現(xiàn)代信號(hào)處理大作業(yè)題目答案

1、研究生“現(xiàn)代信號(hào)處理”課程大型作業(yè)(以下四個(gè)題目任選三題做1. 請(qǐng)用多層感知器(MLP 神經(jīng)網(wǎng)絡(luò)誤差反向傳播(BP 算法實(shí)現(xiàn)異或問(wèn)題(輸入為00;01;10;11X T =,要求可以判別輸出為0或1,并畫出學(xué)習(xí)曲線。其中,非線性函數(shù)采用S 型Logistic 函數(shù)。2. 試用奇階互補(bǔ)法設(shè)計(jì)兩帶濾波器組(高、低通互補(bǔ),進(jìn)而實(shí)現(xiàn)四帶濾波器組;并畫出其頻響。濾波器設(shè)計(jì)參數(shù)為:F p =1.7KHz , F r =2.3KHz , F s =8KHz , A rmin 70dB 。3. 根據(jù)現(xiàn)代數(shù)字信號(hào)處理(姚天任等,華中理工大學(xué)出版社,2001第四章附錄提供的數(shù)據(jù)(pp.352-353,試用如下方法

2、估計(jì)其功率譜,并畫出不同參數(shù)情況下的功率譜曲線:1 Levinson 算法2 Burg 算法3 ARMA 模型法4 MUSIC 算法4. 圖1為均衡帶限信號(hào)所引起失真的橫向或格型自適應(yīng)均衡器(其中橫向FIR 系統(tǒng)長(zhǎng)M =11, 系統(tǒng)輸入是取值為±1的隨機(jī)序列(n x ,其均值為零;參考信號(hào)7(-=n x n d ;信道具有脈沖響應(yīng):12(21cos(1,2,3(20 n n h n W -+=其它式中W 用來(lái)控制信道的幅度失真(W = 24, 如取W = 2.9,3.1,3.3,3.5等,且信道受到均值為零、方差001.02=v (相當(dāng)于信噪比為30dB的高斯白噪聲(n v 的干擾。

3、試比較基于下列幾種算法的自適應(yīng)均衡器在不同信道失真、不同噪聲干擾下的收斂情況(對(duì)應(yīng)于每一種情況,在同一坐標(biāo)下畫出其學(xué)習(xí)曲線:1 橫向/格-梯型結(jié)構(gòu)LMS 算法2 橫向/格-梯型結(jié)構(gòu)RLS 算法并分析其結(jié)果。 圖1 橫向或格-梯型自適應(yīng)均衡器參考文獻(xiàn)1 姚天任, 孫洪. 現(xiàn)代數(shù)字信號(hào)處理M. 武漢: 華中理工大學(xué)出版社, 20012 楊綠溪. 現(xiàn)代數(shù)字信號(hào)處理M. 北京: 科學(xué)出版社, 20073 S. K. Mitra. 孫洪等譯. 數(shù)字信號(hào)處理基于計(jì)算機(jī)的方法(第三版M. 北京: 電子工業(yè)出版社, 20064 S.Haykin, 鄭寶玉等譯. 自適應(yīng)濾波器原理(第四版M.北京: 電子工業(yè)出版

4、社, 20035 J. G. Proakis, C. M. Rader, F. Y. Ling, etc. Algorithms for Statistical Signal Processing M.Beijing: Tsinghua University Press, 2003一、請(qǐng)用多層感知器(MLP神經(jīng)網(wǎng)絡(luò)誤差反向傳播(BP算法實(shí)現(xiàn)異或問(wèn)題(輸入為00;01;10;11,要求可以判別輸出為0或1,并畫出學(xué)習(xí)曲線。其X T中,非線性函數(shù)采用S型Logistic函數(shù)。1、原理:反向傳播(BP算法:(1、多層感知器的中間隱層不直接與外界連接,其誤差無(wú)法估計(jì)。(2、反向傳播算法:從后向前(反向

5、逐層“傳播”輸出層的誤差,以間接算出隱層誤差。分兩個(gè)階段:正向過(guò)程:從輸入層經(jīng)隱層逐層正向計(jì)算各單元的輸出反向過(guò)程:由輸出層誤差逐層反向計(jì)算隱層各單元的誤差,并用此誤差修正前層的權(quán)值。2、流程圖:開始選擇初始值j3、程序:%使用了3層結(jié)構(gòu),第二層隱藏層4個(gè)單元。2,3層都使用Logisitic函數(shù)。%訓(xùn)練xor數(shù)據(jù)。function mlp(f= fopen('XOR.txt'A = fscanf(f, '%g',3 inf;A = A;p = A(1:2, :'%訓(xùn)練輸入數(shù)據(jù)t = A(3, :'%desire outtrain_num , i

