Terminologies - sparse channel, sparse input, compressed sensing

Question

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:

1. Under what conditions is the channel sparse and what is the significance of sparse?

2. Under what conditions is the input information sparse and why?

3. 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

Answer 1

The term sparse, as you mention, refers to the fact that some "signal", usually represented by a vector $x$ contains mostly zero or negligible values and only a few non-zero or significant values. In fact, the situation with negligible vs significant values which is not strictly sparse is often referred to as "compressible", see e.g. Candès & Wakin (2008) - An Introduction To Compressive Sampling (you can easily search for an openly accessible PDF of that paper).

Without knowing the literature you are referring to I can only make some guesses, but in general this is what I would assume:

1. The channel being sparse usually means that its (discrete) impulse response is sparse. In the case of speech recording, my guess is that they assume the impulse response from the speech source to the microphone can be modeled as a few distinct reflections of the sound off the surrounding walls.
2. A signal in general is sparse whenever you can find a sparse representation of it in some dictionary. That is, if we have some signal $x$, it is not necessarily $x$ itself that has to consist of only a few non-zero coefficients and a large number of zeros. Whenever we can find some "dictionary" $D$ such that $$x = D z$$ where $z$ is a sparse vector, we can say that $x$ has a sparse representation and compressed sensing can in principle work.
3. The term compressed sensing, very generally, can be used whenever you are trying to solve an under-determined system of linear equations $$y = M x = M D z = A z,$$ where $A$ is $M \times N$, where $M<N$ and you exploit the fact that you can actually solve it because you know that the number $K$ of non-zero coefficients in $z$ is $K<M$ (simplified - $M$ has to be $O(K \log N)$. It can be applicable to input and channel alike, depending on which one is assumed sparse. The signal model used for reconstruction of the sparse quantity just has to be posed accordingly, whether it is the speech signal or the channel through which it is recorded that is assumed sparse.

It seems you are being confused by the fact you have most often seen compressed sensing applied to the recorded signal so that $x$ above is the speech recorded by the microphone and the impulse response would then be represented by the measurement matrix $M$ above or, using the variable $H$ - for impulse response matrix - instead: $$y = H x$$ In the case you mention it sounds as if the channel may be what they are interested in sensing instead. I that case one could imagine recording a known signal $x$ in order to estimate the channel (impulse response), in which case we can swap the roles of channel and signal and instead use the following signal model: $$y = X h$$ to be able to estimate the (sparse) channel $h$ from measurements $y$ and our known speech signal $x$ used to compose the Hankel matrix $X$.