Signal Processing
Signal Processing Signal Processing
Signal Processing Lecture 1 Sven Nordebo School of Computer Science, Physics and Mathematics Linnæus University, Sweden 4ED044 Signal Processing, September 20, 2012
- Page 2 and 3: Chapter 1: Transform theory ⎧ ⎪
- Page 4 and 5: Transform theory Exercise 1 (a): Th
- Page 6 and 7: Transform theory Exercise 1 (c): Th
- Page 8 and 9: Transform theory The Fourier transf
- Page 10 and 11: Transform theory Important properti
- Page 12 and 13: Transform theory The Parseval’s r
- Page 14 and 15: Transform theory The Parseval’s r
- Page 16 and 17: Transform theory Convolution theore
- Page 18 and 19: Transform theory Convolution theore
- Page 20 and 21: Transform theory Convolution theore
- Page 22 and 23: Transform theory Convolution theore
- Page 24 and 25: Transform theory Time-shift in the
- Page 26 and 27: Transform theory Example: Fourier t
- Page 28: Transform theory Important symmetry
<strong>Signal</strong> <strong>Processing</strong><br />
Lecture 1<br />
Sven Nordebo<br />
School of Computer Science, Physics and Mathematics<br />
Linnæus University, Sweden<br />
4ED044 <strong>Signal</strong> <strong>Processing</strong>, September 20, 2012
Chapter 1: Transform theory<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
The Fourier transform:<br />
Continuous–time aperiodic signals x(t)<br />
∫ ∞<br />
X(f) = x(t)e −i2πft dt<br />
−∞<br />
∫ ∞<br />
x(t) = X(f)e i2πft df<br />
−∞<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
The Fourier series:<br />
Continuous–time T -periodic signals ˜x(t)<br />
Discrete spectrum f k = k/T<br />
X k = 1 ∫<br />
˜x(t)e −i 2π T kt dt<br />
T T<br />
∞∑<br />
˜x(t) = X k e i 2π T kt<br />
k=−∞<br />
⎧<br />
The Discrete-Time Fourier transform:<br />
⎧<br />
The Discrete Fourier Transform (DFT):<br />
Discrete–time aperiodic signals x n<br />
Discrete–time N-periodic signals ˜x n<br />
⎪⎨<br />
1-periodic spectrum:<br />
˜X(ν) =<br />
∞∑<br />
n=−∞<br />
x ne −i2πνn<br />
˜X(ν + 1) = ˜X(ν)<br />
⎪⎨<br />
N-periodic spectrum:<br />
˜Xk+N = ˜X k<br />
N−1 ∑<br />
˜X k = ˜x ne −i 2π N kn , k = 0, . . . , N − 1<br />
n=0<br />
⎪⎩<br />
∫ 1/2<br />
x n = ˜X(ν)e i2πνn dν<br />
−1/2<br />
⎪⎩<br />
˜x n = 1 N<br />
N−1 ∑<br />
k=0<br />
˜X k e i 2π N kn , n = 0, . . . , N − 1<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 2(28)
Transform theory<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
The Fourier transform: Alternative definition ω = 2πf [ rad/s]<br />
Continuous–time aperiodic signals<br />
∫ ∞<br />
X(ω) = x(t)e −iωt dt<br />
−∞<br />
x(t) = 1 ∫ ∞<br />
X(ω)e i2πωt dω<br />
2π −∞<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
