%***********************************************************************************
%
%                              Jim Merriam
%                       University of Saskatchewan
%                          jim.merriam@usask.ca
%                              (June, 1990)
%                            (last Oct 1998)
%
%***********************************************************************************
%
clf
disp('            CLICK ON THE FIGURE WINDOW AND FOLLOW PROMPTS')
clear s;
subplot(211),axis([0 10 0 10]),axis('off')
s(1)={'\bf DISCRETE FOURIER TRANSFORMS I\rm'};
text(0,8,s,'FontName','times','FontSize',16),clear s;
s(1)={'This is a demo on Discrete Fourier Transforms.'};
s(2)={'It demonstrates:'};
s(3)={' '};
s(4)={'HOW TO INTERPRET THE SPECTRUM'};
s(5)={'HOW TO CALCULATE THE FREQUENCY AXIS'};
s(6)={'HOW PHASE EFFECTS THE SPECTRUM'};
s(7)={' '};
text(1,2,s,'FontName','times','FontSize',10),clear s

h=uicontrol('style','pushbutton', 'units','normalized','position',[.8 .01 .1 .05], ...
'string','OK','Callback','OK=''continue '';');

OK='PAUSEaBIT';
while OK=='PAUSEaBIT'
   pause(0.01)
end
   

clf;
i=0:7;
j=0:511;
ft=sin(i*2*pi/4);
FT=fft(ft);
subplot(211)
plot(i,ft,'*',j*8/512,sin(j*2*pi/256),'--');
subplot(212)
axis([0 10 0 10]);axis('off');
s(1)={'Here are two complete cycles of a sine wave, sampled 8 times.'};
s(2)={'Notice that the first sample is at the zero of the sine wave.'};
s(3)={'This is important, \bf where the first sample is located in the sine'};
s(4)={'wave determines the phase of the sinusoid that will be returned'};
s(5)={'by the Discrete Fourier Transform.'};
text(1,7,s,'FontName','times','FontSize',10),clear s

h=uicontrol('style','pushbutton','position',[.8 .01 .1 .05], 'units','normalized', ...
'string','OK','Callback','OK=''continue '';');
OK='PAUSEaBIT';
while OK=='PAUSEaBIT'
   pause(0.01)
end

clf;
axis;
subplot(211);
plot(imag(FT),'*');
xlabel('m')
text(1.1,.8,'f=0');text(2,.8,'fn/4');text(3,-2.9,'2fn/4');text(4,.8,'3fn/4')
text(5,.8,'fn');text(6,.8,'-3fn/4'),text(7,3.1,'-2fn/4'),text(8,.8,'-fn/4')
subplot(212);
axis([0 10 0 10]);axis('off');
s(1)={'Here is the imaginary part of the transform. We had 8 samples of the data, '};
s(2)={'so we get 8 samples of the spectrum. The first one is the DC, the next four '}; 
s(3)={'are positive frequencies, and the last three are negative frequencies. '};
s(4)={'The Nyquist is at N/2+1=the fifth location. Each frequency is separated'};
s(5)={'by fn/(N/2), so you calculate the frequencies by adding this amount to'};
s(6)={'each frequency starting at the lowest positive frequency f=fn/(N/2) and'};
s(7)={'working to the right until you get fn, and then change the sign '};
s(8)={'(ie to -fn) and continue (but now you increment down from N/2, so the '};
s(9)={'frequencies reflect about the Nyquist with a change in sign). Note that we'};
s(10)={'have positive and negative frequencies everywhere except at the Nyquist'};
text(0,5,s,'FontName','times','FontSize',10),clear s

h=uicontrol('style','pushbutton','position',[.8 .01 .1 .05], 'units','normalized', ...
'string','OK','Callback','OK=''continue '';');
OK='PAUSEaBIT';
while OK=='PAUSEaBIT'
   pause(0.01)
end


axis;
clf; 
i=0:31;
f1=sin(i*2*pi/16);
F1=fft(f1);
subplot(211)
plot(i,f1,'*',j*32/512,sin(j*2*pi/256),'--');
subplot(212)
axis([0 10 0 10]);axis('off');
s(1)={'Here is the same signal sampled 32 times.'};
s(2)={'Again, note that the first sample is at the zero of the'};
s(3)={'sine wave. There are four times as many samples of the same two'};
s(4)={'cycles as before, but will the spectrum of these samples reveal'};
s(5)={'anything further about the signal?'};
text(0,10,s,'FontName','times','FontSize',10),clear s

