% MATLAB is frequently used to process signals
% 1. Fourier transforms
x = rand(1,10);
y = fft(x); % Fast Fourier Transform of x
iy = ifft(y); % inverse fft
norm(x-iy)

ans =

  2.2645e-016

% 2. Waveform and Amplitude spectrum
dt = 1/100; % sampling rate
et = 4; % end time
t = 0:dt:et; % defines sampling range
y = 3*sin(4*2*pi*t) + 5*sin(2*2*pi*t);
% now that we have our signal sampled, we plot it
subplot(2,1,1);
plot(t,y); grid on
axis([0 et -8 8]);
xlabel('Time (s)');
ylabel('Amplitude');
% with the fft we find the characteristics
% of the signal
Y = fft(y); % from the time to frequency domain
n = size(y,2)/2; % number of samples/2
amp_spec = abs(Y)/n; 
subplot(2,1,2)
freq = (0:79)/(2*n*dt);
plot(freq,amp_spec(1:80)); grid on;
xlabel('Frequency (Hz) ');
ylabel('Amplitude');
% looking at the second plot, we see the
% essential characteristics of the signal
% 3. filtering noise
noise = randn(1,size(y,2));
ey = y + noise;
eY = fft(ey);
n = size(ey,2)/2;
figure
subplot(2,1,1);
plot(t,ey); grid on;
axis([0 et -8 8]);
xlabel('Time (s)');
ylabel('Amplitude');
subplot(2,1,2);
freq = (0:79)/(2*n*dt);
amp_spec = abs(eY)/n;
plot(freq,amp_spec(1:80));
grid on
xlabel('Frequency (Hz)');
ylabel('Amplitude');
% now we do the restoration of the signal,
% by removal of the low amplitude noise
figure
plot(Y/n,'r+')
hold on
plot(eY/n,'bx');
fY = fix(eY/100)*100; % numbers < 100 are cut
ifY = ifft(fY);
cy = real(ifY);
figure
plot(t,cy); grid on
xlabel('Time (s)');
ylabel('Amplitude');
% compare with the original signal
figure
plot(t,y); grid on
xlabel('Time (s)');
ylabel('Amplitude');
diary off