循环卷积与线性卷积_dsp循环卷积与线性卷积

1 循环卷积

function[yn]=circularconvolution(xn,hn,N);

N=input("N=");

xn=[2 2 1 1 zeros(1,N-4)];

hn=[1 2 0 1 zeros(1,N-4)];

k=[0:1:N-1];

n=[0:1:N-1];

Wn=exp(-j*2*pi/N);

kn=k"*n;

Wn_kn=Wn.^kn;

Xk=xn*Wn_kn;

k=[0:1:N-1];

n=[0:1:N-1];

Wn=exp(-j*2*pi/N);

kn=k"*n;

Wn_kn=Wn.^kn;

Hk=hn*Wn_kn;

Yk=Xk.*Hk;

k=[0:1:N-1];

n=[0:1:N-1];

Wn=exp(-j*2*pi/N);

kn=k"*n;

Wn_kn=Wn.^(-kn);

yn1=(Yk*Wn_kn)/N;

yn=real(yn1);

输入N=4 ，6 ，8 ，10

结果[6 7 6 5][3 6 5 5 4 1][2 6 5 5 4 1 1 0][2 6 5 5 4 1 1 0 0 0]

当N 大于6时，循环卷积与线性卷积结果一致。

2

n=0:47;

xn1 = cos(n*pi/6);

xn2 = cos(n*pi/3);

hn=[-0.0322,-0.0531,-0.0450,0.0000,0.0750,0.1592,0.2251,

0.2500,0.2251,0.1592,0.0750,0.0000,-0.0450,-0.0531,-0.0322]

N=length(hn)+47;

xn=xn1+xn2;

h1n = [hn,zeros(1,47)];

x1n = [xn,zeros(1,14)];

k=[0:1:N-1];

n=[0:1:N-1];

Wn=exp(-j*2*pi/N);

kn=k"*n;

Wn_kn=Wn.^kn;

Xk = x1n*Wn_kn;

k=[0:1:N-1];

n=[0:1:N-1];

Wn=exp(-j*2*pi/N);

kn = k"*n;

Wn_kn = Wn.^kn;

Hk = h1n*Wn_kn;

Yk=Xk.*Hk;

k=[0:1:N-1];

n=[0:1:N-1];

Wn=exp(-j*2*pi/N);

kn=k"*n;

Wn_kn=Wn.^(-kn);

yn1 = (Yk*Wn_kn)/N;

yn = real(yn1);

stem(0:61,yn);

3

n=0:39; k=10;

xn=cos(pi*n/10)+cos(2*n*pi/5); hn=[1,1,1,1,1];

m=length(hn);

subplot(221)

stem(n,xn,".")

xlabel("n"); ylabel("Amplitude"); title("x(n)");

subplot(223)

stem(n,[hn,zeros(1,35)],".")

axis([0,40,0,1.2]);

xlabel("n"); ylabel("Amplitude"); title("h(n)");

xk_n1=xn(1:k); yk_n1=conv(hn,xk_n1);

add_zeros1=zeros(1,length(n)-length(yk_n1)+m-1);

yk_n1=[yk_n1,add_zeros1];

subplot(522)

stem(0:length(yk_n1)-1,yk_n1,".")

xlabel("n"); ylabel("Amplitude"); legend("y0")

xk_n2=xn(1+k:2*k); yk_n2=conv(hn,xk_n2);

add_zeros21=zeros(1,k);

add_zeros22=zeros(1,length(n)-length(yk_n2)-k+m-1);

yk_n2=[add_zeros21,yk_n2,add_zeros22];

subplot(524)

stem(0:length(yk_n2)-1,yk_n2,".")

xlabel("n"); ylabel("Amplitude");legend("y1")

xk_n3=xn(1+2*k:3*k); yk_n3=conv(hn,xk_n3);

add_zeros31=zeros(1,2*k);

add_zeros32=zeros(1,length(n)-length(yk_n3)-2*k+m-1);

yk_n3=[add_zeros31,yk_n3,add_zeros32];

subplot(526)

stem(0:length(yk_n3)-1,yk_n3,".")

xlabel("n"); ylabel("Amplitude"); legend("y2",2)

xk_n4=xn(1+3*k:4*k); yk_n4=conv(hn,xk_n4);

add_zeros4=zeros(1,k*3);

yk_n4=[add_zeros4,yk_n4];

subplot(528)

stem(0:length(yk_n4)-1,yk_n4,".")

xlabel("n"); ylabel("Amplitude"); legend("y3",2)

y1=yk_n1(k+1:k+m-1)+yk_n2(k+1:k+m-1);

y2=yk_n2(2*k+1:2*k+m-1)+yk_n3(2*k+1:2*k+m-1);

y3=yk_n3(3*k+1:3*k+m-1)+yk_n4(3*k+1:3*k+m-1);

yk=[yk_n1(1:k),y1,yk_n2(k+m:k*2),y2,yk_n3(2*k+m:3*k),y3,yk_n4(3*k+m:4*k)]; subplot(5,2,10)

stem(0:length(yk)-1,yk,".")

xlabel("n"); ylabel("Amplitude"); legend("y(n)",3) ；

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