The Discrete-Time Fourier transform: Alternative definition Ω = 2πν<br />
Discrete–time aperiodic signals<br />
2π-periodic spectrum: X(Ω + 2π) = X(Ω)<br />
X(Ω) =<br />
∞∑<br />
x(n)e −iΩn<br />
n=−∞<br />
x(n) = 1 ∫ π<br />
X(Ω)e iΩn dΩ<br />
2π −π<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 3(28)
Transform theory<br />
Exercise 1 (a): The Fourier transform for continuous–time aperiodic signals<br />
{ A 0 ≤ t ≤ Tp<br />
x(t) =<br />
0 otherwise<br />
∫ ∞<br />
X(f) = x(t)e −i2πft dt =<br />
−∞<br />
= A 1 − e−i2πfTp<br />
i2πf<br />
∫ Tp<br />
[ e<br />
Ae −i2πft −i2πft ] Tp<br />
dt = A<br />
0<br />
−i2πf 0<br />
= Ae −iπfTp eiπfTp − e −iπfTp<br />
i2πf<br />
−iπfTp sin(πfTp)<br />
= AT pe<br />
πfT p<br />
Zeros of X(f):<br />
X(f) = 0 ⇔ sin(πfT p) = 0 ⇔ πfT p = mπ ⇔ f = m T p<br />
, m = 0, ±1, ±2, . . .<br />
Draw a sketch of both the signal x(t) and its Fourier transform X(f).<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 4(28)
Transform theory<br />
Exercise 1 (b): The Fourier series for continuous–time T -periodic signals<br />
⎧<br />
A 0 ≤ t < T p<br />
⎪⎨<br />
˜x(t) = 0 T p < t < T<br />
⎪⎩<br />
˜x(t + T ) all t<br />
X k = 1 ∫<br />
˜x(t)e −i 2π T kt dt = 1 ∫ T<br />
˜x(t)e −i 2π T kt dt = 1 ∫ Tp<br />
Ae −i 2π T kt dt<br />
T T<br />
T 0<br />
T 0<br />
{<br />
= same integral as above with f k = k }<br />
= 1 T T ATpe−iπ T k sin(π k Tp T Tp)<br />
π k T Tp<br />
Draw a sketch of both the signal ˜x(t) and its Fourier series X k .<br />
Note that the Fourier series coefficients X k of ˜x(t) are equal to 1 times the Fourier<br />
T<br />
transform of one single period x(t) of the periodic signal ˜x(t), sampled at the<br />
frequencies f = k T<br />
X k = 1 ∫<br />
˜x(t)e −i 2π T kt dt = 1 ∫<br />
x(t)e −i 2π T kt dt = 1 ∫ ∞<br />
x(t)e −i 2π T kt dt<br />
T T<br />
T T<br />
T −∞<br />
= 1 T X(f)| f= k T<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 5(28)
Transform theory<br />
Exercise 1 (c): The Discrete-Time Fourier transform for discrete–time aperiodic signals<br />
{ A 0 ≤ n ≤ Np − 1<br />
x n =<br />
0 otherwise<br />
˜X(ν) =<br />
∞∑<br />
n=−∞<br />
N<br />
∑ p−1<br />
x ne −i2πνn = Ae −i2πνn =<br />
n=0<br />
⎧<br />
⎨<br />
N p−1<br />
⎩ Geom. ser. ∑<br />
n=0<br />
= A 1 − e−i2πνNp<br />
1 − e −i2πν = A e−iπνNp (e iπνNp − e −iπνNp )<br />
e −iπν (e iπν − e −iπν )<br />
= Ae −iπν(Np−1) (eiπνNp − e −iπνNp )/(2i)<br />
(e iπν − e −iπν )/(2i)<br />
q n = 1 − qNp<br />
1 − q<br />
−iπν(Np−1) sin(πνNp)<br />
= Ae<br />
sin(πν)<br />
Zeros of ˜X(ν):<br />
˜X(ν) = 0 ⇔ sin(πνN p) = 0 ⇔ πνN p = mπ ⇔ ν = m N p<br />
, m = 0, ±1, ±2, . . .<br />
⎫<br />
⎬<br />
⎭<br />
