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?