  In this paper, we demonstrate some applications of compressive sensing over
networks. We make a connection between compressive sensing and traditional
information theoretic techniques in source coding and channel coding. Our
results provide an explicit trade-off between the rate and the decoding
complexity. The key difference of compressive sensing and traditional
information theoretic approaches is at their decoding side. Although optimal
decoders to recover the original signal, compressed by source coding have high
complexity, the compressive sensing decoder is a linear or convex optimization.
First, we investigate applications of compressive sensing on distributed
compression of correlated sources. Here, by using compressive sensing, we
propose a compression scheme for a family of correlated sources with a
modularized decoder, providing a trade-off between the compression rate and the
decoding complexity. We call this scheme Sparse Distributed Compression. We use
this compression scheme for a general multicast network with correlated
sources. Here, we first decode some of the sources by a network decoding
technique and then, we use a compressive sensing decoder to obtain the whole
sources. Then, we investigate applications of compressive sensing on channel
coding. We propose a coding scheme that combines compressive sensing and random
channel coding for a high-SNR point-to-point Gaussian channel. We call this
scheme Sparse Channel Coding. We propose a modularized decoder providing a
trade-off between the capacity loss and the decoding complexity. At the
receiver side, first, we use a compressive sensing decoder on a noisy signal to
obtain a noisy estimate of the original signal and then, we apply a traditional
channel coding decoder to find the original signal.