h=uicontrol('style','pushbutton','position',[.8 .01 .1 .05], 'units','normalized', ...
'string','OK','Callback','OK=''continue '';');
OK='PAUSEaBIT';
while OK=='PAUSEaBIT'
   pause(0.01)
end


clf;
subplot(211)
plot(real(F1),'*')
title('THIS IS THE REAL PART OF THE TRANSFORM'),ylabel('AMPLITUDE')
xlabel('THIS IS THE FREQUENCY INDEX m')
hold on;
plot([1 2],[4e-15 3.e-15])
text(3,3.0e-15,'NOTICE THE SCALE!')
subplot(212)
hold off
plot(imag(F1),'*')
title('THIS IS THE IMAGINARY PART OF THE TRANSFORM'),ylabel('AMPLITUDE')
xlabel('THIS IS THE FREQUENCY INDEX m')
text(1,10,'Is there any further information in the spectrum?')

h=uicontrol('style','pushbutton','position',[.8 .01 .1 .05], 'units','normalized', ...
'string','OK','Callback','OK=''continue '';');
OK='PAUSEaBIT';
while OK=='PAUSEaBIT'
   pause(0.01)
end

clf
subplot(211)
plot(imag(F1),'*')
subplot(212)
axis([0 10 0 10]);axis('off');
s(1)={' Here is the imaginary part again.'};
s(2)={' Where is the DC?'};
s(3)={' Where is the Nyquist?'};
s(4)={' What frequencies are represented by the two spikes?'};
s(5)={' '};
s(6)={' The amplitude of the two spikes is -16 and +16, and'};
s(7)={' the input sine was unit amplitude, so this particular'};
s(8)={' FFT has 1/N in front of the reverse transform and 1 in '};
s(9)={' front of the forward transform.'};
text(0,7,s,'FontName','times','FontSize',10),clear s

h=uicontrol('style','pushbutton','position',[.8 .01 .1 .05], 'units','normalized', ...
'string','OK','Callback','OK=''continue '';');
OK='PAUSEaBIT';
while OK=='PAUSEaBIT'
   pause(0.01)
end


axis;
clf
m=-15:16;
plot(i+1,imag(F1),'.')
xlabel('m')
hold on;
plot(i(1)+1,imag(F1(1)),'o')
text(0,3,'The  DC is the first element m=1')
plot(17,imag(F1(17)),'o')
text(17,-3,'Here is the Nyquist m=N/2+1')
plot(3,imag(F1(3)),'o')
text(4,-15,'f=(m-1)*fn/(N/2)=2*fn/16')
plot(31,imag(F1(31)),'o')
text(15,14,'f=-(m-1)*fn/(N/2)=-2*fn/16')

h=uicontrol('style','pushbutton','position',[.8 .01 .1 .05], 'units','normalized', ...
'string','OK','Callback','OK=''continue '';');
OK='PAUSEaBIT';
while OK=='PAUSEaBIT'
   pause(0.01)
end


clf
hold off
f1=sin((i+1)*2*pi/16);
F1=fft(f1);
subplot(211)
plot(i+1,f1,'*',j*32/512,sin((j+1)*2*pi/256),'--');
subplot(212)
axis([0 10 0 10]);axis('off');
s(1)={'Here is the same signal sampled 32 times, but this time'};
s(2)={'the first sample is one position to the right of the'};
s(3)={'zero crossing. The signal represented by the asterisks'};
s(4)={'is no longer a zero phase sine, because the first sample'};
s(5)={'is no longer at a zero crossing of the sine.'};
text(0,9,s,'FontName','times','FontSize',10),clear s


h=uicontrol('style','pushbutton','position',[.8 .01 .1 .05], 'units','normalized', ...
'string','OK','Callback','OK=''continue '';');
OK='PAUSEaBIT';
while OK=='PAUSEaBIT'
   pause(0.01)
end


clf;
subplot(211)
plot(real(F1),'*')
title('THIS IS THE REAL PART OF THE TRANSFORM'),ylabel('AMPLITUDE')
xlabel('THIS IS THE FREQUENCY INDEX m')
subplot(212)
axis([0 10 0 10]);axis('off');
s(1)={'The real part of this spectrum is not zero, as it was in the'};
s(2)={'last spectrum. The phase lead we introduced in the '};
s(3)={'sampling is reflected in larger elements in the real'};
s(4)={'part of the spectrum. Notice that the spikes are at the same'};
s(5)={'frequencies they were at before.'};
text(0,8,s,'FontName','times','FontSize',10),clear s


