Fourier transform of symbolic function

Hi. I would like to see the magnitude and phase spectrum of a symbolic function. I tried using the 'fourier' function on a basic symbolic expression, but it didn't work.

syms t w;

x=sin(2*t);

X=fourier(x);

fplot(abs(X)) // Nothing in this figure

Doing the transform manually works:

T=10;

fX=int(x* exp(1j*w*t), t, -T, T);

fplot(abs(X)/T) // This works

Please can someone let me know how to see the phase and magnitude responses using the 'fourier' function ?

Many thanks

-s

Answer 1

[Paul]

Hi soup,

The Fourier transform of x is

syms t w;
x=sin(2*t);
fourier(x)

ans = 

fplot() ignores Dirac delta functions. Because there is nothing else to plot, fplot() is empty

The "manual transform"

T = 10;  % not sure why T = 10?
simplify(int(x* exp(1j*w*t), t, -T, T),100)

ans = 

isn't the Fourier transform of x. Rather, it's the Fourier transform of the x multiplied by a rectangular window of length 2T.

sympref('FourierParameters',[1 1]);
simplify(fourier(x*rectangularPulse(-T,T,t)),100)

ans = 

[soup]

Hi Paul,

Thanks for the resonse! T=10 was arbitrary really. T=inf (integrating over -inf to inf ) didnt work, (I dont know why), and having some value there made things work..

T=10;

fX=int(x* exp(1j*w*t), t, -T, T)

do you know why if I use T=inf above, it doesnt work?

" fplot() ignores Dirac delta functions"

Does that mean it is impossible to plot the fourier transform magnitude and phase of a sinusoid using 'fourier'?

Thanks for your help.

-s

[Paul]

As you're seeing, the Fourier transform integral doesn't converge for a periodic signal. However, we can define the Fourier transform of a periodic signal as a Dirac delta impulse train scaled by the signal's Fourier series coefficients, which is exactly what we see as the Fourer transform of a sin function. See this link, for example, for further discussion. Matlab's fourier function knows how to deal with basic periodic functions, like sin and complex exponentials.

Any plot of a signal that contains a Dirac delta will be artifcial. The concept of "magnitude" doesn't really apply to a Dirac delta. Can't really plot it, just have to make up some symbol, like a vertical arrow, to signify where the Dirac delta sits on the real line, as is done in the linked reference.

[soup]

Thank you very much. I'll read the article : )

-s