The * denotes convolution and u[n] as the heaviside function.
$x[n]= u[n]α^n$
Determine a sequence $h[n]$ such that-:
$x[n]∗h[n]=α^n(u[n+2]−u[n−2])$
I am trying this problem for quite awhile now. Can someone shed some light as to which is the proper path to take. Im thinking of taking the fourier transform of the L.H.S and then solve it algebrically followed by an inverse fourier transform but I get lost.
We have $x[n] = \alpha^n$ for $n \geq 0$ and
$x[n]*h[n] = \alpha^n-1$ for $n \geq 2$
$x[n]*h[n] = -1$ for $ -2 \leq n \leq 1$
$x[n]*h[n] = 0$ else
You just have to pick clever values of $h[n]$ such as $h[n] = (1-\frac{1}{\alpha^n})$ and multiply by various variants of step functions to match the above requirements.