# Modelling problem

Considering a finite-length impulse response $$h[n]$$ of length $$M$$ (which amounts to considering $$h[n] = 0,$$ for $$n \geq M$$). The data model, with additive disturbance, is then written as: $$y[n] = y_M[n] + \epsilon[n]$$ where

$$y_M[n]= \sum_{k=0 }^{ M-1} h[k]x[n-k].$$

We seek to estimate the impulse response $$h[n]$$, i.e., the parameter vector $$h = \big[h[0], h[1], ..., h[M-1]\big]$$, from the observed data $$y[0], y[1], ..., y[N-1]$$ and known inputs $$x[0], x[1], ..., x[N-1]$$ (we assume that the input signal is causal, i.e., $$x[n] = 0$$ for $$n < 0$$).

1. What is the output of the system $$y[n]$$ when an impulse is placed at the input? Show that we can deduce an estimator $$hri$$ of the impulse response, specifying its bias and covariance matrix.

2. Is the model $$y_M[n]$$ linear with respect to the parameters?

This is what I did, but I don’t know how to write it in a matrix form

$$\cases{ y_M [n] = \displaystyle\sum_{k=0}^{M-1}h[k]\mu[n-k]\\ y[n] = h[n] + \epsilon[n]}$$

• Is this a homework problem? Commented Mar 12, 2023 at 14:33
• You say "estimator hri" -- do you mean something like $h_{ri}$, or is "hri" an abbreviation for something? Please edit your question to put the math into math, or to spell out your abbreviations. Commented Mar 12, 2023 at 18:03
• Is the additive disturbance truly not dependent on $k$? I.e., is $y[k] = y_M[k] + \epsilon$ correct, or is it $y[k] = y_M[k] + \epsilon[k]$? Commented Mar 12, 2023 at 18:04
• @TimWescott I have corrected it Commented Mar 12, 2023 at 19:00