Signum function and sinc function

Signum Function

This function is represented as

sgn(t)={1−1fort>0fort<0$$sgn(t)=\{\begin{array}{ll}1& fort>0\\ -1& fort<0\end{array}$$

It is a power type signal. Its power value and RMS (Root mean square) values, both are 1. Average value of signum function is zero.

Sinc Function

It is also a function of sine and is written as −

SinC(t)=SinΠtΠT=Sa(Πt)$$SinC(t)=\frac{Sin\mathrm{\Pi }t}{\mathrm{\Pi }T}=Sa(\mathrm{\Pi }t)$$

Properties of Sinc function

• It is an energy type signal.
• Sinc(0)=limt→0sinΠtΠt=1$Sinc(0)=\underset{t\to 0}{lim}\frac{\sin \mathrm{\Pi }t}{\mathrm{\Pi }t}=1$
• Sinc(∞)=limt→∞sinΠ∞Π∞=0$Sinc(\mathrm{\infty })=\underset{t\to \mathrm{\infty }}{lim}\frac{\sin \mathrm{\Pi }\mathrm{\infty }}{\mathrm{\Pi }\mathrm{\infty }}=0$(Range of sinπ∞ varies between -1 to +1 but anything divided by infinity is equal to zero)
• If sinc(t)=0=>sinΠt=0$\sin c(t)=0=>\sin \mathrm{\Pi }t=0$
  ⇒Πt=nΠ$\Rightarrow \mathrm{\Pi }t=n\mathrm{\Pi }$
  ⇒t=n(n≠0)