How to calculate Fourier transform of this 2D function?
$\phi(x,y)=2x$
for $-1<=x <= 1 ; and -1<=y<=1$
and
$\phi(x,y)=0$ ; otherwise
I tried like this:
$\psi(u,v) = \int_{-\infty}^\infty \int_{-\infty}^\infty\phi(x,y)e^{-2\pi i(ux+vy)}dxdy$
which can be written as, by taking $\alpha = 2\pi iu$ and $\beta=2\pi iv$:
$\psi(u,v)= [2\int_{-\infty}^\infty xe^{-\alpha x}dx][ \int_{-\infty}^\infty e^{-\beta y} dy$]
The first integral does not converge (I checked this using Mathematica).
I know that we can find FT numerically which I do not want to do right now.
Hint: Note that $\phi(x,y)=0$ if $(x,y)\not\in[-1,1]\times[-1,1]$. Thus,
$$\psi(u,v) = \int_{-\infty}^\infty \int_{-\infty}^\infty\phi(x,y)e^{-2\pi i(ux+vy)}dxdy = \int_{-1}^1\int_{-1}^1\phi(x,y)e^{-2\pi i(ux+vy)}dxdy.$$
I guess you are able to continue from here.