In this lecture you will learn: System impulse response, linear constant-coefficient difference equations, fourier transforms and frequency response. Inviting you refer.
Trang 1Digital Signal Processing, II, Zheng-Hua Tan, 2006
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Digital Signal Processing, Fall 2006
EStudy
-Zheng-Hua Tan
Department of Electronic Systems Aalborg University, Denmark
zt@kom.aau.dk
Lecture 2: Fourier transforms and
frequency response
Course at a glance
Discrete-time signals and systems
Fourier-domain
representation
System structures
Filter
MM1
MM2
MM6
MM4
Sampling and
System analysis System
Trang 2Digital Signal Processing, II, Zheng-Hua Tan, 2006
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Part I: System impulse response
System impulse response
Linear constant-coefficient difference
equations
Fourier transforms and frequency response
FIR systems – reflected in the h[n]
Ideal delay
Forward difference
Backward difference
Finite-duration impulse response (FIR) system
samples.
integer.
positive a
], [
] [
], [
] [
d d
d n n n n h
n n
n x n y
−
=
∞
<
<
∞
−
−
=
δ
] [ ] 1 [ ] [
] [ ] 1 [ ] [
n n
n h
n x n
x n y
δ
δ + −
=
− +
=
] 1 [ ] [ ] [
] 1 [ ] [ ] [
−
−
=
−
−
=
n n n h
n x n x n y
δ δ
n d
0 -1 0
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IIR systems – reflected in the h[n]
Infinite-duration impulse response (IIR) system
Stability
finite in magnitude.
] [ ] [ ] [
] [ ] [
n u k n
h
k x n y
n
k
n
k
=
=
=
∑
∑
−∞
=
−∞
=
…
∞
<
−
=
=
<
=
∑∞
|)
| 1 ( 1
|
|
1
|
| with ] [ ] [
S
a n
u a n h
n n
∞
<
=∑∞
−∞
=
?
| ] [
|
n h n S
Cascading systems
Causality
Ideal delay
0 , 0 ] [
?
<
n h
] [ ] [n n n d
h =δ −
] [ ] 1 [ ]
[
difference Forward
n n
n
] 1 [ ] [ ] [
difference Backward
−
−
= n n n
] 1 [ ] [
delay sameple
-One
−
= n
n
]
[n
x
]
[n
]
[n y
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Cascading systems
Accumulator + Backward difference system
Inverse system:
] [ ]
[
system
r Accumulato
n u n
] [ ] [n n
] 1 [ ] [ ] [
difference Backward
−
−
n
]
[n
x
]
[n
]
[n x
] [ ] [
* ] [ ] [
* ]
Part II: LCCD equations
System impulse response
Linear constant-coefficient difference
equations
Fourier transforms and frequency response
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LCCD equations
An important class of LTI systems: input and output
satisfy an Nth-order LCCD equations
Difference equation representation of the accumulator
] [ ] 1 [
]
[
] 1 [ ] [ ] [ ]
[
]
[
] [ ]
1
[
] [ ]
[
1 1
n x n
y
n
y
n y n x k x n
x
n
y
k x n
y
k x
n
y
n
k
n
k
n
k
=
−
−
− +
= +
=
=
−
=
∑
∑
∑
−
−∞
=
−
−∞
=
−∞
=
∑
∑
=
=
−
=
m m N
k
k y n k b x n m
a
0 0
] [ ]
[
One-sample delay x[n]
y[n-1]
y[n]
Recursive representation
Part III: Fourier transforms
System impulse response
Linear constant-coefficient difference
equations
Fourier transforms and frequency response
Frequency-domain representation of discrete-time
signals and systems
Symmetry properties of the Fourier transform
Fourier transform theorems
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Signal representations
A sum of scaled, delayed impulse
Sinusoidal and complex exponential sequences
frequency and with amplitude and phase determined by
the system
systems.
exponentials
) cos(
] [n =A ω0n+ φ
x
∑∞
−∞
=
−
=
k
k n k x n
n j
e n
∑∞
−∞
=
−
=
k
k n h k x n
Eigenfunctions
Complex exponentials as input to system h[n]
Define
Then
n j
k
k j k
k n j n
j
n j
e e k h
e k h e
T n
y
n e
n
x
ω ω
ω ω
ω
) ] [ (
] [ }
{ ]
[
, ]
[
) (
∑
∑
∞
−∞
=
−
∞
−∞
=
−
=
=
=
∞
<
<
∞
−
=
n j j k
k j j
e e H n y
e k h e
H
ω ω
ω ω
) ( ] [
] [ )
(
=
= ∑∞
−∞
=
−
A – linear operator
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Eigenvalue – called frequency response
Frequency response is generally complex
Frequency response of the ideal delay system
) (
, 1
| ) (
|
).
