A.10  Sinc Function

Our definition of sinc is:

Some authors call it Sa (sample function) and others do not include $\pi$ in the definition. Its first zero occurs when $x\; =\; 1$. Its value $\mathrm{\text{sinc}}(0)\; =\; 1$ at origin can be determined using L’Hospital rule. The sinc is an energy signal with unitary energy $E\; =\; 1$, which can be determined by its Fourier transform and Parseval’s relation. Its scaled version $\mathrm{\text{sinc}}(t∕T_s)$ is widely used in sampling theory and has energy $E\; =T_s$. As discussed in Example C.5, $\mathrm{\text{sinc}}((t\; -\; 5)∕3)$ corresponds to expanding $\mathrm{\text{sinc}}(t)$ by a factor of 3 and then delaying this intermediate result by 5.

The sincs are orthogonal when shifted by integers $m,n\; \in \; \mathbb{Z}$ (e. g., $\mathrm{\text{sinc}}(t\; -\; 3)$ and sinc(t+1) are orthogonal) and, consequently, the scaled sincs $\mathrm{\text{sinc}}(t∕T_s)$ are orthogonal when shifted by multiples of $T_s$, i. e.