基于最小方差法低通FIR的设计论文

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MATLAB相关知识，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（ej ）与理想滤波器的频率响应Hd（ej）之间的最大绝对误差最小。（二）低通FIR滤波器技术指标1 Wp =0.25：（通带截止频率）%=0.35二（阻带截止频率）0 =1dB （通带衰减）氏=40dB （阻带衰减）|H（ej0）|亠、ap = 20 lg jw（通带最大衰减

2、）H（ej c）|H（ej0）as =20lg （阻带最小衰减）|H（ej s）（三）低通FIR滤波器的设计3.1低通FIR滤波器阶数的估计K1-20lg（/）-13N 拓fc4614.6（豹s -每）/2兀由于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k .2匕=H 5阿）卑站）D（站）i=1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lH ） = Q 疋 a k 】cos（wk）k =0l3.3 H co)=Q(co)E

3、 a kcos(wk)式中 Q(co)、a Ik 、L 的确定a QC )的确定由于不同类型 QC ）也就不尽相同，不同类型时 QC ）的表达式如下Q( )=1对于1型Q(co)=sin(co )对于 3 型Q( J =sin( / 2) 对于4型由于我们设计的低通FIR滤波器为1型所以b k 1的确定冋样根据不冋的类型其 k 的表达式也不一样 k =二 k 对于1 型k=kl对于2型k=kl对于3型k=k对于4 型我们选择 k 1=:-k 1,对于 1 型bhM 1,:- 1k2hC L的确定L=M对于1型2M -1L=2对于2型L= M-1对于3型2M -1L=2对于4型IM k 1，1E

4、kEMQ( J=cos( /2) 对于2型Q( ) =1N T 1N -1根据N与M的关系M=，表示取不大于的最大整数，所以M=22,L=22 。-2 2k23.4 八 W(j)H ( J -DC J f 中 D(.)、W( )的确定 i=1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 (co)=1在通带中D (豹)=0在阻带中3.5根据上面式子可以确定 QC )，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0 _k _1由它可生成(L+1)个等式的线性方程组，用来求解k lo我们考虑1型线性相位FIR滤波器的设计。在这种情况下，QC

5、)=1, k k 1且L=22。则均方误差的表达式为_M|=Z 丿W(纠)匡 a klcosjk)-Di) i2亠M.吃 Wi)a kbosik) -Wi)D)! k z0若有W( 1)COS(M JW( 2)COS(M 2)W(Y) W( 1)COSC )W 伸 2)W2)COS2)H =WK)WWK)COSk)W(，k)COS(M K)：-=a 0 a 1 !.a md 二 W( 1)D( 1)W( 2)D( 2).W( k)D( K)T=eTe,式中 e = Ha d。计算如下1-15时W()取1, 16-22时取0将0到0.35二上取均匀的22点最后求的H=1 0.9987 0.995

6、0 0.9888 0.980 0.968 0.955 0.939 0.921 0.900 0.877 0.853 0.825 0.796 0.7650.732 0.697 0.660 0.622 0.582 0.540 0.498 ;1 0.995 0.980 0.955 0.921 0.877 0.825 0.765 0.6970.622 0.540 0.454 0.363 0.268 0.170 0.07 0.028 0.128 0.226 0.322 0.415 0.588;1 0.988 0.9550.900 0.825 0.732 0.622 0.498 0.363 0.219 0.

7、071 0.078 0.226 0.369 0.504 0.627 0.737 0.8290.904 0.957 0.989 0.999 ;1 0.980 0.921 0.825 0.697 0.541 0.363 0.170 0.028 0.226 0.415 0.5880.737 0.856 0.942 0.989 0.998 0.967 0.897 0.792 0.654 0.492 ;1 0.968 0.877 0.732 0.540 0.3150.071 0.177 0.415 0.627 0.800 0.924 0.989 0.99 0.937 0.821 0.655 0.312

8、0.212 0.035 0.282 0.510 ;1 0.955 0.825 0.622 0.363 0.071 0.226 0.504 0.737 0.904 0.989 0.988 0.897 0.727 0.492 0.2120.0085 0.376 0.633 0.833 0.959 0.999 ;1 0.939 0.765 0.498 0.170 0.177 0.504 0.769 0.942 0.9990.937 0.760 0.277 0.163 0.184 0.510 0.774 0.944 0.999 0.934 0.755 0.485;1 0.921 0.697 0.363

9、0.028 0.415 0.736 0.942 0.998 0.897 0.655 0.309 0.008 0.467 0.774 0.959 0.993 0.870 0.6100.254 0.142 0.516; 1 0.900 0.622 0.219 0.226 0.627 0.903 0.999 0.897 0.616 0.212 0.233 0.6330.906 0.999 0.894 0.610 0.205 0.240 0.638 0.909 0.999 ;1 0.877 0.541 0.071 0.415 0.800 0.9890.937 0.655 0.2120.282 0.70

