Signal and Systems Chapter 6: Time-Frequency Characterization of - - PowerPoint PPT Presentation

Signal and Systems

Chapter 6: Time-Frequency Characterization of Systems

• 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

Magnitude and Phase of FT, and Parseval Relation

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 2

( ) 2 2 2 ( ) 2

1 : ( ) ( ) 2 ( ) | ( ) | 1 ParsevalRelation: | ( ) | | ( ) | 2 1 : [ ] ( ) 2 ( ) | ( ) | ParsevalRelation: | [ ]|

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 | ( ) | 2

j

X e d

 

  



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

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 3

Book Chapter 6 Computer Engineering Department, Signal and Systems 4

Log-Magnitude and Phase

Book Chapter 6 Computer Engineering Department, Signal and Systems 5

| ( ) | | ( ) |.| ( ) | 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            

Plotting Log-Magnitude and Phase

Book Chapter 6 Computer Engineering Department, Signal and Systems 6

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

• utput power
• f decibels (dB): (1bel= 10decibels=

=10) input power

Plot for ω ≥ 0, often with a logarithmic scale for frequency in CT

2 10

10log| ( ) | 20log | ( ) | | ( ) | 1 | ( ) | 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                  

A Typical Bode plot for a second-

• rder CT system

Book Chapter 6 Computer Engineering Department, Signal and Systems 7

20log | ( ) | ( ) .log H j and H j vs    

40 dB/decade Changes by -π

A typical plot of magnitude and phase of second order DT frequency response

Book Chapter 6 Computer Engineering Department, Signal and Systems 8

20log | ( ) | ( ) .log H j and H j vs    

Linear phase

Book Chapter 6 Computer Engineering Department, Signal and Systems 9

CT

( ) | ( )| 1 , ( ) ( ) ( ) ( ) ( ) ( )

j time shift j

H j e H j H j Linear in Y j e X j y t x t

 

       

 

          

[ ] [ ] ( ) ( ) ( ) | ( ) | 1, ( )

j n j j j n j 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 ω α, α ≠ integer?

All-Pass Systems

Book Chapter 6 Computer Engineering Department, Signal and Systems 10

Demo:Impulse response and output of an all- pass system with nonlinear phase

Book Chapter 6 Computer Engineering Department, Signal and Systems 11

How do we think about signal delay when the phase is nonlinear? Group Delay

Book Chapter 6 Computer Engineering Department, Signal and Systems 12

When the signal is narrow-band and concentrated near ( ) ~ linear ( ) with , ( ) H j d H j near then instead d H j

• f

reflect the time delay            

( ) ( )

( ) ( ) ( )( ) ( ). ( ) { ( )} ( ) | ( ) | ~| ( ) |

j j j t j t j

H j H j d H j Group Delay d for near H j H j e e e H j e e

       

                  

 

             

Ideal Low pass Filter

Book Chapter 6 Computer Engineering Department, Signal and Systems 13

Nonideal Low pass Filter

Book Chapter 6 Computer Engineering Department, Signal and Systems 14

CT Rational Frequency Responses

Book Chapter 6 Computer Engineering Department, Signal and Systems 15

 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

( ) ( ) ( ) ( ) ( ) ( )

k k k k k k i k i k k i

d j dt b j H 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 j j H j j j j j                                 

— Second-order system, has two energy storing elements, e.g. L and C

DT Rational Frequency Responses

 If the system is described by LCCDE’s (Linear-Constant-

Coefficient Difference Equations), then

Book Chapter 6 Computer Engineering Department, Signal and Systems 16

[ ] ( ) , [ ] ( ) ( ) ( ) ( ) ( ) ( )

j jk j jk jk j k k k j j k k i jk j k i k k k k j i

y n k Y e e x n k X e e b e b e H e H e a e a e H e First or Second order

                

       

    

DT First-Order Systems

Book Chapter 6 Computer Engineering Department, Signal and Systems 17

2 1 1

[ ] [ 1] [ ], | | 1 1 ( ) 1 1 | ( ) | 1 2 cos sin ( ) tan 1 cos [ ] [ ] 1 [ ] [ ]* [ ] [ ] 1

j j j j n n n k k

y n ay n x n a initial rest H e ae Frequency Domain H e a a a H e a Time Domain h n a u n a s n h n u n a u n a

   

  

   

                        



Demo: Unit-sample, unit-step, and frequency response of DT first-order systems

Book Chapter 6 Computer Engineering Department, Signal and Systems 18

DT Second-Order System

Book Chapter 6 Computer Engineering Department, Signal and Systems 19

2 2 2 1 2 1 2 1 2

[ ] 2 cos [ 1] [ 2] [ ], 0 1 1 ( ) 1 (2 cos ) 1 1 . 1 1 1 1 : , 2 sin 2 sin [ ] [ ] [ ] sin( 1)

j j 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 e r e r e re e re e A A re e re e where e e A A j j h n A r e A r e u n r n

              

      

         

                         [ ] sin [ ] [ ]* [ ] u n s n h n u n        

Demo: Unit-sample, unit-step, and frequency response of DT second-order systems

Book Chapter 6 Computer Engineering Department, Signal and Systems 20