1、1Introduction FIR filter:direct design of DT filter with the often added linear-phase requirement (1)Windowed Fourier series approach(10.2)(2)Frequency sampling approach(Problem 10.31,10.32)(3)Computer-based optimization method(10.3)Chap.10 FIR Digital Filter Design2 10.1 Preliminary ConsiderationsF
2、or FIR system:real polynomial approximation1(1)0()011 (10.1)NNNnnH zhhzh Nzh N zh n z0()(10.2)Njj nnH eh n eif a linear phase is desired h nh Nn 10.1.1 Basic Approaches to FIR Digital Filter Design310.1.2 Estimation of the Filter OrderKaisers Formula 1020log()13(10.3)14.6()2psspN For lowpass FIR fil
3、ter design:P397-398 Bellangers Formula 102log(10)1(10.4)3()2psspN Hermanns Formula 2(,)(,)()2(10.5)()2pspsspspDFNParameters see P398.0()Njj nnH eh n e410.2 Design of FIR Filters by Windowing(P400)()jj nddnHeh n eapproximation10.2.1 Least Integral-Squared Error Design of FIR Filters 21()()(10.9)2jjRd
4、H eHed Parseval s relation2 Rdnh nh n 122201 (10.11)Ndddnnn Nh nh nh nh n ,0dh nh nfornNtruncation,()(10.13)0,jcjLPceHe510.2.2 Impulse Responses of Ideal FiltersIdeal linear phase lowpass filter sin(),(10.14)()cLPnhnnn ()Ideal linear phase highpass filter 0,(),cjHPjcHeesin()sin(),(10.16)()()cHPnnhnn
5、nn 6Impulse Responses of Ideal Filters(II)Ideal linear phase bandpass filter 12,()0,otherwisejccjBPeHe21sin()sin(),(10.17)()()ccBPnnhnnnn Ideal linear phase bandstop filter 120,(),otherwiseccjBSjHee12sin()sin()sin(),(10.18)()()()ccBSnnnhnnnnn 7Impulse Responses of Ideal Filters(III)Ideal multiband f
6、ilter 1(),(10.19)1,2,jMLkkkHeAforkL11sin,(10.20)LMLnhnAAnn 10,LLAIdeal discrete-time Hilbert transformer ,0()(10.21),0jHTjHej0,(10.22)2,HTfor n evenhnfor n oddnIdeal discrete-time differentiator (),0(10.23)jDIFHej0,0(10.24)cos,0DIFnhnnnn8Gibbs phenomenon:Oscillatory behavior in the magnitude respons
7、e of causal FIR filters designed utilizing truncation10.2.3 Gibbs Phenomenon dh nh nw n9mainlobesidelobeMainlobe width-truncationperiodic continuous convolution()1()()()2jjjdH eHeW ed 1,0 0,otherwisenNw nsin(1)/2()sin(2)jjNW ee=N/2()je()()()jjdccHeeuu41mN10.2.3 Gibbs Phenomenon(II)10-2/(N+1)1110.2.3
8、 Gibbs Phenomenon(III)12 N oscillate more rapidly,but the amplitudes of the largest ripples=constant(1)For ,N m ,sidelobe,()jW e10.2.3 Gibbs Phenomenon(IV)(2)For the integral ,oscillation will occur at each sidelobe of moves past the discontinuity()1()()2jjdHeW ed()()jW e (3)The methods to reduce Gi
9、bbs phenomenon:-tapering the window smoothly to zero at each end,but m -a smooth transition in magnitude specifications1310.2.4 Fixed Window Functions(1)Hanning window:A=B=1/2,C=0;(2)Hamming window:A=0.54,B=0.46,C=0(3)Blackman window:A=0.42,B=0.5,C=0.08.Rectangular window:wn=un un N 124 coscos ,0nnw
10、 nABCnNNN Hanning,Hamming,Blackman:2/0/2 22/2 0 otherwisen NnNw nn NNnN Bartlett window:triangular14P406 Fig.10.6 Commonly used fixed windows10.2.4 Fixed Window Functions(II)N/2NRectangular Hamming HanningBartlett Blackman n wn 11510.2.4 Fixed Window Functions(III)P407 Fig.10.750N1610.2.4 Fixed Wind
11、ow Functions(IV)()0.5cjH eParameters predicting the performance of a windowmain lobe width relative sidelobe level (dB)MLSLASame ripples in passbandand stopbandwidth of transition bandspML()jH eType of windowRelative SidelobeLevel(dB)Main-lobe widthMinimum Stopband Attenuation(dB)TransitionBandwidth