10、7 0.9590.9770.755 0.349 0.142 0.5990.9090.997 0.841 0.479;10.852 0.454 0.0780.587 0.923 0.9870.7600.309 0.233 0.707 0.9720.9510.649 0.156 0.3820.809 0.997 0.891 0.522 0 0.522 ;1 0.825 0.363 0.226 0.736 0.989 0.897 0.492 0.085 0.633 0.9590.951 0.610 0.0570.516 0.909 0.9850.7170.199 0.389 0.841 0.999;

11、1 0.796 0.268 0.369 0.8560.994 0.727 0.1630.467 0.907 0.9770.6490.057 0.558 0.946 0.9490.5640.049 0.644 0.9750.909 0.473;1 0.765 0.170 0.504 0.940 0.937 0.491 0.184 0.774 0.999 0.755 0.156 0.516 0.9460.932 0.479 0.198 0.783 0.999 0.746 0.142 0.528; 1 0.732 0.071 0.627 0.989 0.821 0.212 0.5100.959 0.

12、8940.3490.349 -0.909 0.9480.4780.247 0.841 0.984 0.599 0.1070.7550.999;0 0 0 00 0 0 0 00 00 0 00 000 00 0 0;0 000 00 0 00 0 00 00 0 00 0 00 00;0 00 0 00 000 00 00 0 0 0 00 00 0 0;0 000 00 0 0 0 000 00 0 00 0 00 00;0 00 0 00 00 0 00 0 00 000 00 00 0;0 0 00 00 0 00 000 00 0 0 0 000 00;0 00 0 00 00 0 0

13、0 0 00 00 0 00 0 00算这个H时算出来的负值取了它的绝对值2 求k 1令 d= 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0然后最小均方解可以根据求解线性方程H T Ha = H Td得到。令 r = H T H n = HTdT a=kl a=n*pinv(r)用MATLAB可以计算出 ac=ac =1.0e+005 *Colu mns 1 through 190.95392.5572-1.4336-0.37351.9754-3.75723.5858-2.6165-2.31652.7508-3.48832.3659-3.81731.9

14、0290.59010000Colu mns 20 through 2200 0二 k 1= 0.95392.5572-1.4336-0.37351.9754-3.75723.5858-2.6165-2.31652.7508-3.48832.3659-3.81731.90290.590100000003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( .) =1 k =用 k =cL=M=22根据表达式 H （）=Q（豹） a k】cos（wk）k =0H ( ) =0.9539 cos( ) +2.5572 cos(2 ) -1.4336cos(3 ) -0.3735cos(4 )

15、 +1.9754cos(5 ) -3.7572 cos(6 -)+3.5858 cos(7 )-2.6165 cos(8 -)-2.3165 cos(9 )+2.7508 cos(10 )-3.4883 cos(11 -)+2.3659 cos(12 )-3.8173 cos(13 )+1.9029cos(14 )+0.5901 COS(15 )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。ji223_3Z 二e求 H(Z)=0.0916( z z )+0.1171( z z )-0.6515 (z Z )-0.186744(Z Z ) +0.9877(z5

16、Z 直)-1.87866 6(z z ) +1.7929(z7z) -1.3082(Z8 Z*) -1.1582(z9z-9) +1.3754(z10zJ0) -1.7441(z11 - zJ1) +1.1829(z12 z2)-1.9086(z1313Z ) +0.951414J415(z - z ) +0.2950 (z z )4计算误差eTe和e二Ha -d可求出误差用MATLAB计算误差如下h=1 0.9987 0.9950 0.9888 0.980 0.968 0.955 0.939 0.921 0.900 0.8770.9950.9800.9550.9210.8770.8250.7

17、650.6970.6220.5400.4540.3630.268 0.170 0.07 0.028 0.128 0.226 0.322 0.415 0.588;10.988 0.9550.9000.8250.7320.6220.4980.3630.2190.0710.0780.2260.3690.5040.627 0.737 0.829 0.904 0.957 0.989 0.999 ;10.9800.9210.8250.6970.5410.3630.1700.0280.2260.4150.5880.7370.8560.9420.9890.9980.967 0.897 0.792 0.654

18、0.492 ;10.9680.8770.7320.5400.3150.0710.1770.4150.6270.8000.9240.98S)0.990.9370.8210.6550.3120.2120.0350.2820.510; 1 0.9550.8250.6220.3630.0710.2260.5040.7370.904 0.989 0.988 0.897 0.727 0.492 0.212 0.0085 0.376 0.633 0.8330.959 0.999 ;10.939 0.765(0.4980.1700.1770.5040.7690.9420.9990.853 0.825 0.79

