# How to make the impulse response sparse? How does one know that the channel is sparse?

I am new to sparse channel estimation algorithms and reading research articles. One such paper is blind sparse channel estimation using a modification of the BOMP technique titled, "Blind Acoustic Source Separation Via System Identification for Leak Detection in Pipelines."

How are the impulse response/coefficients made sparse?

Dankers, Jalilian, and Westwick have explained the way they have done parameterization under their algorithm on page 4 and (10). How does one do parameterization of impulse response to decide for sparse/non-sparse?

In general, how is parameterization applied to make the impulse response to zero and in general how does one know which coefficients are sparse?

How you parameterize your sparsity will depend on your application. The authors of that paper, in a paragraph on page 231 say: which is why they clump the coefficients together in $P$ blocks of start time $t_{B_k}$ of duration $n_{B_k}$.

For one impulse response this is shown in their figure 1(b). The overall sparsity will depend on the reflections and delays that they have when this impulse response is used by the direct signal and all its indirect reflections.

How are the impulse response/coefficients made sparse?

You need to decide which coefficients in the overall impulse response are zero. To do that, you need to decide why they are zero (what is the cause?). Perhaps there is a long dead-time between the onset and the first secondary response.

How does one do parameterization of impulse response to decide for sparse/non-sparse?

Once you've decided on the form, you decide on the structure of the non-sparse parts of the response and how they are separated from each other.

Suppose your non-sparse parts are all defined by $\alpha_k h[n]$, where $\alpha_k$ is the gain of the $k^\mbox{th}$ non-sparse segment and $h[n]$ is the FIR response of all segments (modulo a gain term). $h[n]$ is of duration $H$. Then the sparsity will come about by how you space out the $\alpha_k h[n]$: $$g[n] = \sum_{k=1}^K \alpha_k h[n-t_{B_k}]$$ where $t_{B_k} \gg H$ for sparsity to be true (so there is lots of space between the $\alpha_k h[n]$.

In general, how is parameterization applied to make the impulse response to zero and in general how does one know which coefficients are sparse?

This comes down to how do you choose $t_{B_k}$ and $H$. That will depend on what system you're trying to model.

• Thank you for your answer. I am having really a hard time understanding the terms and points mentioned in your answer mainly because I don't know the background related to the following: Could you please suggest some reading materials where I can dig in to clear these concepts?(1) what is the meaning of gain, $\alpha_k$? In general I have seen that the channel coefficients $h$ are convolved with the input to get the output measurements. But the equation for $g[n]$ is certainly not the output measurements. What is this equation? – Ria George Feb 14 '18 at 20:54
• (2) It is unclear how to choose $t_{B_k}$ and $H$. Is there any guideline or any common popular technique to do so?(3)Are there methods to find out which coefficients are zero by looking at the data alone?If the method is blind estimation, then there is no access to the impulse response or the input to the channel. So how does one know or guess how many coefficients are sparse and what are their positions? I really appreciate if you can kindly elaborate on these points if possible. Thank you for your time and effort. – Ria George Feb 14 '18 at 20:54
• $h[n]$ is the effect of the direct line transmission from transmitter to receiver. There will be others paths from the transmitter to the receiver. These take a longer time to get there because they are not direct, so the $k^{th}$ on is delayed by $t_{B_k}$. The $\alpha_k$ is just a gain change for these indirect paths; these reflections bounce off different things that have different reflection properties. – Peter K. Feb 15 '18 at 12:31
• $g[n]$ is just the net effect of the direct path and all the reflections. – Peter K. Feb 15 '18 at 12:31

Channel/system identification is an important topic in data processing. One wants to retrieve the behavior of systems given some inputs AND hypotheses. In other words: people hear the sound of a gun in some place. They want to know where the gunner is. Must of of have already experienced that we sometimes are mistaken on the real origin of a sound. To cope with that, one generally resorts to physical laws, several listeners, and additional assumptions.

Assuming linear systems, the heard sound (observation) comes for the source called $s$. The listen sound is a linear combination of the source distorted by its propagation through different media $h$: air, reflection on walls (echoes), and unrelated disturbances ($n$, noises). The linearity entails that, often, one can model the observation as a convolution plus noise model:

$$y = h\ast s+n\,.$$

As we don't know the medium precisely, and the noise is stochastic, resolving this equation with a unique solution is infeasible in general. One solution is to suppose additional natural constrains. Those would restrict the size of the set of potential solutions to, hopefully, a unique faithful one. Parametric models for the system is often not known. One of the simplest assumption is that those are easy to describe. This is a pre-scientific concept called sparsity or parsimony. It is often related to Wilhelm of Ockham, under the idea of the law of parsimony: in today's termes, among several models to describe of data set, the "simplest" can have a special favor. There are optimization methods to solve this.

A sparse system/signal has many zero components. An interpretation is taking values in a labyrinth.

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