If for a signal, the quantities are defined only on a discrete set of times, we call it a discrete-time signal. In other words, a discrete-time real (or complex) signal can be seen as a function from the set of integers to the set of real (or complex) numbers.
A continuous-time real (or complex) signal is any real-valued (or complex-valued) function which is defined for all time t in an interval, most commonly an infinite interval.
Analog and digital signals
Less formally than the theoretical distinctions mentioned above, two main types of signals encountered in practice are analog and digital. In short, the difference between them is that digital signals are discrete and quantized, as defined below, while analog signals possess neither property.
Discretization
One of the fundamental distinctions between different types of signals is between continuous and discrete time. In the mathematical abstraction, the domain of a continuous-time (CT) signal is the set of real numbers (or some interval thereof), whereas the domain of a discrete-time (DT) signal is the set of integers (or some interval). What these integers represent depends on the nature of the signal.
DT signals often arise via sampling of CT signals. An audio signal, for example consists of a continually fluxuating voltage on a line that can be digitized by an ADC circuit, wherein the circuit will read the voltage level on the line, say, every 50 µs. The resulting stream of numbers are stored as digital data on a discrete-time signal. Computers and other digital devices are restricted to discrete time.
Quantization
If a signal is to be represented as a sequence of numbers, it is impossible to maintain arbitrarily high precision - each number in the sequence must have a finite number of digits. As a result, the values of such a signal are restricted to belong to a finite set; in other words, it is quantized.
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