6、nput_scale= size(p ;%規(guī)模fclose(f;accumulate_error=zeros(1,3001;alpha = 0.5;%學(xué)習(xí)率threshold = 0.005;% 收斂條件e2 < thresholdwd1=0; wd2=0;bd1=0; bd2=0;circle_time =0;hidden_unitnum = 4; %隱藏層的單元數(shù)w1 = rand(hidden_unitnum,2;%4個(gè)神經(jīng)元,每個(gè)神經(jīng)元接受2個(gè)輸入w2 = rand(1,hidden_unitnum;%一個(gè)神經(jīng)元,每個(gè)神經(jīng)元接受4個(gè)輸入b1 = rand(hidden_unitn

7、um,1;b2 = rand(1,1;while 1temp=0;circle_time = circle_time +1;for i=1:train_num%前向傳播a0 = double ( p(i,:' ;%第i行數(shù)據(jù)n1 = w1*a0+b1;a1 = Logistic(n1;%第一個(gè)的輸出n2 = w2*a1+b2;a2 = Logistic(n2;%第二個(gè)的輸出a = a2;%后向傳播敏感性e = t(i,:-a;accumulate_error(circle_time = temp + abs(e2;temp=accumulate_error(circle_time;s2

8、 = F(a2*e; %輸出層delta值s1 = F(a1*w2'*s2;%隱層delta值%修改權(quán)值wd1 = alpha .* s1*a0'wd2 = alpha .* s2*a1'w1 = w1 + wd1;w2 = w2 + wd2;bd1 = alpha .* s1;bd2 = alpha .* s2;b1 = b1 + bd1;b2 = b2 + bd2;end;%end of forif accumulate_error(circle_time <= threshold| circle_time>3001 %then break;end;%e

9、nd of ifend;%end of whileplot(accumulate_error,'m'grid;xlabel('學(xué)習(xí)次數(shù)'ylabel('誤差'disp('計(jì)算誤差= ',num2str(accumulate_error(circle_time ;disp('迭代次數(shù)= ',num2str(circle_time;%測(cè)試a0 = double (0 0'n1 = w1*a0+b1;a1 = Logistic(n1;n2 = w2*a1+b2;a2 = Logistic(n2;a = a2;di

10、sp('0 0 = ',num2str(a;a0 = double (0 1'n1 = w1*a0+b1;a1 = Logistic(n1;n2 = w2*a1+b2;a2 = Logistic(n2;a = a2;disp('0 1 = ',num2str(a;a0 = double (1 0'n1 = w1*a0+b1;a1 = Logistic(n1;n2 = w2*a1+b2;a2 = Logistic(n2;a = a2;disp('1 0 = ',num2str(a;a0 = double (1 1'n1 =

11、w1*a0+b1;a1 = Logistic(n1;n2 = w2*a1+b2;a2 = Logistic(n2;a = a2;disp('1 1 = ',num2str(a;m=0;%- function a= Logistic(na = 1./(1+exp(-n;%- function result= F(ar,c = size(a;result = zeros(r,r;for i =1:rresult(i,i = (1-a(i*a(i;end;4、實(shí)驗(yàn)結(jié)果:計(jì)算誤差= 0.0049993迭代次數(shù)= 27060 0 = 0.0231820 1 = 0.9631101 0 =

12、 0.9653901 1 = 0.0433745、學(xué)習(xí)曲線圖: 圖1.MLP二、試用用奇階互補(bǔ)法設(shè)計(jì)兩帶濾波器組(高、低通互補(bǔ),進(jìn)而實(shí)現(xiàn)四帶濾波器組;并畫出其頻響。濾波器設(shè)計(jì)參數(shù)為:F p =1.7KHz , F r =2.3KHz , F s =8KHz , A rmin 70dB 。1、設(shè)計(jì)步驟:(1對(duì)Fp 、Fr 進(jìn)行預(yù)畸;(;(''Fs Fr tg Fs Fp tg r p = (2計(jì)算'''*r p c =,判斷'c 是否等于1,即該互補(bǔ)濾波器是否為互補(bǔ)鏡像濾波器(3計(jì)算相關(guān)系數(shù)-=+=+-=-=-=偶數(shù)N 為(;21 奇數(shù)N 為 (;

13、lg /16/1lg(;150152;1121;1;110(110(1213090500''02'''211-min 1.0min 1.0i i u q k N q q q q q k k q k k k k r pAr Ap ;2cos(1(2112(sin(1(2101(21'2=+-+-=m m m m m m m i u Nm q u N m q q ;42=N N ;221N N N -= ;/1(1(2'2'k k v i i i -= 12'1212,1;12N i v i i i =+=- 22'2