Draw a sketch of both the signal x n and its Fourier transform ˜X(ν).<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 6(28)
Transform theory<br />
Exercise 1 (d): The Discrete Fourier Transform (DFT) for discrete–time N-periodic<br />
signals<br />
⎧<br />
A 0 ≤ n ≤ N p − 1<br />
⎪⎨<br />
˜x n = 0 N p ≤ n ≤ N − 1<br />
⎪⎩<br />
˜x n+N all n<br />
N−1 ∑<br />
˜X k =<br />
n=0<br />
N<br />
˜x ne −i 2π ∑ p−1<br />
N kn =<br />
n=0<br />
Draw a sketch of both the signal ˜x n and its DFT ˜X k .<br />
{<br />
Ae −i 2π N kn = same sum as above with ν k = k }<br />
N<br />
= Ae −iπ k N (Np−1) sin(π k N Np)<br />
sin(π k N )<br />
Note that the DFT ˜X k of ˜x n is equal to the Fourier transform of one single period x n<br />
of the periodic signal ˜x n, sampled at the frequencies ν = k N<br />
˜X k =<br />
N−1 ∑<br />
n=0<br />
˜x ne −i 2π N kn =<br />
N−1 ∑<br />
n=0<br />
x ne −i 2π N kn =<br />
∞∑<br />
n=−∞<br />
x ne −i 2π N kn = ˜X(ν)| ν= k<br />
N<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 7(28)
Transform theory<br />
The Fourier transform as the limit of the Fourier series<br />
Let x(t) be an aperiodic continuos-time signal with the property<br />
lim x(t) = 0<br />
|t|→∞<br />
Extend this signal as a periodic signal with period T<br />
so that<br />
˜x(t) =<br />
∞∑<br />
n=−∞<br />
x(t − nT )<br />
lim ˜x(t) = x(t) for t ∈ [−T/2, T/2]<br />
T →∞<br />
It follows that<br />
X k = 1 ∫ T /2<br />
˜x(t)e −i2π T k t dt → 1 ∫ T /2<br />
x(t)e −i2π T k t dt → 1 T −T /2<br />
T −T /2<br />
T X( k T )<br />
as T → ∞. Moreover, let f k = k/T → f and f k+1 − f k = 1/T → df, so that<br />
˜x(t) =<br />
as T → ∞.<br />
∞∑<br />
k=−∞<br />
X k e i2πf kt →<br />
∞∑<br />
k=−∞<br />
∫<br />
1<br />
∞<br />
T X(f k)e i2πfkt → X(f)e i2πft df = x(t)<br />
−∞<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 8(28)
Transform theory<br />
Classical Fourier transform: F : L 2 → L 2<br />
⎧ ∫ ∞<br />
⎪⎨ Fx(t) = x(t)e −iωt dt<br />
⎪⎩<br />
Important properties:<br />
⎧<br />
Duality:<br />
⎪⎨<br />
⎪⎩<br />
−∞<br />
F −1 X(ω) = 1<br />
2π<br />
Time-derivative:<br />
Frequency-derivative:<br />
Time-shift:<br />
∫ ∞<br />
−∞<br />
X(ω)e iωt dω<br />
FFx(t) = 2πx(−t)<br />
F∂ t x(t) = iωX(ω)<br />
Ftx(t) = i∂ ω X(ω)<br />
Fx(t − t 0 ) = e −iωt0 X(ω)<br />
Frequency-shift: Fe iω0t x(t) = X(ω − ω 0 )<br />
Scaling: Fx(at) = 1<br />
|a| X(ω a )<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 9(28)
Transform theory<br />
Important properties:<br />
⎧<br />
Parseval:<br />
⎪⎨<br />
⎪⎩<br />
Parseval:<br />
Time-convolution:<br />
∫ ∞<br />
−∞<br />
∫ ∞<br />
−∞<br />
x(t)y ∗ (t)dt = 1<br />
2π<br />
|x(t)| 2 dt = 1<br />
2π<br />
∫ ∞<br />
