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Check Understanding Impulse & Frequency Response

  1. Feb 24, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the impulse and frequency responses of the following systems:

    1. [itex]y(n) = \frac{1}{N+1}\sum_{k=-N}^{N}(1-\frac{|k|}{N+1})x(n-k)[/itex]

    2. [itex]y(n)=ay(n-1)+(1-a)x(n)[/itex], where 0<a<1



    2. Relevant equations



    3. The attempt at a solution


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

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

    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 [itex] h(w) = \sum_{m=-\infty}^{\infty}h(m)e^{-jwm}[/itex]. 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.
     
  2. jcsd
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