Conceptual questions from signal processing

Question

I don't have a thorough background in Signal processing and require some information for an application pertaining to computer science. Minimum Entropy Blind Signal Deconvolution with Non Minimum Phase FIR Filters paper proposes to use entropy for blind system identification. My questions are the following based on terms that I cannot follow in the paper and shall be grateful for detailed explanation

Q1: What does minimum phase filter mean? In my understanding phase of a signal means the angular displacement or the angle $\theta$ that appears in an equation like $x=A sin (\omega*t + \theta)$

Q2: What is an inverse filter? How is it related to convolution and deconvolution?

Answer 1

We will be talking about linear time-invariant systems.

1) A minimum phase filter is one which is causal and stable and its inverse is causal and stable. In the case of a discrete time system, you have all the poles and zeros of the transfer function within the unit circle.

2) An inverse filter of a filter with transfer function $H(z)$ is a filter $G(z)$ such that $H(z) G(z) = 1$ -- that is, when you cascade the filter with its inverse filter, its the identity filter. In the z-transform domain, multiplication corresponds to convolution in the time domain. So, if your input is $X(z)$, and your filter is $H(z)$ and your output is $Y(z)$, the relation is $Y(z) = H(z) X(z)$ in the z-transform domain, while in the time domain, $y[n] = h[n] * x[n]$ where $*$ denotes convolution and $h$ is the impulse response of the filter. The inverse filter convolves with the impulse response corresponding to $1/H(z)$ (that is, you take the output $y$ and find an input which when convolved with $h$ gives the output -- i.e. deconvolution).

You can find more details in any introductory Digital Signal Processing text like Discrete-Time Signal Processing by Oppenheim, Schafer and Buck (now in its 3rd edition).

Answer 2

Batman has given a great answer. You need to go through the recommended book in order to understand the concepts mentioned. Let me try to simplify it.

BIG PICTURE: De-convolution or inverse filtering is required to retrieve an estimate of the original signal that went through an unknown linear system. Basically, we have a signal which went through an unknown system and we want to get the original back. Example: You shouted something across a room with lot of reverberation (echos) to your brother. Now for your brother to understand it, he needs to remove the reverberation effect. Let us see it in signal processing terminology.

What is convolution: Let $x(k)$ be your speech signal, and $h(k)$ is the unknown response of the room, then speech heard by your brother is $y(k) = h(k)\ast x(k)$ (this is convolution) and is more formally defined as:

$$y(k) = \sum_{n=0}^{N-1} h(n)x(k-n) $$

Basically, your brother is listening to attenuated copies of your sound, i.e., $y(k) = h(0)x(k) + h(1)x(k-1) + ... + h(N-1)x(k-N+1)$. Note here x(k) is the complete speech of yours and $k-i$ denotes shift by $i$th sample, referring to the shifted copies.

What is deconvolution: Now lets say your brother recorded $y(k)$ and he wants to know what did you say? i.e., what is $x(k)$. What he needs to do is design a filter $w(k)$ such that $\delta(k) = w(k)\ast h(k)$, i.e., $w(k)$ is the inverse of $h(k)$. If you know basic signal processing then $\delta$ is the dirac delta function or an impulse. Now if we convolve $y(k)$ with $w(k)$ then $w(k)\ast y(k) = w(k)\ast h(k) \ast x(k) = \delta(k) \ast x(k) = x(k)$. And he can know what you said.

Minimum Phase: deals with the design of $w(k)$. Minimum phase also means that our filter is designed such that it responds much faster than another filter that is non-minimum phase. It has to do with setting of zeros of $w(k)$. This concept will require another discussion, which I'll postpone for now.