−∞<br />
∫ ∞<br />
−∞<br />
Fx(t) ∗ y(t) = X(ω)Y (ω)<br />
Freq.-convolution: Fx(t)y(t) = 1 X(ω) ∗ Y (ω)<br />
2π<br />
Conj. symmetry: x(t) real ⇔ X(ω) = X ∗ (−ω)<br />
Conj. symmetry:<br />
Conj. asymmetry:<br />
Conj. asymmetry:<br />
X(ω) real ⇔ x(t) = x ∗ (−t)<br />
X(ω)Y ∗ (ω)dω<br />
|X(ω)| 2 dω<br />
x(t) imaginary ⇔ X(ω) = −X ∗ (−ω)<br />
X(ω) imaginary ⇔ x(t) = −x ∗ (−t)<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 10(28)
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)
Transform theory<br />
The Parseval’s relations for Fourier transforms<br />
∫ ∞<br />
−∞<br />
∫ ∞<br />
∫ ∞<br />
x(t)y ∗ (t) dt =<br />
−∞<br />
−∞<br />
X(f)Y ∗ (f) df<br />
∫ ∞<br />
|x(t)| 2 dt = |X(f)| 2 df<br />
−∞<br />
Proof: Employ the orthogonality relationship of the basis functions<br />
We have<br />
∫ ∞<br />
e i2π(f−f ′ )t dt = δ(f − f ′ )<br />
−∞<br />
∫ ∞<br />
∫ ∞ ∫ ∞<br />
∫ ∞<br />
x(t)y ∗ (t) dt =<br />
X(f)e i2πft df Y ∗ (f ′ )e −i2πf ′t df ′ dt<br />
−∞<br />
−∞ −∞<br />
−∞<br />
∫ ∞ ∫ ∞<br />
∫ ∞<br />
=<br />
X(f)Y ∗ (f ′ ) e i2π(f−f ′ )t dt df df ′<br />
−∞ −∞<br />
−∞<br />
∫ ∞ ∫ ∞<br />
∫ ∞<br />
=<br />
X(f)Y ∗ (f ′ )δ(f − f ′ ) df ′ df = X(f)Y ∗ (f) df<br />
−∞ −∞<br />
−∞<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 12(28)
Transform theory<br />
The Parseval’s relations for the Discrete Fourier Transform (DFT)<br />
N−1 ∑<br />
n=0<br />
N−1 ∑<br />
n=0<br />
˜x nỹ ∗ n = 1 N<br />
|˜x n| 2 = 1 N<br />
N−1 ∑<br />
k=0<br />
N−1 ∑<br />
k=0<br />
˜X k Ỹ ∗ k<br />
| ˜X k | 2<br />
Proof: Employ the orthogonality relationship of the basis functions<br />
We have<br />
= 1 N<br />
N−1 ∑<br />
n=0<br />
N−1 ∑<br />
˜x nỹn ∗ =<br />
N−1 ∑<br />
N−1 ∑<br />
k=0 k ′ =0<br />
N−1<br />
1 ∑<br />
{<br />
e i 2π N (k−k′)n = ˜δ 1 k = k<br />
kk ′ =<br />
′ + mN<br />
N<br />
0 k ≠ k ′ + mN<br />
n=0<br />
n=0<br />
N−1<br />
1 ∑<br />
˜X k e i 2π N kn 1 N−1 ∑<br />
N<br />
N<br />
k=0<br />
N−1<br />
˜X k Ỹk ∗ 1 ∑<br />
′<br />
N<br />
n=0<br />
k ′ =0<br />
e i 2π N (k−k′ )n = 1 N<br />
Ỹ ∗ k ′e−i 2π N k′ n<br />
N−1 ∑<br />
N−1 ∑<br />
k=0 k ′ =0<br />
˜X k Ỹ ∗ k ′ ˜δ kk ′ = 1 N<br />
N−1 ∑<br />
k=0<br />
˜X k Ỹ ∗ k<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 13(28)
Transform theory<br />
The Parseval’s relations for Discrete-Time Fourier transforms<br />
∞∑<br />
n=−∞<br />
∞∑<br />
n=−∞<br />
x ny ∗ n = ∫ 1/2<br />
−1/2<br />
˜X(ν)Ỹ ∗ (ν) dν<br />
∫ 1/2<br />
|x n| 2 = | ˜X(ν)| 2 dν<br />
−1/2<br />
Proof: Employ the orthogonality relationship of the basis functions<br />
∞∑<br />
e i2π(ν−ν′ )n = ˜δ(ν − ν ′ ) mod 1<br />
n=−∞<br />
We have<br />
∞∑<br />
x nyn ∗ =<br />
∞∑<br />
∫ 1/2<br />
