Ferkans — Interactive Telecom Tutor

The ISAC Signal Serves Two Masters

We now formalize the dual-function signal model. The BS transmits a single waveform; from a communication perspective, it carries data to users; from a sensing perspective, it illuminates a target. Both perspectives share the same transmit signal, same BS precoder, and same RIS phase matrix. The optimization must account for both roles simultaneously.

Definition:
Joint ISAC Signal Model

BS transmits $\mathbf{x}(t) = \mathbf{W} \mathbf{s}(t)$ , where $\mathbf{s}(t) = [s_1(t), \ldots, s_K(t)]^T$ are user data streams.

Communication model: User $k$ receives $y_k = \mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{k} s_k + \sum_{j\neq k} \mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{j} s_j + w_k,$ same as Chapter 5. The SINR is

$\text{SNR}_{\text{comm},k} = \frac{|\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{k}|^2}{\sum_{j \neq k} |\mathbf{h}_{k,\text{eff}}^H \mathbf{v}_{j}|^2 + \sigma^2}.$

Sensing model: The signal illuminates target at $\mathbf{t}$ via BS→RIS→target→RIS→BS path. The received radar signal at the BS is (after matched filtering)

$\mathbf{y}_{\text{sens}} = \alpha_t \sigma_t \cdot \mathbf{H}_{\text{sens}}(\boldsymbol{\Phi}) \mathbf{W} \mathbf{s}(t - \tau_t) + \mathbf{w}_{\text{sens}},$

where $\mathbf{H}_{\text{sens}}(\boldsymbol{\Phi})$ is the RIS- dependent sensing channel (round-trip BS-RIS-target) and $\tau_t = 2 d_t/c$ is the round-trip delay.

The sensing beampattern at angle $\boldsymbol{\theta}$ is

$P_{\text{sens}}(\boldsymbol{\theta}) = |\mathbf{a}^H(\boldsymbol{\theta}) \mathbf{H}_1^H \boldsymbol{\Phi}^H \mathbf{W}|^2,$

i.e., the power radiated toward angle $\boldsymbol{\theta}$ through the combined BS precoder + RIS.

Theorem: Beampattern Matching under RIS

The beampattern-matching problem is

$\min_{\mathbf{W}, \boldsymbol{\Phi}} \int_{\boldsymbol{\theta}} \left|P_{\text{sens}}(\boldsymbol{\theta}; \mathbf{W}, \boldsymbol{\Phi}) - P_{\text{des}}(\boldsymbol{\theta})\right|^2 d\boldsymbol{\theta}$

subject to power and unit-modulus constraints. The $\int$ is discretized in practice into a grid of angles of interest (e.g., the target direction + clutter directions).

With RIS in the loop, the achievable beampatterns are a strict superset of those without: the RIS adds $N$ -dimensional passive control above and beyond the active precoder.

A well-designed radar waveform produces a desired beampattern: high power toward the target, low power toward clutter. Achieving this requires choosing the precoder such that $P_{\text{sens}}(\boldsymbol{\theta})$ matches a target pattern $P_{\text{des}}(\boldsymbol{\theta})$ . The RIS introduces an additional degree of freedom in this matching: tuning $\boldsymbol{\Phi}$ lets us hit $P_{\text{des}}$ with smaller $\mathbf{W}$ or more flexibly.

Proof

Beampattern as quadratic form

$P_{\text{sens}}(\boldsymbol{\theta}) = \mathbf{v}^{H} (\mathbf{a}(\boldsymbol{\theta}) \mathbf{a}^H(\boldsymbol{\theta})) \cdot \mathbf{H}_1^H \boldsymbol{\Phi}^H \cdot \cdot \boldsymbol{\Phi} \mathbf{H}_1 \mathbf{v}$ — quadratic in $\mathbf{v}$ for fixed $\boldsymbol{\Phi}$ , and quartic in $\boldsymbol{\phi}$ for fixed $\mathbf{v}$ .

Achievable region

The set $\{P_{\text{sens}}(\boldsymbol{\theta})\}_{(\mathbf{W},\boldsymbol{\Phi})}$ strictly contains $\{P_{\text{sens}}(\boldsymbol{\theta})\}_{\mathbf{W}}$ (no-RIS). Hence the RIS-aided achievable region is at least as large as no-RIS.

Feasible design

Within this region, pick $(\mathbf{W}, \boldsymbol{\Phi})$ to minimize $\|P_{\text{sens}} - P_{\text{des}}\|^2$ over the discretized grid of angles. Solution via AO: fix $\boldsymbol{\Phi}$ , optimize $\mathbf{W}$ (convex in $\mathbf{v}^{H} \mathbf{v}$ under power constraint); then fix $\mathbf{W}$ , optimize $\boldsymbol{\Phi}$ (non-convex QCQP, use Ch. 6 methods). $\blacksquare$

The Cramér-Rao Bound for RIS-Aided Sensing

Sensing SNR is a single number; the Cramér-Rao bound (CRB) gives a multi-parameter view: lower bounds on estimation errors for target position $\mathbf{t}$ , velocity, and other parameters. Under unbiased estimators, the CRB is

$\text{CRB}(\mathbf{t}) = \mathbf{J}^{-1}(\mathbf{t}; \mathbf{W}, \boldsymbol{\Phi}),$

where $\mathbf{J}$ is the Fisher Information Matrix, which depends on the sensing signal (waveform + precoder + RIS phases). The RIS phases appear inside $\mathbf{J}$ — tuning them reduces the CRB, improving estimation precision. Chapter 14 develops this in detail for localization.

Example: A 2-User RIS-ISAC System

2-user ISAC system with one radar target. $N_t = 4, N = 64, P_t/\sigma^2 = 20\text{ dB}$ . User 1 at angle $30°$ , user 2 at angle $-20°$ , target at angle $60°$ . Describe the optimization goal.

Solution

Communication goal

Maximize per-user SINR for users at angles $\pm 30°, -20°$ . Each user gets their own beam through $\mathbf{v}_{k}$ .

Sensing goal

Form a beam toward target at $60°$ . Detect / estimate target parameters (position, velocity) via backscattered energy.

Joint optimization

Choose $(\mathbf{W}, \boldsymbol{\Phi})$ to:

Concentrate power toward user directions via active beams.
Concentrate power toward target direction via RIS-shaped side-lobe (or dedicated radar beam).
Minimize clutter sidelobes. The RIS adds flexibility: user beams stay with the active precoder; $\boldsymbol{\Phi}$ can be dedicated to sensing alignment.

Practical design

Typical solution: active precoder forms user beams; $\boldsymbol{\Phi}$ focuses backscattered signal from target region into BS antennas. At $N = 64$ , the sensing beam has $\sim 64$ angular selectivity (beamwidth $\sim 1°$ ). $\blacksquare$

RIS-ISAC Beampattern

Visualize the combined BS + RIS beampattern. Show the main beam toward target and the user beams. Vary the tradeoff parameter to see how the beampattern shifts between "radar-focused" and "comm-focused" profiles.

Parameters

RIS elements

N

64

BS antennas

N_t

8

Target angle (deg)45

Sensing-comm tradeoff λ0.5

User 1 angle (deg)20

Joint Sensing-Communication Signal Model