The term sparse in general means that there are more elements that are zero valued or very close to zero in comparison to the number of non-zero. In speech deonvolution research papers, the channel is assumed to be sparse, so the channel has more number of zeros. For instance: the channel is sparse when the speech signal is recorded by one microphone in an enclosed space. The microphone signal is corrupted by reverberation and additive noise. I don't understand why the channel is sparse in this case.
In compressive sensing methods, I have seen that the input signal is sparse. In this case, it is related to nyquist sampling theorum and there should be a large number of measurements. I have the following questions:
Under what conditions is the channel sparse and what is the significance of sparse?
Under what conditions is the input information sparse and why?
When is the term compressed sensing used? Is it applicable to sparse channel or sparse input?
Please correct me where wrong.
I could not find definite answers and information in research articles and book related to these questions, and shall really appreciate a succint explanation. Thank you