Spline-Like Chebyshev Polynomial Representation for Compressed Sensing

Compressed sensing (CS) is a technique for signal sampling below the Nyquist rate, based on the assumption that the signal is sparse in some transform domain. The acquired signal is represented in a compressed form that is appropriate for storage, transmission and further processing. In this paper, use of the Chebyshev polynomials of the first kind on intervals for an efficient representation of one-dimensional, continuous-time signals is proposed. To avoid boundary artifacts, a desired number of derivatives are equalized on each interval end in a spline-like fashion. Unlike splines, the proposed system of equations is underdetermined to provide a necessary degree of freedom for achieving sparsity using the l
₁
optimization. The obtained parametric model fits into the compressed sensing setup and offers a new paradigm for efficient processing of analog data on a digital computer. Simulation results of the proposed measurement system and an example of data processing are given to prove its potential.