Ferkans — Interactive Telecom Tutor

The Two Bounds of a Dual-Purpose Network

A cell-free network that simultaneously serves communication and positioning lives under two fundamental bounds. For communication, the Shannon capacity and its UatF/hardening proxies from TUatF Bound for Uplink Spectral Efficiency measure bits per second per Hertz. For positioning, the Cramer-Rao bound measures meters of RMS position error. This section unifies them: we derive the Position Error Bound as a function of the same parameters (power, bandwidth, AP density) that control the rate, and then expose the rate-PEB tradeoff by plotting the achievable region.

The result is a single Pareto frontier on which every operating point costs one quantity in the other. An ISAC system designer picks a point on the frontier to match the service-level agreement: mission-critical positioning takes a hit on rate, while consumer mobile broadband takes a hit on accuracy.

,

Definition:
Position and Angle Error Bounds

Let $\boldsymbol{\theta}$ be the parameter vector containing the user position $\mathbf{p}$ together with nuisance parameters $\boldsymbol{\eta}$ (channel phases, amplitudes, etc.):

$\boldsymbol{\theta} = [\mathbf{p}^T, \boldsymbol{\eta}^T]^T$

The Fisher information matrix $\mathbf{J}(\boldsymbol{\theta})$ partitions as

$\mathbf{J} = \begin{pmatrix} \mathbf{J}_{\mathbf{p}\mathbf{p}} & \mathbf{J}_{\mathbf{p}\boldsymbol{\eta}} \\ \mathbf{J}_{\boldsymbol{\eta}\mathbf{p}} & \mathbf{J}_{\boldsymbol{\eta}\boldsymbol{\eta}} \end{pmatrix}$

The Equivalent Fisher Information Matrix (EFIM) on position is

$\mathbf{J}_{\mathbf{p}} = \mathbf{J}_{\mathbf{p}\mathbf{p}} - \mathbf{J}_{\mathbf{p}\boldsymbol{\eta}} \mathbf{J}_{\boldsymbol{\eta}\boldsymbol{\eta}}^{-1} \mathbf{J}_{\boldsymbol{\eta}\mathbf{p}}$

which accounts for the loss due to unknown nuisances. The Position Error Bound is

$\text{PEB}(\mathbf{p}) \triangleq \sqrt{\text{tr}(\mathbf{J}_{\mathbf{p}}^{-1})}$

and the Angle Error Bound on a single AOA parameter $\phi$ is

$\text{AEB}(\phi) \triangleq \sqrt{[\mathbf{J}_{\phi}^{-1}]_{11}}$

For any unbiased estimator, $\mathbb{E}[\|\hat{\mathbf{p}} - \mathbf{p}\|^2] \geq \text{PEB}^2$ .

The Schur-complement form shows explicitly how unknown nuisance parameters degrade positioning: the subtracted term is zero only when position and nuisance Fisher information are orthogonal, which almost never holds. In cell-free systems, unknown channel phases are the dominant nuisance — they couple strongly to the TOA parameters and reduce the EFIM by 3-10 dB depending on regime.

Theorem: EFIM for Joint TOA+AOA Cell-Free Positioning

Consider the cell-free model where AP $l$ provides both a TOA measurement with variance $\sigma_{\tau,l}^2 = 1/(8\pi^2 \beta_{\text{rms}}^2 \text{SNR}_{l})$ and an AOA measurement with variance $\sigma_{\phi,l}^2 = 1/(2\pi^2 N \cos^2\phi_l \cdot \text{SNR}_{l} \cdot (D/\lambda)^2)$ , where $N$ is the per-AP antenna count and $D$ is the array aperture. The EFIM on position is

$\mathbf{J}_{\mathbf{p}} = \sum_{l=1}^{L}\!\left[ \frac{\lambda_{\tau,l}}{c^2}\, \mathbf{u}_l \mathbf{u}_l^T + \frac{\lambda_{\phi,l}}{d_l^2}\, \mathbf{u}_l^\perp (\mathbf{u}_l^\perp)^T \right]$

where $\lambda_{\tau,l} = 1/\sigma_{\tau,l}^2$ , $\lambda_{\phi,l} = 1/\sigma_{\phi,l}^2$ , $\mathbf{u}_l$ is the unit vector toward the user, and $\mathbf{u}_l^\perp$ is its 90-degree rotation.

TOA information is directional: it tightens the position only along the line from AP to user. AOA information is perpendicular-directional: it tightens the position only across that line. Together they provide rank-2 contributions per AP, dramatically improving GDOP near the AP where TOA alone is weak in the tangential direction but AOA is strongest. This is the information-theoretic reason cell-free multi-antenna APs are so much better for positioning than single-antenna APs — each AP provides a full-rank information contribution.

Show Hint

Compute $\nabla \tau_l$ and $\nabla \phi_l$ with respect to $\mathbf{p}$ .

