基于最小方差法低通FIR的设计说明书

1、基于最小方差低通FIR滤波器设计说明书一设计目标根据所学的数字信号处理和MATLABf关知识,用最小方差法设计一个低通FIR滤波器。从FIR数字滤波器的系统函数可以看出，极点都是在z平面的原点，而零点的分布是任意的。不同的分布将对应不同的频率响应，最优化设计实际上就是调节这些零点的分布，使得实际滤波器的频率响应Hd与理想滤波器的频率响应Hd之间的最大绝对误差最小。二低通FIR滤波器技术指标1Wp0.25通带截止频率pWs0.35阻带截止频率p1dB通带衰减s40dB阻带衰减ap201gH(ej0)H(ejWc通带最大衰减H(ej0)as201gRj)阻带最小衰减三氐通FIR滤波器的设计3.1低

2、通FIR滤波器阶数的估计N-20-s-134614.6(sp)/2由于N为偶数，所以可以设计一个1型的低通FIR滤波器。3.2对于基于最小方差的线性相位FIR滤波器的设计下面式子为误差的简化为k2W(i)H(i)D(i)i1其中H()是低通FIR的振幅响应，D()是要求的振幅响应，W()是权重函数。由于所有四种类型的线性相位FIR滤波器的振幅响应可以表示为iH()Q()kcosk03.3H()Q()kcos式中Q()、k、L的确定k0aQ()的确定由于不同类型Q()也就不尽相同，不同类型时Q()的表达式如下Q()=1对于1型Q()=cos对于2型Q()=sin对于3型Q()=sin对于4型由于

3、我们设计的低通FIR滤波器为1型所以Q()=1bk的确定同样根据不同的类型其k的表达式也不一样_k=bkk=k一k=dk我们选择k=k,对于1型0hM,k2hMk,1kM的最大整数，所以M=22,L=22。23.4k2W(i)H(i)D(i)中D()、W()的确定i1根据最小方差的相关要求可知W( ) 1在通带中W( ) 0在阻带中D( ) 1在通带中D( ) 0在阻带中3.5根据上面式子可以确定Q(),L的值和k的表达式，由于最小方差是滤波器参数k的一个函数。为了得到的最小值，令由它可生成L+1个等式的线性方程组，用来求解我们考虑1型线性相位fir滤波器的设计。在这种情况下，Q()=1,k=

4、k且L=22。则均方误差的表达式为 TOC o 1-5 h z kM HYPERLINK l bookmark13 o Current Document W(i)kcos(ik)D(i)i1k0M2W(i)kcos(ik)W(i)D(i)k0若有W( 1) W( 1)COS( 1)W( 1)COS(M 1)W(2)W(2)COS(2).W(2)COS(M2)HW(k)W(k)COS(k).W(k)COS(Mk)a0a1.aMTdW(1)D(1)W(2)D(2).W(k)D(k)teTe,式中eHad。计算如下1求H,1-15时川（）取1,16-22时取0将0到0.35上取均匀的22点最后求的H

5、=10.99870.99500.98880.9800.9680.9550.9390.9210.9000.8770.8530.8250.7960.7650.7320.6970.6600.6220.5820.5400.498;10.9950.9800.9550.9210.8770.8250.7650.6970.6220.5400.4540.3630.2680.1700.070.0280.1280.2260.3220.4150.588;10.9880.9550.9000.8250.7320.6220.4980.3630.2190.0710.0780.2260.3690.5040.6270.7370.

6、8290.9040.9570.9890.999;10.9800.9210.8250.6970.5410.3630.1700.0280.2260.4150.5880.7370.8560.9420.9890.9980.9670.8970.7920.6540.492;10.9680.8770.7320.5400.3150.0710.1770.4150.6270.8000.9240.9890.990.9370.8210.6550.3120.2120.0350.2820.510;10.9550.8250.6220.3630.0710.2260.5040.7370.9040.9890.9880.8970.

7、7270.4920.2120.00850.3760.6330.8330.9590.999;10.9390.7650.4980.1700.1770.5040.7690.9420.9990.9370.7600.2770.1630.1840.5100.7740.9440.9990.9340.7550.485;10.9210.6970.3630.0280.4150.7360.9420.9980.8970.6550.3090.0080.4670.7740.9590.9930.8700.6100.2540.1420.516;10.9000.6220.2190.2260.6270.9030.9990.897

8、0.6160.2120.2330.6330.9060.9990.8940.6100.2050.2400.6380.9090.999;1 TOC o 1-5 h z 0.8770.5410.0710.4150.8000.9890.9370.6550.2120.2820.7070.9590.9770.7550.3490.1420.5990.9090.9970.8410.479;10.8520.4540.0780.5870.9230.9870.7600.3090.2330.7070.9720.9510.6490.1560.3820.8090.9970.8910.52200.522;10.8250.3

