Poles and Zeros Frequency

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Poles and Zeros Frequency

• Chapter 12
• Lecture 15

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Frequency response from Poles and Zeros

• Recall z esT e(sj?)T
• And zk ? Hkzk
• So ejOk ? HejOejOk letting s 0, O ?T
• e-jOk ? He-jOe-jOk
• This is the same as
• cosOk ? HejOcos(Ok?HejO)
• This says the response to a sinusoid inputs
  is.

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Example

• Consider yk1 -0.8yk fk1

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Example-cont

• Let fk cosp/6k 0.2
• Hejp/6
• Hejp/6

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So far we have been analyzing system
responses.Lets start designing responses based
on poles and zeros

• We will plot on the unit circle.
• The closer the pole or zero is to the point ej?T,
  the more affect it has on the system response.
• To enhance the amplitude response at freq-?,
  place a pole close to ej?T
• To suppress the amplitude response at freq-?,
  place a zero close to ej?T
• Placing a zero on ej?T, zeros the response (notch
  filter)
• Placing a zero on the Origin, causes a pure delay
• All poles must lie inside the Unit Circle (zeros
  dont matter)

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• Consider H(z) bn(z-z1) (z-z2) (z-z3)
• (z-p1) (z-p2) (z-p3)

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Examples
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