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Estimating an impulse response from input-output measurements is called system identification. When the impulse response of the linear system $h(n)$, is of finite duration (M samples) or can be effectively modeled with a finite impulse response system. System identification, that is, the modeling and identification of a system from knowledge of its input $x(n)$ and output signals $y(n)$ is known as non-blind whereas if the knowledge of the input and the system both are unknown by the receiver end then it is termed as blind system identification. In wireless communications and in signal processing research papers I have seen that the channel impulse response is modeled as a moving average. In communications, there is an additive noise known as the measurement noise $e(n)$. Mathematically expressing, $$y(n) = x(n)*h(n) + e(n)$$

In signal processing and estimation, the objective is to estimate the channel impulse response and recover the input from the noisy observations.

How can I use the above model where the input signal is a sentence. So, a sentence would be represented by assigning each unique word by a symbol. Let's say, I have the sentence "the quick brown fox jumped over the lazy dog" and I have the vocalbulary and mapping as

$the \mapsto 1$

$quick \mapsto 2$

$brown \mapsto 3$

$fox \mapsto 4$

$jumped \mapsto 5$

$over \mapsto 6$

$lazy \mapsto 7$

$dog \mapsto 8$

So, the sentence would be mapped to $[1,2,3,4,5,6,1,7,8]$.

This is the input signal or the input data.

I cannot understand what the role of the channel impulse response would be if

  • I want to estimate after decoding the input speech signal from noisy observations.
  • What would be the noise term and
  • would there be any "channel impulse response" ? What would be the best way to use the FIR model for this kind of data?
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  • $\begingroup$ Your second sentence is incomplete. I have seen that the channel impulse response is modeled as a moving average. That's a very rare case, and often just a gross simplification or an example of an overly simplified channel. $\endgroup$
    – Marcus Müller
    Apr 19, 2017 at 8:00
  • $\begingroup$ This question is like "I want to go into the jungle and end world hunger by dyeing tigers' bum hairs pink. Does this car make me look fat? Here's a list of statements regarding hair dye.". Your approach, your objective, your question and your methods do not match. And, I'd recommend you rethink your attitude towards tigers :) $\endgroup$
    – Marcus Müller
    Apr 19, 2017 at 8:06

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How can I use the above model where the input signal is a sentence.

makes no sense; your whole system is not linear (which is one of the requirements for your whole convolution approach); for example, if you you play back the words "fox" and "jumped" at the same time, you don't hear "lazy".

Not linear means your system can't be represented by an impulse response.

Your signal model is overly simplistic, too. You might map words to numbers as soon as you know the words, but that's only happening after speech recognition (and might be helpful in understanding logically what has been said), but not before.

So, your whole approach shows you haven't dealt with a lot of speech recognition before – which is OK, but it means that you can't just come up with some approach and assume it makes sense from the start.

I'd recommend reading a book on speech recognition. There's certainly a big wealth of literature out there.

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  • $\begingroup$ I did some background reading now and I have realized that I have used some misleading terms but the problem definition is the same. I am considering speech deconvolution application and looking at speech as a sentence consisting of symbols. Ignoring the relationship between successive words that is, like you said that there is no guarantee about the sequence of the appearance of the words for linear channel, I am assuming a simple case study where the signal is a sequence of symbols. $\endgroup$
    – Ria George
    Apr 19, 2017 at 19:35
  • $\begingroup$ The reference papers where I found FIR model for speech are (1) "BLIND DECONVOLUTION OF REVERBERATED SPEECH SIGNALS VIA REGULARIZATION" considers multiple channels (2) KALMAN FILTER AND STATE-SPACE APPROACH TO BLIND DECONVOLUTION by L.-Q. Zhang, A. Cichocki and S. Amari (3) Blind dereverberation of speech signals using independence transform matrix $\endgroup$
    – Ria George
    Apr 19, 2017 at 19:37
  • $\begingroup$ I will appreciate if you can provide few ideas on how to formulate the problem using FIR filter and Kalman state space where the input is a symbol where each symbol is an alias of a word occurring in a sentence so that I can apply these speech deconvolution methods to the application of sentence deconvolution. $\endgroup$
    – Ria George
    Apr 19, 2017 at 19:40
  • $\begingroup$ Can a speech be a sequence of symbols where each symbol is a word in the sentence? Then similar to the papers in which the Authors explain why speech signals need to be estimated, I was thinking to apply the same justification for estimating the sentence / symbol representation of the sentence. I don't know yet which estimation method to use, maybe start off with the simple least square in a non-blind setting. But at the present moment I am stuck in the justification of why I need to estimate sentence, what would the filter coefficients be called as and why FIR model in this case. $\endgroup$
    – Ria George
    Apr 19, 2017 at 20:01

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