Signal Processing
Signal Processing Signal Processing
Transform theory Convolution theorem for the Fourier transform (linear convolution in continuous frequency) Let Then Proof: z(t) = x(t)y(t) Z(ω) = 1 ∫ 1 ∞ X(ω) ∗ Y (ω) = X(ω − ϕ)Y (ϕ) dϕ 2π 2π −∞ { } 1 z(t) = F −1 2π X(ω) ∗ Y (ω) = 1 ∫ ∞ { ∫ 1 ∞ } X(ω − ϕ)Y (ϕ) dϕ e iωt dω 2π −∞ 2π −∞ = {(υ, ϕ) = (ω − ϕ, ϕ), dυ dϕ = dω dϕ} = 1 ∫ ∞ ∫ ∞ (2π) 2 X(υ)Y (ϕ)e i(υ+ϕ)t dυ dϕ −∞ −∞ = 1 ∫ ∞ X(υ)e iυt dυ 1 ∫ ∞ Y (ϕ)e iϕt dϕ = x(t)y(t) 2π −∞ 2π −∞ Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 16(28)
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)
- Page 1 and 2: Signal Processing Lecture 1 Sven No
- Page 3 and 4: Transform theory ⎧ ⎪⎨ ⎪⎩
- Page 5 and 6: Transform theory Exercise 1 (b): Th
- Page 7 and 8: Transform theory Exercise 1 (d): Th
- Page 9 and 10: Transform theory Classical Fourier
- Page 11 and 12: Transform theory The Parseval’s r
- Page 13 and 14: Transform theory The Parseval’s r
- Page 15: Transform theory Convolution theore
- Page 19 and 20: Transform theory Convolution theore
- Page 21 and 22: Transform theory Convolution theore
- Page 23 and 24: Time-shift in the Fourier transform
- Page 25 and 26: Transform theory Time-derivative in
- Page 27 and 28: Transform theory Important symmetry
Transform theory<br />
Convolution theorem for the Fourier transform (linear convolution in<br />
continuous frequency)<br />
Let<br />
Then<br />
Proof:<br />
z(t) = x(t)y(t)<br />
Z(ω) = 1<br />
∫<br />
1 ∞<br />
X(ω) ∗ Y (ω) = X(ω − ϕ)Y (ϕ) dϕ<br />
2π 2π −∞<br />
{ }<br />
1<br />
z(t) = F −1 2π X(ω) ∗ Y (ω) = 1 ∫ ∞ { ∫ 1 ∞<br />
}<br />
X(ω − ϕ)Y (ϕ) dϕ e iωt dω<br />
2π −∞ 2π −∞<br />
= {(υ, ϕ) = (ω − ϕ, ϕ), dυ dϕ = dω dϕ} = 1 ∫ ∞ ∫ ∞<br />
(2π) 2 X(υ)Y (ϕ)e i(υ+ϕ)t dυ dϕ<br />
−∞ −∞<br />
= 1 ∫ ∞<br />
X(υ)e iυt dυ 1 ∫ ∞<br />
Y (ϕ)e iϕt dϕ = x(t)y(t)<br />
2π −∞<br />
2π −∞<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 16(28)
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