$\nabla \tau_l = \mathbf{u}_l/c$ and $\nabla \phi_l = \mathbf{u}_l^\perp/d_l$ .

Sum the rank-1 outer products weighted by the respective inverse variances.

Proof

Gradient of TOA

$\tau_l = \|\mathbf{p} - \mathbf{q}_l\|/c$ , so $\nabla_\mathbf{p} \tau_l = (\mathbf{p} - \mathbf{q}_l)/(c d_l) = \mathbf{u}_l/c$ .

Gradient of AOA

$\phi_l = \text{atan2}(p_y - q_{l,y}, p_x - q_{l,x})$ . Computing the partial derivatives, $\nabla_\mathbf{p} \phi_l = (-\sin\phi_l, \cos\phi_l)/d_l = \mathbf{u}_l^\perp/d_l$ where $\mathbf{u}_l^\perp$ denotes $\mathbf{u}_l$ rotated by $+\pi/2$ .

Assemble the per-AP FIM

Each AP contributes $\lambda_{\tau,l} \nabla \tau_l \nabla \tau_l^T + \lambda_{\phi,l} \nabla \phi_l \nabla \phi_l^T = (\lambda_{\tau,l}/c^2) \mathbf{u}_l \mathbf{u}_l^T + (\lambda_{\phi,l}/d_l^2) \mathbf{u}_l^\perp (\mathbf{u}_l^\perp)^T$ .

Orthogonality gives rank-2

Because $\mathbf{u}_l$ and $\mathbf{u}_l^\perp$ are orthogonal, the two rank-1 contributions sum to a rank-2 matrix — a full-rank contribution in 2D. Summing over APs and taking the Schur complement for nuisance parameters gives the EFIM. $\blacksquare$

,

Key Takeaway

Multi-antenna APs close the positioning gap. A single-antenna cell-free network provides TOA-only information, making the EFIM rank-1 per AP and requiring many APs for a full-rank position fix. Multi-antenna APs add AOA information in the orthogonal direction, giving every AP a rank-2 contribution. For 8-antenna APs at 100 MHz bandwidth, the PEB is typically 3-5 $\times$ smaller than for single-antenna APs at the same aggregate power.

Definition:
Rate-PEB Achievable Region

Let $(\mathbf{x}_{\text{comm}}, \mathbf{x}_{\text{pos}})$ be a decomposition of the transmitted waveform into communication-dedicated and position-dedicated components satisfying the power constraint $\mathbb{E}[\|\mathbf{x}_{\text{comm}}\|^2 + \|\mathbf{x}_{\text{pos}}\|^2] \leq P_t$ . Given this decomposition, the communication rate $R$ and PEB satisfy

$R \leq \log_2\!\left(1 + \frac{\|\mathbf{x}_{\text{comm}}\|^2 \cdot \text{eff. gain}}{\sigma^2}\right)$

$\text{PEB}^2 \geq \text{tr}\!\left(\mathbf{J}_{\mathbf{p}}(\mathbf{x}_{\text{pos}} + \mathbf{x}_{\text{comm}})^{-1}\right)$

(Communication symbols also contribute to positioning since they are observed at the APs.) The rate-PEB region $\mathcal{R}$ is the set of achievable pairs $(R, \text{PEB})$ over all valid decompositions.

The region is convex in $(R, 1/\text{PEB}^2)$ under a fixed power budget, and characterized by the tangent line

$R + \mu \cdot \text{tr}(\mathbf{J}_{\mathbf{p}}^{-1}) \leq f(\mu)$

for $\mu \geq 0$ parameterizing the tradeoff.

Rate-PEB Tradeoff Curve

Sweep the fraction of power allocated to positioning (vs communication) and plot the resulting Pareto frontier of the rate-PEB region. Each point on the curve is achievable by some power split; points inside the curve are suboptimal; points outside are infeasible at the current SNR budget. Users pick the operating point by specifying either a target rate or a target PEB.

Parameters

Number of APs12

Bandwidth (MHz)100

Reference SNR (dB)10

Antennas per AP4

Example: Numerical PEB for a 16-AP Square Deployment

A cell-free network deploys 16 APs uniformly on a $200 \times 200\,\text{m}$ square grid. Each AP has $N = 4$ antennas, and the uplink bandwidth is 100 MHz. Assume the per-AP receive SNR is 10 dB and the path-loss exponent is 3.5. Compute the PEB for a user at the center of the square.

Solution

Per-AP TOA variance

With $\beta_{\text{rms}}^2 \approx W^2/3 = (10^8)^2/3 \approx 3.3 \times 10^{15}\,\text{Hz}^2$ for a flat-spectrum pulse, $\text{SNR} = 10$ : $\sigma_\tau^2 = 1/(8\pi^2 \cdot 3.3 \times 10^{15} \cdot 10) \approx 3.8 \times 10^{-19}\,\text{s}^2$ . Range std: $c \sigma_\tau \approx 0.19$ m.

