Zhongguo Liu_Biomedical Engineering_Shandong Univ. Biomedical Signal processing Chapter 7 Filter Design Techniques Zhongguo Liu Biomedical Engineering School of Control Science and Engineering, Shandong University

2 Chapter 7 Filter Design Techniques  7.0 Introduction  7.1 Design of Discrete-Time IIR Filters From Continuous-Time Filters  7.2 Design of FIR Filters by Windowing  7.3 Examples of FIR Filters Design by the Kaiser Window Method  7.4 Optimum Approximations of FIR Filters  7.5 Examples of FIR Equiripple Approximation  7.6 Comments on IIR and FIR Discrete- Time Filters

3 Filter Design Techniques 7.0 Introduction

4  Frequency-selective filters pass only certain frequencies  Any discrete-time system that modifies certain frequencies is called a filter.  We concetrate on design of causal Frequency-selective filters

5 Stages of Filter Design  The specification of the desired properties of the system.  The approximation of the specifications using a causal discrete-time system.  The realization of the system.  Our focus is on second step  Specifications are typically given in the frequency domain.

6 Frequency-Selective Filters  Ideal lowpass filter

7 Frequency-Selective Filters  Ideal highpass filter

8 Frequency-Selective Filters  Ideal bandpass filter

9 Frequency-Selective Filters  Ideal bandstop filter

10  If input is bandlimited and sampling frequency is high enough to avoid aliasing, then overall system behave as a continuous-time system: Linear time-invariant discrete-time system continuous-time specifications are converted to discrete time specifications by:

11 Example 7.1 Determining Specifications for a Discrete-Time Filter  Specifications of the continuous-time filter:  1. passband  2. stopband

12 Example 7.1 Determining Specifications for a Discrete-Time Filter  Specifications of the continuous-time filter:  1. passband  2. stopband

13 Example 7.1 Determining Specifications for a Discrete-Time Filter Specifications of the discrete-time filter in

14 Filter Design Constraints  Designing IIR filters is to find the approximation by a rational function of z.  The poles of the system function must lie inside the unit circle(stability, causality).  Designing FIR filters is to find the polynomial approximation.  FIR filters are often required to be linear- phase.

15 Filter Design Techniques 7.1 Design of Discrete-Time IIR Filters From Continuous-Time Filters

Design of Discrete-Time IIR Filters From Continuous-Time Filters  The traditional approach to the design of discrete-time IIR filters involves the transformation of a continuous-time filter into a discrete filter meeting prescribed specification.

17 Three Reasons 1.The art of continuous-time IIR filter design is highly advanced, and since useful results can be achieved, it is advantageous to use the design procedures already developed for continuous-time filters.

18 Three Reasons 2.Many useful continuous-time IIR design method have relatively simple closed form design formulas. Therefore, discrete-time IIR filter design methods based on such standard continuous-time design formulas are rather simple to carry out.

19 Three Reasons 3.The standard approximation methods that work well for continuous-time IIR filters do not lead to simple closed-form design formulas when these methods are applied directly to the discrete-time IIR case.

20 Steps of DT filter design by transforming a prototype continuous-time filter  The specifications for the continuous- time filter are obtained by a transformation of the specifications for the desired discrete-time filter.  Find the system function of the continuous-time filter.  Transform the continuous-time filter to derive the system function of the discrete-time filter.

21 Constraints of Transformation  to preserve the essential properties of the frequency response, the imaginary axis of the s-plane is mapped onto the unit circle of the z-plane.

22 Constraints of Transformation  In order to preserve the property of stability, If the continuous system has poles only in the let half of the s-plane, then the discrete-time filter must have poles only inside the unit circle.

