Ferkans — Interactive Telecom Tutor

Pareto Boundary: Two Axes, One Panel

The central ISAC question: for a fixed BS + RIS hardware, what combinations of (comm rate, sensing SNR) are achievable? The Pareto boundary describes this: points on the boundary are efficient — cannot improve sensing without hurting comm, or vice versa. Off-boundary points are inefficient — dominated by some boundary point. This section characterizes the Pareto boundary for RIS-ISAC and describes how to traverse it via the tradeoff parameter $\lambda$ .

Definition:
Scalarized RIS-ISAC Problem

The joint RIS-ISAC optimization can be written as a weighted-sum problem:

$\max_{\mathbf{W}, \boldsymbol{\Phi}} (1 - \lambda) \cdot R_{\text{comm}}(\mathbf{W}, \boldsymbol{\Phi}) + \lambda \cdot R_{\text{sens}}(\mathbf{W}, \boldsymbol{\Phi})$

subject to the usual power and unit-modulus constraints. Here:

$R_{\text{comm}}$ : sum rate (or average per-user rate).
$R_{\text{sens}}$ : sensing "rate" (log-SNR, beampattern fidelity, or $-\text{CRB}$ ).
$\lambda \in [0, 1]$ : tradeoff parameter.

$\lambda = 0$ : pure communication (Chapter 5's problem). $\lambda = 1$ : pure sensing. $\lambda \in (0, 1)$ : balanced dual-function design.

Varying $\lambda$ traces out the Pareto boundary of the achievable (comm, sensing) region.

Theorem: Pareto Boundary of RIS-ISAC

Let $\mathcal{F}^{\text{ISAC}}_{\text{RIS}}$ and $\mathcal{F}^{\text{ISAC}}_{\text{no-RIS}}$ denote the achievable $(R_{\text{comm}}, R_{\text{sens}})$ regions with and without RIS. Then:

$\mathcal{F}^{\text{ISAC}}_{\text{no-RIS}} \subseteq \mathcal{F}^{\text{ISAC}}_{\text{RIS}}$ (RIS is never worse).
For any $\lambda \in [0, 1]$ and solution on the Pareto boundary, the RIS-aided boundary point dominates the no-RIS point.
The Pareto boundary is convex (RIS contributions sum, not compete): the RIS-aided region is a convex expansion of the no-RIS region.

In particular: at any fixed $R_{\text{comm}}^\star$ , the RIS boosts achievable $R_{\text{sens}}$ by a factor up to $N^4$ (via Theorem 13.1). And at any fixed $R_{\text{sens}}^\star$ , the RIS boosts achievable $R_{\text{comm}}$ by up to $N^2$ .

For each $\lambda$ , the scalarized problem's optimum gives one Pareto-boundary point. Sweeping $\lambda$ from 0 to 1 traces the complete Pareto frontier. With the RIS, the frontier expands outward: every $(\lambda, R_{\text{comm}}, R_{\text{sens}})$ achievable without RIS is also achievable with RIS, plus additional points dominated only by the RIS-augmented solution.

Proof

Inclusion

No-RIS $\Leftrightarrow$ $\boldsymbol{\Phi} = \mathbf{I}_N$ (or zero-out the RIS path). This is one feasible $\boldsymbol{\Phi}$ under RIS, so all no-RIS operating points are achievable with RIS.

Expansion

$\boldsymbol{\Phi} \neq \mathbf{I}$ gives more choices: non-trivial RIS phases amplify either comm, sensing, or both. The Pareto boundary includes these augmented operating points.

Convexity

Time-sharing between two boundary points produces intermediate operating points on a line segment; the achievable region is convex under time-sharing. Hence convex expansion. $\blacksquare$

Pareto Frontier: Comm Rate vs. Sensing SNR

Traverse the Pareto frontier by varying $\lambda$ . The RIS-aided frontier is pushed outward compared to no-RIS. Change $N$ to see the outward shift grow.