9、630.2260.7360.9890.8970.4920.0850.6330.9590.9510.6100.0570.5160.9090.9850.7170.1990.3890.8410.999;10.7960.2680.3690.8560.9940.7270.1630.4670.9070.9770.6490.0570.5580.9460.9490.5640.0490.6440.9750.9090.473;10.7650.1700.5040.9400.9370.4910.1840.7740.9990.7550.1560.5160.9460.9320.4790.1980.7830.9990.74

10、60.1420.528;10.7320.0710.6270.9890.8210.2120.5100.9590.8940.3490.349-0.9090.9480.4780.2470.8410.9840.5990.1070.7550.999;0000000000000000000000;0000000000000000000000;0000000000000000000000;0000000000000000000000;0000000000000000000000;0000000000000000000000;0000000000000000000000算这个H时算出来的负值取了它的绝对值2求

11、k令d=1111111111111110000000然后最小均方解可以根据求解线性方程HTHaHTd得到。令rHTHnHTdTa=ka=n*pinv用MATLAB1以计算出ac=ac=1.0e+005*Columns1through190.95392.5572-1.4336-0.37351.9754-3.75723.5858-2.6165-2.31652.7508-3.48832.3659- TOC o 1-5 h z 3.81731.90290.59010000Columns20through22|000k=0.95392.5572-1.4336-0.37351.9754-3.75723.5

12、858-2.6165-2.31652.7508-3.48832.3659-3.81731.90290.590100000003最终H()的结果由于N的限制所以所设计的滤波器为1型所以Q()=1k=k=cL=M=22i根据表达式H()Q()kcosk0H()=0.9539cos()+2.5572cos(2)-1.4336cos(3)-0.3735cos(4)+1.9754cos(5)-3.7572cos(6)+3.5858cos(7)-2.6165cos(8)-2.3165cos(9)+2.7508cos(10)-3.4883cos(11)+2.3659cos(12)-3.8173cos(13)

13、+1.9029cos(14)+0.5901cos(15)16至ij22项为0这里我们只是求出了幅频特性，但由于其相频特性是确定的所以在设计中不考虑其相频特性。 TOC o 1-5 h z j122,33、ze求H(Z)=0.0916Vzz+0.1171Vzz-0.6515(zz)-0.1867(z4z4)+0.9877(z5z5)-1.8786(z6z6)+1.7929(z7z7)-1.3082(z8z8)-1.199101011111212582(zz)+1.3754(zz)-1.7441(zz)+1.1829(zz)-1.9086,1313、/1414、/1515、(zz)+0.9514(

14、zz)+0.2950(zz)4计算误差由eTe和eHad可求出误差用MATLAB算误差如下h=10.99870.99500.98880.9800.9680.9550.9390.9210.9000.8770.8530.8250.7960.7650.7320.6970.6600.6220.5820.5400.498;10.9950.9800.9550.9210.8770.8250.7650.6970.6220.5400.4540.3630.2680.1700.070.0280.1280.2260.3220.4150.588;10.9880.9550.9000.8250.7320.6220.4980

15、.3630.2190.0710.0780.2260.3690.5040.6270.7370.8290.9040.9570.9890.999;10.9800.9210.8250.6970.5410.3630.1700.0280.2260.4150.5880.7370.8560.9420.9890.9980.9670.8970.7920.6540.492;10.9680.8770.7320.5400.3150.0710.1770.4150.6270.8000.9240.9890.990.9370.8210.6550.3120.2120.0350.2820.510;10.9550.8250.6220

16、.3630.0710.2260.5040.7370.9040.9890.9880.8970.7270.4920.2120.00850.3760.6330.8330.9590.999;10.9390.7650.4980.1700.1770.5040.7690.9420.9990.9370.7600.2770.1630.1840.5100.7740.9440.9990.9340.7550.485;10.9210.6970.3630.0280.4150.7360.9420.9980.8970.6550.3090.0080.4670.7740.9590.9930.8700.6100.2540.1420

17、.516;10.9000.6220.2190.2260.6270.9030.9990.8970.6160.2120.2330.6330.9060.9990.8940.6100.2050.2400.6380.9090.999;10.8770.5410.0710.4150.8000.9890.9370.6550.2120.2820.7070.9590.9770.7550.3490.1420.5990.9090.9970.8410.479;10.8520.4540.0780.5870.9230.9870.7600.3090.2330.7070.9720.9510.6490.1560.3820.809

