Signal and Systems Prof. 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

Magnitude and Phase of FT, and Parseval Relation Book Chapter#: Section# Computer Engineering Department, Signal and Systems 2

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

Plotting Log-Magnitude and Phase Book Chapter 6 Computer Engineering Department, Signal and Systems 6 Plot for ω ≥ 0, often with a logarithmic scale for frequency in CT

A Typical Bode plot for a second- order CT system Book Chapter 6 Computer Engineering Department, Signal and Systems 7 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

Linear phase Book Chapter 6 Computer Engineering Department, Signal and Systems 9 CT Result:Linear phase ⇔ simply a rigid shift in time, no distortion Nonlinear phase ⇔ distortion as well as shift DT Question: What about H (e jω ) = 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

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 — 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

DT First-Order Systems Book Chapter 6 Computer Engineering Department, Signal and Systems 17

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

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