>> W1=[1 1 1 1;1 -j -1 j;1 -1 1 -1;1 j -1 -j] W1 =

>> W1=[1 1 1 1;1 -j -1 j;1 -1 1 -1;1 j -1 -j]

W1 =

1.0000 1.0000 1.0000 1.0000 1.0000 0 - 1.0000i -1.0000 0 + 1.0000i 1.0000 -1.0000 1.0000 -1.0000 1.0000 0 + 1.0000i -1.0000 0 - 1.0000i

>> W2=conj(W1)/4

W2 =

0.2500 0.2500 0.2500 0.2500 0.2500 0 + 0.2500i -0.2500 0 - 0.2500i 0.2500 -0.2500 0.2500 -0.2500 0.2500 0 - 0.2500i -0.2500 0 + 0.2500i

>> W1*W2

ans =

1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

function Xk=dft(xn,N)% Computes Discrete Fourier Transform%Xk: DFT coeff. array over 0<=k<=N-1 (column vector)%xn: N-point finite duration sequence (column vector)%N: Length of DFT n=[0:1:N-1]; % row vector for nk=[0:1:N-1]; % row vector for kWN=exp(-j*2*pi/N); % Wn factornk=n'*k; %create a N by N matrix of nk valuesWNnk=WN.^nk; %DFT matrixXk=WNnk*xn;

function xn=idft(Xk,N)% Computes Discrete Fourier Transform%Xk: DFT coeff. array over 0<=k<=N-1 (column vector)%xn: N-point finite duration sequence (column vector)%N: Length of DFT n=[0:1:N-1]; % row vector for nk=[0:1:N-1]; % row vector for kWN=exp(-j*2*pi/N); % Wn factornk=n'*k; %create a N by N matrix of nk valuesWNnk=WN.^(-nk); %IDFT matrixxn=WNnk*Xk/N;

>> x1=[1 2 3 4]';N=4;>> X1=dft(x1,N);>> x2=idft(X1,4)

x2 =

1.0000 - 0.0000i 2.0000 - 0.0000i 3.0000 - 0.0000i 4.0000 + 0.0000i

>> x=[1 1 1 1]';N=4;X=dft(x,N);>> w=linspace(0,2*pi,512);>> plot(w/pi,4*abs(sinc(2*w/pi)./sinc(w/2/pi)))>> hold on; plot([0:N-1]*2/N,abs(X),'ro');>> title('N=4'); hold off

>> x=[1 1 1 1 zeros(1,4)]';N=8;X=dft(x,N);>> w=linspace(0,2*pi,512);>> plot(w/pi,4*abs(sinc(2*w/pi)./sinc(w/2/pi)))>> hold on; plot([0:N-1]*2/N,abs(X),'ro');>> title('N=8'); hold off

>> x=[1 1 1 1 zeros(1,28)]';N=32;X=dft(x,N);>> w=linspace(0,2*pi,512);>> plot(w/pi,4*abs(sinc(2*w/pi)./sinc(w/2/pi)))>> hold on; plot([0:N-1]*2/N,abs(X),'ro');>> title('N=32'); hold off

>> n=0:5000;x=cos(0.48*pi*n)+cos(0.52*pi*n);>> L=10;x1=[x(1:L) zeros(1,1024-L)]';>> X1=dft(x1,1024);>> plot(2*[0:1023]/1024,abs(X1)); title('L=10')

>> n=0:3

n =

0 1 2 3

>> mod(n-2,4)

ans =

2 3 0 1>> x=1:4

x =

1 2 3 4

>> x(mod(n-2,4)+1)

ans =

3 4 1 2

>> n=0:10; x=10*(0.8).^n; >> y=x(mod(-n,11)+1);>> subplot(2,1,1);stem(n,x); axis([-1 11 0 11]);title('Original sequence')>> subplot(2,1,2);stem(n,y); axis([-1 11 0 11]);title('Circularly folded sequence')

>> X=dft(x',11);Y=dft(y',11);>> subplot(2,2,1);stem(n,real(X));title('Real[DFT[x(n)]]');>> subplot(2,2,2);stem(n,imag(X));title('Imag[DFT[x(n)]]');>> subplot(2,2,3);stem(n,real(Y));title('real[DFT[x((-n))11]]');>> subplot(2,2,4);stem(n,imag(Y));title('imag[DFT[x((-n))11]]');

>> W1=[1 1 1 1;1 -j -1 j;1 -1 1 -1;1 j -1 -j] W1 =

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Transcript of >> W1=[1 1 1 1;1 -j -1 j;1 -1 1 -1;1 j -1 -j] W1 =