h=uicontrol('style','pushbutton','position',[.8 .01 .1 .05], 'units','normalized', ...
'string','OK','Callback','OK=''continue '';');
OK='PAUSEaBIT';
while OK=='PAUSEaBIT'
   pause(0.01)
end


clf;axis;
subplot(211)
plot(imag(F1),'*') 
title('THIS IS THE IMAGINARY PART OF THE TRANSFORM')
ylabel('AMPLITUDE')
xlabel('THIS IS THE FREQUENCY INDEX m')
subplot(212);
axis([0 10 0 10]);axis('off');
s(1)={'The imaginary part is not the same as before either,'};
s(2)={'some of the spike amplitude has evidently gone into'};
s(3)={'the real part. Instead of having a purely imaginary'};
s(4)={'spectrum it is now composed of real and imaginary parts.'};
s(5)={'This is because a sine with a phase lead can be '};
s(6)={'expressed as  the sum of a sine and a cosine with zero'};
s(7)={'phase. The real part of a spectrum is a cosine with'};
s(8)={'zero phase, and the imaginary part a sine with zero phase'};
text(0,6,s,'FontName','times','FontSize',10),clear s

h=uicontrol('style','pushbutton','position',[.8 .01 .1 .05], 'units','normalized', ...
'string','OK','Callback','OK=''continue '';');
OK='PAUSEaBIT';
while OK=='PAUSEaBIT'
   pause(0.01)
end


clf

i=0:31;
j=0:511;
f1=sin((i+2)*2*pi/16);
F1=fft(f1);
subplot(211)
plot(i,f1,'*');
subplot(212);axis([0 10 0 10]);axis('off');
s(1)={'Here is the same signal as the last excercise sampled 32 times again'};
s(2)={'except now  the first sample is even further to the right, ie the '};
s(3)={'phase is even greater than before.'};
text(0,6,s,'FontName','times','FontSize',10),clear s

h=uicontrol('style','pushbutton','position',[.8 .01 .1 .05], 'units','normalized', ...
'string','OK','Callback','OK=''continue '';');
OK='PAUSEaBIT';
while OK=='PAUSEaBIT'
   pause(0.01)
end


axis;
clf;
subplot(211)
plot(real(F1),'*')
title('THIS IS THE REAL PART OF THE TRANSFORM'),ylabel('AMPLITUDE')
xlabel('THIS IS THE FREQUENCY INDEX m')
hold on;
s(1)={'Notice that the scale is similar in both, that is'};
s(2)={'that there is equal energy in the real and imaginary'};
s(3)={'parts, and that the peaks appear in both parts. The real'};
s(4)={'describes that part of the signal that looks like a cosine'};
text(4,6,s,'FontName','times','FontSize',10),clear s

subplot(212)
hold off
plot(imag(F1),'*')
title('THIS IS THE IMAGINARY PART OF THE TRANSFORM'),ylabel('AMPLITUDE')
xlabel('THIS IS THE FREQUENCY INDEX m')
s(1)={'and the imaginary part describes that part of the signal'};
s(2)={'that looks like a sine. The real and imaginary parts are equal'};
s(3)={'because the first sample was about 45 deg up the sine wave, so '};
s(4)={'the DFT sees the phase as an equal mix of sine and cosine phase'};
text(.5,10,s,'FontName','times','FontSize',10),clear s

h=uicontrol('style','pushbutton','position',[.8 .01 .1 .05], 'units','normalized', ...
'string','OK','Callback','OK=''continue '';');
OK='PAUSEaBIT';
while OK=='PAUSEaBIT'
   pause(0.01)
end



 clf
axis([0 10 0 10]);axis('off');
s(1)={'\bf DISCRETE FOURIER TRANSFORMS I\rm'};
s(2)={''};
s(3)={'              THE END'};
s(4)={' '};
text(1,9,s,'FontName','times','FontSize',16),clear s

h=uicontrol('style','pushbutton','position',[.8 .01 .1 .05], 'units','normalized', ...
'string','OK','Callback','OK=''continue '';');
%OK='PAUSEaBIT';
%while OK=='PAUSEaBIT'
   load toocool.mat;
FIGAXES=axes('position', [.28 .01 .45 .65]);
image(TOOCOOL);
set(FIGAXES,'visible','off')
TEXTAXES=axes('position',[.05 .45 .9 .52]);
set(TEXTAXES,'visible','off');
text(.28,.47,'COOL, You''re done','fontname','italic','fontsize',14);
clear TOOCOOL

%end
pause(5)
clf