Cell-Free Positioning

Consider a cell-free network with $L$ APs at known positions $\mathbf{q}_l \in \mathbb{R}^2$, each with $N$ antennas, all connected via fronthaul to a central processing unit (CPU). A user at unknown position $\mathbf{p}$ transmits an uplink pilot $s(t)$ of energy $E_s$ and RMS bandwidth $\beta_{\text{rms}}$.

The baseband signal at AP $l$, after carrier frequency and sample-timing downconversion, is

$$\mathbf{y}_l(t) = \alpha_l \, \hat{\mathbf{a}}(\phi_l) \, s(t - \tau_l) + \mathbf{w}_{l}(t)$$

where

• $\tau_l = \|\mathbf{p} - \mathbf{q}_l\|/c$ is the TOA at AP $l$,
• $\phi_l = \text{atan2}(p_y - q_{l,y},\, p_x - q_{l,x})$ is the AOA at AP $l$,
• $\alpha_l \in \mathbb{C}$ is the complex amplitude (path loss and random phase),
• $\hat{\mathbf{a}}(\phi_l) \in \mathbb{C}^N$ is the receive array response,
• $\mathbf{w}_{l}(t) \sim \mathcal{CN}(\mathbf{0}, \sigma^2 \mathbf{I}_N)$.

The CPU receives quantized samples from all $L$ APs via the fronthaul and forms a joint estimate of $\mathbf{p}$.

In multi-RTT positioning, each anchor $l$ transmits a downlink probe, the user echoes it back, and the anchor measures the round-trip time $T_l^{\text{RTT}}$. The estimated distance is

$$\hat{d}_l = \frac{c}{2}\!\left(T_l^{\text{RTT}} - T_l^{\text{UE, proc}}\right)$$

where $T_l^{\text{UE, proc}}$ is a reported processing delay at the UE. The user position is then estimated from $L$ noisy distance measurements via least-squares or ML:

$$\hat{\mathbf{p}} = \arg\min_{\mathbf{p}}\, \sum_{l=1}^{L} \frac{1}{\sigma_l^2}\!\left(\|\mathbf{p} - \mathbf{q}_l\| - \hat{d}_l\right)^2$$

Because RTT cancels any user-side clock offset, multi-RTT requires only inter-AP timing synchronization — not user-to-AP synchronization. In 3GPP 38.305, this is the default cell-based positioning mode for 5G NR.

In UL-TDOA, the user transmits a single uplink pilot, and each AP timestamps the arrival using its local reference clock. Let $\hat{\tau}_l$ denote the timestamp at AP $l$. Choosing AP 1 as reference and forming

$$\Delta\hat{\tau}_{l,1} = \hat{\tau}_l - \hat{\tau}_1 \approx \frac{\|\mathbf{p} - \mathbf{q}_l\| - \|\mathbf{p} - \mathbf{q}_1\|}{c}$$

eliminates the unknown transmit time at the user. The locus of points consistent with one TDOA measurement is a hyperbola with foci at $\mathbf{q}_1$ and $\mathbf{q}_l$. With $L - 1$ independent TDOA measurements, the user position is the intersection of $L - 1$ hyperbolas, and the ML estimator is

$$\hat{\mathbf{p}} = \arg\min_{\mathbf{p}}\, \|\mathbf{r} - \mathbf{h}(\mathbf{p})\|_{\boldsymbol{\Sigma}^{-1}}^2$$

with $\mathbf{r} = [\Delta\hat{\tau}_{2,1}, \ldots, \Delta\hat{\tau}_{L,1}]^T$ and $\mathbf{h}(\mathbf{p})$ the predicted TDOA vector. The TDOA noise covariance $\boldsymbol{\Sigma}$ is not diagonal: because all TDOAs share a common reference, $\text{Cov}(\Delta\hat\tau_{l,1}, \Delta\hat\tau_{l',1}) = \sigma_{\tau,1}^2$ for $l \neq l'$.