Check Understanding Impulse & Frequency Response

1. Feb 24, 2012

Poopsilon

1. The problem statement, all variables and given/known data

Find the impulse and frequency responses of the following systems:

1. $y(n) = \frac{1}{N+1}\sum_{k=-N}^{N}(1-\frac{|k|}{N+1})x(n-k)$

2. $y(n)=ay(n-1)+(1-a)x(n)$, where 0<a<1

2. Relevant equations

3. The attempt at a solution

Ok so for 1. I look at h(n) which is $\frac{1}{N+1}[(1-\frac{N}{N+1}) + (1-\frac{N-1}{N+1}) + ... +1+ (1-\frac{1}{N+1}) + ... + (1-\frac{N}{N+1})]$ And then this is equal to $\frac{1}{N+1}[2N+1 - \frac{2N + 2(N-1) + ... + 2}{N+1}] = 1$. Thus the impulse response is just 1.

Now for 2. I observe the recursive behavior and conclude $y(n) = (1-a)x(n) + a(1-a)x(n-1) + a^2 (1-a)x(n-2)+...$. And thus $h(n)= (1-a)\sum_{n=0}^{\infty} a^n = (1-a)\frac{1}{1-a} = 1$.

Thus both my impulse responses are 1. And now for the frequency response I'm a bit confused, I think I find the frequency response by calculating $h(w) = \sum_{m=-\infty}^{\infty}h(m)e^{-jwm}$. Is that correct? And so then h(m) would just be 1 in both cases? I'm a bit lost and we are using a Fourier analysis text and not a signal processing text so I'm having trouble matching engineering terminology with the appropriate mathematics. Thanks.