Parameters

RIS elements

N

128

BS antennas

N_t

8

Users

K

2

Transmit SNR (dB)10

AO for Scalarized RIS-ISAC

Complexity:

O(T \cdot (C_{\text{active-AO}} + C_{\text{passive}}))

; typical T = 15-25

Input: channels, target angle

\boldsymbol{\theta}_t

,

\lambda \in [0,1]

, power

P_t

.

Output:

(\mathbf{W}^\star, \boldsymbol{\Phi}^\star)

on the Pareto boundary.

1. Initialize

\boldsymbol{\Phi}^{(0)}

, e.g., all ones.

2. Repeat

t = 0, 1, \ldots

:

3.

\quad

Compute effective channels

\mathbf{h}_{k,\text{eff}}

and sensing channel

\mathbf{H}_{\text{sens}}

.

4.

\quad

Active update: solve

\max_{\mathbf{W}} (1-\lambda) R_{\text{comm}} + \lambda R_{\text{sens}}

s.t.

\|\mathbf{W}\|^2 \leq P_t

.

Use WMMSE for comm; sensing contributes a quadratic term.

Reformulate as SOCP or SDP for convexity.

5.

\quad

Passive update: solve

\max_{\boldsymbol{\phi}} (1-\lambda) R_{\text{comm}} + \lambda R_{\text{sens}}

s.t.

|\phi_n| = 1

.

Non-convex; use Chapter 6 methods.

6.

\quad

Check convergence:

|f^{(t+1)} - f^{(t)}| < \epsilon

.

7. return

(\mathbf{W}, \boldsymbol{\Phi})

.

One AO iteration is $\sim 50$ ms at $N = 128, K = 2$ on a modern CPU. Sweeping $\lambda$ for the Pareto frontier takes $\sim$ 10-20 values $\times$ 50 ms = 0.5-1 s total — offline analysis.

Example: Reading a Pareto Frontier

A RIS-ISAC system has Pareto frontier: $R_{\text{comm}}^{\star}(\lambda = 0) = 5$ bits/s/Hz, $R_{\text{sens}}^{\star}(\lambda = 1) = 60\text{ dB}$ sensing SNR. At $\lambda = 0.5$ (balanced), what to expect?

Solution

Fully sensing regime

$\lambda = 1$ : all resources to sensing. Users served only by direct path (if any). Comm rate drops. Sensing SNR at maximum.

Balanced regime

$\lambda = 0.5$ : split. Comm rate drops to $\sim 3$ - $4$ bits/s/Hz (minus 1-2 bits from full-comm optimum). Sensing SNR drops to $\sim 45$ - $50\text{ dB}$ (minus 10-15 dB from full-sensing optimum).

Pareto curve

The tradeoff is smooth, not linear. Small $\lambda$ perturbation near $\lambda = 0$ costs little comm but gains much sensing; large $\lambda$ perturbation near $\lambda = 1$ costs much sensing but gains little comm. The middle of the frontier is the "sweet spot" for dual-function operation. $\blacksquare$

Self-Interference in ISAC

A subtle issue: the BS is both transmitter and radar receiver. Its own transmit signal — strongly radiated — can swamp the (weak) backscattered return. Self-interference must be suppressed, typically by:

Temporal separation: pulse-based radar (listen during silent periods). Loses continuous sensing.
Spatial suppression: RIS-enabled nulling — force the RIS to steer the transmit signal away from the radar receive direction. Achievable when the target direction differs from the strong sidelobes.
Dedicated RF chain: separate radar receiver antenna with analog cancellation. Requires hardware upgrade.

The RIS can help with option 2: $\boldsymbol{\Phi}$ is chosen both to focus toward target AND to null toward the receive antennas of the BS's own radar receiver. Adds one more constraint to the joint optimization.

Common Mistake: The Tradeoff Is Not Linear

Mistake:

"At $\lambda = 0.5$ , comm and sensing should each be at half of their full values."

Correction:

The Pareto tradeoff curve is concave, not linear: small weights $\lambda$ give disproportionately large sensing gains at small comm cost (and vice versa). The middle of the curve is balanced but both objectives are above half their max values. Reading the curve as linear misses the key ISAC benefit: moderate weighting gives good performance on both axes.

The Sensing-Communication Tradeoff