A signal as referred to in communication systems, signal processing, and electrical engineering "is a function that conveys information about the behavior or attributes of some phenomenon".^{[1]} In the physical world, any quantity exhibiting variation in time or variation in space (such as an image) is potentially a signal that might provide information on the status of a physical system, or convey a message between observers, among other possibilities.^{[2]} The IEEE Transactions on Signal Processing elaborates upon the term "signal" as follows:^{[3]}
The term "signal" includes, among others, audio, video, speech, image, communication, geophysical, sonar, radar, medical and musical signals.
Other examples of signals are the output of a thermocouple, which conveys temperature information, and the output of a pH meter which conveys acidity information.^{[1]} Typically, signals are often provided by a sensor, and often the original form of a signal is converted to another form of energy using a transducer. For example, a microphone converts an acoustic signal to a voltage waveform, and a speaker does the reverse.^{[1]}
The formal study of the information content of signals is the field of information theory. The information in a signal is usually accompanied by noise. The term noise usually means an undesirable random disturbance, but is often extended to include unwanted signals conflicting with the desired signal (such as crosstalk). The prevention of noise is covered in part under the heading of signal integrity. The separation of desired signals from a background is the field of signal recovery,^{[4]} one branch of which is estimation theory, a probabilistic approach to suppressing random disturbances.
Engineering disciplines such as electrical engineering have led the way in the design, study, and implementation of systems involving transmission, storage, and manipulation of information. In the latter half of the 20th century, electrical engineering itself separated into several disciplines, specialising in the design and analysis of systems that manipulate physical signals; electronic engineering and computer engineering as examples; while design engineering developed to deal with functional design of man–machine interfaces.
Contents

Signal processing 1

Some definitions 2

Discretetime and continuoustime signals 3

Analog and digital signals 4

Discretization 4.1

Quantization 4.2

Examples of signals 5

Entropy 6

Signals and Systems 7

See also 8

References 9

General references 10
Signal processing
Signal transmission using electronic signals
A typical role for signals is in signal processing. A common example is signal transmission between different locations. The embodiment of a signal in electrical form is made by a transducer that converts the signal from its original form to a waveform expressed as a current (I) or a voltage (V), or an electromagnetic waveform, for example, an optical signal or radio transmission. Once expressed as an electronic signal, the signal is available for further processing by electrical devices such as electronic amplifiers and electronic filters, and can be transmitted to a remote location by electronic transmitters and received using electronic receivers.
Some definitions
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 the context of signal processing, arbitrary binary data streams are not considered as signals, but only analog and digital signals that are representations of analog physical quantities.
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.
In telephone networks, signalling, for example commonchannel signaling, refers to phone number and other digital control information rather than the actual voice signal.
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. Discretetime signals are often referred to as time series in other fields. Continuoustime signals are often referred to as continuous signals even when the signal functions are not continuous; an example is a squarewave signal.
A second important distinction is between discretevalued and continuousvalued. Digital signals are sometimes defined as discretevalued sequences of quantified values, that may or may not be derived from an underlying continuousvalued physical process. In other contexts, digital signals are defined as the continuoustime waveform signals in a digital system, representing a bitstream. In the first case, a signal that is generated by means of a digital modulation method is considered as converted to an analog signal, while it is considered as a digital signal in the second case.
Discretetime and continuoustime signals
Discretetime signal created from a continuous signal by
sampling
Digital signal resulting from approximation to an analog signal, which is a continuous function of time
If for a signal, the quantities are defined only on a discrete set of times, we call it a discretetime signal. A simple source for a discrete time signal is the sampling of a continuous signal, approximating the signal by a sequence of its values at particular time instants.
A discretetime real (or complex) signal can be seen as a function from (a subset of) the set of integers (the index labeling time instants) to the set of real (or complex) numbers (the function values at those instants).
A continuoustime real (or complex) signal is any realvalued (or complexvalued) function which is defined at every 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. The figure shows a digital signal that results from approximating an analog signal by its values at particular time instants. 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 continuoustime (CT) signal is the set of real numbers (or some interval thereof), whereas the domain of a discretetime (DT) signal is the set of integers (or some interval). What these integers represent depends on the nature of the signal.
DT (discrete time) signals often arise via sampling of CT (continuous time) signals, for example, a continually fluctuating voltage on a line that can be digitized by an analogtodigital converter circuit, wherein the circuit will read the voltage level on the line, say, every 50 microseconds. The resulting stream of numbers is stored as digital data on a discretetime 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.
Examples of signals
Signals in nature can be converted to electronic signals by various sensors. Some examples are:

Motion. The motion of an object can be considered to be a signal, and can be monitored by various sensors to provide electrical signals.^{[5]} For example, radar can provide an electromagnetic signal for following aircraft motion. A motion signal is onedimensional (time), and the range is generally threedimensional. Position is thus a 3vector signal; position and orientation of a rigid body is a 6vector signal. Orientation signals can be generated using a gyroscope.^{[6]}

Sound. Since a sound is a vibration of a medium (such as air), a sound signal associates a pressure value to every value of time and three space coordinates. A sound signal is converted to an electrical signal by a microphone, generating a voltage signal as an analog of the sound signal, making the sound signal available for further signal processing. Sound signals can be sampled at a discrete set of time points; for example, compact discs (CDs) contain discrete signals representing sound, recorded at 44,100 samples per second; each sample contains data for a left and right channel, which may be considered to be a 2vector signal (since CDs are recorded in stereo). The CD encoding is converted to an electrical signal by reading the information with a laser, converting the sound signal to an optical signal.^{[7]}

Images. A picture or image consists of a brightness or color signal, a function of a twodimensional location. The object's appearance is presented as an emitted or reflected electromagnetic wave, one form of electronic signal. It can be converted to voltage or current waveforms using devices such as the chargecoupled device. A 2D image can have a continuous spatial domain, as in a traditional photograph or painting; or the image can be discretized in space, as in a raster scanned digital image. Color images are typically represented as a combination of images in three primary colors, so that the signal is vectorvalued with dimension three.