Filter Design by Impulse Invariance  The impulse response of discrete-time system is defined by sampling the impulse response of a continuous-time system. Relationship of frequencies

24 relation between frequencies S plane Z plane - Relationship of frequencies

25 Aliasing in the Impulse Invariance

26 periodic sampling T ： sample period; fs=1/T:sample rate Ωs=2π/T:sample rate Review

27 Time domain ： Complex frequency domain ： Laplace transform Relation between Laplace Transform and Z-transform Review

28 Fourier Transform frequency domain ： Laplace transform Fourier Transform is the Laplace transform when s have the value only in imaginary axis, s=jΩ Since So

29 For discrete-time signal ， 令： z-transform of discrete- time signal Laplace transform the Laplace transform

30 so ：  Laplace transform Laplace transform continuous time signal z-transform z-transform discrete-time signal let ：

31 DTFT : Discrete Time Fourier Transform S plane Z plane -

32 plane

33  If input is bandlimited and f s >2f max, : discrete-time filter design by impulse invariance

34 relation between frequencies S plane Z plane - Relationship of frequencies

35 periodic sampling T ： sample period; fs=1/T:sample rate Ωs=2π/T:sample rate Review

36 proof of T ： sample period; fs=1/T:sample rate;Ωs=2π/T:sample rate Review s(t) 为冲击串序列，周期为 T ，可展开傅立叶级数

37 periodic sampling

38 discrete-time filter design by impulse invariance

39 Steps of DT filter design by transforming a prototype continuous-time filter  Obtain the specifications for continuous- time filter by transforming the specifications for the desired discrete-time filter.  Find the system function of the continuous- time filter.  Transform the continuous-time filter to derive the system function of the discrete- time filter.

40 Transformation from discrete to continuous  In the impulse invariance design procedure, the transformation is  Assuming the aliasing involved in the transformation is neglected, the relationship of transformation is

41 Steps of DT filter design by transforming a prototype continuous-time filter  Obtain the specifications for continuous- time filter by transforming the specifications for the desired discrete-time filter.  Find the system function of the continuous- time filter.  Transform the continuous-time filter to derive the system function of the discrete- time filter.

42 Continuous-time IIR filters  Butterworth filters  Chebyshev Type I filters  Chebyshev Type II filters  Elliptic filters

43 Steps of DT filter design by transforming a prototype continuous-time filter  Obtain the specifications for continuous- time filter by transforming the specifications for the desired discrete-time filter.  Find the system function of the continuous- time filter.  Transform the continuous-time filter to derive the system function of the discrete- time filter.

44 Transformation from continuous to discrete

45 Example 7.2 Impulse Invariance with a Butterworth Filter  Specifications for the discrete-time filter:  Assume the effect of aliasing is negligible

46 Example 7.2 Impulse Invariance with a Butterworth Filter

47 Example 7.2 Impulse Invariance with a Butterworth Filter

48 Example 7.2 Impulse Invariance with a Butterworth Filter Plole pairs:

49 Example 7.2 Impulse Invariance with a Butterworth Filter Plole pairs:

50 Example 7.2 Impulse Invariance with a Butterworth Filter

51 Basic for Impulse Invariance  To chose an impulse response for the discrete-time filter that is similar in some sense to the impulse response of the continuous-time filter.  If the continuous-time filter is bandlimited, then the discrete-time filter frequency response will closely approximate the continuous-time frequency response.  The relationship between continuous-time and discrete-time frequency is linear; consequently, except for aliasing, the shape of the frequency response is preserved.

Bilinear Transformation  Bilinear transformation can avoid the problem of aliasing.  Bilinear transformation maps onto  Bilinear transformation:

Bilinear Transformation

Bilinear Transformation

Bilinear Transformation

56 relation between frequency response of H c (s), H(z)

57 Comments on the Bilinear Transformation  It avoids the problem of aliasing encountered with the use of impulse invariance.  It is nonlinear compression of frequency axis. S plane Z plane -

58 Comments on the Bilinear Transformation  The design of discrete-time filters using bilinear transformation is useful only when this compression can be tolerated or compensated for, as the case of filters that approximate ideal piecewise-constant magnitude-response characteristics.

