2009/01/15 by Ehsan Ardestanizadeh, Ardestanizadeh, Ehsan, Mahdi Cheraghchi +3
Computer Science · Engineering · Mathematics · #Blind Source Separation Techniques #FOS: Computer and information sciences #Information Theory (cs.IT) #Microwave Imaging and Scattering Analysis #Sparse and Compressive Sensing Techniques #cs.IT #math.IT
paper · pdf · doi:10.48550/arxiv.0901.2147
arxiv created 2009/01/15 · openalex publication_date 2009/01/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper studies the stability of some reconstruction algorithms for compressed sensing in terms of the bit precision. Considering the fact that practical digital systems deal with discretized signals, we motivate the importance of the total number of accurate bits needed from the measurement outcomes in addition to the number of measurements. It is shown that if one uses a 2k × n Vandermonde matrix with roots on the unit circle as the measurement matrix, O(ℓ + k log(n/k)) bits of precision per measurement are sufficient to reconstruct a k-sparse signal x ∈ \Rn with dynamic range (i.e., the absolute ratio between the largest and the smallest nonzero coefficients) at most 2^ℓ within ℓ bits of precision, hence identifying its correct support. Finally, we obtain an upper bound on the total number of required bits when the measurement matrix satisfies a restricted isometry property, which is in particular the case for random Fourier and Gaussian matrices. For very sparse signals, the upper bound on the number of required bits for Vandermonde matrices is shown to be better than this general upper bound.