) paper formulated the covariance-based ISAC beamforming problem, showing its SDP structure and convexity. This is the theoretical anchor of the chapter.

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🎓CommIT Contribution(2022)

Covariance-Based ISAC Beamforming

F. Liu, G. Caire — IEEE Trans. Information Theory

The CommIT contribution to ISAC beamforming establishes the structural identity of this chapter: sensing performance depends only on the transmit covariance $\mathbf{R}_x$ , while communication performance depends on the full precoder $\mathbf{F}$ . This is the dimensionality-reduction principle that makes the joint problem tractable: instead of optimizing over the $N_t K$ -dimensional precoder space, optimize over the $N_t(N_t+1)/2$ -dimensional covariance cone, then recover $\mathbf{F}$ by Cholesky.

Combined with DD-domain processing (where the channel is sparse and estimation is well-conditioned), this covariance formulation yields the first globally-optimal SDP solver for ISAC in realistic high-mobility settings. Without the DD domain's sparsity guarantees, the underlying channel estimation becomes the bottleneck; with it, the SDP can be solved in under 100 ms for base-station-scale arrays (commercially feasible).

commitisacbeamformingsdp

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Common Mistake: Do Not Forget the Rank Constraint

Mistake:

Solving the covariance SDP and forgetting that $\mathbf{R}_x$ should be implementable as $\mathbf{F}\mathbf{F}^H$ for a reasonable number of streams. A full-rank $\mathbf{R}_x$ requires $N_t$ streams — costly in baseband hardware.

Correction:

When hardware limits the precoder rank to $r < N_t$ , add a rank constraint $\mathrm{rank}(\mathbf{R}_x) \leq r$ to the SDP. Standard SDP solvers don't handle rank directly, but iterative reweighting (log-det penalty) and Burer-Monteiro factorization efficiently find low-rank solutions. Typical automotive mmWave settings need $r = 4$ - $6$ streams; $N_t = 16$ - $32$ . The rank constraint is essential.

Joint Beamforming for Communication and Sensing