ch5 滤波器模型与仿真

1、1 2 3 4 5 6 7 8 9 10 11 01 (5 1) NN kk k=k= y nb x n-ka y nk 12 10 ( ) 1( )(52) NN kk kk kk= Y za zX zb z 13 01 ( ) ( )1(53) ( ) NN kk kk k=k Y z H zb za z X z exp2 ( )exp2(54) zjfT H zHjfT 14 1,0 (55) 0,0 n n n 15 0 ( )(56) N k k k= H zb z 0 (57) N k k= h nbn-k 16 00 (58) NN k k=k= y nb x n-kh k x

2、n-k 17 18 19 20 21 010 11021 22032 33043 4404 1(59) 1(510) 1(511) 1(512) wnwnb x n wna wnwnb x n wna wnwnb x n wna wnwnb x n wna wnb x n (513) 22 23 24 25 1(5 14) cd nnnx nWF WF WB (5 15)y nnCW 26 0000 11111 2222 3333 4444 10000001000 1000000100 1000000010 1000000001 1000000000 wnwnwnb w nw nw nab w

3、nwnwna w nw nw na wnwnwna 2 3 4 (5 16)x nb b b 27 28 29 30 0510152025303540 -0.05 0 0.05 0.1 0.15 Sample Index Block Processing 0510152025303540 -0.05 0 0.05 0.1 0.15 Sample Index Serial Processing 31 32 33 34 1 ( )(5 17) a t nT H zTLHs 35 ( )(5 18) a a Hs sa 1 ( )( )= exp() ( )(5 19) aa LHsh ta-at

4、u t 36 1 ( )= exp()(520) aa t=nT LHshna-anT u n 1 ( )exp()(521) n n a t=nT n Z LHsa-aTz 1 ( )(522) exp aT H z aT z 37 1 1 ( )( )(523) 1 a t nT H zLHs zs 38 1 ( )(524) a a Hs ss sa 111 ( )(525) a Hs sssa 1 1 ( )exp(526) a t nT LHsu nanT u n s 39 1 0 1 ( )1exp(527) 1 n n n H zaTz z 111 111 ( )(528) 11

5、1 exp H z zzaT z 1 1 1 ( )1(529) 1 exp z H z aT z 1 1 1 exp ( )(530) 1 exp aTz H z aT z 40 11 (1)/(1) ( )( )(531) a s=Czz H zHs 41 1exp (532) 1exp d d s T s s T 42 expexp 2(533) expexp dd a dd jf Tjf T jfC jf Tjf T 2tan(534) ad fCf T 2cot(535) ad Cff T 43 1 (536) ds ff 2(537) ad fCf T 2/2(538) s CTf

6、 44 11 (1)/(1) ( )(539) s=Czz a H z sa 1 11 1 ( )(540) (1)(1) az H z Czaz 1 1 ( )(541) 1 aa z CaCa H z Ca z Ca 45 0.628319(542)aT exp0.533488(543)aT 1 0.628319 ( )(544) 1 0.533488 ii Hz z 1 1 1 0.533488 ( )(545) 1 0.533488 si z Hz z 46 2 (546)C T 1 1 0.2390570.239057 ( )(547) 1 0.5218861 bl z Hz z (

7、548) 2 ad a ff 47 cot(549) 2 aT Ca 40 cot 0.2(550)C 1 1 0.4208080.420808 ( )(551) 1 0.158384 bl z Hz z 48 % File: c5_threefilters.m T = 0.01; f = 0:0.1:50; z = exp(-i*2*pi*f*T); % see (5.4) a0 = 0.239057; a1 = 0.239057; b1 = 0.521886; % bilinear invariant num = a0+a1*z; den = 1-b1*z; ampx = abs(num.

8、/den); a0 = 0.628319; b1 = 0.533488; % impulse invariant num = a0; den = 1-b1*z; ampy = abs(num./den); a0 = 1.0; a1 = 0.533488; b1 = 0.533488; % step invriant num = (a0-a1)*z; den = 1-b1*z; ampz = abs(num./den); plot(f,ampx,f,ampy,f,ampz) xlabel(Frequency - Hz) ylabel(Amplitude Response) % End of sc

9、ript file. 49 50 01020304050 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Frequency - Hz Amplitude Response 双线性z 变换 阶跃不变滤波器 冲激不变滤波器 51 52 % File: c5_ellipexam.m fs = 100;% set sampling frequency fc = 20;% set cuttoff frequency f = 0:0.1:50;% define frequency vector b,a = ellip(5,1,20,2*pi*fc,s); % synthesize ellipti

10、c filter h = freqs(b,a,2*pi*f);% amp. resp. of analog filter bz1,az1 = impinvar(b,a,fs);% impulse invariant digital filter h1 = freqz(bz1,az1,f,fs);% amplitude response of above bz2,az2 = bilinear(b,a,fs);% bilinear z filter (not prewarped) h2 = freqz(bz2,az2,f,fs);% amplitude response of above bz3,

11、az3 = bilinear(b,a,fs,fc);% bilinear z filter (prewarped) h3 = freqz(bz3,az3,f,fs);% amplitude response of above subplot(211)% subplot 1 plot(f,abs(h),f,abs(h1)% plot xlabel(Frequency - Hz)% label x axis ylabel(Amplitude Response)% label y axis subplot(212)% subplot 2 plot(f,abs(h2),f,abs(h3)% plot

