Ferkans — Interactive Telecom Tutor

Negotiating Two Objectives

§2 showed that joint ISAC beamforming reduces to choosing a covariance matrix $\mathbf{R}_x$ . Different choices trace out different operating points in the plane of communication sum rate and sensing distortion. This section characterizes the boundary of that set — the rate-CRB Pareto frontier — and shows what the boundary looks like for concrete MIMO-OTFS ISAC scenarios. The answer reveals an important structural feature: the frontier has a knee, not a smooth slope, and systems should operate at the knee.

Definition:
Rate-CRB Pareto Frontier

For a MIMO-OTFS-ISAC system with power budget $P_t$ and target scene $\Theta$ , the rate-CRB achievable region is $\mathcal{R}(\Theta, P_t) \;=\; \{(R, D) \,:\, \exists \,\mathbf{R}_x \succeq 0,\, \mathrm{tr}(\mathbf{R}_x) \leq P_t,\, R \leq R_{\text{sum}}(\mathbf{R}_x),\, D \geq \mathrm{tr}[J^{-1}(\mathbf{R}_x; \Theta)]\}.$ The Pareto frontier $\mathcal{P}(\Theta, P_t)$ is the upper-right boundary of $\mathcal{R}(\Theta, P_t)$ — points where increasing $R$ strictly increases $D$ .

Three natural operating points:

Comms-only ( $D \to \infty$ ): $\mathbf{R}_x = \sum_k \mathbf{h}_k \mathbf{h}_k^H / P_t$ (MRT). Achieves $R = R_{\max}^{\text{comms}}$ .
Sensing-only ( $R = 0$ ): $\mathbf{R}_x \propto \sum_i \mathbf{a}(\theta_i) \mathbf{a}(\theta_i)^H$ . Achieves $D = D_{\min}^{\text{sens}}$ .
Knee point: the $(R, D)$ minimizing the Euclidean distance to the ideal $(R_{\max}^{\text{comms}}, D_{\min}^{\text{sens}})$ — typically within 90% of both objectives.

,

Theorem: Convexity of the Rate-CRB Frontier

The rate-CRB achievable region $\mathcal{R}(\Theta, P_t)$ is a convex set, and the Pareto frontier is a concave function $D(R)$ decreasing in $R$ .

Consequence. Every Pareto-optimal point is the solution of a linear scalarization $\max_{\mathbf{R}_x \succeq 0, \,\mathrm{tr}(\mathbf{R}_x) \leq P_t} \; \alpha\, R_{\text{sum}}(\mathbf{R}_x) \,-\, (1-\alpha)\, \mathrm{tr}[J^{-1}(\mathbf{R}_x; \Theta)], \qquad \alpha \in [0, 1].$ Sweeping $\alpha$ from $0$ to $1$ traces out the full frontier.

This theorem is the reason the Pareto frontier is numerically tractable: a single convex program, parameterized by one scalar weight $\alpha$ , produces the full frontier. The weight $\alpha$ represents the system designer's relative valuation of comms rate vs sensing fidelity — a policy decision, not an algorithm choice. The convexity guarantees the frontier has no interior Pareto points that the convex sweep misses.

Proof

Convexity of the achievable region

If $(R_1, D_1)$ and $(R_2, D_2)$ are achievable via $\mathbf{R}_x^{(1)}$ and $\mathbf{R}_x^{(2)}$ , then $\mathbf{R}_x^{(\lambda)} = \lambda \mathbf{R}_x^{(1)} + (1-\lambda) \mathbf{R}_x^{(2)}$ is a valid covariance (positive semidefinite, same trace constraint). The sum rate is concave in $\mathbf{R}_x$ (sum of $\log(1 + \text{affine})$ ), and $\mathrm{tr}[J^{-1}]$ is convex in $\mathbf{R}_x$ (Fisher $J$ is linear in $\mathbf{R}_x$ , matrix inverse is matrix-convex on the PSD cone). So $(\lambda R_1 + (1-\lambda) R_2, \lambda D_1 + (1-\lambda) D_2)$ is achievable.

Concavity of the frontier

The frontier $D(R)$ is the upper boundary of a convex set — hence a concave function of $R$ .

Scalarization

Maximizing a linear combination $\alpha R - (1-\alpha) D$ over a convex achievable region yields a Pareto-optimal point for each $\alpha \in [0, 1]$ . As $\alpha$ sweeps, the locus of maximizers traces the frontier. $\blacksquare$

Example: Pareto Frontier for a 16-Antenna BS

Compute the rate-CRB Pareto frontier for the BS in Example 13.2: $N_t = 16$ , $K = 4$ users at $(-45°, -15°, 15°, 45°)$ , $T_{\text{tgt}} = 2$ targets at $(-30°, 30°)$ , $P_t = 10$ W, $\sigma_w^2 = 10^{-6}$ W/Hz.

(a) Plot the frontier for $\alpha \in [0, 1]$ . (b) Identify the knee. (c) Compare with comms-only ( $\alpha = 1$ ) and sensing-only ( $\alpha = 0$ ) endpoints.

Solution

Comms-only endpoint

$\alpha = 1$ : MRT to users. $R_{\text{sum}} \approx 27$ bits/s/Hz, $D \to \infty$ (no energy in target directions).

Sensing-only endpoint

$\alpha = 0$ : $\mathbf{R}_x \propto \mathbf{a}(-30°)\mathbf{a}(-30°)^H + \mathbf{a}(30°)\mathbf{a}(30°)^H$ . $D = \mathrm{tr}[J^{-1}] \approx 10^{-5}$ . $R_{\text{sum}} \approx 0$ (user channels do not align with sensing directions).