14、2,1;12N i v ii i =+= (4互補(bǔ)鏡像濾波器的數(shù)字實(shí)現(xiàn) ;22i i i A +-=;22i i i B +-= 1221,1;1(N i Z A Z A Z H i i i =+=- 22212,1;1(N i Z B Z B Z Z H i ii =+=- ;(21(21Z H Z H Z H L += 2、程序:function filter2(Fp=1700;Fr=2300;Fs=8000;Wp=tan(pi*Fp/Fs;Wr=tan(pi*Fr/Fs;Wc=sqrt(Wp*Wr;k=Wp/Wr;k1=sqrt(sqrt(1-k2;q0=0.5*(1-k1/(1+k1;

15、q=q0+2*q05+15*q09+150*q013;N=11;N2=fix(N/4;M=fix(N/2;N1=M-N2;for jj=1:Ma=0;for m=0:5a=a+(-1m*q(m*(m+1*sin(2*m+1*pi*jj/N;%N is odd, u=j endab=0;for m=1:5b=b+(-1m*q(m2*cos(2*m*pi*jj/N;endbW(jj=2*q0.25*a/(1+2*b;V(jj=sqrt(1-k*W(jj2*(1-W(jj2/k;endfor i=1:N1alpha(i=2*V(2*i-1/(1+W(2*i-12;endfor i=1:N2beta(

16、i=2*V(2*i/(1+W(2*i2;endfor i=1:N1a(i=(1-alpha(i*Wc+Wc2/(1+alpha(i*Wc+Wc2;endfor i=1:N2b(i=(1-beta(i*Wc+Wc2/(1+beta(i*Wc+Wc2;endw=0:0.0001:0.5;LP=zeros(size(w;HP=zeros(size(w;for n=1:length(wz=exp(j*w(n*2*pi;H1=1;for i=1:N1H1=H1*(a(i+z(-2/(1+a(i*z(-2 ;endH2=1/z;for i=1:N2H2=H2*(b(i+z(-2/(1+b(i*z(-2;e

17、ndLP(n=abs(H1+H2/2;HP(n=abs(H1-H2/2;endplot(w,LP,'k',w,HP,'m'%hold on;xlabel('數(shù)字頻率'ylabel('幅度'3、實(shí)驗(yàn)結(jié)果: 圖2.兩帶濾波器4、四帶濾波器組程序:function filterfourFp=1700;Fr=2300;Fs=8000;Wp=tan(pi*Fp/Fs;Wr=tan(pi*Fr/Fs;Wc=sqrt(Wp*Wr;k=Wp/Wr;k1=sqrt(sqrt(1-k2;q0=0.5*(1-k1/(1+k1;q=q0+2*q05+15

18、*q09+150*q013;N=11;N2=fix(N/4;M=fix(N/2;N1=M-N2;for jj=1:Ma=0;for m=0:5a=a+(-1m*q(m*(m+1*sin(2*m+1*pi*jj/N; % N is odd, u=j endb=0;for m=1:5b=b+(-1m*q(m2*cos(2*m*pi*jj/N;endW(jj=2*q0.25*a/(1+2*b;V(jj=sqrt(1-k*W(jj2*(1-W(jj2/k;endfor i=1:N1alpha(i=2*V(2*i-1/(1+W(2*i-12;endfor i=1:N2beta(i=2*V(2*i/(1+

19、W(2*i2;endfor i=1:N1a(i=(1-alpha(i*Wc+Wc2/(1+alpha(i*Wc+Wc2;endfor i=1:N2b(i=(1-beta(i*Wc+Wc2/(1+beta(i*Wc+Wc2;endw=0:0.0001:0.5;LLP=zeros(size(w;LHP=zeros(size(w;HLP=zeros(size(w;HHP=zeros(size(w;for n=1:length(wz=exp(j*w(n*2*pi;H1=1;for i=1:N1H1=H1*(a(i+z(-2/(1+a(i*z(-2 ;endH21=1;for i=1:N1H21=H21

20、*(a(i+z(-4/(1+a(i*z(-4 ;endH2=1/z;for i=1:N2H2=H2*(b(i+z(-2/(1+b(i*z(-2;endH22=1/(z2;for i=1:N2H22=H22*(b(i+z(-4/(1+b(i*z(-4;endLP=(H1+H2/2;HP=(H1-H2/2;LLP(n=abs(H21+H22/2*LP;LHP(n=abs(H21-H22/2*LP;HHP(n=abs(H21+H22/2*HP;HLP(n=abs(H21-H22/2*HP;endplot(w,LLP,'k',w,LHP,'m',w,HLP,'g

21、',w,HHP,'b'xlabel('數(shù)字頻率'ylabel('幅度'5、實(shí)驗(yàn)結(jié)果: 圖3.四帶濾波器組三、根據(jù)現(xiàn)代數(shù)字信號(hào)處理(姚天任等,華中理工大學(xué)出版社,2001第四章附錄提供的數(shù)據(jù)(pp.352-353,試用如下方法估計(jì)其功率譜,并畫出不同參數(shù)情況下的功率譜曲線:1Levinson算法2Burg算法3ARMA模型法4MUSIC算法1、Levinson算法分析:(1、由輸入數(shù)據(jù)估計(jì)自相關(guān)函數(shù),一種漸近無(wú)偏估計(jì)(稱之為取樣自相關(guān)函數(shù)的公式為:-=-+=mN n xxN m n m x n x Nm R 10*1,(1(2、根據(jù)估計(jì)所得