59 Bilinear Transformation of

60 Comparisons of Impulse Invariance and Bilinear Transformation  The use of bilinear transformation is restricted to the design of approximations to filters with piecewise-constant frequency magnitude characteristics, such as highpass, lowpass and bandpass filters.  Impulse invariance can also design lowpass filters. However, it cannot be used to design highpass filters because they are not bandlimited.

61 Comparisons of Impulse Invariance and Bilinear Transformation  Bilinear transformation cannot design filter whose magnitude response isn’t piecewise constant, such as differentiator. However, Impulse invariance can design an bandlimited differentiator.

62  Butterworth Filter,  Chebyshev Approximation,  Elliptic Approximation Example of Bilinear Transformation

63 Example 7.3 Bilinear Transformation of a Butterworth Filter

64 Example 7.3 Bilinear Transformation of a Butterworth Filter

65 Locations of Poles Plole pairs:

66 Example 7.3 Bilinear Transformation of a Butterworth Filter Plole pairs:

67 Ex. 7.3 frequency response of discrete-time filter

68 Example 7.4 Butterworth Approximation (Hw)

69 Example 7.4 frequency response

70 Chebyshev filters C Chebyshev filter (type I) 1 Chebyshev polynomial Chebyshev filter (type II) 1

71 Example 7.5 Chebyshev Type I, II Approximation Type I Type II

72 Example 7.5 frequency response of Chebyshev Type I Type II

73 E elliptic filters Elliptic filter 1 Jacobian elliptic function

74 Example 7.6 Elliptic Approximation

75 Example 7.6 frequency response of Elliptic

76 *Comparison of Butterworth, Chebyshev, elliptic filters: Example -Given specification -Order Butterworth Filter : N=14. ( max flat) Chebyshev Filter : N=8. ( Cheby 1, Cheby 2) Elliptic Filter : N=6 ( equiripple) B C E

77 -Pole-zero plot (analog) -Pole-zero plot (digital) BC1C2E BC1C2E (14)(8)

78 -Magnitude -Group delay B C1 C2 E B C1 C2 E

Design of FIR Filters by Windowing  FIR filters are designed based on directly approximating the desired frequency response of the discrete- time system.  Most techniques for approximating the magnitude response of an FIR system assume a linear phase constraint.

80 Window Method  An ideal desired frequency response  Many idealized systems are defined by piecewise-constant frequency response with discontinuities at the boundaries. As a result, these systems have impulse responses that are noncausal and infinitely long.

81 Window Method  The most straightforward approach to obtaining a causal FIR approximation is to truncate the ideal impulse response.

82 Windowing in Frequency Domain  Windowed frequency response  The windowed version is smeared version of desired response

83 Window Method  If

84 Choice of Window  is as short as possible in duration. This minimizes computation in the implementation of the filter.  approximates an impulse.

85 Window Method  then would look like, except where changes very abruptly.  If is chosen so that is concentrated in a narrow band of frequencies around

86 Rectangular Window  for the rectangular window has a generalized linear phase.  As M increases, the width of the “main lobe” decreases.  While the width of each lobe decreases with M, the peak amplitudes of the main lobe and the side lobes grow such that the area under each lobe is a constant.

87 Rectangular Window  will oscillate at the discontinuity.  The oscillations occur more rapidly, but do not decrease in magnitude as M increases.  The Gibbs phenomenon can be moderated through the use of a less abrupt truncation of the Fourier series.

88 Rectangular Window  By tapering the window smoothly to zero at each end, the height of the side lobes can be diminished.  The expense is a wider main lobe and thus a wider transition at the discontinuity.

