Simultaneous Localization and Mapping

The joint posterior of radio-SLAM has a natural factor graph representation that reveals the conditional independence structure and enables scalable inference.

The factor graph contains:

• Variable nodes for each user state $\mathbf{s}_t$ and each VA position $\mathbf{f}_k$.
• Factor nodes connecting variables:
  • Transition factors $\psi_t(\mathbf{s}_{t-1}, \mathbf{s}_t) = p(\mathbf{s}_t | \mathbf{s}_{t-1})$ encode the motion model (e.g., constant-velocity with process noise).
  • Measurement factors $\phi_{t,k}(\mathbf{s}_t, \mathbf{f}_k) = p(\mathbf{z}_{t,k} | \mathbf{s}_t, \mathbf{f}_k)$ encode the likelihood of observing MPC $k$ at time $t$ given the user state and VA position.
  • Data association factors handle the unknown correspondence between detected MPCs and VAs (a combinatorial challenge analogous to multi-target tracking).

Belief propagation on this graph iteratively passes messages between variable and factor nodes. For the user state, the messages from all associated VAs are combined to form a position belief. For each VA, messages from all time steps refine its position estimate. The algorithm naturally handles:

• Appearing/disappearing features (VAs that become visible or occluded as the UE moves)
• Unknown number of features (via a Poisson or negative-binomial prior on $K$)
• Measurement origin uncertainty (which detected MPC corresponds to which VA)

The computational cost per time step scales as $O(K_t \cdot P)$ where $P$ is the number of particles used to represent the beliefs and $K_t$ is the number of detected MPCs.

Channel-SLAM (Gentner et al., 2016) takes the radio-SLAM idea to its logical extreme: it operates with a single physical base station and treats each resolvable MPC as an independent virtual anchor. The measurements from each MPC --- delay $\tau_k$, possibly angle $\theta_k$ and Doppler shift $\nu_k$ --- are used to estimate both the UE trajectory and the VA positions.

The key insight is that as the UE moves, each MPC traces a characteristic trajectory in the delay-angle-Doppler space that depends on the geometry of the reflection. By tracking these trajectories over time, Channel-SLAM can:

1. Resolve the VA positions from the temporal evolution of the MPC parameters (similar to how a moving observer can triangulate a static target)
2. Use the resolved VAs as anchors for subsequent position estimation, progressively refining both the map and the trajectory

This approach is particularly powerful at mmWave and sub-THz frequencies, where the large bandwidth resolves individual MPCs in delay, and the large antenna arrays resolve them in angle. In rich indoor environments with 5--20 resolvable MPCs, Channel-SLAM can achieve sub-metre accuracy with a single BS --- a capability impossible with conventional single-BS methods.

An alternative to model-based positioning is fingerprinting, which learns a mapping from radio measurements to positions using a database of labelled training samples. The approach has two phases:

Offline phase (training): A survey is conducted in which RSS values from all hearable BSs are measured at known positions on a grid spanning the area of interest. The collection $\{(\mathbf{p}_j, \mathbf{r}_j)\}_{j=1}^{M}$ forms the fingerprint database (or radio map), where $\mathbf{r}_j = [P_{r,1}^{(j)}, \ldots, P_{r,N}^{(j)}]^T$ is the RSS vector at position $\mathbf{p}_j$.

Online phase (positioning): The UE measures its current RSS vector $\hat{\mathbf{r}}$ and finds the closest match(es) in the database:

$$\hat{\mathbf{p}} = \sum_{j \in \mathcal{K}} w_j \, \mathbf{p}_j, \qquad w_j \propto \exp\!\left(-\frac{\|\hat{\mathbf{r}} - \mathbf{r}_j\|^2}{2\sigma^2}\right)$$

where $\mathcal{K}$ is the set of $k$ nearest neighbours in RSS space.

Modern approaches replace the explicit database with a neural network (DNN, CNN, or transformer) that learns the mapping $\hat{\mathbf{p}} = f_\theta(\hat{\mathbf{r}})$ from training data, achieving 1--3 m accuracy in indoor WiFi environments. Channel state information (CSI) fingerprinting further improves accuracy by exploiting amplitude and phase across subcarriers, achieving sub-metre accuracy in controlled environments.