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MODULE-9 - MODULE 9 DIGITAL FILTER DESIGN II IIR Filter...

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MODULE 9 DIGITAL FILTER DESIGN II IIR Filter Design Analog Prototypes Impulse Invariance Bilinear Transformation Lowpass-to-Otherpass Transformations Page 9.1

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INDEX Infinite Impulse Response (IIR) Filter Design Advantages of IIR Filters Analog Prototype-Based Digital Filter Design Butterworth Filter Type 1 Chebyshev Filter Type 2 Chebyshev Filter Elliptical Filter IIR Filter Design by Impulse Invariance Ideal Frequency Relationship Implementation of Impulse Invariance Design Stability of Impulse Invariance Design Example - Impulse Invariance Design IIR Filter Design by Bilinear Transformation Stability of Bilinear Transformation Bilinear Frequency Mapping Frequency Warping Selection of the Warping Parameter Prewarping Comments on Bilinear Transformation Example - Bilinear Transformation Design Lowpass-to-Otherpass Transformations MAIN INDEX Page 9.2
9. DIGITAL FILTER DESIGN II index READ : Appendix B, Section 7.1 of Oppenheim & Schafer. Work as many related problems as possible. Infinite Impulse Response (IIR) Filter Design Here it is assumed that H ( z ) = k K k m M m z k z m a b - = - = - ) ( 1 ) ( 1 0 Problem : Given a desired response D ( e j ϖ ), FIND : { a ( m ) ; m = 0 ,..., M } and { b ( k ) ; k = 1 ,..., K } . IIR filters are often called recursive for obvious reasons. However, IIR filters can be implemented non-recursively (via a bank of "resonators"). Primarily of theoretical interest. We'll always assume recursive implementation in practice . Page 9.3

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Advantages of IIR Filters index A lower-order filter is generally sufficient to satisfy the stated specifications - so fewer coefficients are needed: - less storage - less computation Disadvantages of IIR Filters They are not necessarily stable . Transients may have an infinite duration . Quantization noise can be a problem . Cannot be designed to have exact linear phase when causal . Page 9.4
Analog Prototype-Based Digital Filter Design index IIR digital filter design usually involves starting with an analog filter prototype - from which a digital filter is created via a transformation . We will study two such transformations: - impulse invariance - bilinear transformation The steps involved are typically as shown (and can be combined in practice, once each is understood): transformation s z e j ϖ | H a ( ) | 2 H a ( s ) H d ( e j ϖ ) analog Laplace digital prototype representation filter        . Basic knowledge of the Laplace transform is assumed . Page 9.5

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Obtaining the Laplace Representation index The following series of steps ensues: (1) Start with an analog prototype H a ( ). Usually just the magnitude spectrum is specified: | H a ( ) | 2 . (2) Go to the Laplace domain : replace j with s in | H a ( ) | 2 to get H a ( s ) H a ( -s ). Recall that the continuous Fourier and Laplace transforms are related by: H a ( ) = H a ( s )   | s = j .
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MODULE-9 - MODULE 9 DIGITAL FILTER DESIGN II IIR Filter...

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