∫ 1/2<br />
˜X(ν)e i2πνn dν Ỹ ∗ (ν ′ )e −i2πν′n dν ′<br />
n=−∞<br />
n=−∞ −1/2<br />
−1/2<br />
∫ 1/2 ∫ 1/2<br />
=<br />
˜X(ν)Ỹ ∗ (ν ′ )<br />
∞∑<br />
e i2π(ν−ν′ )n dν dν ′<br />
−1/2 −1/2<br />
n=−∞<br />
∫ 1/2 ∫ 1/2<br />
∫ 1/2<br />
=<br />
˜X(ν)Ỹ ∗ (ν ′ )˜δ(ν − ν ′ ) dν ′ dν = ˜X(ν)Ỹ ∗ (ν) dν<br />
−1/2 −1/2<br />
−1/2<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 14(28)
Transform theory<br />
Convolution theorem for the Fourier transform (linear convolution in<br />
continuous time)<br />
Let<br />
Then<br />
Proof:<br />
∫ ∞<br />
z(t) = x(t) ∗ y(t) = x(t − τ)y(τ) dτ<br />
−∞<br />
Z(ω) = X(ω)Y (ω)<br />
∫ ∞<br />
∫ ∞ {∫ ∞<br />
}<br />
Z(ω) = z(t)e −iωt dt =<br />
x(t − τ)y(τ) dτ e −iωt dt<br />
−∞<br />
−∞ −∞<br />
∫ ∞ ∫ ∞<br />
= {(u, τ) = (t − τ, τ), du dτ = dt dτ} =<br />
x(u)y(τ)e −iω(u+τ) du dτ<br />
−∞ −∞<br />
∫ ∞<br />
∫ ∞<br />
= x(u)e −iωu du y(τ)e −iωτ dτ = X(ω)Y (ω)<br />
−∞<br />
−∞<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 15(28)
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)
Transform theory<br />
Convolution theorem for the Fourier series (periodic convolution in<br />
continuous time)<br />
Let<br />
Then<br />
Proof:<br />
˜z(t) = 1 T ˜x(t) ⊗ ỹ(t) = 1 ∫<br />
˜x(t − τ)ỹ(τ) dτ<br />
T T<br />
Z k = X k Y k<br />
Z k = 1 ∫<br />
˜z(t)e −i 2π T kt dt = 1 ∫ T { ∫ 1 T<br />
}<br />
˜x(t − τ)ỹ(τ) dτ e −i 2π T kt dt<br />
T T<br />
T 0 T 0<br />
= {(u, τ) = (t − τ, τ), du dτ = dt dτ} = 1 ∫ T ∫ T −τ<br />
T 2 ˜x(u)ỹ(τ)e −i 2π T k(u+τ) du dτ<br />
0 −τ<br />
= 1 ∫<br />
˜x(u)e −i 2π T ku du 1 ∫<br />
ỹ(τ)e −i 2π T kτ dτ = X k Y k<br />
T T<br />
T T<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 17(28)
Transform theory<br />
Convolution theorem for the Fourier series (linear convolution in discrete<br />
frequency)<br />
Let<br />
Then<br />
Proof:<br />
˜z(t) = ˜x(t)ỹ(t)<br />
Z k = X k ∗ Y k =<br />
∞∑<br />
q=−∞<br />
X k−q Y q<br />
⎧<br />
⎫<br />
∞∑<br />
∞∑ ⎨ ∞∑ ⎬<br />
˜z(t) = Z k e i 2π T kt =<br />
X<br />
⎩<br />
k−q Y q<br />
⎭ ei 2π T kt = {(p, q) = (k − q, q)}<br />
k=−∞<br />
k=−∞ q=−∞<br />
∞∑ ∞∑<br />
∞∑<br />
=<br />
X pY qe i 2π T (p+q)t = X pe i 2π ∑ ∞<br />
T pt Y qe i 2π T qt = ˜x(t)ỹ(t)<br />
p=−∞ q=−∞<br />
p=−∞<br />
q=−∞<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 18(28)
Transform theory<br />
Convolution theorem for the Discrete-Time Fourier transform (linear<br />
convolution in discrete time)<br />
Let<br />
Then<br />
z n = x n ∗ y n =<br />
˜Z(ν) =<br />
∞∑<br />
q=−∞<br />
˜X(ν)Ỹ (ν)<br />
x n−q y q<br />
Proof:<br />
⎧<br />
⎫<br />
∞∑<br />
∞∑ ⎨ ∞∑ ⎬<br />
˜Z(ν) = z ne −i2πνn =<br />
x n−q y q<br />
⎩<br />
⎭ e−i2πνn<br />
n=−∞<br />
