Reconstruction of Piecewise-Constant Sparse Signals for Modulo Sampling
Haruka Kobayashi, Ryo Hayakawa
TL;DR
This paper addresses the challenge of unfolding signals from modulo samples without clipping, where conventional residual-difference reconstruction suffers from error propagation. It proposes directly reconstructing the residual $z[n]=f_\lambda[n]-f[n]$ by solving a fused sparse optimization that promotes sparsity in both $z$ and its first-order difference, solved efficiently with an ADMM algorithm that exploits FFT-based diagonalization for $O(N\log N)$ per-iteration complexity. The method, Fused Sparse Reconstruction (FSR), uses the real-domain observation model $\bm{F}_\lambda^{\mathrm{R}} \approx \bm{V}^{\mathrm{R}} \bm{z}$ with two $\ell_1$-regularizers and a circulant difference matrix $\mathbf{D}$, achieving improved NMSE over the prior LASSO-$B^2R^2$ approach in simulations under noise. This approach offers a practical, robust path to high-fidelity signal recovery in modulo sampling, with reduced error propagation and computational efficiency suitable for real-time or power-constrained settings.
Abstract
Modulo sampling is a promising technology to preserve amplitude information that exceeds the observable range of analog-to-digital converters during the digitization of analog signals. Since conventional methods typically reconstruct the original signal by estimating the differences of the residual signal and computing their cumulative sum, each estimation error inevitably propagates through subsequent time samples. In this paper, to eliminate this error-propagation problem, we propose an algorithm that reconstructs the residual signal directly. The proposed method takes advantage of the high-frequency characteristics of the modulo samples and the sparsity of both the residual signal and its difference. Simulation results show that the proposed method reconstructs the original signal more accurately than a conventional method based on the differences of the residual signal.