CE 40763 Digital Signal Processing Design of Digital

CE 40763 Digital Signal Processing Design of Digital IIR Filters Hossein Sameti Department of Computer Engineering Sharif University of Technology

IIR vs. FIR Filters FIR IIR Achieving a linear phase is always possible Difficult to control the linear-phase property. Almost no particular technique is available. Can be unstable Filter order: less Always stable Filter order: higher Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 2

Design of IIR Digital Filters We focus on IIR filters with a rational transfer function: P and Q are polynomials in z. Filter Design: To determine the values of a(n) and b(n) such that specs given to us are met. Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 3

Design of IIR Digital Filters IIR Filter Design Optimization techniques Pole-zero placement Impulse Invariance Bilinear Transformation Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 4

Transformation Techniques 1. 2. 3. 4. A set of specs for the digital (discrete-time) filter is given. We transform the specs from the D. T. to C. T. (z s) Design a C. T. IIR filter : s z Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 5

Why Transform to Analog IIR Filter Design? The art of CT IIR filter design is highly advanced. Many CT IIR methods have relatively closed-form design formulas. Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 6

Desirable Properties of Transformations 1) Causal/stable analog filter should be transformed to a causal stable DT filter. Causal and stable Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 7

Desirable Properties of Transformations 2) jΏ axis in the s-plane (CT) needs to be transformed to the unit circle in the z-domain. * Needed to translate the specs from discrete to analog domain 3) Rational transfer function in the s-domain should be transformed into a rational transfer function in the zdomain. Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 8

Bilinear Transformation Proposal: T: arbitrary parameter • Does this transformation satisfy the desirable properties that we just discussed? • Does a rational analog filter lead to a rational digital filter? A rational analog filter translated into a rational digital filter. Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 9

Properties of Bilinear Transformation • Is jΏ axis in the s-plane (CT) transformed to the unit circle in the z-domain? Let: Mehrdad Fatourechi, Electrical and Computer Engineering, University of British Columbia, Summer 2011 10

Properties of Bilinear Transformation • Does a causal and stable analog filter lead to a causal and stable digital filter? • We need to show that LHP in the s-domain is mapped into inside the unit circle in the z-domain. 11

Example of Bilinear Transformation • Using bilinear transformation, design a low-pass IIR filter that satisfies the above spec. Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 12

Steps to follows 1. 2. 3. 4. A set of specs for the digital (discrete-time) filter is given. We transform the specs from the D. T. to C. T. (z s) Design a C. T. IIR filter : s z Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 13

Translating the Specifications 14

Filter Specifications in C. T. Domain Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 15

C. T. Low-pass IIR Filters Butterworth filter: Monotonic in stopband passband Cut-off frequency Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 16

Butterworth Filters Poles of : Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 17

Butterworth Filters Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 18

Chebyshev and Elliptic filters Chebyshev: ripple in either pass-band or stop-band Elliptic: ripple in both pass-band or stop-band See Appendix B in the textbook for related formulae. Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 19

Filter Specifications in C. T. Domain - Which one would you choose? Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 20

Example- Continued • Butterworth filter of order N: Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 21

Example of Bilinear Transformation 22

Example of Bilinear Transformation Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 23

Frequency response of the filter Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 24

Group delay of the filter Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 25

Another example • Using bilinear transformation, design a low-pass IIR filter that satisfies the above spec. Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 26

Translating the Specifications Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 27

Filter Specifications in C. T. Domain Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 28

Example- Continued • Butterworth filter of order N: Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 29

Example of Bilinear Transformation Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 30

Example of Bilinear Transformation Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 31

Frequency Transformation of Low-pass IIR Filters Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 32

Design of High-pass digital IIR Filters (Approach 1 - Design in the Analog Domain) 1. 2. 3. 4. 5. 6. Start with spec. in D. T. For HPF. Translate the filter specs from D. T. to C. T. specs of a HPF in C. T. (using bilinear transform. ) Translate specs of C. T. HPF to C. T. LPF Design the LPF (Butterworth) Transform C. T. LPF C. T. HPF Transform C. T. HPF D. T. HPF (using bilinear transform. ) C. T. LPF Ha(s 1) C/C C/D C. T. HPF Ha’(s 2) D. T. HPF Hd(z) Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 33

