数字信号处理英文版课件：Chap 10 FIR Digital Filter Design

1、DSP group 2007 chap10-ed11 Chapter 10 FIR Digital Filter DesignBasic approaches to FIR Digital filter designFIR Digital filter design based on Windowed Fourier Series (*)Frequency sampling approachComputer-aided Design of FIR digital filtersDSP group 2007 chap10-ed12 IntroductionCharacteristics of F

2、IR digital filter: always stable always possible to design FIR filters with exact linear-phase. 10.1.1 Basic Approaches to FIR Digital Filter Design 10.1 Preliminary ConsiderationsDSP group 2007 chap10-ed13 10.1.1 Basic Approaches to FIR Digital Filter Design Design of FIR filter: Find either the im

3、pulse response hn (0 n N) , or N+1 samples of its frequency response DTFT H(e j). For a linear phase FIR digital filter: Windowed Fourier series approach;Frequency sampling approach. DSP group 2007 chap10-ed14Kaisers Formula: Bellangers Formula Hermanns Formula 10.1.2 Estimation of Lowpass FIR Filte

4、r OrderLowpass FIR digital filter: ppeak passband ripple;speak stopband ripple;ppassband edge frequency;sstopband edge frequency. See page 398.DSP group 2007 chap10-ed1510.2.1 Least Integral-Squared Error Design 10.2 FIR Filter Design Based on Windowed Fourier SeriesHd(ej): is the desired frequency

5、response (usually it is piecewise constant ).hdn: is the desired impulse response (usually it is of infinite length and noncausal).DSP group 2007 chap10-ed16 10.2.1 Least Integral-Squared Error Design of FIR Filters H(ej): is the frequency response of FIR filter to be designed, the DTFT of hn.hn: is

6、 the impulse response of FIR filter to be designed, it is of finite length and causal. R: is the integral-squared error:DSP group 2007 chap10-ed17 10.2.1 Least Integral-Squared Error Design of FIR Filters Using Parsevals relation:The integral-squared error is minimum when truncation of the desired i

7、mpulse response.Linear-phase hn= hNnDSP group 2007 chap10-ed18 10.2.2 Impulse response of ideal filters Linear-phase FIR digital filters Lowpass digital filterscc|HLP( e j )|10 ()DSP group 2007 chap10-ed19 Impulse response of ideal filters (10.2.2) Highpass digital filterscc|HHP( e j )|10DSP group 2

8、007 chap10-ed110 Bandpass digital filter 10.2.2 Impulse response of ideal filters|HBP( e j )|10c2c1c1c2 Bandstop digital filter |HBS( e j )|10c2c1c1c2DSP group 2007 chap10-ed111 Truncation 10.2.3 Gibbs Phenomenon reasonwhereDTFTmainlobesidelobeNN Mainlobe width12We (e j() = c 10.2.3 Gibbs Phenomenon

9、 reason Convolution theoremWe (e j)2/(N+1)ccWe(e j() c1314 Windowing effectsDSP group 2007 chap10-ed115 10.2.3 Gibbs PhenomenonOscillatory behavior in the magnitude responseDSP group 2007 chap10-ed116 For W(e j), N m , sidelobe ; But the area under each lobe remains constant. (2) For the integral os

10、cillation will occur at each sidelobe of W(e j() moves past the discontinuity. 10.2.3 Gibbs Phenomenon Explanation With N increasing, ripples in H(e j) around the point of discontinuity occur more closely but with no decrease in amplitude. Gibbs phenomenonDSP group 2007 chap10-ed117 Methods to reduc

11、e Gibbs phenomenon: 10.2.3 Gibbs Phenomenon ExplanationUsing a window that tapers smoothly to zero at each end, but m providing a smooth transition from the passband to the stopband in magnitude specifications The height of sidelobes diminish, but m DSP group 2007 chap10-ed118Hanning window: A= B =0

12、.5, C=0; (Hann)Hamming window: A=0.54, B = 0.46, C=0Blackman window: A=0.42, B = 0.5, C = 0.08. Rectangular : wn= un un (N + 1) Bartlett (triangular) 10.2.4 Fixed Window FunctionsDSP group 2007 chap10-ed119 10.2.4 Fixed Window FunctionsP469 Fig. 10.6 Commonly used fixed windowsBartlett BlackmanHanni

13、ng Hamming Rectangular N/2N n wn 102050N 10.2.4 Fixed Window FunctionsP. 470Fig. 10.721 Incorporation of Linear PhaseAll windows discussed above are symmetricthat is As a result, their Fourier transforms are of the formWe(ej) a real even function of . If the desired impulse response is also symmetri

14、c or antisymmetric,DSP group 2007 chap10-ed122 10.2.4 Fixed Window FunctionsSame ripples in passband and stopbandwidth of transition bandDSP group 2007 chap10-ed123Parameters predicting the performance of a FIR filter 10.2.4 Fixed Window Functions Transition bandwidth peak ripple of passband and sto

15、pband mainlobe width ML . relative sidelobe level Asl (dB). They are two contradictory requirements.Type of windowRelative Sidelobe LevelMain-lobe widthMinimum Stopband AttenuationTransition Bandwidth Rect.13.3dB4/(N+1)20.9dB1.84 /NBartlett26.5dB4 /NHanning31.5dB8 /N43.9dB6.22 /NHamming42.7dB8 /N54.

