Suggestions on how to use FIR model in sentence retrieval task 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?
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.