数字信号处理英文版课件：7-4 Design of FIR Filter by Windowing

Method

Windowing(窗函数法)

Frequencysampling（频率采样法）

Optimumapproximation（最优逼近法）§7.4DesignofFIRFilterbyWindowing一、DesignFIRbyWindowing1、PrincipalIdealsystem:FIRsystem:(infinite,noncausal)(finite,causal)approximateTruncate?shapelengthSetawindow窗函数设计法是从时域出发，把理想的hd(n)用一定形状的窗函数截取成有限长的h(n),以h(n)来近似hd(n),从而使频率响应函数

近似理想频率响应1)Causal、finitei.e.2）LinearphaseThewindowsLinearphaseTheidealfilterand

Suppose

TheFIRfilterSoh[n]isalinearphaseFIRsystem

where,FortheidealLPfilter:3)Propertiesof(designaLPFIR)Fortheobtainedfilter:Forthewindow:,

,

So,themagnituderesponseoftheobtainedfilter:main-lobeside-lobemainlobewidth

istheareaofbetween

and

Whenisaboutthewholeareaofa).When,aboutahalfareaofinside,sob),and

occur

overshot.c)TheoverlapareaisthebiggestatorWhenor,theoverlapareaisthesmallest,andoccurundershot.Theleftsidelobeofwill

sweptthroughthepassband.Thentheconvolutionwillfluctuatearoundzero.d）Theleftsidelobeofwill

sweptthroughthepassband.Thentheconvolutionwillfluctuatearound1.e）f)AsMincrease,theoscillatorybehavioratismorerapid,butthesizeoftheripplesdoesnotdecrease4)EffectsofthedifferencewindowingThewidthoftransition(intervalbetweenovershootandundershoot)equaltothewidthofthemainlobe.b)Tothesametypeofw[n],Misbigger,thetransitionisnarrower.Tothedifferenttypeofw[n],thewidthofthemainlobecanbedifferentwiththesameM.c)Thesidelobe(旁瓣)determinesthepeakandtheripple――>attenuationd)Theripplesinpassbandequaltostopbandapproximately.i.e.Gibbsphenomenon

改变M，只能改变窗谱的主瓣宽度，但不能改变主瓣与旁瓣的相对比例，这个相对比例是取决于窗函数的形状。因此，增加M，只能使“过渡带”变窄，但不能改变肩峰的相对值。

比如，对于矩形窗，不管M如何改变，最大相对肩峰总是8.95%，这种现象称为吉布斯(Gibbs)效应。5)ImprovementIncreaseM,makethetransitonnarrower.)b)Selectw[n],depressthemagnitudeofsidelobe,focustheenergyonthemainlobe.Itmeansdepressthepeak,improvethetolerance.【说明】

实际设计中，窄的过渡带和小纹波是相互矛盾的，故常以牺牲过渡带来换取较小的阻带纹波和较好的通带平坦性。RectangularBartlett(triangular)二、Commonlyusedwindows1)TypeofWindowsHanningHammingBlackman二阶余弦窗Raised-cosinewindows

2)TimedomainBlackmanHamming

Hanning

BartlettRectangular3)Frequencydomain（M=50）RectangularHammingBlackmanHanning

a)ComparetheamplituderesponseofthewindowN=8RectangularHanning

HammingBlackmanN=64

矩形窗主瓣最窄，旁瓣峰值最大随着窗形状的变化，旁瓣峰值逐渐下降，主瓣宽度也逐渐加宽RectangularHanningHammingBlackmanb)ComparetheapproximationtotheidealfilterN=8IdealfilterRectangularHanningHammingBlackmanIdealfilterN=16随着阶数的增大，FIR越来越逼近理想滤波器矩形加窗后的滤波器过渡带最窄，但肩峰和纹波最大N=8N=16N=64N=32HammingN增大时可以改变主瓣的宽度，不能改变主瓣与旁瓣的相对比例c)Comparethemain-to-sideloberatioRectangularN=64Hanning

N=64HammingN=64BlackmanN=64主瓣与旁瓣的相对比例只由窗函数形状决定4)ComparisonofCommonlyUsedWindows

阻带最小衰减只由窗形状决定，与M无关

过渡带宽与窗形状和窗长M+1都有关ThetransitionoftheFIRfilter0.45×4π/(M+1)0.525×8π/M0.775×8π/M0.825×8π/M0.917×12π/Mwhere

:thezeroth-ordermodifiedBesselfunctionofthefirstkind:shapeparameterTwoparametersM+1:lengthVary(M+1)and,thewindowlengthandshapecanbeadjustedtotradeside-lobeamplitudeformain-lobewidth.三、Kaiserwindowp475(M=20)

Increasing

whileholdingM,thewindowistaperedmore,andthesidelobesaresmaller,butthemainlobeiswider.

IncreasingMwhileholding,themainlobedecreaseinwidth,butthesidelobesareinvariant.EmpiricalEquation:GivenThenwindowingmethod:transitionwidth:Thecutofffrequency:四、StepofFIRFilterDesignbyWindowing1)Specifications:2)Givenidealfilter:

LP:HP:BP:BS:Idealfrequency-selectivefilters 1)noncausal,IIR, 2)notcomputationallyrealizable 3)zero-phase1.Lowpass2.Highpass3、Bandpass4.Bandstop3)SelectbythegivenattenuationandcalculateMbytransition.

(a)Fixedwindows:(b)Kaiserwindow:4)DesigntheDFFIR:5)Verify(实际设计时考虑):——TypeIMiseven——TypeIIMisoddh[n]=h[M-n]LP/BP——TypeIMiseven——TypeIIMisoddHP/BSFIR加窗法特点：

设计过程简单、方便，但边界频率不易精确控制。Example1DesignaLinearPhaseFIRLPFbyWindowingSpecifications:Solution:1)SpecificationsofDF

attenuationinstopband2)Determine3)Determinethewindoww[n]Hammingwindow(-53dB)So,Fromtable7.1,4)5)StructureTypeⅡ,Example2KaiserWindowDesignofaLPFSpecifications:

Solution:

1)Determine2)DeterminetheparametersoftheKaiserwindowChooseSo,then,3)TypeII

4)Linear-phasestructureExample3KaiserWindowDesignofaHPFSpecifications:

Solution:1)Determine

where2)DeterminetheparametersoftheKaiserwindowChooseSo,

then,

Note:cannotchooseM=25forHP4)Verify(M=26)(M=25)highlyunsatisfactory3)Example4KaiserWindowDesignofaBPFilter.Specifications:Solution:1)Specifications:2)Determine3)DeterminetheparametersoftheKaiserwindowChooseSo,then,

4)Example5KaiserWindowDesignofaBSFSpecifications:Solution:1)Specifications:2)Determine3)DeterminetheparametersoftheKaiserwindowChooseSo,then,4)【Appendix】1.Differentiator2.IdealMultiplepassbandsandstopbandsfilter——TypeIIIMiseven——TypeIVMisoddThedesigneddifferentiator:Theidealdifferentiatorwithgeneralizedlinearphase:h[n]=-h[M