So I have this system: $$ y(n) = \frac{1}{N+1}\sum_{k=-N}^{N}\left(1-\frac{|k|}{N+1}\right)x(n-k) $$

Now for the impulse response I'm getting:

\begin{cases}\frac{1}{N+1}\left(1-\frac{|k|}{N+1}\right) &\text{for}\quad|n|\leq N\\ 0& \text{otherwise} \end{cases}

But now I'm supposed to find the frequency response and supposedly a closed form expression can be found for it.

So the frequency response is just the Fourier transform of the impulse response so I have this function:

$$ F(s)=\frac{1}{p(N+1)}\sum_{k=-N}^{N}\left(1-\frac{|k|}{N+1}\right)e^{\frac{-2\pi isn}{p}}$$

From here I used Euler's formula and the $isin$ parts cancel themselves and then I was able to use Legendre's summation formulas to find a super messy closed form but I think that can't be right. I think instead I have set up the frequency response wrong. Could some one help me? Thanks.


1 Answer 1


Is this for homework?

If so I won't directly give an answer, just a hint about the general approach that can be used to solve such questions without a single derivation:

Plot the impulse response. How does it look? Try building this by combining or modifying (through addition, dilatation, translation, multiplication, convolution, derivation, integration...) very simple functions the DFT of which is known (complex exponentials, constants, rectangles, diracs, Heaviside step). From then, use identities on the DFT to get the DFT of your impulse response.


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