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In the fields of communications, signal processing, and in electrical engineering more generally, a signal is any time-varying quantity. Signals are often scalar-valued functions of time (waveforms), but may be vector valued and may be functions of any other relevant independent variable.
The concept is broad, and hard to define precisely. Definitions specific to subfields are common. For example, in information theory, a signal is a codified message, that is, the sequence of states in a communication channel that encodes a message. In a communication system, a transmitter encodes a message into a signal, which is carried to a receiver by the communications channel. For example, the words "Mary had a little lamb" might be the message spoken into a telephone. The telephone transmitter converts the sounds into an electrical voltage signal. The signal is transmitted to the receiving telephone by wires; and at the receiver it is reconverted into sounds.
Signals can be categorized in various ways. The most common distinction is between discrete and continuous spaces that the functions are defined over, for example discrete and continuous time domains. Discrete-time signals are often referred to as time series in other fields. Continuous-time signals are often referred to as continuous signals even when the signal functions are not continuous; an example is a square-wave signal.
A second important distinction is between discrete-valued and continuous-valued. Digital signals are discrete-valued, but are often derived from an underlying continuous-valued physical process.
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.
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.
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 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. For instance, sensors output data continuously, but since a continuous stream may be difficult to record, a discrete-time signal is often used as an approximation. Computers and other digital devices are restricted to discrete time.
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.
Signals are often analyzed or modeled in terms of their frequency spectrum. Frequency domain techniques are applicable to all signals, both continuous-time and discrete-time. If a signal is passed through an LTI system, the frequency spectrum of the resulting output signal is the product of the frequency spectrum of the original input signal and the frequency response of the system.
Another important property of a signal (actually, of a statistically defined class of signals) is its entropy or information content.
Shannon, C. E., 2005 [1948], "A Mathematical Theory of Communication," (corrected reprint), accessed Dec. 15, 2005. Orig. 1948, Bell System Technical Journal, vol. 27, pp. 379-423, 623-656.