Design of FIR Filters by Windowing Method  To design an ilowpass FIR Filters Review

Properties of Commonly Used Windows  Rectangular  Bartlett (triangular)

Properties of Commonly Used Windows  Hanning  Hamming

Properties of Commonly Used Windows  Blackman

Properties of Commonly Used Windows

94 Frequency Spectrum of Windows (a) Rectangular, (b) Bartlett, (c) Hanning, (d) Hamming, (e) Blackman, (M=50) (a)-(e) attenuation of sidelobe increases, width of mainlobe increases.

Properties of Commonly Used Windows biggest ， high oscillations at discontinuity smallest ， the sharpest transition Table 7.1

Incorporation of Generalized Linear Phase  In designing FIR filters, it is desirable to obtain causal systems with a generalized linear phase response.  The above five windows are all symmetric about the point,i.e.,

Incorporation of Generalized Linear Phase  Their Fourier transforms are of the form

Incorporation of Generalized Linear Phase

99 Frequency Domain Representation

100 Example 7.7 Linear-Phase Lowpass Filter  The desired frequency response is

101 magnitude frequency response

Properties of Commonly Used Windows smallest ， the sharpest transition biggest ， high oscillations at discontinuity

The Kaiser Window Filter Design Method Trade side-lobe amplitude for main-lobe width

104 Figure 7.24 As  increases, attenuation of sidelobe increases, width of mainlobe increases. As M increases, attenuation of sidelobe is preserved, width of mainlobe decreases. M=20 (a) Window shape, M=20, (b) Frequency spectrum, M=20, (c) beta=6  =6

105 Table 7.1 Transition width is a little less than mainlobe width

106  Increasing M wile holding  constant causes the main lobe to decrease in width, but does not affect the amplitude of the side lobe. Comparison  If the window is tapered more, the side lobe of the Fourier transform become smaller, but the main lobe become wider. M=20  =6 M=20

107 Filter Design by Kaiser Window

108 Filter Design by Kaiser Window M=20

109 Example 7.8 Kaiser Window Design of a Lowpass Filter

110 Example 7.8 Kaiser Window Design of a Lowpass Filter

111 Example 7.8 Kaiser Window Design of a Lowpass Filter

112 Ex. 7.8 Kaiser Window Design of a Lowpass Filter

Examples of FIR Filters Design by the Kaiser Window Method  The ideal highpass filter with generalized linear phase

114 Example 7.9 Kaiser Window Design of a Highpass Filter  Specifications:  By Kaiser window method

115 Example 7.9 Kaiser Window Design of a Highpass Filter  Specifications:  By Kaiser window method

Discrete-Time Differentiator

117 Example 7.10 Kaiser Window Design of a Differentiator  Since kaiser’s formulas were developed for frequency responses with simple magnitude discontinuities, it is not straightforward to apply them to differentiators.  Suppose

118 Group Delay  Phase:  Group Delay:

119 Group Delay  Phase:  Group Delay:  Noninteger delay

Optimum Approximations of FIR Filters  Goal: Design a ‘best’ filter for a given M  In designing a causal type I linear phase FIR filter, it is convenient first to consider the design of a zero phase filter.  Then insert a delay sufficient to make it causal.

Optimum Approximations of FIR Filters

Optimum Approximations of FIR Filters  Designing a filter to meet these specifications is to find the (L+1) impulse response values  In Packs-McClellan algorithm, is fixed, and is variable.  Packs-McClellan algorithm is the dominant method for optimum design of FIR filters.

Optimum Approximations of FIR Filters

Optimum Approximations of FIR Filters

Optimum Approximations of FIR Filters

126 Minimax criterion  Within the frequency interval of the passband and stopband, we seek a frequency response that minimizes the maximum weighted approximation error of

127 Other criterions

128  Let denote the closet subset consisting of the disjoint union of closed subsets of the real axis x. Alternation Theorem  is an r th-order polynomial.  denotes a given desired function of x that is continuous on  is a positive function, continuous on  The weighted error is  The maximum error is defined as

129 Alternation Theorem  A necessary and sufficient condition that be the unique rth-order polynomial that minimizes is that exhibit at least (r+2) alternations; i.e., there must exist at least (r+2) values in such that  and such that for