Per-AP AOA variance

For $N = 4$ and half-wavelength spacing, $(D/\lambda)^2 \approx 2.25$ , so $\sigma_\phi^2 \approx 1/(2\pi^2 \cdot 4 \cdot 1 \cdot 10 \cdot 2.25) \approx 1.1 \times 10^{-3}\,\text{rad}^2$ . AOA std: $0.033$ rad $\approx 1.9^\circ$ .

Assemble EFIM at center

By symmetry, the 16 $\mathbf{u}_l \mathbf{u}_l^T$ terms sum to $8 \mathbf{I}_2$ and the 16 $\mathbf{u}_l^\perp \mathbf{u}_l^{\perp T}$ terms sum to $8 \mathbf{I}_2$ . Distances range from $\sqrt{2} \cdot 33$ m to $\sqrt{2} \cdot 100$ m. Using average $d \approx 70$ m: $\mathbf{J}_\mathbf{p} \approx (8/c^2/\sigma_\tau^2 + 8/d^2/\sigma_\phi^2) \mathbf{I}_2$ . Numerically $\approx (2.2 \times 10^2 + 1.5 \times 10^3) \mathbf{I}_2 \approx 1700 \mathbf{I}_2\,\text{m}^{-2}$ .

Compute PEB

$\text{PEB} = \sqrt{\text{tr}(\mathbf{J}_\mathbf{p}^{-1})} = \sqrt{2/1700} \approx 0.034$ m $\approx 3.4$ cm. At the center of a well-designed cell-free cluster with 100 MHz and multi-antenna APs, centimeter-level accuracy is achievable — matching the 3GPP Release 17 industrial IoT requirement.

,

🎓CommIT Contribution(2023)

Optimal Rate-Distortion Tradeoff for ISAC under Gaussian Channels

F. Liu, Y. Xiong, K. Wan, T. X. Han, Y. Cui, G. Caire — IEEE Trans. Information Theory, vol. 69, no. 9

This CommIT result establishes the information-theoretic tradeoff between communication rate and sensing distortion (which includes position error as a special case) under Gaussian channels. The authors show that the optimal input distribution is Gaussian on both the communication and sensing parts, and that the Pareto frontier of the rate-distortion region has a water-filling-like structure: communication uses the strong eigendirections of the effective channel, while sensing pilots occupy the weak ones. When specialized to cell-free positioning, the theorem predicts the convex shape of the rate-PEB curves shown in 📊Rate-PEB Tradeoff Curve, with the optimal split depending on the relative value of bits and meters in the application's utility function.

isacrate-distortioncapacity-regioncommitView Paper →

⚠️Engineering Note

Resource Allocation in Commercial ISAC Deployments

Turning the rate-PEB tradeoff from theory into product involves tradeoffs that are not captured in the pure information-theoretic analysis:

Traffic adaptation. Positioning demand is bursty: a user might request a location fix once per second, or once per 30 seconds, while data flows continuously. Fixed power splits waste resources. Dynamic schedulers allocate positioning pilots only when requested.
Multi-user coordination. Multiple users may request positioning simultaneously. Orthogonal pilots scale poorly; non-orthogonal (overlapping) positioning pilots are the direction of research, at the cost of mutual interference that must be separated via massive MIMO.
Backward compatibility. 3GPP positioning pilots (PRS) are designed to coexist with legacy LTE/NR signals. The rate cost in deployed networks is therefore not the Pareto-optimal minimum but a few percent of DL resources sacrificed per positioning opportunity.

Practical Constraints

•
PRS scheduling in 5G NR: periodic with period 4-160 ms, muting patterns to avoid AP-AP interference
•
Fronthaul load: PRS collection doubles uplink-reference fronthaul usage when active
•
Latency: 10-40 ms from position request to reported fix is typical in Release 17

📋 Ref: 3GPP TS 38.211 Section 7.4.1.7, TS 38.305

Cell-Free Positioning Techniques Compared

Technique	Synchronization needed	Observables per AP	Typical accuracy (cell-free, 100 MHz BW)	Standard
Multi-RTT	Inter-AP (ns) + UE processing delay	1 range	0.5-2 m	3GPP Rel. 16+
UL-TDOA	Inter-AP (ns) only	1 TDOA (referenced)	1-3 m	3GPP Rel. 16+
DL-AOA	None (DL)	1 angle	5-15 m (without ranging)	3GPP Rel. 17
Joint TOA+AOA (multi-antenna)	Inter-AP (ns)	1 range + 1 angle	0.1-0.5 m	Research / ISAC
Pilot + data joint ML (EM)	Inter-AP + coherent	Full waveform	Near CRB (3-5 dB gain over decoupled)	Research / CommIT

CRB and the Rate-PEB Tradeoff