n=−∞ q=−∞<br />
∞∑ ∞∑<br />
= {(p, q) = (n − q, q)} =<br />
x py qe −i2πν(p+q)<br />
=<br />
∞∑<br />
p=−∞<br />
p=−∞ q=−∞<br />
x pe −i2πνp<br />
∞ ∑<br />
q=−∞<br />
y qe −i2πνq =<br />
˜X(ν)Ỹ (ν)<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 19(28)
Transform theory<br />
Convolution theorem for the Discrete-Time Fourier transform (periodic<br />
convolution in continuous frequency)<br />
Let<br />
Then<br />
Proof:<br />
z n = x ny n<br />
˜Z(ν) = ˜X(ν) ⊗ Ỹ (ν) = ∫ 1/2<br />
−1/2<br />
˜X(ν − α)Ỹ (α) dα<br />
{ } ∫ { 1/2 ∫ } 1/2<br />
z n = F −1 ˜X(ν) ⊗ Ỹ (ν) =<br />
˜X(ν − α)Ỹ (α) dα e i2πνn dν<br />
−1/2 −1/2<br />
= {(β, α) = (ν − α, α), dβ dα = dν dα}<br />
∫ 1/2 ∫ 1/2−α<br />
=<br />
˜X(β)Ỹ (α)ei2π(β+α)n dβ dα<br />
α=−1/2 β=−1/2−α<br />
∫ 1/2<br />
∫ 1/2<br />
= ˜X(β)e i2πβn dβ Ỹ (α)e i2παn dα = x ny n<br />
−1/2<br />
−1/2<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 20(28)
Transform theory<br />
Convolution theorem for the Discrete Fourier Transform (periodic<br />
convolution in discrete time)<br />
Let<br />
Then<br />
Proof:<br />
N−1 ∑<br />
˜z n = ˜x n ⊗ ỹ n = ˜x n−q ỹ q<br />
q=0<br />
˜Z k = ˜X k Ỹ k<br />
⎧<br />
⎫<br />
N−1 ∑<br />
N−1<br />
˜Z k = ˜z ne −i 2π ∑ ⎨N−1<br />
∑ ⎬<br />
N kn =<br />
˜x n−q ỹ q<br />
⎩<br />
⎭ e−i 2π N kn<br />
n=0<br />
n=0 q=0<br />
= {(p, q) = (n − q, q)} =<br />
N−1 ∑<br />
q=0<br />
N−1−q ∑<br />
p=−q<br />
˜x pỹ qe −i 2π N k(p+q)<br />
N−1 ∑<br />
N−1<br />
= ˜x pe −i 2π ∑<br />
N kp ỹ qe −i 2π N kq = ˜X k Ỹ k<br />
p=0<br />
q=0<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 21(28)
Transform theory<br />
Convolution theorem for the Discrete Fourier Transform (periodic<br />
convolution in discrete frequency)<br />
Let<br />
Then<br />
Proof:<br />
{ 1<br />
˜z n = DFT −1 ˜X k ⊗<br />
N Ỹk<br />
˜z n = ˜x nỹ n<br />
˜Z k = 1 ˜X k ⊗<br />
N Ỹk = 1 N−1 ∑<br />
˜X k−q Ỹ q<br />
N<br />
q=0<br />
}<br />
= 1 N<br />
⎧<br />
⎫<br />
N−1 ∑ ⎨ N−1<br />
1 ∑ ⎬<br />
˜X<br />
⎩<br />
k−q Ỹ q<br />
N<br />
⎭ ei 2π N kn<br />
k=0 q=0<br />
= {(p, q) = (k − q, q)} = 1 N−1 ∑ N−1−q ∑<br />
N 2<br />
= 1 N<br />
q=0<br />
N−1 ∑<br />
p=0<br />
p=−q<br />
˜X pe i 2π N pn 1 N<br />
˜X p Ỹ qe i 2π N (p+q)n<br />
N−1 ∑<br />
q=0<br />
Ỹ qe i 2π N qn = ˜x nỹ n<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 22(28)
Time-shift in the Fourier transform<br />
Transform theory<br />
Proof:<br />
F{x(t − t 0 )} = e −iωt 0<br />
X(ω)<br />
∫ ∞<br />
F{x(t − t 0 )} = x(t − t 0 )e −iωt dt = {u = t − t 0 , du = dt}<br />
∫ ∞<br />
−∞<br />
∫ ∞<br />
= x(u)e −iω(u+t0) du = e −iωt 0<br />
x(u)e −iωu du = e −iωt 0<br />
X(ω)<br />
−∞<br />
−∞<br />
Frequency-shift in the Fourier transform<br />
