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Transform theory Important properties: ⎧ Parseval: ⎪⎨ ⎪⎩ Parseval: Time-convolution: ∫ ∞ −∞ ∫ ∞ −∞ x(t)y ∗ (t)dt = 1 2π |x(t)| 2 dt = 1 2π ∫ ∞ −∞ ∫ ∞ −∞ Fx(t) ∗ y(t) = X(ω)Y (ω) Freq.-convolution: Fx(t)y(t) = 1 X(ω) ∗ Y (ω) 2π Conj. symmetry: x(t) real ⇔ X(ω) = X ∗ (−ω) Conj. symmetry: Conj. asymmetry: Conj. asymmetry: X(ω) real ⇔ x(t) = x ∗ (−t) X(ω)Y ∗ (ω)dω |X(ω)| 2 dω x(t) imaginary ⇔ X(ω) = −X ∗ (−ω) X(ω) imaginary ⇔ x(t) = −x ∗ (−t) Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 10(28)
Transform theory The Parseval’s relations for Fourier series ∫ 1 T T ∫ 1 T T ˜x(t)ỹ ∗ (t) dt = |˜x(t)| 2 dt = ∞∑ k=−∞ ∞∑ k=−∞ X k Y ∗ k |X k | 2 Proof: Employ the orthogonality relationship of the basis functions We have ∫ { 1 e i 2π T (k−k′)t 1 k = k ′ dt = δ kk ′ = T T 0 k ≠ k ′ ∫ 1 ˜x(t)ỹ ∗ (t) dt = 1 ∫ ∞∑ X k e i 2π ∑ ∞ T kt Y ∗ 2π k T T T ′e−i T k′t dt T k=−∞ k ′ =−∞ ∞∑ ∞∑ ∫ = X k Y ∗ 1 ∞∑ ∞∑ k ′ e i 2π T (k−k′)t dt = X k Yk ∗ T ′δ kk ′ = ∑ ∞ X k Yk ∗ k=−∞ k ′ =−∞ T k=−∞ k ′ =−∞ k=−∞ Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 11(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: Transform theory Classical Fourier
Page 13 and 14: Transform theory The Parseval’s r
Page 15 and 16: 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

Signal Processing

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