In previous chapters we considered the problem of reconstructing a continuous-time signal from its discrete set of samples. Consequently, the recovery systems we designed were of hybrid nature with discrete-time input and continuous-time output. In this chapter we demonstrate that sampling theory also plays a crucial role in the design of fully discrete-time algorithms. In particular, we treat applications in which one would like to obtain a sequence of samples of a signal from a different set of samples of the same signal. In such settings, both the input and the output are digital.

One example is sampling rate conversion. Consider, for instance, digital audio files, which nowadays populate personal computers and media devices in enormous amounts. The sources of these files are diverse, and consequently a very common situation is that files are recorded at different sampling rates. When an audio file is played, the samples representing the recording are passed through a DAC. The analog output is then amplified and fed into a speaker. A typical DAC supports only a small number of sampling (and reconstruction) rates. If the audio samples do not correspond to one of these rates, then a preprocessing step is required in order to change their rate. This operation is performed by appropriate digital signal processing and is termed sampling rate conversion. Rate conversion is always required if two audio files with different sampling rates are played simultaneously, even when both sampling rates are supported by the DAC, since a standard DAC operates at a single rate.