第五章 数字滤波器 FIR数字滤波器 窗函数设计法
三、 窗函数方法：WindowingMethod设计原理：-->8H0，w—w—πsinw.nejundw =h(n2元元-wc-W—\=CH.CI元Nsin w.n2?wjwn2daeeN-2元元|20N-1ha(n)nh(n) = ha(n)R(n) =0，其他
设计原理： 1 2 1 2 1 , | | ( ) 0 , | | 1 sin ( ) 2 , | | ( ) 0 , | | 1 sin 1 2 ( ) 2 1 2 ( ( ) ( ) ( ) c c c c j c d c j n c d Nj j c d c N c j j n d d d N H e n h n e d n H e e N n h n e e d N n h h n h n R n w w w w w w w w w w w w w w p w w p p w w w w p w w p p - - - - - - ìï £ ¢ = í ï î ¢ = = ìï £ = í ï î - ç - ÷ = = - ç - ÷ = = ò ò 其他 ) , 0 1 0 , ìï n n N - í ï î
Iwl≤w-lwl8元We-wcwsinw.nrwnhd.oe元wnh(n)向右平移（N-1)/2N=21N:windowlengthN=211nEXAMPLE：截止频率为w。的线性相位理想低通滤波器：h(n）偶对称，奇数点
EXAMPLE： c 1 , | | ( ) 0 , | | j c d c H e w w w w w p ìï £ ¢ = í ï î w n n h(n) 向右平移 (N-1)/2 N：window length -c c 1 2 sin ( ) c c j n c d n h n e d n w w p w w w - p ¢ = ò = N = 21 10 20 N=21 h(n)
we-wcnh(n)向右平移N/2-1N=20N:windowlengthN=20=nEXAMPLE：截止频率为w的线性相位理想低通滤波器：h(n）偶对称，偶数点
EXAMPLE： c w n n -c c 向右平移 N/2-1 N：window length N = 20 9 19 N=20 h(n) h(n)
矩形窗截断的影响h(n) = ha(n)R(n)Wr(e)台 R(n)H,(ej)Wr ej(w-0) daHe2元wNsin2e-jwaWe8sin2wNsinN-2Mαw2sin2
矩形窗截断的影响： ( ) ( ) 1 2 ( ) ( ) ( ) ( ) 1 ( ) ( ) 2 sin 2 ( ) ( ) sin 2 sin 1 2 , ( ) 2 sin 2 d N j R N j j j d R N j j j R R R h n h n R n W e R n H e H e W e d N W e e W e N N W w p w w w q p w w wa q p w w w w a w w - - - - ç ÷ - = Û = ìï ç ÷ ï = = í ï ç ÷ - = = î ò
矩形窗截断的影响：Ha(e i~) = H(w)e-jwa-w-1.H.(w) =，——1H.(0)e-joaWr(w - 0)e i(w-)d0H(2元1= e-jwqHa(0)Wr(w - 0)d02元7-jwa= H(w)e1H(w)H.(0)Wr(w - 0)d2元
矩形窗截断的影响： ( ) ( ) ( ) 1 , | | ( ) , 0 , | | 1 ( ) ( ) ( ) 2 1 ( ) ( ) 2 ( ) 1 ( ) ( ) ( ) 2 j j d d c d c j j j d R j d R j d R H e H e H H e H e W e d e H W d H e H H W d w wa p w qa w q p p wa p wa p p w w w w w w p q w q q p q w q q p w w q w q q p - - - - - - - - ìï = í ìï £ ï = í ï î î = - = - = = - ò ò ò
截断效应：吉布斯现象用旭育技术减小数断效应Wr(- 0)1Ha(0))e-jaa10≤0N-1=H, (o)e-jo,αH,(ej)=2元/N22元/N0，0≤0≤元AO01[0|≤。weWeH,(o)=N:窗长Ha(0)0,0,≤0≤元9WRW/e-jao - Wr (a)e-jaasin(/2)(ej)W, [ej(o- jdeH(e2Ha(0)Wr(w-0)L.(0)Wr(co-0)doe-Jao=H(0)e-jaoH6W-2元/N0.089:Ha(0) AN. = 10WR(W-0)N=200h0w.+2元/N40.0895H(w0.5随着截断长度增加，过渡带变窄，起伏振荡变0.10.50.0468密，但最大肩峰却总近+0.0468似为1.0895，阻带性能并w无实质改善！！！Wew00.0895
随着截断长度增加，过 渡带变窄，起伏振荡变 密，但最大肩峰却总近 似为1.0895，阻带性能并 无实质改善！！！ 截断效应：吉布斯现象 用加窗技术减小截断效应 N：窗长
Window Functions for FIR Filter DesignWindow TypeTime-DomainSequence[1, 0≤n≤Mw[n] =RectangularLo, otherwise2n/M,Bartlett0≤n≤M/22-2n/M,(Triangular)w[n] =M/2<n≤MLo,otherwise0.5 - 0.5cos(2元n/M),0≤ n≤MHanningw[n] =Lo,otherwise0.54 - 0.46cos(2元n/M), 0 ≤ n≤MHammingw[n] =0,otherwiseBlackman0.42 - 0.5cos(2元n/M) + 0.08cos(4元n/M), 0 ≤n≤Mw[n] =0otherwiseKaiserIo[β(1 - (n -α)/α)2)1/2]/I(β), 0 ≤ n≤ M, α = M/2w[n] =.0,otherwiseIo(.)is zero order modified Bessel function of the firstkind,β is window shape parameter.M=N-1
Window Type Time-Domain Sequence Rectangular w[n] = 1, 0 n M 0, otherwise Bartlett 2n/M, 0 n M/2 (Triangular) w[n] = 2-2n/M, M/2 < n M 0, otherwise Hanning w[n] = 0.5 – 0.5cos(2pn/M), 0 n M 0, otherwise Hamming w[n] = 0.54 – 0.46cos(2pn/M), 0 n M 0, otherwise Blackman w[n] = 0.42 – 0.5cos(2pn/M) + 0.08cos(4pn/M), 0 n M 0, otherwise Kaiser w[n] = I0 [b(1 - {(n – a)/a}2) 1/2]/I0 (b), 0 n M, a = M/2 0, otherwise I0 (.) is zero order modified Bessel function of the first kind, b is window shape parameter. M = N-1
Shape of commonly used window functions.Rectangularw[n]1.0HammingHanningBlackman0.8Bartlett0.60.40.2MM02
主瓣宽度v.S.副瓣高度RectanqularHammingN=51202o0)M1 01801 02N=51Im/2)M1 01o1 0Z4040606080510000.2m0.4m0.6m0.8m-1000.2m0.4m0.6m0.8mRadian frequency ()Radian frequency ()BlackmanBartlett20N=512N=51m/2)10180102(ma)Ml400180106060808000400-1001000.4m0.6m0.8m0.2mS00.2#0.4m0.6#0.8mRadian frequency ()Radian frequency ()KaiserHanning20N=511/2)M/0180102405060-7510100.6w0.8m0.2w0.4m0.6m0.2m0.4m0.8mRadian frequency (o)Radian frequency ()