12、 Rect.13.34/(N+1)20.91.84/NBartlett26.58/NHanning31.58/N43.96.22/NHamming42.78/N54.56.64/NBlackman58.112/N75.311.12/N1710.2.4 Fixed Window Functions(V)P408 Table 10.21810.2.4 Fixed Window Functions(VI)Example to illustrate the effect of windows N=50 P409 sidelobe level stopband attenuation()2cps1910
13、.2.4 Fixed Window Functions(VII)(1)Compute impulse response of the desired filter(according to the inverse Fourier equation)1()2jj nddh nHeed(2)Determine the suitable window by the minimum stopband attenuation and(3)Determine the length of FIR by the transition width(4)Obtain the designed FIR filter
14、:dh nh nw nSteps for FIR filter design:20Example 10.6 Page 410 Design an FIR lowpass digital filter with specifications:(1)the attenuation of the stopband should more than 40dB;(2).2)According to Table 10.2,we could select Hanning,hamming,Blackman window,then the bandwidth of the transition band sho
15、uld satisfy(for Hanning)Type I:N=32;Type II:N=330.3p10.2.4 Fixed Window Functions(VIII),0.5s(0.30.5)/20.4c1)6.220.2Ni.e.31.1N 32N Please select a suitable window function and determine the smallest length of the window.2110.2.4 Fixed Window FunctionsExample Show that the ideal highpass transformer w
16、ith a frequency response defined by (1)Determine the impulse response hn，the relation of and N?(2)What type of linear-phase FIR filter?(3)Write the impulse response hn using the Hann windows-base method.Solution:ccjcjjdjejeeH00)()()(2210.2.4 Fixed Window Functions()()()()()()()()11 2222112()2()12()c
17、ccccccjjnjjndjj njj njj njj njj nj nh njeedjeedjjeedeedjjeeeej nj neeen )()()()()()()1212()22cos()22()2()(1)sin()/2()cccjj nj njj njnj nj ncncceeeneeeeennnnn 2310.2.4 Fixed Window Functions 0 0ddNh nnNh nh n Rnotherwise（2）If N is even/2Nwhen,the filter has linear phase is integer is integer，h hdn is
18、 anti-symmetries n is anti-symmetries，and hn=-hN-n,the filter is type III.and hn=-hN-n,the filter is type III./2N If N is odd isn isnt integert integer，h hdn is symmetries n is symmetries，and hn=hN-n,the filter is type II.and hn=hN-n,the filter is type II./2N12(1)sin()/2()ncch nnn 2410.2.4 Fixed Win
19、dow Functions（3）21 2(1)sin()/21 cos()()dnccNh nh n W nnn RnnN 25with =N/2.controls the side-lobe amplitudes(attenuation)controls the main lobe width Prediction formula:attenuation s=20 log10s transition region width =s p together with attenuation s N(10.39)10.2.5 Adjustable Window Functions(P410)Kai
20、ser windowN26036Amplitude1.20.90.60.3051015200.40.1102(8.7),50,0.5842(21)0.07886(21),2150,0,21.8 (2.285ssssssssandN)p(10.41)(10.42)10.2.5 Adjustable Window Functions(II)27 Kaiser window design example100.420log40,0.50.30.20.5842(21)0.07886(21)3.395,and(8)/(2.285)22.324.sssssN(1)Determine the window
21、function Kaiser window:,Ni.e.,s=0.01,Assume:0.3,0.5,spQuestion:Is it suitable for N to be 23?28 Kaiser window design example(II)/2/2/20 ,(),.Suppose the freq.response of lowpass filter is,()0 ,.then ()-(),sin(/2)cjHPj Ncj Ncjlpcjj NjHPlpHPHeeeHeHeeHenNhnsin(/2).(/2)(/2)cnNnNnN(2)The desired impulse
