Signal Processing

Transform theory
Convolution theorem for the Fourier series (periodic convolution in
continuous time)
Let
Then
Proof:
˜z(t) = 1 T ˜x(t) ⊗ ỹ(t) = 1 ∫
˜x(t − τ)ỹ(τ) dτ
T T
Z k = X k Y k
Z k = 1 ∫
˜z(t)e −i 2π T kt dt = 1 ∫ T { ∫ 1 T
}
˜x(t − τ)ỹ(τ) dτ e −i 2π T kt dt
T T
T 0 T 0
= {(u, τ) = (t − τ, τ), du dτ = dt dτ} = 1 ∫ T ∫ T −τ
T 2 ˜x(u)ỹ(τ)e −i 2π T k(u+τ) du dτ
0 −τ
= 1 ∫
˜x(u)e −i 2π T ku du 1 ∫
ỹ(τ)e −i 2π T kτ dτ = X k Y k
T T
T T
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 17(28)