Signal Processing
Signal Processing
Signal Processing
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Transform theory<br />
Convolution theorem for the Discrete Fourier Transform (periodic<br />
convolution in discrete time)<br />
Let<br />
Then<br />
Proof:<br />
N−1 ∑<br />
˜z n = ˜x n ⊗ ỹ n = ˜x n−q ỹ q<br />
q=0<br />
˜Z k = ˜X k Ỹ k<br />
⎧<br />
⎫<br />
N−1 ∑<br />
N−1<br />
˜Z k = ˜z ne −i 2π ∑ ⎨N−1<br />
∑ ⎬<br />
N kn =<br />
˜x n−q ỹ q<br />
⎩<br />
⎭ e−i 2π N kn<br />
n=0<br />
n=0 q=0<br />
= {(p, q) = (n − q, q)} =<br />
N−1 ∑<br />
q=0<br />
N−1−q ∑<br />
p=−q<br />
˜x pỹ qe −i 2π N k(p+q)<br />
N−1 ∑<br />
N−1<br />
= ˜x pe −i 2π ∑<br />
N kp ỹ qe −i 2π N kq = ˜X k Ỹ k<br />
p=0<br />
q=0<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 21(28)
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