Ferkans — Interactive Telecom Tutor

From Either-Or to Both

Radar and communications historically compete for the same spectrum, waveforms, and hardware. Integrated Sensing and Communications (ISAC) refuses to choose: one waveform, transmitted from a multi-antenna base station, should simultaneously convey data to a user and illuminate targets for the base station to estimate their range, angle, and velocity. This dual purpose introduces a fundamental tradeoff --- waveforms optimised for data (typically near-Gaussian, high-entropy) are good communicators but poor radars; waveforms optimised for sensing (deterministic, low-entropy, tightly designed) are good radars but poor communicators.

The right way to quantify this tradeoff is via the Cramer-Rao bound for the sensing parameters --- delay, angle, Doppler --- combined with the achievable communication rate. For each choice of transmit sample covariance $\mathbf{R}_s$ , one gets a rate $R(\mathbf{R}_s)$ and a CRB matrix $\mathrm{CRB}(\mathbf{R}_s)$ . The boundary of the achievable (rate, sensing-accuracy) region is traced by sweeping a Lagrange multiplier that trades one against the other. CommIT and adjacent research groups have spent the last five years mapping this region for progressively richer system models; the section that follows presents the core tools.

Definition:
ISAC Signal Model (Monostatic)

A base station with $N_t$ transmit antennas and $N_r$ co-located receive antennas transmits $T$ samples of a matrix signal $\mathbf{S} \in \mathbb{C}^{N_t\times T}$ . A single point target with complex reflectivity $\alpha$ , azimuth $\theta$ , delay $\tau$ , and Doppler $\nu$ reflects the signal, producing received samples $\mathbf{Y} \;=\; \alpha\,\mathbf{a}_r(\theta)\mathbf{a}_t(\theta)^T\, \mathbf{D}_\nu \,\mathbf{S}_\tau + \mathbf{W},$ where $\mathbf{a}_t,\mathbf{a}_r$ are transmit/receive steering vectors, $\mathbf{D}_\nu$ is a diagonal Doppler phase ramp, $\mathbf{S}_\tau$ is the delayed signal, and $\mathbf{W}$ is AWGN with per-entry variance $\sigma^2$ . Meanwhile, a communication user with a single antenna receives $y^{(c)}_t \;=\; \mathbf{h}^H \mathbf{s}_t + w^{(c)}_t,$ where $\mathbf{h}$ is the downlink channel to the user and $\mathbf{s}_t$ is the $t$ -th column of $\mathbf{S}$ .

The transmitter's only lever is the joint distribution of $\mathbf{s}_1,\ldots,\mathbf{s}_T$ , parametrised by the sample covariance $\mathbf{R}_s = \tfrac{1}{T}\sum_t \mathbb{E}[\mathbf{s}_t \mathbf{s}_t^H]$ , which controls both the beampattern and the communication rate.

Theorem: CRB for Joint Delay-Angle-Doppler

Under the ISAC signal model with a single target and AWGN, assuming $\alpha$ is known and we estimate $\boldsymbol\eta = (\theta,\tau,\nu)$ , the Fisher information matrix is $\mathbf{J}(\boldsymbol\eta) \;=\; \frac{2|\alpha|^2}{\sigma^2}\, \mathrm{Re}\!\left\{\partial_{\boldsymbol\eta}\mathbf{G}^H \partial_{\boldsymbol\eta}\mathbf{G}\right\},$ where $\mathbf{G}(\boldsymbol\eta) = \mathbf{a}_r(\theta)\mathbf{a}_t(\theta)^T \mathbf{D}_\nu \mathbf{S}_\tau$ is the noise-free received template, and $\partial_{\boldsymbol\eta}$ stacks the partial derivatives with respect to angle, delay, and Doppler. The CRB for each parameter is $[\mathbf{J}^{-1}]_{ii}$ , and the overall sensing error is governed by $\mathrm{tr}(\mathbf{J}^{-1})$ .

The FIM is quadratic in the transmit covariance $\mathbf{R}_s$ through the waveform $\mathbf{S}$ . Angular precision is driven by $\mathbf{R}_s$ aligning energy with $\mathbf{a}_t(\theta)$ ; delay precision by the effective bandwidth of the waveform; Doppler precision by the effective duration. The three parameters couple through the off-diagonal entries of the FIM.