22、到的自相關(guān)函數(shù),用下面的迭代公式估算AR 模型參數(shù):(0(*1,2i R a R ki i k k=+=-+=ki k i k k a i k R a D 00,0,1(21k kk D =+221211(k k k +-=k i a a a i k k k i k i k ,2,1,*1,1,1 =-=-+11,1+-=k k k a (3、對(duì)于AR (p 模型,按以上述遞推公式迭代計(jì)算直到p k =+1時(shí)為止。將算出的模型參數(shù)代入下式即可得到功率譜估計(jì)值:21,21(=-+=pi jwii p pj xxe a e S 程序:function sigma2,a=levinson(signa

23、l_source,p%階數(shù)由p 確定N=length(signal_source; %確定自相關(guān)函數(shù) for m=0:N-1R(m+1=sum(conj(signal_source(1:N-m.*signal_source(m+1:N/N; end%設(shè)置迭代初值 a1=-R(2/R(1;sigma2=(1-abs(a12*R(1; gamma=-a1; %開始迭代 for k=2:psigma2(k=R(1+sum(a1.*conj(R(2:k; D=R(k+1+sum(a1.*R(k:-1:2; gamma(k=D/sigma2(k;sigma2(k=(1-abs(gamma(k2*sigm

24、a2(k;a1=a1-gamma(k*conj(fliplr(a1,-gamma(k;enda=1 a1;%計(jì)算功率譜估計(jì)值sigma2=real(sigma2;p=15;%p分別為15、30、45、60sigma2,a=Levinson(signal_in_complex,p;%計(jì)算功率譜f1=linspace(-0.5,0.5,512;%從-0.5到0.5生成512個(gè)等間隔數(shù)據(jù)for k=1:512S1(k=10*log10(sigma2(end/(abs(1+sum(a(2:end.*exp(-j*2*pi*f1(k*1:p 2; %公式(2.3.7并以dB表示end;實(shí)驗(yàn)結(jié)果: 圖4.

25、Levinson 算法2、 Burg 算法分析:(1、設(shè)輸入數(shù)據(jù)序列為10(-N n n x ,對(duì)前后向預(yù)測(cè)誤差之和求偏導(dǎo),得反射系數(shù)-=-+-=-+-+-=121211*111(1(2N kn k k N k n k k k n e n en e n e 前后向預(yù)測(cè)誤差遞推公式: - -= -+-+1(11(11*n e n e n e n e k k kk k k 1,.,3,2,1,0,1,1,1,=-=-k i k k k i k i k a k i a a a (2、重復(fù)以上步驟直至k =p ,根據(jù)迭代得到的AR 模型參數(shù)計(jì)算功率譜,計(jì)算功率譜的公式同上面算法。程序:function

26、 sigma2,a=burg(signal_source,p N=length(signal_source;ef=signal_source; eb=signal_source;sigma2=sum(signal_source*signal_source'/N; a=; for k=1:pefp=ef(k+1:end; ebp=eb(k:end-1;gamma(k=2*efp*ebp'/(efp*efp'+ebp*ebp' sigma2(k+1=(1-abs(gamma(k2*sigma2(k; ef(k+1:end=efp-gamma(k*ebp; eb(k+

27、1:end=ebp-gamma(k'*efp; a=a-gamma(k*conj(fliplr(a,-gamma(k;end; a=1 a;sigma2=real(sigma2;實(shí)驗(yàn)結(jié)果: 圖5.Burg 算法3、 ARMA 算法分析:(1、用x (n 通過(guò)A (z ,得到y(tǒng) (n 。(2、用一無(wú)窮階的AR 模型近似MA 模型。用Burg 算法可得到此近似AR 模型的參數(shù)以及激勵(lì)白噪聲的功率。一般此AR 模型的階數(shù)應(yīng)大于MA 模型階數(shù)的兩倍,以獲得較好的近似效果。(3、可以證明,將上一步求出的近似AR 模型參數(shù)視為時(shí)間序列,則MA 模型就可視為一線性預(yù)測(cè)濾波器,按最小均方誤差準(zhǔn)則就可以求出MA 模型參數(shù),方法同樣可用Burg 算法。這樣,ARMA 模型的參數(shù)就全部估計(jì)出來(lái)了,根據(jù)以下公式即可算出功率譜:2121211(=-=-+=pi jwii qi jwii pj xx e a e b e S 程序:function a,b,sigma2=arma(signal_sourc