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, ie, the sequence of states in a communications channel that encodes a message. In a communications 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.