Lesson_220 - EEL 3135: Dr. Fred J. Taylor, Professor Lesson...

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EEL 3135: Dr. Fred J. Taylor, Professor Lesson Title: IIR Systems Lesson Number: 22 [Section 8-4 to 8-6] Background: In the previous lesson infinite impulse response or IIR system. These systems are characterized by the discrete-time difference equation: - + - = = = M 0 m m N 1 m m 0 m] x[k b m] y[k a a 1 y[k] 1 where y[k] is the system’s output time-series to an input x[k] time series. If a 0 =1, the system is called monic . An IIR with all initial conditions zero is said to be at-rest . The response of an LTI to an arbitrary input time series x[k] is formally defined by the convolution theorem is stated below. [ ] [ ] [ ] [ ] [ ] m k x m h m x m k h k y m m - = - = = = 0 0 2 where h[k] is the IIR’s impulse response. A key tool in quantifying the IIR filter’s impulse response is the z -transform. In Lesson 21 the convolution theorem was re-introduced and it stated that: [ ] [ ] [ ] ( 29 ( 29 ( 29 z X z H z Y k x k h k y Z = → = 3. The mechanics and relationships developed in Chapter 7 can be directly applied to the studies from Chapter 8. Consider again the IIR described in Equation 1. The z-transform of the monic version of this Equation is: ( 29 ( 29 ( 29 + = → - + - = = = - = = M 0 m -m m N 1 m m m M 0 m m N 1 m m z X z b z a z Y m] x[k b m] y[k a y[k] z Y Z 4. After a small amount of manipulation: ( 29 ( 29 ( 29 ( 29 ( 29 = = - = = - = - = = M 0 m -m m N 1 m m m M 0 m -m m N 1 m m m z X z b z a 1 z Y z X z b z a z Y z Y 5. As a final step, the transfer function of the at-rest system is given by: 1
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EEL 3135: Dr. Fred J. Taylor, Professor ( 29 ( 29 ( 29 - = = = - = N 1 m m m M 0 m -m m z a 1 z b z X z Y z H 6. The inverse z-transform of H(z), if we knew how to perform one, would produce the system’s impulse response. That is: Z -1 (H(z) = h[k] 7. For an IIR, the transfer function H(z) is the ratio of two polynomials in z . As such, an IIRs transfer function is also defined in terms of poles and zeros . The transfer function given in Equation 6 can, therefore, also be expressed as: ( 29 ( 29 ( 29 = = = - = - - = - = = n m m M m m p z z z k 0 0 N 1 m m m M 0 m -m m ) ( ) ( z a 1 z b z X z Y z H 8. where z m is a filter zero and p m is a filter pole. One of an LTI system’s attributes that historically been connect to the pole-zero locations is stability. Stability The authors motivate the concept of stability in the context of a suite of simple example beginning with a difference equation: y[k]=0.5y[k-1] + 2x[k] 9. which translates into a transfer function given by:
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Lesson_220 - EEL 3135: Dr. Fred J. Taylor, Professor Lesson...

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