1. Signal Processing First
Lecture 16
IIR Filters: Feedback
and H(z)
4/5/2019
© 2003, JH McClellan & RW Schafer

2. © 2003, JH McClellan & RW Schafer
READING ASSIGNMENTS
This Lecture:
Chapter 8, Sects. 8-1, 8-2 & 8-3
Other Reading:
Recitation: Ch. 8, Sects 8-1 thru 8-4
POLES & ZEROS
Next Lecture: Chapter 8, Sects & 8-6
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© 2003, JH McClellan & RW Schafer

3. © 2003, JH McClellan & RW Schafer
LECTURE OBJECTIVES
INFINITE IMPULSE RESPONSE FILTERS
Define IIR DIGITAL Filters
Have FEEDBACK: use PREVIOUS OUTPUTS
Show how to compute the output y[n]
FIRST-ORDER CASE (N=1)
Z-transform: Impulse Response h[n]  H(z)
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© 2003, JH McClellan & RW Schafer

4. © 2003, JH McClellan & RW Schafer
THREE DOMAINS
Z-TRANSFORM-DOMAIN
POLYNOMIALS: H(z)
FREQ-DOMAIN
TIME-DOMAIN
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© 2003, JH McClellan & RW Schafer

5. Quick Review: Delay by nd
IMPULSE RESPONSE
SYSTEM FUNCTION
FREQUENCY RESPONSE
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© 2003, JH McClellan & RW Schafer

6. © 2003, JH McClellan & RW Schafer
LOGICAL THREAD
FIND the IMPULSE RESPONSE, h[n]
INFINITELY LONG
IIR Filters
EXPLOIT THREE DOMAINS:
Show Relationship for IIR:
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© 2003, JH McClellan & RW Schafer

7. © 2003, JH McClellan & RW Schafer
ONE FEEDBACK TERM
ADD PREVIOUS OUTPUTS
PREVIOUS
FEEDBACK
FIR PART of the FILTER
FEED-FORWARD
CAUSALITY
NOT USING FUTURE OUTPUTS or INPUTS
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© 2003, JH McClellan & RW Schafer

8. © 2003, JH McClellan & RW Schafer
FILTER COEFFICIENTS
ADD PREVIOUS OUTPUTS
MATLAB
yy = filter([3,-2],[1,-0.8],xx)
SIGN CHANGE
FEEDBACK COEFFICIENT
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© 2003, JH McClellan & RW Schafer

9. © 2003, JH McClellan & RW Schafer
COMPUTE OUTPUT
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© 2003, JH McClellan & RW Schafer

10. © 2003, JH McClellan & RW Schafer
COMPUTE y[n]
FEEDBACK DIFFERENCE EQUATION:
NEED y[-1] to get started
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© 2003, JH McClellan & RW Schafer

11. © 2003, JH McClellan & RW Schafer
AT REST CONDITION
y[n] = 0, for n<0
BECAUSE x[n] = 0, for n<0
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© 2003, JH McClellan & RW Schafer

12. © 2003, JH McClellan & RW Schafer
COMPUTE y[0]
THIS STARTS THE RECURSION:
SAME with MORE FEEDBACK TERMS
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© 2003, JH McClellan & RW Schafer

13. © 2003, JH McClellan & RW Schafer
COMPUTE MORE y[n]
CONTINUE THE RECURSION:
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© 2003, JH McClellan & RW Schafer

14. © 2003, JH McClellan & RW Schafer
PLOT y[n]
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15. © 2003, JH McClellan & RW Schafer
IMPULSE RESPONSE
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© 2003, JH McClellan & RW Schafer

16. © 2003, JH McClellan & RW Schafer
IMPULSE RESPONSE
DIFFERENCE EQUATION:
Find h[n]
CONVOLUTION in TIME-DOMAIN
IMPULSE RESPONSE
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© 2003, JH McClellan & RW Schafer
LTI SYSTEM

17. © 2003, JH McClellan & RW Schafer
PLOT IMPULSE RESPONSE
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© 2003, JH McClellan & RW Schafer

18. Infinite-Length Signal: h[n]
POLYNOMIAL Representation
SIMPLIFY the SUMMATION
APPLIES to
Any SIGNAL
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© 2003, JH McClellan & RW Schafer

19. © 2003, JH McClellan & RW Schafer
Derivation of H(z)
Recall Sum of Geometric Sequence:
Yields a COMPACT FORM
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© 2003, JH McClellan & RW Schafer

20. H(z) = z-Transform{ h[n] }
FIRST-ORDER IIR FILTER:
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© 2003, JH McClellan & RW Schafer

21. H(z) = z-Transform{ h[n] }
ANOTHER FIRST-ORDER IIR FILTER:
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© 2003, JH McClellan & RW Schafer

22. © 2003, JH McClellan & RW Schafer
CONVOLUTION PROPERTY
MULTIPLICATION of z-TRANSFORMS
CONVOLUTION in TIME-DOMAIN
IMPULSE RESPONSE
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© 2003, JH McClellan & RW Schafer

23. STEP RESPONSE: x[n]=u[n]
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24. © 2003, JH McClellan & RW Schafer
DERIVE STEP RESPONSE
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© 2003, JH McClellan & RW Schafer

25. © 2003, JH McClellan & RW Schafer
PLOT STEP RESPONSE
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© 2003, JH McClellan & RW Schafer