add all m files

0712Gaussian_Fourier/Fouriergauss.m

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+function f_hat=Fouriergauss(w,mu,sigma)
+%w: 
+%mu:mean; 
+%sigma: variance;
+%n=length(w);
+f_hat=exp(-0.5*w.^2*sigma-1i*w*mu);
+end

0712Gaussian_Fourier/gauss.m

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+function f=gauss(x,mu,sigma)
+%x: n-1 vector
+%mu:n-1 vector mean; 
+%sigma: n-n matrix variance;
+%n=length(x);
+%f=1/((2*pi)^(n/2)*sqrt(det(sigma)))*exp(-0.5*(x-mu)'*sigma^(-1)*(x-mu));
+n=1;
+f=1/((2*pi)^(n/2)*sqrt(det(sigma)))*exp(-0.5*(x-mu).^2*sigma^(-1));
+end

0712Gaussian_Fourier/main.m

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+close all
+clear
+clc
+
+mu=0;  sigma=1;
+N=128;                %sample number
+t1=-5;t2=5;
+T=(t2-t1);            % Sampling interval
+F0=1/T;               % Minimum frequency interval
+Fs=(N-1)*F0;          % Sampling frequency
+
+t=[t1:(t2-t1)/(N-1):t2]'; % time series
+g=gauss(t,mu,sigma);  % sampled values of the  Gaussian distribution 
+xx=t1:0.01:t2;
+yy=gauss(xx,mu,sigma);
+figure(1);
+plot(xx,yy,'b-');
+hold on
+stem(t,g)
+xlabel('t(s)');ylabel('Amplitude')
+title('Gaussian Function and Sampling');
+legend('Gaussina Function','Sampling'); 
+
+w=2*pi*Fs*(0:N-1)/N;
+wshift=(-N/2:N/2-1)*2*pi*Fs/N;
+G=Fouriergauss(wshift,mu,sigma);  % sampled values of the continuous Fourier transform
+G_fft=abs(fft(g));
+G_fftshift=fftshift(abs(fft(g)))/Fs;
+G_DFTgauss=fftshift(abs(DFT(g)))/Fs;
+G_myfft=fftshift(abs(myfft(g)))/Fs;
+
+figure(2);
+%1
+subplot(2,2,1);
+plot(w,G_fft,'o');
+xlabel('w');ylabel('Amplitude');
+title('FFT N=128')
+grid on;
+%
+subplot(2,2,2);
+plot(w(1:N/2),G_fft(1:N/2),'o');
+xlabel('w');ylabel('Amplitude');
+title('FFT Before Nyquist Frequency')
+grid on;
+%
+subplot(2,2,3);
+plot(wshift,G,'o');
+xlabel('w');ylabel('Amplitude');
+title('Sample of the Continuous Fourier Transform')
+grid on;
+%
+subplot(2,2,4);
+plot(wshift',G_fftshift,'o');
+xlabel('w');ylabel('Amplitude');
+title('FFTSHIFT and Mdified Amplitude')
+grid on;
+
+figure(3)
+index=find((wshift>-5)&(wshift<5));
+w_continuous=t1:0.01:t2;
+G_continuous=Fouriergauss(w_continuous,mu,sigma);
+plot(w_continuous,G_continuous);
+hold on 
+plot(wshift(index),G(index),'o','markersize',10);
+hold on
+plot(wshift(index),G_fftshift(index),'*','color','r');
+xlabel('w(Angular frequency)');
+ylabel('Amplitude');
+legend('Fourier of Gaussian Function','Sample of Continuous FT ','FFT')
+title('Comparisons of using FFT')
+grid on 
+
+figure(4)
+index=find((wshift>-5)&(wshift<5));
+w_continuous=t1:0.01:t2;
+G_continuous=Fouriergauss(w_continuous,mu,sigma);
+plot(w_continuous,G_continuous);
+hold on 
+plot(wshift(index),G(index),'o','markersize',10);
+hold on
+plot(wshift(index),G_DFTgauss(index),'+','color','r');
+xlabel('w(Angular frequency)');
+ylabel('Amplitude');
+legend('Fourier of Gaussian Function','Sample of Continuous FT','DFT')
+title('Comparisons of using DFT')
+grid on 
+
+figure(5)
+index=find((wshift>-5)&(wshift<5));
+w_continuous=t1:0.01:t2;
+G_continuous=Fouriergauss(w_continuous,mu,sigma);
+plot(w_continuous,G_continuous);
+hold on 
+plot(wshift(index),G(index),'o','markersize',10);
+hold on
+plot(wshift(index),G_DFTgauss(index),'.','markersize',10,'color','r');
+xlabel('w(Angular frequency)');
+ylabel('Amplitude');
