Signal Processing
Signal Processing Signal Processing
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 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 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 />
The Parseval’s relations for Fourier series<br />
∫<br />
1<br />
T T<br />
∫<br />
1<br />
T T<br />
˜x(t)ỹ ∗ (t) dt =<br />
|˜x(t)| 2 dt =<br />
∞∑<br />
k=−∞<br />
∞∑<br />
k=−∞<br />
X k Y ∗ k<br />
|X k | 2<br />
Proof: Employ the orthogonality relationship of the basis functions<br />
We have<br />
∫<br />
{<br />
1<br />
e i 2π T (k−k′)t 1 k = k ′<br />
dt = δ kk ′ =<br />
T T<br />
0 k ≠ k ′<br />
∫<br />
1<br />
˜x(t)ỹ ∗ (t) dt = 1 ∫ ∞∑<br />
X k e i 2π ∑ ∞<br />
T kt Y ∗ 2π k<br />
T T<br />
T<br />
′e−i T k′t dt<br />
T<br />
k=−∞<br />
k ′ =−∞<br />
∞∑ ∞∑<br />
∫<br />
=<br />
X k Y ∗ 1<br />
∞∑ ∞∑<br />
k ′ e i 2π T (k−k′)t dt =<br />
X k Yk ∗ T<br />
′δ kk ′ =<br />
∑ ∞ X k Yk<br />
∗<br />
k=−∞ k ′ =−∞<br />
T<br />
k=−∞ k ′ =−∞<br />
k=−∞<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 11(28)
Change language