Finding Impulse Response of Frequency Response

  • Thread starter Fizzill
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Homework Statement



Consider the frequency response H(e^jw) = e^j3w(5 + cos(w) + 2cos(w)).

Find the impulse response h[n].

Homework Equations



H(e^jw) = Ʃ(from k = 0 to M) bk * e^-jwk = Ʃ(from k = 0 to M) h[k] * e^-jwk

The Attempt at a Solution



So i'm not sure at all how to get about solving this for sure, I believe I did it but then again after going through my book and notes I wasn't able to find anything like it which makes me doubt my answer.

H(e^jw) = e^j3w(e^-j2w + 1/2e^-jw + 5 + 1/2e^jw + e^j2w)
= e^jw + 1/2e^j2w + 5e^j3w + 1/2e^j4w + e^j5w

which i believe would equate to

h[n] = ∂[n + 1] + 1/2∂[n + 2] + 5∂[n + 3] + 1/2∂[n + 4] + ∂[n + 5]
 

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