Chapter 7 Filter Design Techniques. Filter Design Techniques

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1 Chapter 7 Filter Design Techniques Page 1

2 Outline 7.0 Introduction 7.1 Design of Discrete Time IIR Filters 7.2 Design of FIR Filters Page 2

3 7.0 Introduction Definition of Filter Filter is a system that passes certain frequency components and totally rejects all others, but in a broader context any system that modifies certain frequencies relative to others is called a filter. Page 3

4 The Design of Filter 1 The specification of the desired properties of the system. 2 The approximation of the specification using a causal discrete-time system. 3 The realization of the system. In this chapter, we focus on the second step. Page 4

5 The relationship between specifications of the discrete-time filter and the effective continuous-time filter When a discrete-time filter is to be used for discrete-time processing of continuous-time filter and the effective continuous-time filter are typically given in the frequency domain. Page 5

6 If a effective continuous-time system has the frequency response. Basic system for discrete-time filtering of continuous-time signals. Page 6

7 In such cases, it is straightforward to convert from specifications on the effective continuous-time filter through the relation ω = ΩT.That is, H(e jω ) is specified over one period by the equation : Page 7

8 Example Consider a discrete-time filter that is to be used to lowpass filter a continuous-time signal using the basic configuration. Specifically, we want the overall system to have the following properties when the sampling rate is 10 4 samples/s (T=10-4 s) : (1) The gain H eff (jω) should be within 0.01 (0.086dB) of unity (zero db) in the frequency band 0 Ω 2π(2000). Page 8

9 (2) The gain should be no greater than (-60dB) in the frequency band 2π(3000) Ω Such a set of lowpasss pecifications on H eff (jω) can be depicted where the limits of tolerable approximation error are indicated by the shaded horizontal lines. Page 9

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13 7.1 Design of Discrete-time IIR Filters form Continuous-Time Filters The Transformation of a continuous-time filter into a discrete-time filter meeting prescribed specifications. The Reasons for Using this Method: - The art of continuous-time IIR filter design has developed and many results can be used. - Many continuous-time IIR filter design methods have relatively simple closed form design formulas, therefore it is easy to carry out. Page 13

14 - The standard approximation methods for continuous-time IIR filters can not be directly used in discrete-time filter design. 3. Processes of design: - Specifications transformation; - Continuous-time filter design; - Mapping continuous-time filter into discrete-time filter (From s-plane to z- plane). Page 14

15 7.1.1 Filter Design by Impulse Invariance If h c (t) is the impulse response of continuoustime filter, and h c (nt d ) is equally spaced samples of it. The frequency response : Page 15

16 If the continuous-time filter is bandlimited, so that then Page 16

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18 Example Assume that the specifications for the designed discrete-time filter are shown in next slide with,δ 1 = , δ 2 = , ω p = 0.2π and ω s = 0.3π. Τhe maximum gain in stopband is -15dB (20log ), The maximum deviation of 1dB below 0dB gain in passband (20log 10 (1) 20log 10 ( ) =-1 db). In this case the band pass tolerance is between 1- δ 1 and 1. Page 18

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21 The impulse invariance transformation from CT to DT : Page 21

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23 Example Consider the design of a lowpass discretetime filter by applying impulse invariance to an appropriate Butterworth continuous-time filter. The specifications for the discrete-time filter are : Page 23

24 Choose T d =1 so that ω=ω Continuous-time Butterworth filter with magnitude function H c (jω) Page 24

25 Let Ω p = 0.2π and Ω s = 0.3π The magnitude squared function of a Btterworth filter Page 25

26 So that the filter design process consists of determining the parameters N and Ω c to meet the desired specification. Page 26

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28 Since N must be integer N=6 substuting N=6 in equation slide 26. We have Ω c = Page 28

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31 Find the poles : Page 31

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38 7.1.2 Bilinear Transformation In order to avoid the aliasing in impulse invariance, we introduce another method of transformation bilinear transformation, which use an algebraic transform between the variables s and z. This transform is Page 38

39 In the transformation, - Ω maps onto -π ω π,the transformation between the continuous-time and discretetime frequency variables must be nonlinear. Therefore the use of this technique is restricted to the situation where the corresponding warping of the frequency axis is acceptable. Page 39

40 To develop the properties of the algebraic transformationwe solve for z to obtain : Substituting s = σ+jω, we obtain : Page 40

41 If σ<0 then z <1 for any value of Ω. Similarly, If σ>0 then z >1for all Ω.That is if a pole of H c (s) is in the left half s-plane, its image in the z-plane will be inside the unit circle. Therefore causal stable continuous-time filters map into causal stable discrete-time filters. Page 41

42 To show that the jω-axis of the s-plane maps onto the unit circle, we substitute s=jω : It is clear that z =1 for all value of s on the jω-axis Page 42

43 To derive the relationship between the variables ω and Ω, we substituting z= e jω. or Page 43

44 Equating real and imaginary parts on both sides leads to the relations σ=0 or Page 44

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46 The bilinear transformation avoids the problem of aliasing encountered with the use of impulse invariance because it maps the entire imaginary axis of the s-plane onto the unit circle in the z-plane. The price paid for this, however, is the nonlinear compression the frequency axis. Page 46

47 If we transform a lowpass filter from continuous-time form into discrete-time form, the warping of bilinear transformation can be demonstrated in next slide. Page 47

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50 If the critical frequencies (such as the passband and stopband edge frequencies) of continuous-time filter are prewaped according the equation Page 50

51 then when the continuous-time filter is transformed to the discrete-time filter the discrete-time filter will meet the desired specifications. Page 51

52 Example Consider the specification on the discretetime filter : Using the bilinear transformation, the critical frequencies of the discrete-time filter must be prewarped to the corresponding continuous-time frequency Page 52

53 For convenience we choose T d =1, since Butterworth filter has a monotonic magnitude response, so from above equations we obtain : Page 53

54 The form of the magnitude-squared function for the Butterworth filter is : Solving for N and Ω c, we obtain : and Page 54

55 N = log[ 1/ / 1/ ] 2log[tan 0.15 /tan 0.1 ] The result are N = , and take N=6, substituting N = 6 and Ω c = For this value of Ω c, the passband specifications are exceeded and the stopband specifications are met exactly. Page 55

56 In the s-plane, the 12 poles are uniformly distributed in angle on a circle of radius Page 56

57 The system function of the continuoustime filter by selecting the left-plane poles is Page 57

58 The magnitude, log magnitude, and group delay of the frequency response of the discrete-time filter are shown in next slides At ω=0.2π, the log magnitude is -0.56dB, and at ω=0.3π, log magnitude is exactly -15dB. Page 58

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62 From above example, we know Nth-order Butterworth filter has the following form Page 62

63 Homework We wish to design a lowpass digital filter to meet the specifications : δ 1 = 0.01, δ 2 = 0.001, ω p = 0.4π and ω s = 0.6π. Page 63

64 1. Butterworth filter design by impulse invariance. 2. Butterworth filter design by bilinear transformation 3. Chebyshev filter design by bilinear transformation Page 64

Digital Signal Processing

COMP ENG 4TL4: Digital Signal Processing Notes for Lecture #25 Wednesday, November 5, 23 Aliasing in the impulse invariance method: The impulse invariance method is only suitable for filters with a bandlimited