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
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