Convolutional Compressed Sensing Using Deterministic Sequences

Abstract

Abstract-In this paper, a new class of circulant matrices built from deterministic sequences is proposed for convolutionbased 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-ZadoffChu (FZC) sequence, the 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.

Code properties analysis for the implementation of a modulated wideband converter

“…The IDFT of a Zadoff-Chu code with prime length is also a Zadoff-Chu code so both temporal and frequency definition would lead to the same matrice structure, which verifies the hypothesis in [10]. Zadoff-Chu sequences are complex-valued and constant envelope codes known in communications for synchronization in Long-Term Evolution (LTE) mobile communications systems due to their perfect cyclic autocorrelation function.…”

“…They found no significant differences with usual sparse signals thus suggesting less randomness would suffice as well. Furthermore, the authors in [10] point out that randomly sampled deterministic sensing matrices generated from the inverse DFT of an unimodular sequence σ σ σ with perfect (or nearly perfect in a certain extent) autocorrelation guarantee, for signals sparse in time or frequency, better recovery than random filters, which target no specific sparsity domain. Universality guarantees are a common target in compressive sensing but this proves that there might be better options if we have knowledge of the sparsity domain.…”

Construction of Deterministic Measurements Matrix Using Decimated Legendre Sequences

“…After years of researching, we know that a structured sensing matrix can play a better role in reconstruction. There're lots of methods to construct a structured sensing matrix, one of them is to construct from a partial circulant matrix [8]. (1) Where is a sampling operator, and A is an circulant matrix, which can be expressed as follows: (2) According to convolutional Compressed Sensing, matrix A can be factorized into:…”

“…When is random, is also random, it is proved that if , the RIP of will be satisfied. When is deterministic, is a deterministic sampling operator; if [8], the RIP of will be held for any that is orthonormal. In this paper, we construct a new kind of by using a decimated binary Legendre sequence [9].…”

“…Theorem 1: [8] For the matrix in (1), where is a random sampling operator and is unimodular. If (6) Then all k-sparse signals in the canonical basis ( ) or the FFT basis ( ) can be stably recovered as long as .…”