Rectangular Window

The simplest method to bound the ideal derivative reconstruction filter to some finite extend is to truncate it. This is tantamount to multiplying it with a rectangular function which is 1 inside some extend and 0 outside.

$$\textrm{Rect}(x,\tau)=\left \{ \begin{array}{cl} 1 & \textrm{$\vert x\vert < \tau$}\\ 0 & \textrm{else} \end{array} \right.$$ (5)

Although the frequency response approximates the ideal one quite good below $\pi$, it is significantly different from zero in form of bumps above, which is due to the truncation and particularly annoying in gradient reconstruction because it introduces artifacts even more perceivable than in function reconstruction. To avoid these bumps, other windows trade them against the rather good approximation below $\pi$.