signal and systems prof. h. sameti chapter 6: magnitude/phase of transforms and frequency responses...
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Signal and SystemsProf. H. Sameti
Chapter 6:• Magnitude/Phase of Transforms and Frequency
Responses• Linear and Nonlinear Phase• Ideal and Nonideal Frequency-Selective Filters• CT & DT Rational Frequency Responses• DT First- and Second-Order Systems
Book Chapter#: Section#
2
Magnitude and Phase of FT, and Parseval Relation
Computer Engineering Department, Signal and Systems
( )
2 2
2
( )
2
1: ( ) ( )
2
( ) | ( ) |
1Parseval Relation: | ( ) | | ( ) |
2
1: [ ] ( )
2
( ) | ( ) |
Parseval Relation: | [ ] |
j
j t
j X j
Energy density in
j j n
j j j X e
n
CT x t X j e d
X j X j e
x t dt X j d
DT x n X e e d
X e X e e
x n
2
2
1| ( ) |
2jX e d
Book Chapter#: Section#
3
Effect of Phase Not on signal energy distribution as a function of
frequency
Can have dramatic effect on signal shape/character Constructive/Destructive interference
Is that important? Depends on the signal and the context
Computer Engineering Department, Signal and Systems
Book Chapter 6
4Computer Engineering Department, Signal and Systems
Book Chapter 6
5
Log-Magnitude and Phase
Computer Engineering Department, Signal and Systems
| ( ) | | ( ) | . | ( ) |
log | ( ) | log | ( ) | log | ( ) |
( ) ( ) ( )
Y j H j X j
Y j H j X j
Y j H j X j
1 2
1 2
log | ( ) | log | ( ) | log | ( ) |
( ) ( ) ( )
H j H j H j
H j H j H j
Book Chapter 6
6
Plotting Log-Magnitude and Phase
Computer Engineering Department, Signal and Systems
a) For real-valued signals and systems
| ( ) | | ( ) |
( ) ( )
b) In DT, need only plot for 0 (with linear scale)
c) For historical reasons, log-magnitude is usually plotted in u
H j H j
H j H j
nits
output power of decibels (dB): (1bel = 10decibels = =10)
input power
Plot for ω ≥ 0, often with a logarithmic scale for frequency in CT
21010log| ( ) | 20 log | ( ) |
| ( ) | 1 0
| ( ) | 2 ~ 3
| ( ) | 2 ~ 6
| ( ) | 10 20
| ( ) | 100 40
power magnitude
H j H j
H j dB
H j dB
H j dB
H j dB
H j dB
Book Chapter 6
7
A Typical Bode plot for a second-order CT system
Computer Engineering Department, Signal and Systems
20log | ( ) | ( ) .logH j and H j vs
40 dB/decade
Changes by -π
Book Chapter 6
8
A typical plot of magnitude and phase of second order DT frequency response
Computer Engineering Department, Signal and Systems
20log | ( ) | ( ) .logH j and H j vs
Book Chapter 6
9
Linear phase
Computer Engineering Department, Signal and Systems
CT
( ) | ( ) | 1 , ( ) ( )
( ) ( ) ( ) ( )
j
time shiftj
H j e H j H j Linear in
Y j e X j y t x t
0
0
0
0
[ ] [ ] ( ) ( )
( ) | ( ) | 1, ( )
j nj j
j nj j j
y n x n n Y e e X e
H e e H e H e n
Result: Linear phase ⇔ simply a rigid shift in time, no distortion Nonlinear phase ⇔ distortion as well as shift
DT
Question: What about H (ejω) = e-j ωα, ≠α integer?
Book Chapter 6
10
All-Pass Systems
Computer Engineering Department, Signal and Systems
Book Chapter 6
11
Demo:Impulse response and output of an all-pass system with nonlinear phase
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Book Chapter 6
12
How do we think about signal delay when the phase is nonlinear? Group Delay
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0
When the signal is narrow-band and
concentrated near ( ) ~ linear
( )with ,
( )
H j
d H jnear then instead
dH j
of reflect the time delay
0
0
0 0 0 0
0
( )0
( )
( ) ( ) ( )( ) ( ).
( ) { ( )}
( ) | ( ) |
~| ( ) |
jj
j tj t j
H j H j
dH j Group Delay
dfor near
H j H j e e
e H j e e
Book Chapter 6
13
Ideal Low pass Filter
Computer Engineering Department, Signal and Systems
Book Chapter 6
14
Nonideal Low pass Filter
Computer Engineering Department, Signal and Systems
Book Chapter 6
15
CT Rational Frequency Responses
Computer Engineering Department, Signal and Systems
CT: If the system is described by LCCDEs, then
Prototypical System First-order system, has only one energy storing element, e.g. L or C
( )
( )( ) ( )
( )
( )
kk
k
kk
kik
ikk
i
dj
dt
b jH j H j
a j
H j First or Second order factors
1
2
2 2 2 2
1( )
1
1( )
( ) 2 ( )2 1
n
n n
n n
H jj
H jj j
j j
— Second-order system, has two energy storing elements, e.g. L and C
Book Chapter 6
16
DT Rational Frequency Responses
If the system is described by LCCDE’s (Linear-Constant-Coefficient Difference Equations), then
Computer Engineering Department, Signal and Systems
[ ] ( ) , [ ] ( )
( )( ) ( )
( )
( )
j jk j jk
jk j kk kj jk k
ijk j kik kk k
ji
y n k Y e e x n k X e e
b e b eH e H e
a e a e
H e First or Second order
Book Chapter 6
17
DT First-Order Systems
Computer Engineering Department, Signal and Systems
2
1
1
0
[ ] [ 1] [ ], | | 1
1( )
1
1| ( ) |
1 2 cos
sin( ) tan
1 cos
[ ] [ ]
1[ ] [ ]* [ ] [ ]
1
jj
j
j
n
nnk
k
y n ay n x n a initial rest
H eae
Frequency Domain
H ea a
aH e
a
Time Domain
h n a u n
as n h n u n a u n
a
Book Chapter 6
18
Demo: Unit-sample, unit-step, and frequency response of DT first-order systems
Computer Engineering Department, Signal and Systems
Book Chapter 6
19
DT Second-Order System
Computer Engineering Department, Signal and Systems
2
2 2
1 2
1 2
1 2
[ ] 2 cos [ 1] [ 2] [ ], 0 1 0
1( )
1 (2 cos )
1 1.
1 1
1 1:
,2 sin 2 sin
[ ] [ ] [ ]
sin( 1)
jj j
j j j j
j j j j
j j
n jn n jn
n
y n r y n r y n x n r and
H er e r e
re e re eA A
re e re ewhere
e eA A
j j
h n A r e A r e u n
r n
[ ]
sin
[ ] [ ]* [ ]
u n
s n h n u n
Book Chapter 6
20
Demo: Unit-sample, unit-step, and frequency response of DT second-order systems
Computer Engineering Department, Signal and Systems