Videos. A video signal is a sequence of images. A point in a video is identified by its twodimensional position and by the time at which it occurs, so a video signal has a threedimensional domain. Analog video has one continuous domain dimension (across a scan line) and two discrete dimensions (frame and line).

Biological neurons generally have different potentials at different points. These signals have very low energies, but are enough to make nervous systems work; they can be measured in aggregate by the techniques of electrophysiology.
Entropy
Another important property of a signal (actually, of a statistically defined class of signals) is its entropy or information content.
Signals and Systems
In Electrical engineering programs, a class and field of study known as "signals and systems" (S and S) is often seen as the "cut class" for EE careers, and is dreaded by some students as such. Depending on the school, undergraduate EE students generally take the class as juniors or seniors, normally depending on the number and level of previous linear algebra and differential equation classes they have taken.^{[8]}
The field studies input and output signals, and the mathematical representations between them known as systems, in four domains: Time, Frequency, s and z. Since signals and systems are both studied in these four domains, there are 8 major divisions of study. As an example, when working with continuous time signals (t), one might transform from the time domain to a frequency or s domain; or from discrete time (n) to frequency or z domains. Systems also can be transformed between these domains like signals, with continuous to s and discrete to z.
Although S and S falls under and includes all the topics covered in this article, as well as Analog signal processing and Digital signal processing, it actually is a subset of the field of Mathematical modeling. The field goes back to RF over a century ago, when it was all analog, and generally continuous. Today, software has taken the place of much of the analog circuitry design and analysis, and even continuous signals are now generally processed digitally. Ironically, digital signals also are processed continuously in a sense, with the software doing calculations between discrete signal "rests" to prepare for the next input/transform/output event.
In past EE curricula S and S, as it is often called, involved circuit analysis and design via mathematical modeling and some numerical methods, and was updated several decades ago with Dynamical systems tools including differential equations, and recently, Lagrangians. The difficulty of the field at that time included the fact that not only mathematical modeling, circuits, signals and complex systems were being modeled, but physics as well, and a deep knowledge of electrical (and now electronic) topics also was involved and required.
Today, the field has become even more daunting and complex with the addition of circuit, systems and signal analysis and design languages and software, from MATLAB and Simulink to NumPy, VHDL, PSpice, Verilog and even Assembly language. Students are expected to understand the tools as well as the mathematics, physics, circuit analysis, and transformations between the 8 domains.
Because mechanical engineering topics like friction, dampening etc. have very close analogies in signal science (inductance, resistance, voltage, etc.), many of the tools originally used in ME transformations (Laplace and Fourier transforms, Lagrangians, sampling theory, probability, difference equations, etc.) have now been applied to signals, circuits, systems and their components, analysis and design in EE. Dynamical systems that involve noise, filtering and other random or chaotic attractors and repellors have now placed stochastic sciences and statistics between the more deterministic discrete and continuous functions in the field. (Deterministic as used here means signals that are completely determined as functions of time).
EE taxonomists are still not decided where S&S falls within the whole field of signal processing vs. circuit analysis and mathematical modeling, but the common link of the topics that are covered in the course of study has brightened boundaries with dozens of books, journals, etc. called Signals and Systems, and used as text and test prep for the EE, as well as, recently, computer engineering exams.^{[9]} The Hsu general reference given below is a good example, with a new edition scheduled for late 2013/ early 2014.
See also
References

^ ^{a} ^{b} ^{c} Roland Priemer (1991). Introductory Signal Processing. World Scientific. p. 1.

^ Some authors do not emphasize the role of information in the definition of a signal. For example, see Priyabrata Sinha (2009). Speech processing in embedded systems. Springer. p. 9.

^ "Aims and scope". IEEE Transactions on Signal Processing (

^ T. H. Wilmshurst (1990). Signal Recovery from Noise in Electronic Instrumentation (2nd ed.). CRC Press. pp. 11 ff.

^ For an example from robotics, see K Nishio and T Yasuda (2011). "Analog–digital circuit for motion detection based on vertebrate retina and its application to mobile robot". In BaoLiang Lu, Liqing Zhang, James Kwok. Neural Information Processing: 18th International Conference, Iconip 2011, Shanghai, China, November 1317, 2011. Springer. pp. 506 ff.

^ For example, see M. N. Armenise, Caterina Ciminelli, Francesco Dell'Olio, Vittorio Passaro (2010). "§4.3 Optical gyros based on a fiber ring laser". Advances in Gyroscope Technologies. Springer. p. 47.

^ The optical reading process is described by Mark L. Chambers (2004). CD & DVD Recording for Dummies (2nd ed.). John Wiley & Sons. p. 13.

^ David McMahon (2007). Signals & Systems Demystified. New York: McGraw Hill.

^ M.J. Roberts (2011). Signals and Systems: Analysis Using Transform Methods & MATLAB. New York: McGraw Hill.
General references

Hsu, P. H. Schaum's Theory and Problems: Signals and Systems, McGrawHill 1995, ISBN 0070306419

Lathi, B.P., Signal Processing & Linear Systems, BerkeleyCambridge Press, 1998, ISBN 0941413357

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, 623656.
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