It is set.
${\ displaystyle \ Theta (0) = {\ tfrac {1} {2}}}$

The Fourier transformation of the rectangular function results in the sinc function :
${\ displaystyle \ operatorname {sinc} (x) = \ sin (\ pi x) / (\ pi x)}$

Here it is important to use the first definition of the rectangular function, for the last equation is wrong.
${\ displaystyle \ operatorname {rect_ {d}}}$

Shifting and scaling

A rectangular function centered at and having a duration of is expressed by
${\ displaystyle t_ {0}}$${\ displaystyle T}$

As a discontinuous function, the rectangular function is neither differentiable in the classical sense nor is it weakly differentiable . However, it is possible to derive a distribution through the Dirac delta distribution :
${\ displaystyle \ delta}$

^ Hans Dieter Lüke: Signal transmission. Basics of digital and analog communication systems . 6th, revised and expanded edition. Springer, Berlin et al. 1995, ISBN 3-540-58753-5 , pp.2 .