The Fourier transform usually transforms a mathematical function of time, f(t), into a new function usually denoted by F($\mathrm{\ω}$) whose arguments is frequency with units of cycles/sec (hertz) or radians per second. This new function is known as the Fourier transform. The Fourier transform is a mathematical transformation used within many applications in physics and engineering. The term "Fourier transform" refers to both the transform operation and to the complex-valued function it produces. 

The rectangular function can often be seen in signal processing as a representation of different signals. The sinc function, defined as $\frac{\sin \left(t\right)}{t}$, and the rectangular function form a Fourier transform pair.

The Fourier transform of F(t) = $\mathrm{Rect}\left(\frac{t}{\mathrm{\tau }}\right)$ is:  

 The bandwidth or the range of frequency of the function is ≈ $\frac{2\cdot \mathrm{\π}}{\mathrm{\τ}}$$$$$