sin(
) (
), cos(
) (
d j
j
d j
I
d j
R
n e
H
e H
n e
H
n e
H
ω
ω ω
ω ω ω ω
−
=
∠
=
−
=
=
d
n j n
n j d j
e e n n e
−∞
=
−
= [ ] )
(
d
d d
n j j
n j n j n n j n j
e e
H
e e e
e T
n
y
ω ω
ω ω ω
ω
−
−
−
=
=
=
=
)
(
} { ]
) (
| ) (
|
) ( ) ( ) (
ω
ω
ω ω
ω
j e H j j
j I j R j
e e H
e jH e H e
H
∠
=
+
=
describes changes in magnitude and phase
] [ ] [ ] [ ]
[n x n n d h n n n d
Frequency response
The frequency response of discrete-timeLTI
systems is always a periodic function of the
frequency variable w with period
) ( ]
[ ]
[ )
n
n j n j
n
n j j
e H e
e n h e
n h e
−∞
=
−
−
∞
−∞
=
+
− +
π ω
π < ≤
−
π
± π 2
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Ideal frequency-selective filters
For which the frequency response is unity over a
certain range of frequencies, and is zero at the
remaining frequencies
Ideal frequency-selective filters
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Sinusoidal response of LTI systems
Sinusoidal input Æ sinusoidal response with the
same frequency and with amplitude and phase
determined by the system
) cos(
| ) (
| ] [ then
) ( 2
) ( 2
]
[
2 2
) cos(
]
[
0
0
0
0 0 0
0
0 0
θ φ ω
φ ω
ω
ω ω φ ω
ω φ
ω φ ω
φ
+ +
=
+
=
+
=
+
=
−
−
−
−
−
n e
H A n y
e e H e
A e
e H e
A
n
y
e e A e
e A
n A
n
x
j
n j j j n
j j j
n j j n
j j
θ ω ω
ω
− =H e = H e e = H e e
e
H
n
h
j e
H j
j
| ) (
|
| ) (
| ) ( ) (
then real, is ]
[
if
0 0
0 0
Signal representation
More than sinusoids, a broad class of signals can be
represented as a linear combination of complex
exponentials:
If x[n] can be represented as a superposition of
complex exponentials, output y[n] can be computed
∑
=
k
n j
k e k n
∑
=
∴
k
n j j k
k
k e e H n
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Frequency-domain representation of x[n]
By Fourier transforms Æ Fourier representation:
In general, Fourier transform is complex
) (
| ) (
|
) ( ) ( ) (
ω
ω
ω ω
ω
j e X j j
j I j R j
e e X
e jX e X e X
∠
=
+
=
ω π
ω ω
d e e
X
x[n]
n j j
)
(
2
1
sinusoids complex
small mally infinitesi of
ion superposit
a
as represent
n j j
n j j
e n x e
X
d e e X n
x
ω ω
π π
ω
π
−
∞
∞
−
−
∑
∫
=
=
] [ ) (
where
) ( 2
1 ] [
w is continuous-time variable
Fourier transform
n is discrete-time variable
Fourier spectrum Spectrum Magnitude spectrum Amplitude spectrum Phase spectrum
Frequency and impulse responses
Are a Fourier transform pair
Fourier transform is periodic with period
n j j
e n x e
X ω − ω
∞
∞
−
∑
) (
ω π
π π
ω ω
ω ω
d e e H n
h
e n h e
H
n j j n
n j j
∫
∑
−
∞
−∞
=
−
=
∴
=
) ( 2
1 ] [
] [ ) (
Recall
π
2
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Sufficient condition for Fourier transform
Condition for the convergence of the infinite sum
X[n] is absolutely summable, then its Fourier
transform exists (sufficient condition)
∞
<
≤
≤
=
∑
∑
∑
∞
∞
−
−
∞
∞
−
−
∞
∞
−
| ] [
|
|
||
] [
|
| ] [
| | ) (
|
n x
e n x
e n x e
X
n j
n j j
ω
ω ω
Example: ideal lowpass filter
⎩
⎨
⎧
≤
<
<
=
π ω ω
ω ω
ω
|
| 0
|
| , 1 ) (
c
c j
lp e H
∞
<
<
∞
−
=
n
n d
e n
lp
c c
, sin 2
1 ]
[
π
ω ω π
ω ω ω
∑∞
−∞
=
−
=
n
n j c j
n
n e
π
ω
sin )
(
ω
Trang 12Digital Signal Processing, II, Zheng-Hua Tan, 2006
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Example: Fourier transform of a constant
Its Fourier transform is defined as the periodic
impulse train
1 ] [n =
x
) 2 ( 2 )
X
r
jω ∑∞ πδ ω π
−∞
=
+
=
n j j
n j j
e n x e
X
d e e X n
x
ω ω
π π
ω
π
−
∞
∞
−
−
∑
∫
=
=
] [ ) ( where
) ( 2
1 ] [
Part III: Fourier transforms
System impulse response
Linear constant-coefficient difference
