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

© 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. 8-4 8-5 & 8-6 4/5/2019 © 2003, JH McClellan & RW Schafer

© 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) 4/5/2019 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer THREE DOMAINS Z-TRANSFORM-DOMAIN POLYNOMIALS: H(z) FREQ-DOMAIN TIME-DOMAIN 4/5/2019 © 2003, JH McClellan & RW Schafer

Quick Review: Delay by nd IMPULSE RESPONSE SYSTEM FUNCTION FREQUENCY RESPONSE 4/5/2019 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer LOGICAL THREAD FIND the IMPULSE RESPONSE, h[n] INFINITELY LONG IIR Filters EXPLOIT THREE DOMAINS: Show Relationship for IIR: 4/5/2019 © 2003, JH McClellan & RW Schafer

© 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 4/5/2019 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer FILTER COEFFICIENTS ADD PREVIOUS OUTPUTS MATLAB yy = filter([3,-2],[1,-0.8],xx) SIGN CHANGE FEEDBACK COEFFICIENT 4/5/2019 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer COMPUTE OUTPUT 4/5/2019 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer COMPUTE y[n] FEEDBACK DIFFERENCE EQUATION: NEED y[-1] to get started 4/5/2019 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer AT REST CONDITION y[n] = 0, for n<0 BECAUSE x[n] = 0, for n<0 4/5/2019 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer COMPUTE y[0] THIS STARTS THE RECURSION: SAME with MORE FEEDBACK TERMS 4/5/2019 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer COMPUTE MORE y[n] CONTINUE THE RECURSION: 4/5/2019 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer PLOT y[n] 4/5/2019 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer IMPULSE RESPONSE 4/5/2019 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer IMPULSE RESPONSE DIFFERENCE EQUATION: Find h[n] CONVOLUTION in TIME-DOMAIN IMPULSE RESPONSE 4/5/2019 © 2003, JH McClellan & RW Schafer LTI SYSTEM

© 2003, JH McClellan & RW Schafer PLOT IMPULSE RESPONSE 4/5/2019 © 2003, JH McClellan & RW Schafer

Infinite-Length Signal: h[n] POLYNOMIAL Representation SIMPLIFY the SUMMATION APPLIES to Any SIGNAL 4/5/2019 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer Derivation of H(z) Recall Sum of Geometric Sequence: Yields a COMPACT FORM 4/5/2019 © 2003, JH McClellan & RW Schafer

H(z) = z-Transform{ h[n] } FIRST-ORDER IIR FILTER: 4/5/2019 © 2003, JH McClellan & RW Schafer

H(z) = z-Transform{ h[n] } ANOTHER FIRST-ORDER IIR FILTER: 4/5/2019 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer CONVOLUTION PROPERTY MULTIPLICATION of z-TRANSFORMS CONVOLUTION in TIME-DOMAIN IMPULSE RESPONSE 4/5/2019 © 2003, JH McClellan & RW Schafer

STEP RESPONSE: x[n]=u[n] 4/5/2019 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer DERIVE STEP RESPONSE 4/5/2019 © 2003, JH McClellan & RW Schafer

© 2003, JH McClellan & RW Schafer PLOT STEP RESPONSE 4/5/2019 © 2003, JH McClellan & RW Schafer