Proof

Gaussian likelihood

Vectorising $\mathbf{Y}$ and stacking, $\mathbf{y} = \alpha\,\mathbf{g}(\boldsymbol\eta) + \mathbf{w}$ with $\mathbf{w} \sim \mathcal{CN}(\mathbf{0},\sigma^2\mathbf{I})$ . The log-likelihood is $\ell(\boldsymbol\eta;\mathbf{y}) = -\tfrac{1}{\sigma^2} \|\mathbf{y} - \alpha\mathbf{g}(\boldsymbol\eta)\|^2 + \text{const}$ .

Score and Fisher computation

The score is $\partial_{\eta_i}\ell = \tfrac{2}{\sigma^2} \mathrm{Re}\{\alpha^* \partial_{\eta_i}\mathbf{g}^H(\mathbf{y} - \alpha\mathbf{g})\}$ . Taking the covariance under the true model eliminates the residual and leaves $[\mathbf{J}]_{ij} = \tfrac{2|\alpha|^2}{\sigma^2} \mathrm{Re} \{\partial_{\eta_i}\mathbf{g}^H \partial_{\eta_j}\mathbf{g}\}$ , which is exactly the stated form in matrix notation.

Recovering standard radar bounds

Specialising to a uniform linear array and narrowband waveform, the angular CRB reduces to the classical $\sigma^2/(|\alpha|^2\,6 N_t^2 T)$ formula; specialising to a single-antenna wideband waveform, the delay CRB reduces to $\sigma^2/(8\pi^2 B_{\text{rms}}^2|\alpha|^2 T)$ ; specialising to a narrowband long-duration waveform, the Doppler CRB reduces to $\sigma^2/(8\pi^2 T_{\text{rms}}^2 |\alpha|^2 T)$ . The joint FIM recovers each of these as a marginal.

Definition:
Rate-CRB Achievability Region

The rate-CRB region is the set of achievable (rate, sensing accuracy) pairs $\mathcal{R}_{\text{ISAC}} \;=\; \bigcup_{\mathbf{R}_s \succeq \mathbf{0},\, \mathrm{tr}(\mathbf{R}_s) \leq P_t} \Big\{(R, \epsilon^2) : R \leq \log\!\left(1 + \mathbf{h}^H\mathbf{R}_s\mathbf{h}/\sigma^2\right),$ $\quad \epsilon^2 \geq \mathrm{tr}(\mathbf{J}^{-1}(\mathbf{R}_s))\Big\}.$ The Pareto boundary of this region is traced by the beampattern design problem: maximise $R$ subject to a CRB constraint, or minimise $\mathrm{tr}(\mathbf{J}^{-1})$ subject to a rate constraint.

Both problems are non-convex in $\mathbf{R}_s$ because the FIM is quadratic in $\mathbf{S}$ , and hence the CRB involves the inverse of a matrix affine in $\mathbf{R}_s$ . Semidefinite-relaxation and majorisation-minimisation methods are the standard workhorses.

Theorem: Convex Relaxation of the Rate-CRB Problem

Consider the ISAC problem $\min_{\mathbf{R}_s \succeq \mathbf{0}} \;\mathrm{tr}(\mathbf{J}^{-1}(\mathbf{R}_s))$ subject to $\mathbf{h}^H \mathbf{R}_s \mathbf{h} \geq \gamma\, \sigma^2$ (rate constraint) and $\mathrm{tr}(\mathbf{R}_s) \leq P_t$ (power constraint). Introducing a slack matrix $\mathbf{U}\succeq \mathbf{J}^{-1}(\mathbf{R}_s)$ and applying the Schur complement, the problem becomes the SDP $\min \;\mathrm{tr}(\mathbf{U}) \quad \text{s.t.}\quad \begin{bmatrix} \mathbf{U} & \mathbf{I}\\ \mathbf{I} & \mathbf{J}(\mathbf{R}_s) \end{bmatrix} \succeq \mathbf{0},\; \mathbf{h}^H \mathbf{R}_s \mathbf{h} \geq \gamma\,\sigma^2,\; \mathrm{tr}(\mathbf{R}_s) \leq P_t.$ This is a convex program (the FIM is linear in $\mathbf{R}_s$ , the Schur condition is LMI, the rate and power constraints are linear).

The non-convex CRB minimisation is transformed into a convex SDP by the Schur complement. The price is an auxiliary variable $\mathbf{U}$ whose trace upper-bounds $\mathrm{tr}(\mathbf{J}^{-1})$ . At the optimum, the Schur inequality holds with equality and $\mathrm{tr}(\mathbf{U})$ equals the sensing cost.

