Prove of the CSFT of the signals

Yuanjun Wang

Below are CSFT of signals. The general way we solve CSFT questions is to guess its Fourier Transform, then prove it by taking the inverse F.T. of the signals.

1. $ f(x,y)=\frac{ sin(\pi x)}{\pi x} \frac{ sin(\pi y)}{\pi y} $

guess: $ F(u,v) = rect(u) rect(v) $ \\

prove: $ F^{-1}(u,v) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} rect(u) rect(v) e^{j2\pi (ux+vy)} dx dy $

because we know that $ rect(u) = \left\{ \begin{array}{ll} 1, & \text{ if } |t|<\frac{1}{2}\\ 0, & \text{ else} \end{array} \right. $

$ F^{-1}(u,v) = \int_{-\frac{1}{2}}^{\frac{1}{2}} rect(v) \int_{-\frac{1}{2}}^{\frac{1}{2}} e^{j2\pi ux} du e^{j2\pi vy} dy $

$ = \int_{-\frac{1}{2}}^{\frac{1}{2}} rect(v) \frac{e^{j\pi x} - e^{-j\pi x}}{j\pi x} e^{j2\pi vy} dy $

$ = \frac{ sin(\pi x)}{\pi x} \int_{-\frac{1}{2}}^{\frac{1}{2}} rect(v) e^{j2\pi vy} dy $

$ = \frac{ sin(\pi x)}{\pi x} \frac{ sin(\pi y)}{\pi y} $

so $ f(x,y) = \frac{ sin(\pi x)}{\pi x} \frac{ sin(\pi y)}{\pi y} $

so CSFT (f(x,y)) = rect(u) rect(v)