ch7 随机信号的产生与处理

1、 2 3 4 5 6 7 ,cos 2(7 1) ii x tAft 8 ,1cos 2(72) ii x tAuft 9 % File: c7_sinewave.m f = 1;% frequency of sinusoid fs = 100;% sampling frequency t = (0:200)/fs;% time vector for i=1:20 x(:,i) = cos(2*pi*f*t+rand(1)*2*pi); y(:,i) = cos(2*pi*f*t+rand(1)*pi/2); z(:,i) = (1+rand(1)*cos(2*pi*f*t); end sub

2、plot(3,1,1); plot(t,x,k); ylabel(x(t) subplot(3,1,2); plot(t,y,k); ylabel(y(t) subplot(3,1,3); plot(t,z,k); ylabel(z(t) % End of script file. 10 11 00.511.52 -1 0 1 x(t) 00.511.52 -1 0 1 y(t) 00.511.52 -2 0 2 z(t) 12 00.20.40.60.811.21.41.61.82 -1 0 1 x(t) 00.20.40.60.811.21.41.61.82 -1 0 1 y(t) 00.

3、20.40.60.811.21.41.61.82 -2 0 2 z(t) 13 14 x=random binary(nbits,nsamples)+i*random binary (nbits,nsamples) ; 15 020406080 -1 0 1 Sample Index xd 020406080 -1 0 1 Sample Index xq 16 0100200300400500600700800 -1.5 -1 -0.5 0 0.5 1 1.5 Sample Index xd 0100200300400500600700800 -1.5 -1 -0.5 0 0.5 1 1.5

4、Sample Index xq 17 1 mod( )(73) ii xaxcm 18 19 1 mod( )(74) ii xaxcm 20 21 22 1 1 (714) m a k m 1 ,1,2,3,2(715) i a kim m 23 24 25 26 % File: c7_LCDemo1.m m = 2048; c = 1; seed = 1;% default values of m and c a1 = 65; a2 = 1229;% multiplier values ix1 = seed; ix2 = seed;% initialize algorithm x1 = z

5、eros(1,m); x2 = zeros(1,m); % initialize arrays for i=1:m ix1 = rem(ix1*a1+c),m); x1(i) = ix1/m; ix2 = rem(ix2*a2+c),m); x2(i) = ix2/m; end subplot(1,2,1) y1 = x1(1,2:m),x1(1,1); plot(x1,y1,.)% plot results for a1 subplot(1,2,2) y2 = x2(1,2:m),x2(1,1); plot(x2,y2,.)% plot results for a2 % End of scr

6、ipt file. 2700.51 0 0.2 0.4 0.6 0.8 1 00.51 0 0.2 0.4 0.6 0.8 1 28 2 2 2 1 1 1 (7 19) 1 N n N n NX nX n D NXn 22 22 (720) x DEXYE XEXYa 29 2 1(721)YXZ 2 2222 2 1 12 111(722)DEXXZZ 30 % File: c7_Durbin function D = durbin(x) N = length(x);% length of input vector y = x-mean(x);% remove dc ydiff = y(2

7、:N)-y(1:(N-1);% numerator summand Num = sum(ydiff.*ydiff);% numerator factor of D Den = sum(y.*y);% denominator factor of D D = Num/Den;% Durbin factor % End of function file. 2 222 2 11 (723)D 31 % File: c7_LCDemo2 m = 2048; c = 1; seed = 1; a1 = 65; a2 = 1229; ix1 = 1; ix2 = 1; x1 = zeros(1,m); x2

8、 = zeros(1,m); for i=1:m ix1 = rem(ix1*a1+c),m); x1(i) = ix1; ix2 = rem(ix2*a2+c),m); x2(i) = ix2; end D1 = c7_Durbin(x1); D2 = c7_Durbin(x2); % calculate Durbin parameters rho1 = 1-D1/2; rho2 = 1- D2/2;% calculate correlation text1 = The value of D1 is ,num2str(D1), and rho1 is ,num2str(rho1),.; te

