Real world application of signal sparsity?
Sparsity concept is extensively being used in computer vision and image processing. The Idea is that natural image can be pretty sparse when it is transformed to different bases. this bases can be predefined, e.g. FFT,DCT or can be learned from the image, e.g. sparse coding.

Here are a few well know example of algorithms that uses the sparsity assumption on images:


Jpeg image compression: Uses DCT transformation. assumes that high frequencies are low in natural image, so they are being removed in order to preserve space for compression. see more information on Wikepedia
Sparse coding: is a technique being used that assumes that an image has a lot of redundant information and because of that can be reconstructed from an overcomplete dictionary. each patch in the image should be sparse combination in the dictionary. see instroduction to sparse representation presentation
inverse problems in computer vision like recovering image embedded in noise and recovering sharpe image from blurred image are also using sparse assumption, e.g. wavelete shrinkage is being used for image denoising. See application to wavelet trasform in here