Recovery of Audio Signals with Compressed Sensing

Compressed sensing describes a system in which the sampling and compression of a digital signal can be done simultaneously. This means that, when dealing with an analog signal such as audio, the signal must first pass through an ADC and then be sampled. This means that the system still has to capture the entire signal before it can be compressed, negating a large portion of the benefit of using compressed sensing.

Fortunately, there has been some research done on methods to directly compress the analog signal and create samples which can be recovered using standard CS techniques. One of these systems is described below.

The system described by Kirolos et. al. uses a three step process to generate the samples. The analog signal is first spread through multiplication with a PN sequence which must alternate faster than the Nyquist rate of the signal. The “demodulated” signal is then passed through a low-pass filter and sampled at a rate M based on the sparsity. This signal y|m| can be characterized as a linear transformation of the CS samples (α). For this system, the sampling matrix is composed of two operators: Ψ which maps y to α and Φ which maps the original analog signal x to the discrete signal y. The sampling matrix is described by the equation

where v_m,n ∈ V is the element in the m^th row and n^th column of the sampling matrix, ψ is the sparsity inducing basis, p_c is the PN sequence and h is the low pass filter.

Based on this filtering, the original signal x can be recovered by solving the standard ℓ₁ minimization problem using V as the sampling matrix.