Proof

Schur complement

The matrix inequality $\mathbf{U}\succeq \mathbf{J}^{-1}$ is equivalent to $\begin{bmatrix}\mathbf{U}&\mathbf{I}\\\mathbf{I}&\mathbf{J}\end{bmatrix} \succeq \mathbf{0}$ provided $\mathbf{J}\succ 0$ , by the standard Schur complement lemma. The LMI is linear in $(\mathbf{U}, \mathbf{R}_s)$ because $\mathbf{J}$ is linear in $\mathbf{R}_s$ .

Affine-in-$\mathbf{R}_s$ FIM

From the FIM formula, each entry is $[\mathbf{J}]_{ij} = \mathrm{Re}\,\mathrm{tr}(\mathbf{A}_{ij} \mathbf{R}_s)$ for fixed matrices $\mathbf{A}_{ij}$ built from the steering-vector derivatives. Hence the full FIM is affine in $\mathbf{R}_s$ and the LMI is indeed linear.

Rate constraint is linear

The communication rate $R = \log(1+\mathbf{h}^H\mathbf{R}_s \mathbf{h}/\sigma^2)$ is a monotone function of $\mathbf{h}^H\mathbf{R}_s\mathbf{h}$ , which is linear in $\mathbf{R}_s$ . Fixing a minimum rate $R \geq R_0$ is equivalent to $\mathbf{h}^H\mathbf{R}_s\mathbf{h} \geq (e^{R_0}-1) \sigma^2$ , a linear constraint.

Convexity and solution

The resulting problem is a semidefinite program, solvable by interior-point methods in polynomial time. Sweeping the Lagrange multiplier of the rate constraint traces out the Pareto boundary of the rate-CRB region.

,

SDP-Based ISAC Beampattern Design

Complexity: Each SDP solves in O((ntx^2)^3) time by interior-point; K SDPs total

Input: steering vectors a_t(theta), a_r(theta), delay/Doppler operators,

user channel h, noise variance sigma2, power budget P,

grid of rate constraints {R_1,...,R_K}

Output: achievable (rate, CRB) Pareto frontier

Compute partial derivatives of G w.r.t. theta, tau, nu at nominal point

Build the linear maps A_{ij}(R_s) = partial_i G^H partial_j G

for each R_k in rate grid:

Solve SDP:

minimize tr(U)

subject to [[U, I], [I, FIM(R_s)]] >= 0

h^H R_s h >= (exp(R_k) - 1) * sigma2

tr(R_s) <= P

R_s >= 0

record crb_k = tr(U) from the optimiser

(optional) extract transmit covariance, draw N samples s_t ~ CN(0, R_s)

return {(R_k, crb_k)}_{k=1..K}

In practice, warm-starting each SDP with the previous $\mathbf{R}_s$ dramatically accelerates the sweep. For very large arrays, ADMM or block-coordinate methods replace CVX/Mosek.

Rate-CRB Tradeoff Region

Sweep the Lagrange multiplier $\lambda$ between a pure communication objective ( $\lambda=0$ , maximise rate, ignore sensing) and a pure sensing objective ( $\lambda=\infty$ , minimise CRB, ignore rate). The resulting Pareto frontier shows the achievable (rate, angular-CRB) pairs for a 16-element ULA with a known user channel and target direction.

Parameters

N_{\text{tx}}

16

Number of transmit antennas

Communication SNR (dB)10

Sensing SNR (dB)5

Target-user angular separation (deg)45

Transmit Beampattern on the Pareto Frontier

At each Lagrange multiplier, the optimal $\mathbf{R}_s$ produces a specific transmit beampattern $P(\theta) = \mathbf{a}_t(\theta)^H \mathbf{R}_s \mathbf{a}_t(\theta)$ . Move the slider to see how the pattern morphs from a pencil beam toward the user (pure communication) to a pattern with energy at both the user and the target (ISAC) to a pencil beam toward the target (pure sensing).

Parameters

N_{\text{tx}}

16

\lambda

0.5

Sensing/comm weight (0 = pure comm, 1 = pure sensing)

User angle (deg)-20

Target angle (deg)30

Morphing from Comm-Only to Sensing-Only Beampattern

Animated sweep of the Lagrange weight

\lambda

from

0

(pure communication pencil beam toward the user) through

0.5

(ISAC pattern illuminating both user and target) to

1

(pure sensing pencil beam toward the target).

⚠️Engineering Note

ISAC in 6G Standardisation

ISAC is one of the 2024 ITU IMT-2030 usage scenarios for 6G. The standardisation problem is not waveform optimisation per se --- the air interface is largely inherited from 5G --- but resource allocation: which subcarriers, time slots, and spatial streams to dedicate to sensing, and under what QoS guarantees for the communication users. The rate-CRB region is the theoretical benchmark against which practical schedulers are measured, and the SDP formulation above is a building block of every published ISAC scheduler.