18、0.9970.8910.52200.522;1 TOC o 1-5 h z 0.8250.3630.2260.7360.9890.8970.4920.0850.6330.9590.9510.6100.0570.5160.9090.9850.7170.1990.3890.8410.999;10.7960.2680.3690.8560.9940.7270.1630.4670.9070.9770.6490.0570.5580.9460.9490.5640.0490.6440.9750.9090.473;10.7650.1700.5040.9400.9370.4910.1840.7740.9990.7

19、550.1560.5160.9460.9320.4790.1980.7830.9990.7460.1420.528;10.7320.0710.6270.9890.8210.2120.5100.9590.8940.3490.349-0.9090.9480.4780.2470.8410.9840.5990.1070.7550.999;0000000000000000000000;0000000000000000000000;0000000000000000000000;0000000000000000000000;0000000000000000000000;0000000000000000000

20、000;0000000000000000000000a=0.95392.5572-1.4336-0.37351.9754-3.75723.5858-2.6165-2.31652.7508-3.48832.3659-3.81731.90290.59010000000;s=h*ad=1111111111111110000000;c=s-de=c*ce=Columns1through193.62352.4192-0.7942-1.72131.15432.20881.91280.4541 TOC o 1-5 h z 1.29032.47103.52100.6483-4.8485-0.3134-9.87

21、2700002.41921.6152-0.5303-1.14920.77071.47471.27710.30320.86151.64982.35080.4329-3.2372-0.2093-6.59160000-0.7942-0.53030.17410.3773-0.2530-0.4841-0.4193-0.0995-0.2828-0.5416-0.7718-0.14211.06280.06872.16400000-1.7213-1.14920.37730.8177-0.5483-1.0493-0.9086-0.2157-0.6129-1.1738-1.6726-0.30802.30320.1

22、4894.689901.15430.7707-0.2530-0.54830.36770.70360.60930.41100.78711.12160.2065-1.5445-0.0998-3.1450 TOC o 1-5 h z 0002.20881.4747-0.4841-1.04930.70361.34641.16600.78651.50632.14630.3952-2.9556-0.1911-6.01820001.91281.2771-0.4193-0.90860.60931.16601.00970.68111.30441.85860.3422-2.5594-0.1655-5.211600

23、00.45410.3032-0.0995-0.21570.14470.27680.23970.16170.30970.44130.0813-0.6076-0.0393-1.23730001.29030.8615-0.2828-0.61290.41100.78650.68110.45950.87991.25380.2309-1.7265-0.1116-3.51560002.47101.6498-0.5416-1.17380.78711.50631.30440.87991.68502.40110.4421-3.3064-0.2137-6.73250003.52102.3508-0.7718-1.6

24、7261.12162.14631.85861.25382.40113.42140.6300-4.7114-0.3046-9.59340000.64830.4329-0.1421-0.30800.20650.39520.34220.23090.44210.63000.1160-0.8675-0.0561-1.7665000-4.8485-3.23721.06282.3032-1.5445-2.9556-2.5594-1.7265-3.3064-4.7114-0.86756.48780.419413.2106000-0.3134-0.20930.06870.1489-0.0998-0.1911-0

25、.16550.144700.276800.239700.056900.161700.309700.441300.08130-0.60760-0.03930-0.1116-0.2137-0.3046-0.05610.41940.02710.8540-9.8727-6.59162.16404.6899-3.1450-6.0182-5.2116-1.2373 TOC o 1-5 h z -3.5156-6.7325-9.5934-1.766513.21060.854026.8998000000000000000000000000000000000000000000000000000000000000

26、00000000000000000000000000000000000000000000000000000000000000000000000000000Columns20through22000000000000000000000000000000000000000000000000000000000000四用直接型和级联型这两种结构实现1直接型直接型是利用输入信号x(n)和滤波器单位脉冲响应h(n)的线性卷积来描述输出信号y(n)。虽然有22阶但16到22阶H()为0所以直接型结构如下五用FDATOOL析设定相关参数后但到其幅频特性如下0510152。Frequency tkHz)OO相频特性曲线如下01015Frequericy (kHz)20零极图如下11后1.11i一:01/1%I1,11、n*-1於1I1I11115。)111n*!,1二o111111*t#J1d1:A1111一.J.11-1I*1_产一士外11,2-1.5-1-Q,5C0.511石2RealPart六误差分析误差产生的原因因为在数字信号处理中，需要将输入的离散信号和系统的参数进行量化，而量化的结果必然与原来的数值之间存在误差，误差的大小要依据计算机的字长而定。