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
10 Page 10 Digital Signal Processing
11 Page 11 Digital Signal Processing
12 Page 12 Digital Signal Processing
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
17 Page 17 Digital Signal Processing
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
19 Page 19 Digital Signal Processing
20 Page 20 Digital Signal Processing
21 The impulse invariance transformation from CT to DT : Page 21
22 Page 22 Digital Signal Processing
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
27 Page 27
28 Since N must be integer N=6 substuting N=6 in equation slide 26. We have Ω c = Page 28
29 Page 29 Digital Signal Processing
30 Page 30 Digital Signal Processing
31 Find the poles : Page 31
32 Page 32
33 Page 33 Digital Signal Processing
34 Page 34 Digital Signal Processing
35 Page 35 Digital Signal Processing
36 Page 36 Digital Signal Processing
37 Page 37 Digital Signal Processing
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
45 Page 45 Digital Signal Processing
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
48 Page 48
49 Page 49 Digital Signal Processing
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
59 Page 59 Digital Signal Processing
60 Page 60 Digital Signal Processing
61 Page 61 Digital Signal Processing
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
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