F{x(t)e iω0t } = X(ω − ω 0 )<br />
Proof:<br />
∫ ∞<br />
∫ ∞<br />
F{x(t)e iω0t } = x(t)e iω0t e −iωt dt = x(t)e −i(ω−ω0)t dt = X(ω − ω 0 )<br />
−∞<br />
−∞<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 23(28)
Transform theory<br />
Time-shift in the Discrete-Time Fourier transform<br />
F{x n−n0 } = e −i2πνn 0 ˜X(ν)<br />
Proof:<br />
∞∑<br />
F{x n−n0 } = x n−n0 e −i2πνn = {p = n − n 0 }<br />
n=−∞<br />
∞∑<br />
=<br />
p=−∞<br />
x pe −i2πν(p+n0) = e −i2πνn 0<br />
∞∑<br />
p=−∞<br />
x pe −i2πνp = e −i2πνn 0 ˜X(ν)<br />
Frequency-shift in the Discrete-Time Fourier transform<br />
F{x ne i2πν0n } = ˜X(ν − ν 0 )<br />
Proof:<br />
F{x ne i2πν0n } =<br />
∞∑<br />
n=−∞<br />
x ne i2πν0n e −i2πνn =<br />
∞∑<br />
n=−∞<br />
x ne −i2π(ν−ν0)n = ˜X(ν−ν 0 )<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 24(28)
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)
Transform theory<br />
Example: Fourier transform of the Gaussian pulse<br />
{ √<br />
F e −at2} π ω2<br />
= e− 4a<br />
a<br />
Proof: Notice that x(t) = e −at2 is a solution to<br />
∂x(t)<br />
∂t<br />
+ 2atx(t) = 0<br />
Take the Fourier transform and use F{ ∂x(t) } = iωX(ω) and F{tx(t)} = i ∂X(ω)<br />
∂t<br />
∂ω<br />
iωX(ω) + 2ai ∂X(ω)<br />
∂ω<br />
Parseval’s theorem<br />
= 0 ⇒ ∂X(ω) ( ) 1<br />
∂ω + 2 ωX(ω) = 0 ⇒ X(ω) = Ce − 4a 1 ω2<br />
4a<br />
∫ ∞<br />
−∞<br />
∫ ∞<br />
x 2 (t) dt = e −2at2 dt = 1<br />
−∞<br />
2π<br />
= {ω = 2at, dω = 2a dt} = C2<br />
2π<br />
∫ ∞<br />
−∞<br />
∫ ∞<br />
−∞<br />
∫<br />
X 2 (ω) dω = C2 ∞<br />
e − ω2<br />
2a dω<br />
2π −∞<br />
e −2at2 2a dt ⇒ 1 = C2<br />
2π 2a ⇒ C = √ π<br />
a<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 26(28)
Transform theory<br />
Important symmetry properties of the Fourier Transform<br />
{ |X(ω)| = |X(−ω)|<br />
x(t) real ⇔ X(ω) = X ∗ (−ω) ⇔<br />
arg X(ω) = − arg X(−ω)<br />
Proof:<br />
∫ ∞<br />
∫ ∞<br />
X(ω) = X ∗ (−ω) ⇔ x(t)e −iωt dt = x ∗ (t)e −iωt dt<br />
−∞<br />
−∞<br />
⇔ x(t) = x ∗ (t) ⇔ x(t) real<br />
Proof:<br />
X(ω) real ⇔ x(t) = x ∗ (−t)<br />
x(t) = x ∗ (−t) ⇔ 1 ∫ ∞<br />
X(ω)e iωt dω = 1 ∫ ∞<br />
X ∗ (ω)e iωt dω<br />
2π −∞<br />
2π −∞<br />
⇔ X(ω) = X ∗ (ω) ⇔ X(ω) real<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 27(28)
Transform theory<br />
Important symmetry properties of the Discrete Fourier Transform (DFT)<br />
˜x n real ⇔ ˜X k = ˜X ∗ −k = ˜X ∗ N−k ⇔ { | ˜Xk | = | ˜X −k | = | ˜X N−k |<br />
arg ˜X k = − arg ˜X −k = − arg ˜X N−k<br />
Proof: Similar as above.<br />
˜X k real ⇔ ˜x n = ˜x ∗ −n = ˜x∗ N−n<br />
Sven Nordebo, School of Computer Science, Physics and Mathematics, Linnæus University, Sweden. 28(28)