Hello; I'm using the symbolic maths toolbox, specifically the fourier function. How do I access the positive and negative frequency components of the result separately ?

Hello; I'm using the symbolic maths toolbox, specifically the fourier function. How do I access the positive and negative frequency components of the result separately ? I need to multiply the positive frequency components by -i and the negative frequency components by i, to compute the Hilbert transform. Note that Matlab's Hilbert function does not work with the symbolic maths toolbox.

Thank you.

Answer 1

[Star Strider]

The fourier function will give you the symbolic expression for the Fourier transform of your time- or spatial-domain function. You then are free to calculate and plot it for the frequency values you wish.

Also consider using simplify (with the 'Steps',10 name-value pair) and matlabFunction to create a function you can use in numerical computations.

[Star Strider]

‘Hello,

yes, the fourier function gives me the symbolic expression for the FT of my symbolic function. But how do I access the positive and negative frequency components separately ? I want to multiply the former by -i and the latter by i, to implement a Hilbert transform.

thank you’

I was thinking of something like this:

syms f t w 
x = exp(-t) * sin(t);                                           % Function
X = int(x*exp(1i*2*pi*w*t), t, 0, 1);                           % Fourier Transform
X = simplify(X, 'Steps',10);
Xfcn = matlabFunction(X);

wvn = linspace(-2*pi,0,50);                                     % Negative Frequency Vector
wvp = linspace( 0,2*pi,51);                                     % Positive Frequency Vector

Xout = [1i*Xfcn(wvn) -1i*Xfcn(wvp(2:end))];                     % Function Evaluation

figure(1)
plot([wvn wvp(2:end)], real(Xout),  [wvn wvp(2:end)], imag(Xout),  [wvn wvp(2:end)], abs(Xout))
grid
legend('\Re\{X\}', '\Im\{X\}', '|X|')

[Gordon Cooper]

Thanks for all the effort, but I'm using Matlab's symbolic Fourier transform function 'fourier' and I'm battling to manipulate the results.

[Star Strider]

This is my alternative approach. I don’t know what function you’re transforming, but in my experience, coding my own transform — as I did here — generally produceds the result I want.

The ‘problem’ with the fourier function is that it integrates from -Inf to +Inf. If your time-domain function has other than infinite support, this will cause problems. That’s the reason I generally code my transforms myself.

If I know the function has infinite support, fourier produces the appropriate result.

Answer 2

[Gordon Cooper]

Hello, yes, the fourier function gives me the symbolic expression for the FT of my symbolic function. But how do I access the positive and negative frequency components separately ? I want to multiply the former by -i and the latter by i, to implement a Hilbert transform.

thank you

Answer 3

[Walter Roberson]

Multiply the symbolic output by the Heaviside function on the frequency to get the positive components, and by Heaviside of negative of the frequency to get the negative components.

[Gordon Cooper]

Thanks, that helps a lot.