TNO – Time Delay Estimation: Hardware efficient computations #SWI2026

Coherent distributed wireless systems, from cell-free massive Multiple-Input Multiple-Output
(MIMO) to networked radar arrays, require precise synchronization to achieve their theoretical
performance gains. This has applications for the synchronization of drone swarms as well as other
groups of communicating devices. Figure 1 gives a broad sketch of such use case, highlighting key
issues and outcomes. However, these results rely on sophisticated hardware and assume favorable
channel conditions. The core challenge is maintaining such precision in realistic environments:
multipath-rich, low-Signal to Noise Ratio (SNR) scenarios using low cost hardware. Imagine trying to
synchronize multiple radios in a city canyon with strong reflections. The direct signal is weak, while
multiple delayed echoes arrive with comparable strength. Traditional cross-correlation
sees only a blurred peak, unable to distinguish closely spaced echoes. Super-resolution
algorithms like Multiple Signal Classification (MUSIC) and Estimation of Signal Parameters via
Rotational Invariance Techniques (ESPRIT) can theoretically separate these echoes, but they
require heavy matrix computations, e.g., the SVD, that are not a natural fit for low-power
Field-Programmable Gate Arrays (FPGAs) and ARM-processors, typically found in low cost software
defined radios (SDRs). Simplified recursive versions become unstable when implemented
with limited numerical precision, errors gradually accumulate and the tracked subspace
collapses.

Problem definition on CFR samples

Let \(X[k]\) and \(Y[k]\) denote the known transmitted and received complex values on OFDM subcarrier frequency \(f_k\)$ within an L-LTF. The channel frequency response (CFR) estimate is
\begin{equation}
H[k] \;=\; \frac{Y[k]}{X[k]}
\;=\; \sum_{p=1}^{P}\alpha_p\,e^{-j2\pi f_k \tau_p} + n[k],
\qquad k\in K,
\label{eq:cfr_model}\tag{1}
\end{equation}
with complex path gains \(\alpha_p\), continuous delays \(\tau_p\), and additive noise \(n[k]\). Collect \(M=|K|\) used subcarriers into
\[\mathbf{H}=[H[k_1],\ldots,H[k_M]]^\top.
\]
\begin{equation}
\mathbf{H} = \mathbf{V} {\beta} + \mathbf{n}\tag{2}
\end{equation}
with
\begin{equation}
\mathbf{V}=
\begin{bmatrix}
1 & 1 & \cdots & 1 \\
e^{-j2\pi \Delta\tau_1} & e^{-j2\pi \Delta\tau_2} & \cdots & e^{-j2\pi \Delta\tau_P} \\
\vdots & \vdots & \ddots & \vdots \\
e^{-j2\pi (M-1)\Delta\tau_1} & e^{-j2\pi (M-1)\Delta\tau_2} & \cdots & e^{-j2\pi (M-1)\Delta\tau_P}
\end{bmatrix}\tag{3}
\end{equation}
and \(\beta = [\alpha_1e^{j2\pi f_0\tau_1},\alpha_2e^{j2\pi f_0\tau_2},\cdots,\alpha_Pe^{j2\pi f_0\tau_P}]^T \in \mathbb{C}^{P\times 1}\), \(\mathbf{n}=[n_1,n_2,\cdots,n_M]^T\in\mathbb{C}^{M\times 1}\)
where \(\mathbf{V}\) has a Vandermonde structure.

Forward-Backward Frequency Smoothing (FBFS) addresses the rank deficiency in the covariance matrix caused by highly correlated (coherent) signals. By averaging forward and conjugate-reversed snapshots, FBFS decorrelates signals, restores full rank, and stabilizes the subspace estimation for algorithms like MUSIC and ESPRIT. To perform FBFS, choose an integer subarray length \(L\) with \(1< L< M\). Construct overlapping forward snapshots
\begin{equation}
\mathbf{H}_i \;=\; [\,H[i],\,H[i+1],\,\ldots,\,H[i+L-1]\,]^\top,\quad i=1,\ldots,M-L+1,
\label{eq:fwd_subarrays}\tag{4}
\end{equation}
and, for forward–backward averaging, conjugate-reversed counterparts
\begin{equation}
\widetilde{\mathbf{H}}_i \;=\;
[\,\overline{H[M-i+1]}, \overline{H[M-i]}\ldots,\overline{H[M-i-L]} \,]^\top,
\quad i=1,…, M-L+1
\label{eq:bwd_subarrays}\tag{5}
\end{equation}
Form the sample covariance matrix by averaging outer products over the union of forward and backward snapshots:
\begin{equation}
\mathbf{R} \;=\; \frac{1}{|S|}\sum_{\mathbf{x} \in S}\mathbf{x}\,\mathbf{x}^H,\qquad
S = \bigcup_{i=1}^{M-L+1}\{\widetilde{\mathbf{H}}_i, \mathbf{H}_i\}.
\label{eq:cov_fs}\tag{6}
\end{equation}

To ensure high resolution time delay estimation (TDE), typically, subspace methods, like MUSIC,
ESPRIT and the matrix pencil method, are used. These methods employ the SVD/EVD of the
sample covariance matrix as an integral part of their approach.

Problem Statements

• Find a suitable algorithm for this SVD/EVD for the matrix \(\mathbf{R}\) in (6) using the hardware-friendly operations for FPGA logic. Balancing resolution/accuracy and space/time complexity.
• Another option is to find an alternative solution (non-SVD) to the problem of estimating
  parameters (${\tau }_p$ or ${\alpha }_p$) of (un)damped complex exponentials, while balancing resolution/accuracy and space/time complexity.

Performance Evaluation
We will provide synthetic and experimental data sets on which algorithms can be tested on the
spot using a Python-based test harness. We can use this to evaluate the resolution/accuracy and
space/time complexity of the submitted algorithms.