I am trying to evaluate the following expression
$$\mathcal{F}\{\mathrm{rect}_{L_{x}}(x)\mathrm{rect}_{L_{y}}(y)\}$$
which denotes the 2-dimensional Fourier transform (reciprocal variables $k_x$, $k_y$) of a product of two rectangular functions defined by
$$\mathrm{rect}_{L_{x}}(x)=\begin{cases}1, & -L_x<x<L_x \\ 0, & \text{otherwise}\end{cases}$$
This is what I tried so far ("*" denotes convolution):
$$\mathcal{F}\{\mathrm{rect}_{L_{x}}(x)\mathrm{rect}_{L_{y}}(y)\}
= \mathcal{F}\{\mathrm{rect}_{L_{x}}(x)\}*\mathcal{F}\{\mathrm{rect}_{L_{y}}(y)\}\\
={L_xL_y\over\pi^2}\mathrm{sinc}(L_xk_x)*\mathrm{sinc}(L_yk_y)\\
={L_xL_y\over\pi^2}\int_{-\infty}^{\infty}\mathrm{d}k_x'\,\mathrm{sinc}(L_xk_x)\int_{-\infty}^{\infty}\mathrm{d}k_y'\mathrm{sinc}(L_y(k_y-k_y'))
$$
Now both these integrals are Dirichlet integrals and evaluate to ${\pi \over L_x}$ and ${\pi \over L_y}$ respectively which yields total result of $1$.
I suspect there's a mistake somewhere, since this is not the result I am expecting to see. Moreover when I try to do the calculation in Mathematica
FourierTransform[UnitBox[x] UnitBox[y], {x, y}, {a, b}]
I get
$${\mathrm{sinc}(a/2)\mathrm{sinc}(b/2)\over 2\pi}$$
which makes much more sense for the bigger problem I am solving.
Where am I wrong?
Best Answer
The two dimensional Fourier Transform of a function $f(x,y)$ is
$$\mathscr{F}\{f\}(k_x,k_y)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x,y)e^{ik_xx}e^{ik_yy}dxdy$$
For $f(x,y)=\text{rect}_{L_x}(x)\text{rect}_{L_y}(y)$, the Fourier Transform is
$$\mathscr{F}\{f\}(k_x,k_y)=\int_{-L_y}^{L_y}\int_{-L_x}^{L_x}e^{ik_xx}e^{ik_yy}dxdy=2L_x\text{sinc}(k_xL_x)2L_y\text{sinc}(k_yL_y)$$