Calculate impulse response of a dynamic system

Question

I am given an exercise where I need to find the impulse response of the system $$ y_n = 1.02(y_{n-1} + x_n). $$

I have looked online but haven't been able to find an example with a system with memory.

Any help would be much appreciated!!

Answer 1

Seeing this difference equation suggests using the Z-transform. This is because the transform of $x_{n-n_0}$ is $X(z)z^{-n_0}$, where $X(z)$ is the Z-transform of $x_n$.

You want to find the sequence $h_n$ such that $y_n=x_n \ast h_n$. In the Z domain we have that $Y(z)=X(z)H(z)$.

In the Z domain, you have the difference equation $$Y(z)=1.02(Y(z)z^{-1}+X(Z)) \Longrightarrow H(z)=\frac{Y(z)}{X(z)}=\frac{1.02}{1-1.02z^{-1}}$$

You can take a look at the list of properties and common transforms and see that the antitransform of the latter gives

$$h_n=\begin{cases}1.02^{n+1}&\text{if }n\geq 0 \\ 0 &\text{otherwise}\end{cases}\,.$$