This is a difference equation to a causal LTI system:
$y[n] = ay[n - 1] + x[n] - a^Nx[n - N]$
Where N is a positive integer. I need to determine the impulse response of the system, so I have the equation:
$h[n] = ah[n-1] + \delta[n] - a^N\delta[n - N]$
h[n] is simple to find for the case where n < 0, n = 0, or n = 1. However, the response could be different for the case n = 2, depending on the value of N (in this case, N = 1 or N = 2):
$h = ah + 0 - a^1\delta[2-1] = a^2$
$h = ah + 0 - a^2\delta[2-2] = 0$
So which one is it? Intuition tells me the top one as the impulse response will be decaying, but we only know that N is positive.