Signal recovery via compressive sensing

Abstract

Compressed Sensing (CS) has applications in many areas of signal processing such as data

compression, data acquisition and dimensionality reduction. CS ensures faithful recovery of

certain signals or images using a small number of samples or observations than traditional

methods use. Many natural signals have sparse representations when expressed in a proper basis.

Sparse signal can be recovered from the observation vector using convex optimization

teclmiques like !!-minimization (Basis Pursuit). For a faster recovery, greedy algorithms such as

Orthogonal Matching Pursuit (OMP), Regularized Orthogonal Matching Pursuit (ROMP), Stagewise

Orthogonal Matching Pursuit (St- OMP), Backtracking-based Adaptive Orthogonal

Matching Pursuit (BAOMP), etc. can be used . In my experiments OMP algoritlun is used to

recover the original signal as it less complex and computationally inexpensive.

The initial part of the project deals with understanding the recovery of sum of sine/cosine waves

via compressive sensing using OMP algorithm by applying proper basis functions so that signal

is represented with good sparsity. The second and third part of the project is aimed at recovering

sine wave using different basis functions like DCT , DFT and WARPED DFT. This was

challenging because there were multiple peaks and spectral leakages in the frequency spectrum

which imposed difficulty while recovering the data with few measurements. The last part of the

project involved recovering twin sine wave which was also challenging because of less

frequency separation between two sine waves. This imposed problems while picking the location

of the peak from the projection matrix ofOMP.

This report shows the different plots of Mean Squared Error versus Number of Measurements

for different basis functions which helps in determining the best basis function that can be

applied to given signal so that the signal becomes nearly sparse or exactly sparse and can be

recovered with less number of measurements.