Practical Constraints

•
Per-user rate constraints turn the SDP into a multi-constraint problem
•
Clutter and multi-target scenarios require extending the FIM to block-diagonal form
•
Robust ISAC with imperfect CSI replaces fixed $\mathbf{h}$ with a worst-case set

📋 Ref: ITU-R Recommendation M.2160 (IMT-2030)

🎓CommIT Contribution(2023)

CRB-Rate Region for Multi-User ISAC

G. Caire, Y. Xiong, F. Liu — IEEE Transactions on Information Theory

This CommIT contribution extends the single-user rate-CRB region to the multi-user broadcast channel. Each communication user demands a minimum rate, and the radar task is to estimate the parameters of multiple targets, each with its own reflectivity. The fundamental result is that the rate-CRB region has the same convex structure as the single-user case, but the Pareto boundary decomposes into an angular-allocation sub-problem (which users and which targets to serve simultaneously) and an SDP beampattern problem per angular group. The paper gives closed-form expressions for two-user, two-target cases and numerical sweeps for larger instances.

isacmulti-usercrb-rate-tradeoff

Common Mistake: Random vs. Deterministic Signal in the FIM

Mistake:

Treating $\mathbf{S}$ as deterministic in the FIM derivation while simultaneously treating it as random for the communication rate.

Correction:

The rate depends on $\mathbf{R}_s = \mathbb{E}[\mathbf{s}_t \mathbf{s}_t^H]$ , a second-moment quantity; the FIM, as written here, assumes deterministic $\mathbf{S}$ with empirical covariance $\mathbf{R}_s$ . A coherent treatment averages the FIM over the random $\mathbf{S}$ (expected FIM) --- the distinction matters when $T$ is small. For $T$ large enough that empirical and ensemble covariances agree, the two formulations coincide.

Historical Note: A Research Explosion

2018-present

ISAC as a distinct research agenda crystallised around 2018-2020 through a sequence of papers by Fan Liu, Christos Masouros, Bruno Clerckx and collaborators, who reframed long-standing radar-comm cohabitation questions as joint waveform design problems. The Caire-Kobayashi 2018 paper on information-theoretic ISAC fundamentals gave the field its canonical rate-distortion formulation. By 2024 the topic was one of the top-three submission tracks at IEEE Transactions on Signal Processing.

Quick Check

On the Pareto frontier of the rate-CRB region, what happens to the transmit beampattern as the rate constraint is loosened (i.e., as we allow lower rate in exchange for better sensing)?

It remains a pencil beam toward the user; only the power scales.

It shifts monotonically from the user toward the target.

It becomes isotropic.

It collapses to the target direction with zero side-lobe.

Correction:

It shifts monotonically from the user toward the target.

As the rate constraint relaxes, the optimal $\\mathbf{R}_s$ shifts energy from the user direction to the target direction; the pattern morphs continuously, illuminating both at intermediate $\\lambda$ .

ISAC (Integrated Sensing and Communications)

A system paradigm in which a single waveform is transmitted from a multi-antenna transmitter to simultaneously convey data and probe the environment for sensing parameters (angle, delay, Doppler).

Rate-CRB Region

The closure of achievable pairs (rate, sensing CRB) over all feasible transmit covariances in an ISAC system. Its Pareto boundary is traced by SDP-based beampattern design.

Estimation in ISAC Systems

From Either-Or to Both

Definition: ISAC Signal Model (Monostatic)

Theorem: CRB for Joint Delay-Angle-Doppler

Gaussian likelihood

Score and Fisher computation

Recovering standard radar bounds

Definition: Rate-CRB Achievability Region

Theorem: Convex Relaxation of the Rate-CRB Problem

Schur complement

Affine-in-$\mathbf{R}_s$ FIM

Rate constraint is linear

Convexity and solution

SDP-Based ISAC Beampattern Design

Rate-CRB Tradeoff Region

Parameters

Transmit Beampattern on the Pareto Frontier

Parameters

Morphing from Comm-Only to Sensing-Only Beampattern

ISAC in 6G Standardisation

CRB-Rate Region for Multi-User ISAC

Common Mistake: Random vs. Deterministic Signal in the FIM

Historical Note: A Research Explosion

Quick Check

ISAC (Integrated Sensing and Communications)

Rate-CRB Region

Definition:
ISAC Signal Model (Monostatic)

Definition:
Rate-CRB Achievability Region