How to correctly make FFT of sound set?

During 7 sec 7 tones plays with frequences (all in Hz), one tone -- one sec. Code:

    Fs=44100;f=[261.63 293.67 329.63 349.23 392.00 440 493.88];
    duration=7;octava=2;f=f/octava;Df=duration*Fs;FF=2*pi/Fs*f;
len = length(f);tau=Df/len;
    n=1:Df;y=zeros(1,Df);
    for i = 1:len
      amplitude=(n>(i-1)*tau)&(n<i*tau);
      y = y + sin(FF(i)*n).*amplitude;
    end
  soundsc(y, Fs);

This question rather for mathematicians. Now I have to see Fourier tranformation. Perhaps it will not accurate vertical Dirac bars but what?

Code:

Y1=fft(y);plot(abs(Y1))

Two bar on edges? How to make beauty picture? And what frequencies will be?

Answer 1

[Rick Rosson]

Maybe this will help:

%%Parameters:

playAudio = false;

%%Time domain:

Fs = 44100;
dt = 1/Fs;
StartTime = 0;
StopTime = 1;

t = (StartTime:dt:StopTime-dt)';

%%Cosine waves:

Fc = [261.63 293.67 329.63 349.23 392.00 440 493.88];
y = cos(2*pi*t*Fc);
N = size(y,2);

%%Reshape signal to a single column vector:
y = y(:);

%%Reformulate time domain:

StopTime = N*StopTime;
t = (StartTime:dt:StopTime-dt)';
M = size(t,1);

%%Frequency domain:

dF = Fs/M;
f = -Fs/2:dF:Fs/2-dF;

Y = (N/M)*fftshift(fft(y));

%%Play the music:

if playAudio
   sound(y,Fs);
end

%%Plot time domain:

figure;
plot(t,y);

%%Plot frequency domain:

figure;
plot(f,abs(Y));

HTH.

Rick

[Igor]

The best -- y=y(:) :-)

At zoom on figure I see 7 peaks symmetrically..

And I have to understand this result, and I shell push "Accept"-button!

[Rick Rosson]

If you have any questions about this code or why it works, please let me know.

[Rick Rosson]

Also, there are a few things about this script that are not quite optimal. Can you figure out what they are and how to fix them?

[Danilo Persico]

Hello Mr. Rosson, I would like if you could give an explaination of the code, thank you!

Answer 2

[Daniel Shub]

The fft assumes that the signal is cyclical. You need to match up your edges (or zero pad) to get delta functions. The fft also returns both positive and negative frequencies. I would read the documentation about fft and fftshift.

Answer 3

[Rick Rosson]

Please try the following:

dt = 1/Fs;
len = duration*Fs;

n = dt*(0:len-1);

Df = Fs/len;

f = -Fs/2:Df:Fs/2-Df;

Also:

Y1 = fftshift(fft(y));
plot(abs(Y1));

HTH.

[Igor]

Not clear...

1st part -- replace my 2nd row? but what is the sense of *f*?

how does the 2nd part refer to 1st part?

(because of, y-array is unchanged data)