The Sinc Function

Figure: The sinc function $\protect$sinc$(x) \isdef \sin (\pi x)/(\pi x)$.

The sinc function, or cardinal sine function, is the famous ``sine x over x'' curve, and is illustrated in Fig.D.2. For bandlimited interpolation of discrete-time signals, the ideal interpolation kernel is proportional to the sinc function

   sinc$$(f_st) \isdef \frac{\sin(\pi f_st)}{\pi f_st}.$$

where $f_s$ denotes the sampling rate in samples-per-second (Hz), and $t$ denotes time in seconds. Note that the sinc function has zeros at all the integers except 0, where it is 1. For precise scaling, the desired interpolation kernel is $f_s$sinc$(f_st)$, which has a algebraic area (time integral) that is independent of the sampling rate $f_s$.