This Animation will tell you everything about the connection between time and frequency domain of a signal

Think about a signal, lets say a sine with 5Hz frequency and a sine with 15Hz of frequency with amplitudes of 0.8 and 0.4. They look like this (black and red) and combined like the green one in following figure.

So, how about the Frequency Domain of these signals? Looks like a FFT (=Discrete Fast Fourier Transform) does the right thing for us, right?

Connection between time and frequency domain of a signal

Matlab Code of this FFT Animation

%% Animation zum Zusammenhang zwischen Signal im Zeit und Frequenzbereich
% Paul Balzer
% Motorblog | www.cbcity.de

%% Signal generieren
%
dt = 0.001;
t=0:dt:1;

f1 = 5;
s1 = 0.8*sin(2*pi*f1*t);

f2 = 15;
s2 = 0.4*sin(2*pi*f2*t);

s=s1+s2;

% figure(1)
% subplot(3,1,1)
% plot(t,s1,'-k')
% ylabel('Signal 1')
% ylim([-1 1])
% grid on
% 
% subplot(3,1,2)
% plot(t,s2,'-r')
% ylabel('Signal 2')
% ylim([-1 1])
% grid on
% 
% subplot(3,1,3)
% plot(t,s,'-g','linewidth',2)
% xlabel('Time [s]')
% ylabel('Resulting Signal')
% ylim([-1 1])
% grid on

%% FFT
%
Y = fft(s);

n = numel(t);            % Anzahl Werte
m = floor(n/2);          % Spektrumbegrenzung aus Nyquist Frequenz

fspec = t.* 1/(dt*n) * 1/dt;    % Frequenzvektor
ampspec = 2.*abs(Y)/n;          % Amplitude

figure(2)
set(gcf,'color',[1 1 1],'position',[10 10 770 520])
hf = stem3(zeros(1,31),fspec(1:31), ampspec(1:31),'fill');
grid on
hold on
hs1 = plot3(t(1:m), f1.*ones(1,m), s1(1:m),'-k');
hs2 = plot3(t(1:m), f2.*ones(1,m), s2(1:m),'-r');

hs = plot3(t(1:m), zeros(1,m), s(1:m),'-g','linewidth',5);

legend('Amplitude Spectrum',['Signal 1 w/' num2str(f1) 'Hz'],...
    ['Signal 2 w/' num2str(f2) 'Hz'],'Resulting Signal',...
    'location','east')

xlabel('Time [s]','fontsize',18)
xlim([0 max(t(1:m))])
ylabel('Frequency [Hz]','fontsize',18)
ylim([0 max(f1,f2)*2])
zlabel('Amplitude [-]','fontsize',18)
box off
set(gca,'fontsize',16)

%% 3D Darstellung erzeugen
%
view(0,0)      % Frequenzspektrum
axis vis3d      % so skalieren, dass beim Drehen nichts mehr skaliert
title('Signal in time Domain','fontsize',18)
drawnow

% GIF erzeugen
mygif = 'Time-2-Frequency-Domain-FFT.gif'; 
f = getframe(gcf); 
[im,map]=rgb2ind(f.cdata,256,'nodither'); 
imwrite(im,map,mygif,'Delaytime',5,'Loopcount',inf) 

for i = 0:0.5:90
    
    % Smooth View change
    hann=0.5*(1-cos(2*pi*i/90));
    view(90*(3*(i/90)^2-2*(i/90)^3),40*sind(2*i)*hann)

    f = getframe(gcf); 
    [im,map]=rgb2ind(f.cdata,256,'nodither'); 
    imwrite(im,map,mygif,'writemode','append','Delaytime',.02)

end

title('Amplitude Spectrum of the Signal (FFT)','fontsize',18)
set(hs,'visible','off')

for i = -1:.02:0
    zlim([i 1])

    f = getframe(gcf); 
    [im,map]=rgb2ind(f.cdata,256,'nodither'); 
    imwrite(im,map,mygif,'writemode','append','Delaytime',.02)
    
end

legend('off')
annotation('textbox', [.7 0.0 .1 0.1], 'String', 'CC-BY-SA2.0 Paul Balzer | Motorblog');

drawnow
f = getframe(gcf); 
[im,map]=rgb2ind(f.cdata,256,'nodither'); 
imwrite(im,map,mygif,'writemode','append','Delaytime',5)
    

FFT with Excel

  (Tutorial) FFT with Excel

FFT with Matlab

  (Tutorial) FFT with Matlab