equations
Fourier transforms and frequency response
Frequency-domain representation of discrete-time
signals and systems
Symmetry properties of the Fourier transform
Fourier transform theorems
Trang 13Digital Signal Processing, II, Zheng-Hua Tan, 2006
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Symmetry properties of Fourier transform
Any , x [n]
] [ ]
x e = e −
] [ ]
x o = − o −
] [ ] [ ] [
] [ ]) [ ] [ ( 2
1 ] [
] [ ]) [ ] [ ( 2
1 ] [ define
]) [ ] [ ( 2
1 ]) [ ] [ ( 2
1 ] [
*
*
*
*
*
*
n x n x n x
n x n x n x n x
n x n x n x n x
n x n x n x n x n x
o e
o o
e e
+
=
∴
−
−
=
−
−
=
−
=
− +
=
−
− +
− +
=
) ( )) ( ) ( ( 2
1
)
(
) ( )) ( ) ( ( 2
1
)
(
where
) ( ) ( )
(
*
*
*
*
ω ω
ω ω
ω ω
ω ω
ω ω
ω
j o j
j j
o
j e j j
j
e
j o j e j
e X e
X e X e
X
e X e
X e X e
X
e X e X
e
X
−
−
−
−
−
=
−
=
= +
=
+
=
Symmetry properties of Fourier transform
Table 2.1
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Part III: Fourier transforms
System impulse response
Linear constant-coefficient difference
equations
Fourier transforms and frequency response
Frequency-domain representation of discrete-time
signals and systems
Symmetry properties of the Fourier transform
Fourier transform theorems
Linearity of the Fourier Transform
) ( )
( ]
[ ]
[
) ( ]
[
) ( ]
[
2 1
2 1
2 2
1 1
ω ω
ω ω
j j
F
j F
j F
e bX e
aX n bx n
ax
e X n
x
e X n
x
+
↔ +
↔
↔
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Time shifting and frequency shifting
) (
] [
) ( ]
[
) ( ]
[
) ( 0
ω
ω ω
ω
−
−
↔
↔
−
↔
j F n
j
j n j F d
j F
e X n x
e
e X e n
n
x
e X n
x
d
Time reversal
) ( ] [
real.
is ] [
if
) ( ] [
) ( ]
[
* ω
ω ω
j F
j F j F
e X n
x
n x
e X n
x
e X n
x
↔
−
↔
−
↔
−
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Differentiation in frequency
ω
ω ω
d
e dX j n
nx
e X n
x
j F
j F
) ( ]
[
) ( ]
[
↔
↔
Parseval’s theorem
ω π
π π ω
ω
∫
∞
−∞
=
=
=
↔
d e X n
x
E
e X
n
x
j n
j F
2 2
| ) (
| 2
1
| ] [
|
) ( ]
[
Trang 17Digital Signal Processing, II, Zheng-Hua Tan, 2006
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The convolution theorem
) ( ) ( )
(
] [
* ] [ ] [ ] [ ]
[
) ( ]
[
) ( ]
[
ω ω ω
ω ω
j j j
k
j F
j F
e X e H e
Y
n h n x k n h k x n
y
e H
n
h
e X
n
x
=
=
−
=
↔
↔
∑∞
−∞
=
The modulation or Windowing theorem
∫−
−
=
=
↔
↔
π π
θ ω θ ω
ω ω
θ
e
Y
n w n
x
n
y
e W
n
w
e X
n
x
j j j
j F
j F
) ( ) ( 2
1 )
(
] [ ] [
]
[
) ( ]
[
) (
]
[
) (
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Fourier transform theorems
Table 2.2
Fourier transform pairs
Table 2.3
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Example
Determining a Fourier transform using Tables 2.2
and 2.3
ω
ω ω
ω
ω ω
ω ω
ω ω
j
j j
n j
j j
j j
n j j
F n
ae
e a e X
n u a n
x a n
x
ae
e e
X e e X
n u a n
x n x
ae e
X n u a n x
−
−
−
−
−
−
−
−
=
−
=
=
−
=
=
−
=
−
=
−
=
↔
=
1 ) (
]) 5 [ (i.e.
] [ ] [
1 ) ( )
(
]) 5 [ (i.e.
] 5 [ ] [
1
1 ) ( ] [ ] [
5 5 2 5
5 1
5 2
5 1
2
1 1
] 5 [ ]
Summary
System impulse response
Linear constant-coefficient difference
equations
Fourier transforms and frequency response
Frequency-domain representation of discrete-time
signals and systems
Symmetry properties of the Fourier transform
Fourier transform theorems
Trang 20Digital Signal Processing, II, Zheng-Hua Tan, 2006
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Course at a glance
Discrete-time signals and systems
Fourier-domain
representation
DFT/FFT
System structures
Filter structures Filter design
Filter
z-transform
MM1
MM2
MM9,MM10
MM3
MM6
MM4
Sampling and
System analysis System