Signal Processing
Signal Processing
Signal Processing
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Transform theory<br />
The Parseval’s relations for the Discrete Fourier Transform (DFT)<br />
N−1 ∑<br />
n=0<br />
N−1 ∑<br />
n=0<br />
˜x nỹ ∗ n = 1 N<br />
|˜x n| 2 = 1 N<br />
N−1 ∑<br />
k=0<br />
N−1 ∑<br />
k=0<br />
˜X k Ỹ ∗ k<br />
| ˜X k | 2<br />
Proof: Employ the orthogonality relationship of the basis functions<br />
We have<br />
= 1 N<br />
N−1 ∑<br />
n=0<br />
N−1 ∑<br />
˜x nỹn ∗ =<br />
N−1 ∑<br />
N−1 ∑<br />
k=0 k ′ =0<br />
N−1<br />
1 ∑<br />
{<br />
e i 2π N (k−k′)n = ˜δ 1 k = k<br />
kk ′ =<br />
′ + mN<br />
N<br />
0 k ≠ k ′ + mN<br />
n=0<br />
n=0<br />
N−1<br />
1 ∑<br />
˜X k e i 2π N kn 1 N−1 ∑<br />
N<br />
N<br />
k=0<br />
N−1<br />
˜X k Ỹk ∗ 1 ∑<br />
′<br />
N<br />
n=0<br />
k ′ =0<br />
e i 2π N (k−k′ )n = 1 N<br />
Ỹ ∗ k ′e−i 2π N k′ n<br />
N−1 ∑<br />
N−1 ∑<br />
k=0 k ′ =0<br />
˜X k Ỹ ∗ k ′ ˜δ kk ′ = 1 N<br />
N−1 ∑<br />
k=0<br />
˜X k Ỹ ∗ k<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 13(28)