Let denote the th sample of the original sound , where is time in seconds. Thus, ranges over the integers, and is the sampling interval in seconds. The sampling rate in Hertz (Hz) is just the reciprocal of the sampling period, i.e.,
To avoid losing any information as a result of sampling, we must assume is bandlimited to less than half the sampling rate. This means there can be no energy in at frequency or above. We will prove this mathematically when we prove the sampling theorem in §D.3 below.
Let denote the Fourier transform of , i.e.,
The reconstruction of a sound from its samples can thus be interpreted as follows: convert the sample stream into a weighted impulse train, and pass that signal through an ideal lowpass filter which cuts off at half the sampling rate. These are the fundamental steps of digital to analog conversion (DAC). In practice, neither the impulses nor the lowpass filter are ideal, but they are usually close enough to ideal that one cannot hear any difference. Practical lowpass-filter design is discussed in the context of bandlimited interpolation .
Continuous-Time Aliasing Theorem
The Sinc Function