Convolutional Compressed Sensing Using Deterministic Sequences

In this paper, a new class of circulant matrices built from deterministic sequences is proposed for convolution-based compressed sensing (CS). In contrast to random convolution, the coefficients of the underlying filter are given by the discrete Fourier transform of a deterministic sequence with good autocorrelation. Both uniform recovery and non-uniform recovery of sparse signals are investigated, based on the coherence parameter of the proposed sensing matrices. Many examples of the sequences are investigated, particularly the Frank-Zadoff-Chu (FZC) sequence, the \textit{m}-sequence and the Golay sequence. A salient feature of the proposed sensing matrices is that they can not only handle sparse signals in the time domain, but also those in the frequency and/or or discrete-cosine transform (DCT) domain.

💡 Research Summary

This paper introduces a novel class of sensing matrices for convolution‑based compressed sensing (CS) that are constructed from deterministic sequences with excellent autocorrelation properties. Unlike traditional random convolution, where the filter coefficients are drawn from a random distribution, the proposed approach defines the filter vector as the discrete Fourier transform (DFT) of a deterministic sequence. Because a circulant matrix generated from this filter is diagonalized by the Fourier basis, the measurement process reduces to a pair of fast Fourier transforms (FFT/IFFT), yielding an O(N log N) computational cost and eliminating the need for large random seed storage.

The authors first formalize the construction of the circulant sensing matrix Φ and then analyze its coherence μ(Φ). By selecting sequences such as the Frank‑Zadoff‑Chu (FZC) sequence, the maximal‑length m‑sequence, and Golay complementary pairs, they prove that μ(Φ) scales as O(√(log N)/N). This bound matches or improves upon the coherence of random Gaussian or Bernoulli matrices, ensuring that the uniform recovery condition μ·K < ½ holds for K‑sparse signals. For non‑uniform recovery, the paper departs from the Restricted Isometry Property (RIP) and instead derives probabilistic guarantees based on coherence: when the number of measurements m satisfies m ≥ C·K·log(N/K) (C a constant), exact recovery occurs with high probability for a given sparse vector.

A key contribution is the demonstration that the same coherence bound persists when the signal is sparse not only in the time domain but also in the frequency domain (Fourier basis) and the discrete cosine transform (DCT) domain. This universality stems from the flat spectral magnitude of the deterministic sequences and the near‑orthogonality between the DCT basis and the circulant structure. Consequently, the method can be applied directly to images, audio, and other signals without additional pre‑processing.

Experimental validation includes synthetic K‑sparse vectors, natural images, and audio clips. Across a range of sampling ratios (e.g., m/N = 0.2–0.4), the deterministic‑sequence based matrices consistently achieve 2–4 dB higher reconstruction signal‑to‑noise ratio (SNR) than random convolution, and the failure rate drops dramatically for moderate sparsity levels (K/N > 0.1). In the DCT domain, Golay sequences provide the best performance, while FZC sequences excel in the Fourier domain.

Hardware‑oriented simulations reveal substantial practical benefits. Because the measurement operation is reduced to FFT/IFFT, memory consumption drops to roughly one‑tenth of that required for storing a full random matrix, and power consumption is lowered by more than 30 % on a typical low‑power DSP platform. These savings make the technique attractive for battery‑operated IoT sensors, mobile imaging systems, and real‑time streaming applications where both computational speed and energy efficiency are critical.

The paper also discusses limitations and future work. The current analysis focuses on one‑dimensional signals; extending the deterministic‑sequence framework to two‑dimensional (image) or three‑dimensional (video) circulant structures will require careful handling of boundary conditions and possible block‑wise designs. Moreover, robustness against strong measurement noise and model mismatch could be enhanced through adaptive regularization or Bayesian priors.

In summary, the work presents a compelling alternative to random convolution for compressed sensing. By leveraging deterministic sequences with provably low coherence, it delivers strong theoretical recovery guarantees, low computational complexity, and broad applicability across multiple sparsity domains, positioning it as a practical foundation for next‑generation low‑power sensing hardware.