Signal Processing
Signal Processing Signal Processing
Transform theory Time-shift in the Discrete-Time Fourier transform F{x n−n0 } = e −i2πνn 0 ˜X(ν) Proof: ∞∑ F{x n−n0 } = x n−n0 e −i2πνn = {p = n − n 0 } n=−∞ ∞∑ = p=−∞ x pe −i2πν(p+n0) = e −i2πνn 0 ∞∑ p=−∞ x pe −i2πνp = e −i2πνn 0 ˜X(ν) Frequency-shift in the Discrete-Time Fourier transform F{x ne i2πν0n } = ˜X(ν − ν 0 ) Proof: F{x ne i2πν0n } = ∞∑ n=−∞ x ne i2πν0n e −i2πνn = ∞∑ n=−∞ x ne −i2π(ν−ν0)n = ˜X(ν−ν 0 ) Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 24(28)
Transform theory Time-derivative in the Fourier transform F { } ∂x = iωX(ω) ∂t Proof: ∂x ∂t = ∂ ∫ 1 ∞ X(ω)e iωt dω = 1 ∫ ∞ ∂ ∂t 2π −∞ 2π −∞ ∂t {X(ω)eiωt } dω = 1 ∫ ∞ iωX(ω)e iωt dω = F −1 {iωX(ω)} 2π −∞ Frequency-derivative in the Fourier transform Proof: F{tx(t)} = i ∂X(ω) ∂ω i ∂X(ω) ∂ω = i ∂ ∫ ∞ ∫ ∞ x(t)e −iωt ∂ dt = i ∂ω −∞ −∞ ∂ω {x(t)e−iωt } dt ∫ ∞ ∫ ∞ = i (−it)x(t)e −iωt dt = tx(t)e −iωt dt = F{tx(t)} −∞ −∞ Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 25(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 and 16: Transform theory Convolution theore
- Page 17 and 18: Transform theory Convolution theore
- Page 19 and 20: Transform theory Convolution theore
- Page 21 and 22: Transform theory Convolution theore
- Page 23: Time-shift in the Fourier transform
- Page 27 and 28: Transform theory Important symmetry
Transform theory<br />
Time-derivative in the Fourier transform<br />
F<br />
{ } ∂x<br />
= iωX(ω)<br />
∂t<br />
Proof:<br />
∂x<br />
∂t = ∂ ∫<br />
1 ∞<br />
X(ω)e iωt dω = 1 ∫ ∞ ∂<br />
∂t 2π −∞<br />
2π −∞ ∂t {X(ω)eiωt } dω<br />
= 1 ∫ ∞<br />
iωX(ω)e iωt dω = F −1 {iωX(ω)}<br />
2π −∞<br />
Frequency-derivative in the Fourier transform<br />
Proof:<br />
F{tx(t)} = i ∂X(ω)<br />
∂ω<br />
i ∂X(ω)<br />
∂ω<br />
= i ∂ ∫ ∞<br />
∫ ∞<br />
x(t)e −iωt ∂<br />
dt = i<br />
∂ω −∞<br />
−∞ ∂ω {x(t)e−iωt } dt<br />
∫ ∞<br />
∫ ∞<br />
= i (−it)x(t)e −iωt dt = tx(t)e −iωt dt = F{tx(t)}<br />
−∞<br />
−∞<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 25(28)
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