Basic question on the FFT in matlab

[O.J.]

Homework Statement

From what I understand, the array output of the FFT of a function in time represents the frequency contributions of different frequencies with the first output coming from 0 freqeuncy, 2nf from the 1st fundamental frequency and so on.. if so, then why does this code(which is for plotting the fft against frequency) give contributions from various frequencies to the cosine function when we learned in Fourier theory that the only frequency contributiong to a sinusoid are those of its own frequency:
t = -20:20;
x= cos (6*pi*t/13+pi/3);
y = fft(x);
tn = (-length(y)+1)/2 : (length(y)-1)/2;
stem (tn, y);

Homework Equations

The Attempt at a Solution

 

[MATLABdude]

Homework Statement

From what I understand, the array output of the FFT of a function in time represents the frequency contributions of different frequencies with the first output coming from 0 freqeuncy, 2nf from the 1st fundamental frequency and so on.. if so, then why does this code(which is for plotting the fft against frequency) give contributions from various frequencies to the cosine function when we learned in Fourier theory that the only frequency contributiong to a sinusoid are those of its own frequency:
t = -20:20;
x= cos (6*pi*t/13+pi/3);
y = fft(x);
tn = (-length(y)+1)/2 : (length(y)-1)/2;
stem (tn, y);

Homework Equations

The Attempt at a Solution

Howdy!

How many cycles does a regular sinusoid have? In contrast, how long is yours?

 

[O.J.]

sup!
well, a regular sinusoid has... infinitely many cycles. and mine has almost 3 cycles in that range of t.

 

[O.J.]

Ok, after playing with it, I see what you're saying, the larger the range of t, the closer it is to the real thing (although I don't see why). but, how to relate the values on the x-axis to frequency?

 

[MATLABdude]

Ok, after playing with it, I see what you're saying, the larger the range of t, the closer it is to the real thing (although I don't see why). but, how to relate the values on the x-axis to frequency?

Precisely! When you're doing a Fourier Transform, you're breaking down your function and trying to represent it using sinusoids of infinite duration. Since you only have a small snippet of a sinusoid, you need a bunch of sinusoids to add together so that you get the sinusoid segment, and 0 everywhere else. And of course, the longer your snippet, the closer it is to an infinite-duration sinusoid, and the higher its frequency purity (i.e. the power of the signal comes almost entirely from its fundamental frequency, and none of the higher ones).

Regarding your second question, if you know the period T (which you can read off of the graph), then you just take 1/T as the frequency (this assumes that you're not working in degrees, or radian frequency):
http://en.wikipedia.org/wiki/Frequency

What blew my mind--this finite duration of sinusoid snippets and their spectral purity--is that this seemingly mathematical quirk actually has ramifications in physics, especially in regards to the coherence length of light, and why there's always some spectral width in any light source.

 

[nada]

Any one could help me with , numerical Fourier transform of bessel's function using FFT in C language

 

[nada]

also anyone could help me with,invistigate of bessel'e equation using runge-kutta

 

[MATLABdude]

Welcome to Physicsforums, Nada!

In the future, please make a new thread for a topic unrelated to the one at hand. I'm not a computer modeling guy, but you might be able to find something in the book Numerical Recipes in C, probably available in your library (you can also probably find a cheap used copy):
http://www.nr.com/

 

[chroot]

OJ,

Making your sample size bigger increases your resolution. In other words, with more samples, you can resolve smaller details on the plot.

The real problem you're having is that your sample is a non-integer number of wavelengths. The FFT acts on your input as if it were copied end-to-end for all time, so its beginning and end better match up. If the ends do not match up, the discontinuity will appear as high-frequency garbage, which will ruin your plot.

What you need to do is window your samples, so that they go to zero at both ends. Then you can safely use the FFT. Look up the Hamming window, for example:

http://en.wikipedia.org/wiki/Hamming_window#Hamming_window_.5Bnote_1.5D]Hamming window

- Warren

 

[nada]

hello

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could you please help me with
numerical solution by runge-kutta for bessel's equation with C
Also
how i can write programme to compute the series(bessel function) Jn(x) for n=0.1,4 for range of x with C
also numerical Fourier transfom of bessel function using FFT with C too

I would be so grateful if you could if not how can I ask by this website