9、xt2 = The value of D2 is ,num2str(D2), and rho2 is ,num2str(rho2),.; disp(text1) disp(text2) % End of script file. 32 c71.CDemo2 The value of D1 is 1.9925 and rhol is 0.0037273. The value of D2 is 1.6037 and rho2 is 0.19814. 33 34 xi+1(16807xi)mod(2147483647) （7-24） ( )cos210000 10000cos210001 10000

10、(725)x ttt 35 1 1 1 171mod 30269(726) 170mod 30307(727) 172mod 30323(728) ii ii ii xx yy zz 36 mod 1(729) 302693030730323 iii i xyz u 16555425264690(730)a 13 30269 30307 303232.7817 10(731)m 37 38 39 40 41 42 (732) X UFX 1 (733) X XFU 43 Pr(734) X FxXx 44 1 PrPr(735) XXX FxFUxUFxFx 45 exp -( ), 0(73

11、6) X fxx u x 1,0 ( )=(737) 0,0 x u x x 0 exp -exp(738) x X Fxy dyx exp(739)XU 46 exp(740)XU 1 ln(741)XU 47 % File: c7_uni2exp clear all% be safe n = input(Enter number of points ); b = 3;% set pdf parameter u = rand(1,n);% generate U y_exp = -log(u)/b;% transformation N_samp,x = hist(y_exp,20);% get

12、 histogram parameters subplot(2,1,1) bar(x,N_samp,1)% plot histogram ylabel(Number of Samples) xlabel(Independent Variable - x) subplot(2,1,2) y = b*exp(-3*x);% calculate pdf del_x = x(3)-x(2);% determine bin width p_hist = N_samp/n/del_x;% probability from histogram plot(x,y,k,x,p_hist,ok)% compare

13、 ylabel(Probability Density) xlabel(Independent Variable - x) legend(true pdf,samples from histogram,1) % End of script file. 48 00.511.52 0 10 20 30 Number of Samples Independent Variable - x 00.511.52 0 1 2 3 Probability Density Independent Variable - x true pdf samples from histogram 49 00.511.52

14、2.53 0 500 1000 Number of Samples Independent Variable - x 00.511.522.53 0 1 2 3 Probability Density Independent Variable - x true pdf samples from histogram 50 2 22 =exp(742) 2 R rr fru r 22 222 0 =expexp(743) 22 r R yyr Frdy 51 2 2 exp(744) 2 R U 2 2 exp(745) 2 R U 2 2ln(746)RU 52 53 02468 0 500 1

15、000 1500 Number of Samples Independent Variable - x 02468 0 0.2 0.4 Probability Density Independent Variable - x true pdf samples from histogram 54 % File: c7_uni2ray clear all% be safe n = input(Enter number of points ); varR = 3;% set pdf parameter u = rand(1,n);% generate U y_exp = sqrt(-2*varR*l

16、og(u);% transformation N_samp,r = hist(y_exp,20);% get histogram parameters subplot(2,1,1) bar(r,N_samp,1)% plot histogram ylabel(Number of Samples) xlabel(Independent Variable - x) subplot(2,1,2) term1 = r.*r/2/varR;% exponent ray = (r/varR).*exp(-term1);% Rayleigh pdf del_r = r(3)-r(2);% determine

17、 bin width p_hist = N_samp/n/del_r;% probability from histogram plot(r,ray,k,r,p_hist,ok)% compare results ylabel(Probability Density) xlabel(Independent Variable - x) legend(true pdf,samples from histogram,1) % End of script file. 55 56 11 Pr(747) iiiiii Pxxxcxx 11 ( )(748) Xiii FxFcxx 1 11 1 Pr(74

18、9) i iij j FXxP 11 ()(750) Xiii FXUFcXx 11 1 (751) ii i XxUF c 57 1 (752) ii FUF 58 for all(753) XX Mgxfx x / ,0 (754) 0, X bM axa Mgx others 59 /max(755) X bM afx 60 61 0 Pr accepted (756) a X i fx dx V ab 11 Pr accepted (757) i V abM 62 11 11 1 Pr +,accepted Pr );% set N fx = zeros(1,N);% array of

