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1. Linear and Nonlinear Phase

2. Ideal and Nonideal Frequency-SelectiveFilters

3. CT & DT Rational Frequency Responses

4. DT First- and Second-Order Systems

Signals and SystemsFall 2003

Lecture #12

16 October 2003

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Linear Phase

Result: Linear phase ⇔ simply a rigid shift in time, no distortion

 Nonlinear phase ⇔ distortion as well as shift

CT

Question:

DT

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All-Pass Systems

CT

DT

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Demo: Impulse response and output of an all-pass

system with nonlinear phase

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

Group Delay

φ

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Ideal Lowpass Filter

CT

← → ⎯ 

• Noncausal h(t <0) ≠ 0• Oscillatory Response — e.g. step response Overshoot by 9%,

Gibbs phenomenon

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Nonideal Lowpass Filter

• Sometimes we don’t want a sharp cutoff, e.g.

• Often have specifications in time and frequency domain ⇒ Trade-offs

Step responseFreq. Response

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CT Rational Frequency Responses

CT: If the system is described by LCCDEs, then

Prototypical

Systems  — 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 

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DT Rational Frequency Responses

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

Difference Equations), then

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DT First-Order Systems

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Demo: Unit-sample, unit-step, and frequency response

of DT first-order systems

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DT Second-Order System

oscillations

decaying

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Demo: Unit-sample, unit-step, and frequency response of 

DT second-order systems