Given the input and the output, how to determine the impulse response?

Question

I would like to find the impulse response, $h[n]$, of an LTI system given the input

$$x[n] = [1,-3,2]$$

and the output

$$y[n] = [1,-1,-4,4]$$

I know that $y[t]=x[t]*h[t]$, but I am having hard time to figure out the right way to calculate the impulse response. I know very little about signal processing, so if you don't mind giving an easy explanation, then I appreciate it. Or, if it's possible to do an example, that's better.

Answer 1

$\mathcal Z$-transforming the input and the output and then dividing them, we obtain the transfer function

$$\begin{array}{rl} H (z) &= \dfrac{1 - z^{-1} - 4 z^{-2} + 4 z^{-3}}{1 - 3 z^{-1} + 2 z^{-2}}\\\\ &= \dfrac{z^3 - z^2 - 4 z + 4}{z (z^2 - 3 z + 2)}\\\\ &= \dfrac{(z-1) (z-2) (z+2)}{z (z-1) (z-2)}\\\\ &= \dfrac{z+2}{z} = \color{blue}{1 + 2 z^{-1}}\end{array}$$

Computing the inverse $\mathcal Z$-transform of $H (z)$, we finally obtain the impulse response.

Answer 2

From the lengths of the input and the output, we conclude that the impulse response has length $2$. Thus, the output is given by

$$y_k = h_0 x_k + h_1 x_{k-1}$$

We then have an overdetermined system of $4$ linear equations in $h_0$ and $h_1$

$$\begin{bmatrix} x_0 & 0\\ x_1 & x_0\\ x_2 & x_1\\ 0 & x_2\end{bmatrix} \begin{bmatrix} h_0\\ h_1\end{bmatrix} = \begin{bmatrix} y_0\\ y_1\\ y_2\\ y_3\end{bmatrix} $$

Using Python + SymPy:

>>> from sympy import *
>>> X = Matrix([[ 1, 0],
                [-3, 1],
                [ 2,-3],
                [ 0, 2]])
>>> y = Matrix([1,-1,-4,4])
>>> h = (X.T * X)**-1 * X.T * y
>>> h
Matrix([[1],
        [2]])

This is the least-squares solution. It is easy to verify that it is also a solution of the original linear system (otherwise the problem would be ill-posed). Hence,

$$\boxed{y_k = x_k + 2 x_{k-1}}$$