19、 output samples u1 = rand(1,N); u2 = rand(1,N);% generate u1 and u2 v1 = R*u1;% generate v1 v2 = (M/R)*rand(1,N);% generate v2 (g(x) kpts = 0;% initialize counter for k=1:N if v2(k)(M/(R*R)*sqrt(R*R-v1(k)*v1(k); kpts=kpts+1;% increment counter fx(kpts)=v1(k); % save output sample end end fx = fx(1:k

20、pts); N_samp,x = hist(fx,20);% get histogram parameters subplot(2,1,1) bar(x,N_samp,1)% plot histogram ylabel(Number of Samples) xlabel(Independent Variable - x) subplot(2,1,2) yt = (M/R/R)*sqrt(R*R-x.*x);% calculate pdf del_x = x(3)-x(2); % determine bin width p_hist = N_samp/kpts/del_x;% probabili

21、ty from histogram plot(x,yt,k,x,p_hist,ok)% compare ylabel(Probability Density) xlabel(Independent Variable - x) legend(true pdf,samples from histogram,3) text = The number of points accepted is ,. num2str(kpts,15), and N is ,num2str(N,15),.; disp(text) % End of script file. 65 02468 0 100 200 Numbe

22、r of Samples Independent Variable - x 02468 0 0.1 0.2 Probability Density Independent Variable - x true pdf samples from histogram 66 02468 0 1000 2000 Number of Samples Independent Variable - x 02468 0 0.1 0.2 Probability Density Independent Variable - x true pdf samples from histogram 67 68 2 2 1

23、( )exp1(761) 22 x X yx FxdyQ 2 exp/2(762) x Q xydy 69 1/2(763) N i i=0 YBU - 70 1/2(764) N i i=0 E YBE U- 2 var1/2(765) i Ux dx 2 var(766) N yi i BU 2 (767) y NB 71 72 (768) y BN (769) 2 y N N 73 74 75 76 22 22 22 22 11 ( , )expexp 2222 1 exp(770) 22 XY xy fx y xy 77 222 (771)xyr 1 tan(772) y x ,(77

24、3) RRXYXY frdAfx y dA cos sin ,(774) XY RXY x r R y r dA frfx y dA 78 /, /, cossin (775) sincos XY R dx drdx dx y dA dy drdy ddAr r r r 2 22 ,exp, 22 0,02(777) R rr fr r 79 22 2 2222 0 expexp, 222 0(778) R rrrr frd r 2 22 0 1 exp, 222 02(779) rr fdr 80 cos(780)XR sin(781)YR 2 12 2lncos2(782)XUU 2 12

25、 2lnsin2(783)YUU 81 % File: c7_boxmul.m function out1,out2=c7_boxmul(N) u1 = rand(1,N);% generate first uniform RV u2 = rand(1,N);% generate second uniform RV ray = sqrt(-2*log(u1);% generate Rayleigh RV out1 = ray.*cos(2*pi*u2);% first Gaussian output out2 = ray.*sin(2*pi*u2);% second Gaussian outp

26、ut % End of function file. 82 83 84 Pr10.2146(784) circle box A reject A 85 86 87 88 89 2 1(785)ZXY 2 22 22222 1 211(786) Z EXY E XE XYE Y 222222 1(787) Z 90 22 (788)E XZ 2 2 (789) XZ XZ E XZ 91 2 0 /2(790) ns N f 2 0 2(791) sn fN 2 ( )( )(792) YX SfH fSf 92 93 2 ( )(793) Y SfH fK ( )/(794) Y H fSfK

27、 94 1 01 1 1 (795) 1 n n n n bb zb z H z a za z 95 0 0 0| (796) /| ff Hf Kfff 96 00.20.40.60.81 0 2 4 6 Normalized Frequency Magnitude Response 00.20.40.60.81 0 2 4 6 Normalized Frequency Magnitude Response 97 2 1/ 1/,| (797) 0, dd ffff S f others 1/4 2 1/,| (798) 0, dd ffff Hf others 98 99 00.020.040.060.08 -0.5 0 0.5 Time Impulse Response -1000-50005001000 0 0.5 1 1.5 x 10 -3 Frequency PSD 100 -5000500 -40 -20 0 Frequency PSD 0.10.150.20.25 -20 -10 0 Time