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 •

Book Chapter#: Section# Magnitude and Phase of FT, and Parseval Relation  1      j t : ( ) ( ) CT x t X j e d  2       ( ) j X j ( ) | ( ) | X j X j e   1      2 2 ParsevalRelation: | ( ) | | ( ) | x t dt X j d  2    Energy density in 1      j j n : [ ] ( ) DT x n X e e d   2 2      j ( ) j j j X e ( ) | ( ) | X e X e e  1      2 j 2 ParsevalRelation: | [ ]| | ( ) | x n X e d   2 2  n Computer Engineering Department, Signal and Systems 2

Book Chapter#: Section# 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 3

Book Chapter 6 Computer Engineering Department, Signal and Systems 4

Book Chapter 6 Log-Magnitude and Phase     | ( ) | | ( ) |.| ( ) | Y j H j X j      log | ( ) | log | ( ) | log | ( ) | Y j H j X j         ( ) ( ) ( ) Y j H j X j      log | ( )| log | ( )| log | ( )| H j H j H j 1 2        ( ) ( ) ( ) H j H j H j 1 2 Computer Engineering Department, Signal and Systems 5

Book Chapter 6 Plotting Log-Magnitude and Phase a) For real-valued signals and systems       | ( ) | | ( ) | H j H j Plot for ω ≥ 0 , often with a        logarithmic scale for  ( ) ( ) H j H j frequency in CT     b) In DT, need only plot for 0 (with linear scale) c) For historical reasons, log-magnitude is usually plotted in u nits output power of decibels (dB): (1bel= 10decibels= =10) input power    2 10log| ( ) | 20log | ( ) | H j H j 10 power magnitude    | ( ) | 1 0 H j dB    | ( ) | 2 ~ 3 H j dB    | ( ) | 2 ~ 6 H j dB    | ( ) | 10 20 H j dB    | ( ) | 100 40 H j dB Computer Engineering Department, Signal and Systems 6

Book Chapter 6 A Typical Bode plot for a second- order CT system     20log | ( ) | ( ) .log H j and H j vs 40 dB/decade Changes by - π Computer Engineering Department, Signal and Systems 7

Book Chapter 6 A typical plot of magnitude and phase of second order DT frequency response     20log | ( ) | ( ) .log H j and H j vs Computer Engineering Department, Signal and Systems 8

Book Chapter 6 Linear phase CT              j ( ) | ( )| 1 , ( ) ( ) H j e H j H j Linear in           j time shift ( ) ( ) ( ) ( ) Y j e X j y t x t Result: Linear phase ⇔ simply a rigid shift in time, no distortion Nonlinear phase ⇔ distortion as well as shift DT          j j n j [ ] [ ] ( ) ( ) y n x n n Y e e X e 0 0             j j n j j ( ) | ( ) | 1, ( ) H e e H e H e n 0 0 Question: What about H (e j ω ) = e -j ω α , α ≠ integer? Computer Engineering Department, Signal and Systems 9

Book Chapter 6 All-Pass Systems Computer Engineering Department, Signal and Systems 10

Book Chapter 6 Demo: Impulse response and output of an all- pass system with nonlinear phase Computer Engineering Department, Signal and Systems 11

Book Chapter 6 How do we think about signal delay when the phase is nonlinear? Group Delay When the signal is narrow-band and   concentrated near ( ) ~ linear H j   ( ) d H j    with , near then instead  0 d   ( ) H j  of reflect the time delay                   ( ) ( ) ( )( ) ( ). H j H j 0 0 0 0 d        ( ) { ( )} H j Group Delay  d   for near 0         ( ) j j ( ) | ( ) | H j H j e e 0 0         j ( t ) j t j ~| ( ) | e H j e e 0 Computer Engineering Department, Signal and Systems 12

Book Chapter 6 Ideal Low pass Filter Computer Engineering Department, Signal and Systems 13

Book Chapter 6 Nonideal Low pass Filter Computer Engineering Department, Signal and Systems 14

Book Chapter 6 CT Rational Frequency Responses  CT: If the system is described by LCCDEs, then k d   k ( ) j k dt   k ( ) b j k      ( ) k ( ) H j H j   i k ( ) a j i k k   ( ) H j First or Second order factors i  Prototypical System  First-order system, has only one energy storing element, e.g. L or C 1   ( ) H j 1   1 j — Second-order system,  2 1    ( ) n H j has two energy storing       2 2 2 2       ( ) 2 ( ) j j    n n   2   1 elements, e.g. L and C j j       n n Computer Engineering Department, Signal and Systems 15

Book Chapter 6 DT Rational Frequency Responses  If the system is described by LCCDE ’ s (Linear-Constant- Coefficient Difference Equations), then           j jk j jk [ ] ( ) , [ ] ( ) y n k Y e e x n k X e e       jk j k ( ) b e b e       k k j j ( ) k k ( ) H e H e       i jk j k ( ) a e a e i k k k k   j ( ) H e First or Second order i Computer Engineering Department, Signal and Systems 16

Book Chapter 6 DT First-Order Systems     [ ] [ 1] [ ], | | 1 y n ay n x n a initial rest 1   j ( ) H e    j 1 ae Frequency Domain 1   j | ( ) | H e    2 1 2 cos a a    sin a      1 j ( ) tan H e       1 cos a Time Domain  n [ ] [ ] h n a u n   1 n n 1 a     k [ ] [ ]* [ ] [ ] s n h n u n a u n  1 a  0 k Computer Engineering Department, Signal and Systems 17

Book Chapter 6 Demo: Unit-sample, unit-step, and frequency response of DT first-order systems Computer Engineering Department, Signal and Systems 18

Book Chapter 6 DT Second-Order System             2 [ ] 2 cos [ 1] [ 2] [ ], 0 1 0 y n r y n r y n x n r and 1   j ( ) H e        2 2 j j 1 (2 cos ) r e r e 1 1  .          j j j j 1 1 re e re e A A   1 2          j j j j 1 1 re e re e : where    j j e e   , A A   1 2 2 sin 2 sin j j      n jn n jn [ ] [ ] [ ] h n A r e A r e u n 1 2     n sin( 1) r n    [ ] u n    sin  [ ] [ ]* [ ] s n h n u n Computer Engineering Department, Signal and Systems 19

Book Chapter 6 Demo: Unit-sample, unit-step, and frequency response of DT second-order systems Computer Engineering Department, Signal and Systems 20