Reconstruction from Samples--Pictorial Version

Figure G.1 shows how a sound is reconstructed from its samples. Each sample can be considered as specifying the scaling and location of a sinc function. The discrete-time signal being interpolated in the figure is a digital rectangular pulse:

$$x = [\dots, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, \dots]$$

The sinc functions are drawn with dashed lines, and they sum to produce the solid curve. An isolated sinc function is shown in Fig. G.2. Note the ``Gibb's overshoot'' near the corners of the continuous rectangular pulse in Fig. G.1 due to bandlimiting. (A true continuous rectangular pulse has infinite bandwidth.)

Figure: How sinc functions sum up to create a continuous waveform from discrete-time samples.

Notice that each sinc function passes through zero at every sample instant but the one it is centered on, where it passes through 1.

Figure: The sinc function.

The sinc function is the famous ``sine x over x'' curve, defined by

   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. The sinc function has zeros at all the integers except 0, where it equals 1.