Signal Processing

Transform theory
The Parseval’s relations for Discrete-Time Fourier transforms
∞∑
n=−∞
∞∑
n=−∞
x ny ∗ n = ∫ 1/2
−1/2
˜X(ν)Ỹ ∗ (ν) dν
∫ 1/2
|x n| 2 = | ˜X(ν)| 2 dν
−1/2
Proof: Employ the orthogonality relationship of the basis functions
∞∑
e i2π(ν−ν′ )n = ˜δ(ν − ν ′ ) mod 1
n=−∞
We have
∞∑
x nyn ∗ =
∞∑
∫ 1/2
∫ 1/2
˜X(ν)e i2πνn dν Ỹ ∗ (ν ′ )e −i2πν′n dν ′
n=−∞
n=−∞ −1/2
−1/2
∫ 1/2 ∫ 1/2
=
˜X(ν)Ỹ ∗ (ν ′ )
∞∑
e i2π(ν−ν′ )n dν dν ′
−1/2 −1/2
n=−∞
∫ 1/2 ∫ 1/2
∫ 1/2
=
˜X(ν)Ỹ ∗ (ν ′ )˜δ(ν − ν ′ ) dν ′ dν = ˜X(ν)Ỹ ∗ (ν) dν
−1/2 −1/2
−1/2
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 14(28)