12、xlabel(Frequency - Hz)% label x axis ylabel(Amplitude Response)% label y axis 53 54 01020304050 0 0.5 1 1.5 Frequency - Hz Amplitude Response 01020304050 0 0.5 1 1.5 Frequency - Hz Amplitude Response 模拟原型 冲激不变法 双线性z 变换 带预歧变 未预歧变 55 56 1(1, )(552)y ny nnn (1, )(553) 2 T nnx nx n 1(554) 2 T y ny nx nx

13、 n 57 1 1 ( )1 ( )(555) ( )2 1 Y zTz H z X zz 1 1 2 1 1 ( )( )(556)z a s=T z H zHs 1 1 2 1 1 ( )( )(557)z a s= T z z H zHs z 58 59 60 61 1,| 0.1,0.250.35 ( )(558) 0,0.120.23,0.371 d ff Af ff 62 63 ： 64 00.20.40.60.81 -2000 -1000 0 1000 Normalized Frequency ( rad/sample) Phase (degrees) 00.20.40.60.8

14、1 -100 -50 0 50 Normalized Frequency ( rad/sample) Magnitude (dB) 65 66 67 0 ()()()(559) N k y nTh kT xnk T 00 (560) NN k kk y nh k x nkb x nk 0 ( )( )(561) N k k k Y zb zX z 68 69 70 71 72 22 00 (562) N kk h kh k 0 (563) others kN w k 73 74 75 76 1 2 0 exp2(564) M jfT n H eh njnfT 2 exp2 ()(565) L

15、jfT kL H eh kLjkL fT 77 2 exp2exp2(566) L jfT kL H ejLfTh kLjkfT 22 1 exp2(567) jfTjfT H ejLfT He 2 11 exp2(568) L jfT kL Heh kjkfT 78 1 ( )exp2exp2(569) L kL A fjmfTh kjmk fT 1 ( , ) ( )exp2(570) s s L f /2 -f /2 kL h k I m kA fjmfT df ( , )exp2(571) s s L f /2 -f /2 kL I m kjmk fT df 79 sin ( , )(

16、572) mk I m km-k Tm-kT 1 ( )exp2(573) s s f /2 -f /2 h mTA fjmfT dfLmL /2 ( )(574) s ff A f others 80 1 (1)exp2(575) s s f /2 -f /2 h mTjmfT df 1 11 exp2exp2(576) 2 ss h mTjm f Tjm f T mT j 1 sin (577) m h m m 81 82 83 cos(578) n w nLnL L 84 0102030405060 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 Samp

17、le index Impulse response 85 00.20.40.60.81 -2000 -1000 0 Normalized Frequency ( rad/sample) Phase (degrees) 00.20.40.60.81 -200 -100 0 100 Normalized Frequency ( rad/sample) Magnitude (dB) 86 00.20.40.60.81 0 0.5 1 1.5 Frequency (normalized to the Nyquist frequency = fs/2) |H(f)| (unwindowed) 00.20

18、.40.60.81 0 0.5 1 1.5 Frequency (normalized to the Nyquist frequency = fs/2) |H(f)| (windowed) 87 ( )(579) 1/ n c A f ff ()(580) 1/ k n kc A f ff 88 % File: c5_firbutter.m order = 30;fc =5;% set filter parameters fmax = 100;% set max frequency npts = 256;% set number of samples f = (0:(npts-1)*(fmax

19、/(npts-1);% frequency vector nn = 2*npts;% size ifft H = zeros(1,nn);% initialize vector Ha = 1./(sqrt(1+(f/fc).order);% amplitude response H = Ha 0 fliplr(Ha(2:npts);% even amplitude response cimp_resp = ifft(H,nn);% complex impulse response imp_resp = real(cimp_resp);% take real part aa = imp_resp

20、(1:npts);% time = 0 bb = imp_resp(npts+1):nn);% time 0 reimpulse = bb aa;% real and even imp. resp. plot(reimpulse)% plot result 89 90 0100200300400500600 -0.1 0 0.1 0.2 0.3 0.4 Sample Index Impulse response 0100200300400500600 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 Sample Index Impulse respons

21、e 91 ( )(581) k k d tdt -kT 92 ( )(582) k k x td p t -kT 1 ,0 2 111 ( )1 cos,(583) 2222 1 0, 2 Tf T TT P fff TTT f T 93 2 22 cos/sin/ ( )(584) /14/ t Tt T p t t TtT (585) ss s nTmkTtn m TkTk 94 2 2 cos( / )sin( / ) 2(5 86) ( / ) 1 4( / ) n kmn km p nnm n km n km 1(587)y np nx np n 95 1 2 2 cos1( / )

22、sin1( / )4( / ) 4(589) 1 16( / ) SQRC n kmn kmn km pn Tn km 1 2 22 cos1/sin1/4/ ( )4(588) 1 16/ SQRC t Tt Tt T pt TtT 96 11(590) SQRCSQRC pnnpn 97 0246810 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 Time Amplitude 98 2345678 -1 0 1 2 Time hn-1 6789101112 -5 0 5 10 Time conv(hn-1,hn) 99 100 101 102 103 104 105 106 0200400600 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Time Sample Index Filter Impulse Response 0200400600 0 0.02 0.04 0.0