Transformation of LPF to HPF Proposal: k: positive constant Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 34

Checking the Desirable Properties Is jΏ 1 axis in the s 1 -plane (CT) transformed to the jΏ 2 axis in the s 2 -plane (CT)? Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 35

Checking the Desirable Properties • Does a causal and stable analog LPF filter lead to a causal and stable analog HPF? • Does LHP in the s 1 -domain (CT) map into the LHP in the s 2 -domain (CT)? • It is easy to prove the above statement. • It is also easy to show that a rational transfer function is mapped into another rational transfer function. Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 36

Translating the Specs Frequency response is symmetric. Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 37

Translating the Specs Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 38

Frequency Transformations for Analog Filters *Assumption: prototype lowpass filter has band edge frequency Type of Transformation Band edge frequencies of the new filter Lowpass Highpass Bandstop Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 39

Design of High-pass digital IIR Filters (Approach 1 - Design in the Analog Domain) 1. 2. 3. 4. 5. 6. Start with spec. in D. T. For HPF. Translate the filter specs from D. T. to C. T. specs of a HPF in C. T. (bilinear transformation) Translate specs of C. T. HPF to C. T. LPF Design the LPF (Butterworth) Transform C. T. LPF C. T. HPF Transform C. T. HPF D. T. HPF (bilinear transformation) C. T. LPF Ha(s 1) C/C C/D C. T. HPF Ha’(s 2) D. T. HPF Hd(z) Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 40

Design of High-pass digital IIR Filters (Approach 2 - Design in the Digital Domain) 1. 2. Start with spec. in D. T. For HPF. Translate the filter specs from D. T. HPF to D. T. LPF (using the transformation discussed shortly) 3. 4. Design the LPF : (a) translate the DT LPF specs CT LPF specs; (b) Design CT LPF; (c) Transform CT LPF to DT LPF. Transform D. T. LPF D. T. HPF C. T. LPF Ha(s) C/D D. T. LPF H(z 1) D. T. HPF Hd(z 2) Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 41

D/D Transformation Proposal: It can be shown that this transformation has the 3 properties that we usually investigate for transformations: 1 - A rational transfer function is transformed to a rational transfer function. 2 - Unit circle in one domain is mapped into the unit circle in the other domain. 3 - Inside of the unit circle in one domain is mapped to the inside of the unit circle in the other domain. Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 42

D/D Transformation Proposal: Proof of the second property: Unit circle in one domain is mapped into the unit circle in the other domain. Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 43

D/D Transformation Proposal: Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 44

Design of High-pass digital IIR Filters (Approach 2 - Design in the Digital Domain) 1. 2. Start with spec. in D. T. For HPF. Translate the filter specs from D. T. HPF to D. T. LPF (using the transformation discussed earlier) 3. 4. Design the LPF : (a) translate the DT LPF specs CT LPF specs; (b) Design CT LPF; (c) Transform CT LPF to DT LPF. Transform D. T. LPF D. T. HPF C. T. LPF Ha(s) C/D D. T. LPF H(z 1) D. T. HPF Hd(z 2) Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 45

Frequency Transformations for Digital Filters Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 46

Frequency Transformations for Digital Filters Mehrdad Fatourechi, Electrical and Computer Engineering, University of British Columbia, Summer 2011 47

Example Suppose we have designed a filter that has met the following specs: We have designed a Chebyshev lowpass filter with the following system function: Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 48

Example (cont. ) Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 49

Example (cont. ) To transfer this filter to a highpass filter with passband edge frequency of : This results in the following high-pass filter: Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 50

Example (cont. ) Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 51

Summary IIR filters generally have lower order compared to FIR filters, however, linear-phase cannot be guaranteed. The most popular technique is the transformation technique, although other methods such as pole-zero placement also exist. Using transformation techniques, a low-pass prototype filter can be transformed into HP, BP and BS filters. Hossein Sameti, Dept. of Computer Eng. , Sharif University of Technology 52