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