+legend('Fourier of Gaussian Function','Sample of Continuous FT','My FTT')
+title('Comparisons of using My FFT')
+grid on 
+
+tic
+Gt_fft=fft(g)/Fs;
+t_fft=toc
+tic
+Gt_DFT=DFT(g)/Fs;
+t_dft=toc
+tic
+Gt_myfft=myfft(g)/Fs;
+t_myfft=toc
+
+%%
+
+clc
+clear
+close all
+
+N=128;
+mu=[4.5,0];
+sigma=[0.05 0; 0 0.05];
+t1=-5;t2=5;
+T=(t2-t1);            % Sampling interval
+F0=1/T;               % Minimum frequency interval
+Fs=(N-1)*F0;          % Sampling frequency
+wx=(-N/2:N/2-1)*2*pi*Fs/N;
+wy=wx;
+[WX,WY]=meshgrid(wx,wy);
+[X,Y]=meshgrid((-N/2:N/2-1)*T/N,(-N/2:N/2-1)*T/N);
+p=mvnpdf([X(:) Y(:)],mu,sigma);  %the joint probability density, the Z-axis.
+p=reshape(p,size(X));            
+figure(1)
+set(gcf,'Position',get(gcf,'Position').*[1 1 1.3 1])
+subplot(2,3,[1 2 4 5])
+mesh(X,Y,p),axis tight,title('2-D Gaussian Distribution')
+xlabel('x1');ylabel('x2');
+subplot(2,3,3)
+mesh(X,Y,p),view(2),axis tight,title('Projection on XOY Plane')
+xlabel('x1');ylabel('x2');zlabel('Amplitude');
+subplot(2,3,6)
+mesh(X,Y,p),view([0 0]),axis tight,title('Projection on XOZ Plane')
+xlabel('x1');ylabel('z');
+
+for i=1:N 
+    for j=1:N
+        WXY=[WX(i,j),WY(i,j)];
+        G_2d(i,j)=exp(-0.5*WXY*sigma*WXY'-1i*WXY*mu');
+        G_2d(i,j)=abs(G_2d(i,j)); 
+    end 
+end
+
+G_2dfft=abs(fft2(p))/Fs^2;
+G_2dfftshift=fftshift(G_2dfft);
+
+figure(2)
+subplot(2,2,1);
+mesh(WX,WY,G_2d);
+xlabel('w1');ylabel('w2');zlabel('Amplitude');
+title('Sample of Continuous FT')
+
+subplot(2,2,2);
+%imagesc(G_2d);
+surf(WX,WY,G_2d),view(2),axis tight,title('Top View of Sample of Continuous FT')
+shading interp
+xlabel('w1');ylabel('w2');title('Top View of Sample of Continuous FT')
+colorbar
+
+subplot(2,2,3);
+mesh(WX,WY,G_2dfft);
+xlabel('w1');ylabel('w2');zlabel('Amplitude');
+title('FFT of the 2-D Gaussian Distribution')
+
+subplot(2,2,4);
+surf(WX,WY,G_2dfft),view(2),title('Top View')
+shading interp
+%imagesc(G_2dfft);
+xlabel('w1');ylabel('w2');title('Top View of FFT')
+colorbar
+
+figure(3)
+index=find((wx>-5)&(wx<5));
+set(gcf,'Position',get(gcf,'Position').*[1 1 1.3 1])
+subplot(2,3,[1 2 4 5])
+% surf(WX,WY,G_2dfftshift);
+% shading interp
+for i=index(1):index(end)
+    for j=index(1):index(end)
+        plot3(WX(i,j),WY(i,j),G_2dfftshift(i,j),'.','markersize',10);
+        hold on
+    end
+end
+ xlabel('w1');ylabel('w2');title('Shift FFT of 2-D Guassian')
+
+subplot(2,3,3)
+w_continuous=t1:0.02:t2;
+[W_C1,W_C2]=meshgrid(w_continuous,w_continuous);
+for i=1:length(w_continuous) 
+    for j=1:length(w_continuous)
+        W_C=[W_C1(i,j),W_C2(i,j)];
+        G_2d_C(i,j)=exp(-0.5*W_C*sigma*W_C'-1i*W_C*mu');
+        G_2d_C(i,j)=abs(G_2d_C(i,j)); 
+    end 
+end
+surf(W_C1,W_C2,G_2d_C),view(2);
+xlabel('w1');ylabel('w2');title('Top View of Continuous 2-D Fourier')
+shading interp
+colorbar
+
+subplot(2,3,6)
+surf(WX(index,index),WY(index,index),G_2dfftshift(index,index)),view(2);
+xlabel('w1');ylabel('w2');title('Top View of 2-D Shift FFT');
+shading interp
+colorbar

0712Gaussian_Fourier/myfft.m

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+function y=myfft(x)
+N=length(x);
+if N==1
+    y=x;
+else
+    u=x(1:2:end);
+    v=x(2:2:end);
+
+    yu=myfft(u);
+    yv=myfft(v);
+    w=exp(-2*pi*1i/N);
+
+    y=zeros(N,1);
+    y(1:N/2)=yu(1:N/2)+yv(1:N/2).*((w.^(0:N/2-1)).');
+    y(N/2+1:N)=yu(1:N/2)-yv(1:N/2).*((w.^(0:N/2-1)).');
+end
+end