Impulse response of an infinite-length system I have a system which is expressed as following:
$$y[n] = x[n] - \alpha y[n-N]$$ where $y[n]$ is the output and $x[n]$ is the input.
When I take the z-transform of both sides, I've found the transfer function as $$H(z)=\frac{1}{1+\alpha z^{-N}}$$.
However, I need to find the impulse response of this system but I couldn't find $h[n]$. Can anybody help me how to solve this problem?
Cconsider the following pair of signals and their transforms:
If $g[n]$ has a Z-transform $G(z)$ then the expanded (zero stuffed) signal $h[n]$ has the Z-transform $H(z) = G(z^N)$, where
$$ h[n] = \begin{cases} {g[\frac{n}{N}] ~~~,~~~ n = m N  \\ 0 ~~~~~~~~~,~~~ \text{otherwise}  \tag{1}}\end{cases} $$
Looking at the given Z-transform :
$$H(z) = \frac{ 1}{ 1 - \alpha z^{-N} }$$
it can be seen that if $H(z) = G(z^N)$ where
$$G(z)= \frac{ 1}{ 1 - \alpha z^{-1} }$$
and $g[n] = \alpha^n u[n] $.
Then you can simply conlcude that :
$$ h[n] = \alpha^{n/N} ~u[n/N] =  \begin{cases} {\alpha^{n/N}  ~~ ,~~ n = m N, m = 0,1,..  \\ 0 ~~~~~~~~~,~~ \text{otherwise} }\end{cases} $$