16、5dB6.64 / NBlackman58.1dB12 /N75.3dB11.12 / N 10.2.4 Fixed Window Functions Table 10.2Replace P. 471DSP group 2007 chap10-ed125(2) The attenuation of the stopband should be more than 40dB. 10.2.4 Fixed Window FunctionsExample 10.6Design an FIR lowpass digital filter with specifications :Solution:(1)

17、 With the attenuation of the stopband, we could select Hanning、Hamming、Blackman window.According to Table 10.2,DSP group 2007 chap10-ed126 10.2.4 Fixed Window FunctionsHere, we use Hanning (for example)(2) With the transition bandwith, =0.50.3 = 0.2 (3) The impulse response: Cutoff frequency27 10.2.

18、4 Fixed Window Functions(1) Determine the suitable window by the minimum stopband attenuation(2) Determine the length of FIR by the transition width (3) Compute impulse response of the desired filter (according to the IDTFT) with c(4) Obtain the designed FIR filter:28with = N/2. controls the side-lo

19、be amplitudes (attenuation) controls the main lobe width Prediction formula: attenuation s = 20 log10s transition region width = sp together with attenuation s N 10.2.5 Adjustable Window Functions Kaiser window29Amplitude1.20.90.60.305101520 10.2.5 Adjustable Window FunctionsP412DSP group 2007 chap1

20、0-ed130and 10.2.5 Adjustable Window Functions ExampleExample A1Solution:(1) Determine the parameters of the window:DSP group 2007 chap10-ed131Question: Is it suitable for N to be 23? 10.2.5 Adjustable Window Functions Example(2) Find the ideal impulse response:DSP group 2007 chap10-ed132(3) The FIR

21、filter designedWhere N=24, =3.395 Type I linear phase FIR 10.2.5 Adjustable Window Functions Example33Approximation methods:(2) Interpolation Frequency sampling approachLeast Integral-Squared approximation Windowed Fourier Series approach(3) Chebyshev approximation Equiripple approximation,Parks-McC

22、lellan Algorithm 10.3 CAD of Equiripple Linear-Phase FIR Filters34 10.3 CAD The Parks-McClellan Algorithm Weighted error function: For typical filter design:35 10.3 CAD The Parks-McClellan Algorithm by manipulation: For Type I linear phase FIR filter:DSP group 2007 chap10-ed136 10.3 CAD The Parks-Mc

23、Clellan AlgorithmOther types FIR filter see page 417 Eq.(10.63)Eq.(10.66); Weighted error function:Find ak to minimize the peak absolute value E() minimax criterionDSP group 2007 chap10-ed137 Let R be a union of disjoint closed subsets of Let a desired function D(x) and weighting function W(x) be co

24、ntinuous on R Define the error function E (x) = W (x) PL(x)D(x) Maximum error 10.3 CAD Alternation theoremLet:38necessary and sufficient condition for PL(x) being the unique Lth order polynomial under the minimax criterion can be expressed by the alternation theorem: E(x) has at least L + 2 alterati

25、ons on R , i.e. xi , i = 1, . . . , L L + 2 such that xi xi+1, E(xi) = E(xi+1)= Emax , for i = 1, . . . , L 1 10.3 CAD Alternation theoremDSP group 2007 chap10-ed1391. initialize i to some values2. compute and A(i ), where is the ripple corresponding to the alternation frequencies; 3. interpolate a

26、polynomial between the alternation points4. find the maximum/minimum values of the error 5. if |E()| max : stop else compute new i, as the extreme of E(), and go to 2 (else recursive) 10.3 CAD Remez exchange algorithmDSP group 2007 chap10-ed140Order Estimation:kaiord() Kaisers Formulabellangord() Be

27、llangers Formularemezord() Hermanns Formulakaiserord() filter order for Kaiser window-based design 10.5 FIR Digital Filter Design Using Matlab41remez() equiripple FIR filter design using Parks-McClellan algorithmExample10.15 Design an equiripple FIR filter withspecifications: 10.5 FIR Digital Filter

28、 Design Using Matlab Solution:042 10.5 FIR Digital Filter Design Using Matlab B = REMEZ(N, fpts,mag,wt)fedge=800 1000; mval=1 0;dev=0.0559 0.01; FT=4000;N, fpts,mag,wt=remezord(fedge,mval,dev,FT); Matlab codes: N the approximate order; fptsnormalized frequency band edges; Magfrequency band magnitudes; wtweights43Gain (dB) Gain (dB) Gain (dB) Gain (dB) Equiripple FIR Lowpass FilterPassband DetailsN=28N=30DSP group 2007 chap10-ed144 10.5 FIR Digital Filter Design Using MatlabExample10.16 Design an equiripple FIR filter withspecifications:DSP group 2007 chap10-ed145 Solution:N=26;