Knee point

$\alpha \approx 0.7$ : $\mathbf{R}_x$ partially aligned with both user and target directions. $R_{\text{sum}} \approx 23$ bits/s/Hz, $D \approx 2 \times 10^{-4}$ . Sensing accuracy: $\sqrt{D} \approx 0.014$ rad = $0.8°$ — adequate for target tracking.

Summary

The knee is within 10% of both single-objective optima. A well-designed ISAC system operates at the knee, obtaining near- ideal comms and sensing from a single beam design.

Rate-CRB Pareto Frontier Sweep

Sweep $\alpha$ from $0$ (sensing-only) to $1$ (comms-only) and plot the resulting $(R_{\text{sum}}, \mathrm{tr}[J^{-1}])$ locus. The knee is highlighted. Sliders control BS antenna count, number of users, and SNR.

Parameters

N_t

16

K

users4

T_{\text{tgt}}

2

SNR (dB)15

Theorem: Knee Fraction

For a MIMO-OTFS-ISAC system with $K$ communication users in directions $\{\phi_k\}$ and $T_{\text{tgt}}$ sensing targets in directions $\{\theta_i\}$ that are angularly separated by at least $\Delta\theta \geq 2/N_t$ radians (beamwidth), the knee of the Pareto frontier satisfies $\frac{R_{\text{knee}}}{R_{\max}^{\text{comms}}} \;\geq\; 1 - \frac{T_{\text{tgt}}}{K + T_{\text{tgt}}}, \qquad \frac{D_{\text{knee}}}{D_{\min}^{\text{sens}}} \;\leq\; 1 + \frac{K}{K + T_{\text{tgt}}}.$ Interpretation. The system loses at most a fraction $T_{\text{tgt}}/(K + T_{\text{tgt}})$ of its comms rate and pays at most a factor $(1 + K/(K + T_{\text{tgt}}))$ in sensing distortion. For typical deployments ( $K \gg T_{\text{tgt}}$ ), this is a few percent on each side.

Angular separation makes joint beamforming cheap: comms and sensing do not compete for the same spatial resources. The fractional penalty is the "interference coupling" between the two — small when the directions are well-separated (typical mmWave ISAC), larger for dense arrays of co-located targets.

This result is the quantitative form of the design heuristic "comms and sensing in separate beams." It gives a closed-form bound for the tax the joint design pays versus single-objective designs.

Proof

Proof sketch

Decompose $\mathbf{R}_x = \mathbf{R}_x^c + \mathbf{R}_x^s$ where $\mathbf{R}_x^c$ is supported in user-direction subspace and $\mathbf{R}_x^s$ in target-direction subspace. By angular separation, the two subspaces are nearly orthogonal. The sum rate is dominated by $\mathbf{R}_x^c$ and the CRB by $\mathbf{R}_x^s$ . Pareto-optimal allocation splits power in the ratio of dimensionalities: $K : T_{\text{tgt}}$ .

Upper and lower bounds

When $K$ of $P_t$ power goes to comms, $R_{\text{sum}} \geq \log(1 + K P_t / K \sigma_w^2) = R_{\max}^{\text{comms}} - \log(1 + T/K) \cdot 1/\ln 2$ . Similar for CRB. $\blacksquare$

Key Takeaway

The Pareto frontier has a practical knee, not a smooth curve. For angularly separated comms and sensing, the knee operating point retains more than 85% of both single-objective performance. The design heuristic is simple: treat comms and sensing as parallel tasks on non-overlapping beams, then merge. The DD-domain framework does not change this heuristic — but it lets the sensing side work under arbitrary Doppler (which OFDM cannot match).

🔧Engineering Note

Choosing the Right Knee

Different ISAC applications have different tolerances:

Application	$\alpha$ (comms weight)	Sensing $D$	Rate $R$
V2X safety (sensing-first)	0.3	0.0001 (tight)	15 bits/s/Hz
Automotive infotainment	0.7	0.001 (mod)	23 bits/s/Hz
UAV backhaul + ATC	0.6	0.001 (mod)	22 bits/s/Hz
Urban cellular + sensing	0.9	0.01 (loose)	26 bits/s/Hz
Healthcare monitoring	0.4	0.0005 (tight)	18 bits/s/Hz

System designer picks $\alpha$ based on application priorities. The SDP solves the rest. Unlike classical TF-domain beamforming, the DD-domain formulation does not need re-tuning as channel mobility changes — the optimization depends only on the instantaneous DD-channel $\mathbf{H}_{DD}$ and target scene $\Theta$ , both of which are stable over multiple frames.

Practical Constraints

•
α ∈ [0, 1] is the policy knob
•
Knee typically α ≈ 0.6-0.8 for mixed ISAC
•
SDP solves in < 100 ms on server-grade CPU

Common Mistake: The Achievable Frontier Assumes Perfect CSI

Mistake:

Treating the Pareto frontier computed with perfect channel state information as the achievable operating point in deployment. In practice, $\mathbf{R}_x^\star$ is computed from estimated channels, and the gap between estimated and true channels degrades both comms and sensing performance.

Correction:

Robust ISAC beamforming replaces the SDP by a min-max formulation: maximize worst-case sum rate subject to worst-case CRB under a bounded channel estimation error. This shrinks the achievable region but guarantees performance in deployment. Rule of thumb: at 10 dB pilot SNR, the knee point loses about 1-2 dB relative to perfect CSI; at 0 dB pilot SNR, the loss can be 5-10 dB. Chapter 14 covers sensing-assisted channel estimation, which reduces this gap.

The Rate-CRB Pareto Frontier