Inverse Fourier transform of unilateral exponential function - ifourier MATLAB

Hi everyone. How can I get the inverse Fourier tranform of unilateral exponential? I tried with the following code:

    close all
    clear all
    syms t f
    a=sym('a','positive');
    y=ifourier(1/(a + pi*f*2i),t)

the result is Y=(exp(-(a*t)/(2*pi))*(sign(t) + 1))/(4*pi). Instead I'd like to get Y=exp(-a*t)*heaviside(t).

Thank you for your time.

Answer 1

[Paul]

syms t f
a = sym('a','positive');
Y(f) = 1/(a + pi*f*2i)

Y(f) = 

The default parameters in sympref assume that the transfomr variable is angular frequency (e.g., w, rad/sec) as opposed to ordinary frequency (e.g., f, Hz). So we can either change the sympref or operate with angular frequency. Choosing the latter yields

syms w
y(t) = ifourier(Y(w/2/sym(pi)),w,t)

y(t) = 

Now rewrite in terms of heaviside

y = rewrite(y,'heaviside')

y(t) = 

That rewrite worked with the default settings of sympref. However, if we have a different HeavisideAtOrigin we get an incorrect result for the inverse Fourier transform of Y(f) (in the sense that fourier(y) won't recover Y(w) )

sympref('HeavisideAtOrigin',10);
y(t) = ifourier(Y(w/2/sym(pi)),w,t);
y = rewrite(y,'heaviside')

y(t) = 

which is a consequence of ifourier returning the result in terms of sign instead of heaviside. However, going the other way, fourier does not care about the HeavisideAtOrigin

Y(f) = fourier(exp(-a*t)*heaviside(t),t,2*sym(pi)*f) % same as above

Y(f) =