19、6 0.765 0.732 0.697 0.660 0.622 0.582 0.5400.498 ;10.937 0.760 0.277 0.163 0.184 0.510 0.774 0.944 0.999 0.934 0.7550.485;1 0.921 0.697 0.363 0.028 0.415 0.736 0.942 0.998 0.897 0.655 0.309 0.008 0.467 0.774 0.959 0.993 0.870 0.610 0.254 0.142 0.516;1 0.900 0.622 0.219 0.226 0.627 0.903 0.999 0.897

20、0.616 0.212 0.2330.633 0.906 0.999 0.894 0.610 0.205 0.240 0.638 0.909 0.999;10.8770.5410.0710.4150.8000.9890.9370.6550.2120.2820.7070.9590.9770.755 0.349 0.142 0.599 0.909 0.997 0.841 0.479;1 0.852 0.454 0.0780.5870.9230.9870.7600.3090.2330.7070.9720.9510.6490.1560.3820.8090.9970.8910.5220 0.522;10

21、.8250.3630.2260.7360.9890.8970.4920.0850.6330.9590.9510.6100.0570.5160.9090.9850.7170.1990.389 0.841 0.999;1 0.796 0.268 0.369 0.856 0.994 0.727 0.163 0.4670.9070.9770.6490.0570.5580.9460.9490.5640.0490.6440.9750.9090.473;1 0.765 0.170 0.504 0.940 0.937 0.491 0.184 0.774 0.999 0.7550.156 0.516 0.946

22、 0.932 0.479 0.198 0.783 0.999 0.746 0.142 0.528;1 0.732 0.071 0.627 0.989 0.821 0.212 0.510 0.959 0.894 0.349 0.349-0.909 0.948 0.478 0.247 0.841 0.984 0.599 0.107 0.755 0.999;0 0 00 0 0 0 00 000 000 00 0 00 0 0;0 000 000 000 00 0 000 0 0 0 00 0;00 000 00 0 00 0 00 000 000 000;00 0 000 0 0 0 00 000

23、 000 00 0 00;0 00 000 000 000 00 0 000 0 0 0 0;0 000 000 00 0 00 0 00 000 000 0;00 00 0 000 0 0 0 00 000 000 00 0a=0.95392.5572 -1.4336-0.37351.9754 -3.75723.5858 -2.6165 -2.3165 2.7508 -3.4883 2.3659 -3.81731.90290.5901000000 0；s=h*ad= 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0;c=s-de=c*ce=Colu mn

24、s 1 through 193.62352.4192-0.7942-1.72131.15432.20881.91280.45411.29032.47103.52100.6483-4.8485-0.3134-9.8727000 02.41921.6152-0.5303-1.14920.77071.47471.27710.30320.86151.64982.35080.4329-3.2372-0.2093-0.7942 -0.53030.17410.3773 -0.2530 -0.4841 -0.4193-0.0995-0.2828-0.5416-0.7718-0.14211.06280.0687

25、2.164000 0 0-1.7213-1.14920.37730.8177-0.5483-1.0493-0.9086-0.2157-0.6129-1.1738-1.6726-0.30802.30320.14894.689900 0 01.15430.7707-0.2530-0.54830.36770.70360.60930.14470.41100.78711.12160.2065-1.5445-0.0998-3.1450000 02.20881.4747-0.4841-1.04930.70361.34641.16600.27680.78651.50632.14630.3952-2.9556-

26、0.1911-6.0182000 01.91281.2771-0.4193-0.90860.60931.16601.00970.23970.68111.30441.85860.3422-2.5594-0.1655-5.2116000 00.45410.3032-0.0995-0.21570.14470.27680.23970.05690.16170.30970.44130.0813-0.6076-0.0393-1.2373000 01.29030.8615-0.2828-0.61290.41100.78650.68110.16170.45950.87991.25380.2309-1.7265-

27、0.1116-3.5156000 02.47101.6498-0.5416-1.17380.78711.50631.30440.30970.87991.68502.40110.4421-3.3064-0.2137-6.7325000 03.52102.3508-0.7718-1.67261.12162.14631.85860.44131.25382.40113.42140.6300-4.7114-0.3046-9.5934000 00.64830.4329-0.1421-0.30800.20650.39520.34220.08130.23090.44210.63000.1160-0.8675-0.0561-1.76650000-4.8485 -3.23721.06282.3032 -1.5445 -2.9556 -2.55940.4194-0.16550.0271-5.21160.8540-0.6076 -1.7265 -3.3064 -4.7114 -0.86756.487813.21060000-0.3134-0.20930.06870.1489-0.0998-0.1911-0.0393-0.1116-0.2137-0.304