22、response0.42spc29 Kaiser window design example(III)(3)The FIR filter designed2001(),where ,()and 2,0dnIh nh n w nw nIN/nNWhere N=24,=3.395Type I linear phase FIR3010.3 CAD of Equiripple Linear-Phase FIR Filters0()Njj nnH eh n e()jj nddnHeh n eapproximationApproximation methods:(2)Least Integral-Squa
23、red approximationWindowed Fourier Series approach(1)InterpolationFrequency sampling approach(3)Chebyshev approximationEquiripple approximationParks-McClellan Algorithm()jjeeH3110.3 CAD of Equiripple Linear-Phase FIR Filters(II)()()()()EWHDWeighted error function:(10.47)1,(),psin passbandWin stopband
24、or,()1,spin passbandWin stopband()()()HQA(10.62)()()()()()(10.67)()DEWQAQ(10.68)()()()WAD3210.3 CAD of Equiripple Linear-Phase FIR Filters(III)Chebyshev or Minimax criterion:equiripple FIR filterMinimize the peak absolute value of()E min max|()|h nRELinear-phase FIR filters obtained by the criterion
25、0()(cos)()LkkWkD0()cos()cos()(cos)LkkAa kkkT0()()cos()()LkEWa kkD polynomial approximation3310.3 CAD of Equiripple Linear-Phase FIR Filters(IV)Alternation Theorem:0()LkLkkP xa xl Let R be a union of disjoint closed subsets of l Let a desired function D(x)and weighting function W(x)be continuous on R
26、 l Define the error function E(x)=W(x)PL(x)D(x)l Maximum error maxmax()x FEE x3410.3 CAD of Equiripple Linear-Phase FIR Filters(V)necessary 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
27、 L+2 alterations on F,i.e.xi,i=1,.,L L+2 such that xi xi+1,E(xi)=E(xi+1),for i=1,.,L1 and E(xi)=Emax,for i=1,.,Li1iix1ix3510.3 CAD of Equiripple Linear-Phase FIR Filters(VI)Parks-McClellan AlgorithmIterative method to determine the alternation frequencies i and the ripple 1.initialize i to some pute
28、 the ripple corresponding to the alternation frequencies3.interpolate a polynomial between the alternation points4.find the maximum/minimum values of the error5.if|E()|:stop else compute new i as extreme of E(),and go to 2 (else recursive)3610.5 FIR Digital Filter Design Using MatlabOrder Estimation
29、:kaiord()-Kaisers Formulabellangord()-Bellangers Formularemezord()-Hermanns Formulakaiserord()-filter order for Kaiser window-based design3710.5 FIR Digital Filter Design Using Matlab(II)Equiripple linear-phase FIR filter design:remez()-equiripple FIR filter design using Parks-McClellan algorithmExa
30、mple10.15 Design an equiripple FIR filter with specifications:0.8,1,4,0.5,40,0.0559,0.01psTpspsFKHz FKHz FKHzdBdB3810.5 FIR Digital Filter Design Using Matlab(III)3910.5 FIR Digital Filter Design Using Matlab(IV)118.7psdBdB4010.5 FIR Digital Filter Design Using Matlab(V)4110.5 FIR Digital Filter Des
31、ign Using Matlab(VI)42Windowing method for FIR filter design:fir1()and fir2()Example10.15 Design a FIR lowpass filter using a kaiser window with specifications:0.3,0.4,0.003162pss 10.5 FIR Digital Filter Design Using Matlab(VI)4310.1,10.2,10.3-estimation formula10.4-multiband filter impulse response
32、10.5 -truncation approximation10.6,10.7-ideal digital Hilbert transformation10.8 -ideal digital differentiator10.9 -delay-complementary pair10.10,10.11,10.12,10.18-inverse DTFT10.15,10.16,10.17-windowing method design10.20-fractional delay FIR filter10.21-ideal comb filter 10.27,10.28-different fitting algorithm10.29-filter sharpening10.3110.35-frequency sampling method 10.40-WDFT10.3610.38-Parks-McClellan algorithm weighting functionExercises
