In the context of digital signal processing (DSP), a digital signal is a discrete-time signal for which not only the time but also the amplitude has discrete values; in other words, its samples take on only values from a discrete set (a countable set that can be mapped one-to-one to a subset of integers). If that discrete set is finite, the discrete values can be represented with digital words of a finite width. Most commonly, these discrete values are represented as fixed-point words (either proportional to the waveform values or companded) or floating-point words.
Discrete cosine waveform with frequency of 50 Hz and a sampling rate of 1000 samples/sec, easily satisfying the sampling theorem for reconstruction of the original cosine function from samples
The process of analog-to-digital conversion produces a digital signal. The conversion process can be thought of as occurring in two steps:
sampling, which produces a continuous-valued discrete-time signal, and
quantization, which replaces each sample value by an approximation selected from a given discrete set (for example by truncating or rounding).
It can be shown that for signal frequencies strictly below the Nyquist limit that the original continuous-valued continuous-time signal can be almost perfectly reconstructed, down to the (often very low) limit set by the quantisation.
Common practical digital signals are represented as 8-bit (256 levels), 16-bit (65,536 levels), 24-bit (16.8 million levels) and 32-bit (4.3 billion levels). But the number of quantization levels is not necessarily limited to powers of two. A floating point representation is used in many DSP applications.