130 Example 7.11 Alternation Theorem and Polynomials  Each of these polynomials is of fifth order.  The closed subsets of the real axis x referred to in the theorem are the regions

Optimal Type I Lowpass Filters  For Type I lowpass filter  The desired lowpass frequency response  Weighting function

Optimal Type I Lowpass Filters  The weighted approximation error is  The closed subset is or

Optimal Type I Lowpass Filters  The alternation theorem states that a set of coefficients will correspond to the filter representing the unique best approximation to the ideal lowpass filter with the ratio fixed at K and with passband and stopband edge and if and only if exhibits at least (L+2) alternations on, i.e., if and only if alternately equals plus and minus its maximum value at least (L+2) times.  Such approximations are called equiripple approximations.

Optimal Type I Lowpass Filters  The alternation theorem states that the optimum filter must have a minimum of (L+2) alternations, but does not exclude the possibility of more than (L+2) alternations.  In fact, for a lowpass filter, the maximum possible number of alternations is (L+3).

Optimal Type I Lowpass Filters  Because all of the filters satisfy the alternation theorem for L=7 and for the same value of, it follows that and/or must be different for each,since the alternation theorem states that the optimum filter under the conditions of the theorem is unique.

136 Property for type I lowpass filters from the alternation theorem  The maximum possible number of alternations of the error is (L+3)  Alternations will always occur at and  All points with zero slop inside the passband and all points with zero slop inside stopband will correspond to alternations; i.e., the filter will be equiripple, except possibly at and

Optimal Type II Lowpass Filters  For Type II causal FIR filter:  The filter length (M+1) is even, ie, M is odd  Impulse response is symmetric  The frequency response is

Optimal Type II Lowpass Filters

Optimal Type II Lowpass Filters  For Type II lowpass filter,

The Park-McClellan Algorithm  From the alternation theorem, the optimum filter will satisfy the set of equation

The Park-McClellan Algorithm  Guessing a set of alternation frequencies and

The Park-McClellan Algorithm

The Park-McClellan Algorithm  For equiripple lowpass approximation  Filter length: (M+1)

Examples of FIR Equiripple Approximation Lowpass Filter

145 Comments  M=26, Type I filter  The minimum number of alternations is (L+2)=(M/2+2)=15  7 alternations in passband and 8 alternations in stopband  The maximum error in passband and stopband are and , which exceed the specifications.

Lowpass Filter  M=27,, Type II filter, zero at z=-1  The maximum error in passband and stopband are and , which exceed the specifications.  The minimum number of alternations is (L+2)=(M-1)/2+2=15  7 alternations in passband and 8 alternations in stopband

147 Comparison  Kaiser window method require M=38 to meet or exceed the specifications.  Park-McClellan method require M=27  Window method produce approximately equal maximum error in passband and stopband.  Park-McClellan method can weight the error differently.

Comments on IIR and FIR Discrete-Time Filters  What type of system is best, IIR or FIR?  Why give so many different design methods?  Which method yields the best result?

Comments on IIR and FIR Discrete-Time Filters Closed- Form Formulas Generalized Linear Phase Order IIRYesNoLow FIRNoYesHigh

Properties of Commonly Used Windows  Their Fourier transforms are concentrated around  They have a simple functional form that allows them to be computed easily.  The Fourier transform of the Bartlett window can be expressed as a product of Fourier transforms of rectangular windows.  The Fourier transforms of the other windows can be expressed as sums of frequency-shifted Fourier transforms of rectangular windows.(Problem7.34)

151 Homework  Simulate the frequency response (magnitude and phase) for Rectangular, Bartlett, Hanning, Hamming, and Blackman window with M=21 and M=51

Zhongguo Liu_Biomedical Engineering_Shandong Univ. Chapter 5 HW  7.2, 7.4, 7.15, 上一页下一页 返 回