Signal (information Theory)

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## Definitions

## Classification

### Analog and digital signals

#### Analog signal

#### Digital signal

### Energy Signals and Power Signals^{[14]}

### Deterministic and Random Signals

### Even and Odd Signals

### Periodic and Aperiodic Signals

#### Time discretization

### Amplitude quantization

## Examples of signals

## Signal processing

## Signals and systems

## See also

## References

## Further reading

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Signal Information Theory

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In signal processing, a **signal** is a function that conveys information about a phenomenon.^{[1]} In electronics and telecommunications, it refers to any time varying voltage, current or electromagnetic wave that carries information. A signal may also be defined as an observable change in a quantity.^{[2]}

Any physical quantity that exhibits variation in space or time can be a signal used, among other possibilities, to share messages between observers.^{[3]} According to the *IEEE Transactions on Signal Processing*, a signal can be audio, video, speech, image, sonar and radar-related and so on.^{[4]} In a later effort of redefining a signal,^{[2]} anything that is only a function of space, such as an image, is excluded from the category of signals. Also, it is stated that a signal may or may not contain any information.

In nature, signals can be actions done by an organism to alert other organisms, ranging from the release of plant chemicals to warn nearby plants of a predator, to sounds or motions made by animals to alert other animals of food. Signalling occurs in all organisms even at cellular levels, with cell signaling. Signaling theory, in evolutionary biology, proposes that a substantial driver for evolution is the ability for animals to communicate with each other by developing ways of signaling. In human engineering, signals are typically 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]}

Information theory serves as the formal study of signals and their content, and the information of a signal is often accompanied by noise. The term "noise" refers to unwanted signal modifications, but is often extended to include unwanted signals conflicting with desired signals (crosstalk). The reduction of noise is covered in part under the heading of signal integrity. The separation of desired signals from background noise is the field of signal recovery,^{[5]} 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 user-machine interfaces.

Definitions specific to sub-fields 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, signals are analog and digital representations of analog physical quantities.

In terms of their spatial distributions, signals may be categorized as point source signals (PSSs) and distributed source signals (DSSs).^{[2]}

In a communication system, a *transmitter* encodes a *message* to create 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 signal. The signal is transmitted to the receiving telephone by wires; at the receiver it is reconverted into sounds.

In telephone networks, signaling, for example common-channel 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. Discrete-time signals are often referred to as *time series* in other fields. Continuous-time signals are often referred to as *continuous signals*.

A second important distinction is between discrete-valued and continuous-valued. Particularly in digital signal processing, a digital signal may be defined as a sequence of discrete values, typically associated with an underlying continuous-valued physical process. In digital electronics, digital signals are the continuous-time waveform signals in a digital system, representing a bit-stream.

Another important property of a signal is its entropy or information content.

In Signals and Systems, signals can be classified according to many criteria, mainly: according to the different feature of values, classified into analog signals and digital signals; according to the determinacy of signals, classified into deterministic signals and random signals; according to the strength of signals, classified into energy signals and power signals.

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 *quantized*, while analog signals are continuous.

An analog signal is any continuous signal for which the time varying feature of the signal is a representation of some other time varying quantity, i.e., *analogous* to another time varying signal. For example, in an analog audio signal, the instantaneous voltage of the signal varies continuously with the sound pressure. It differs from a digital signal, in which the continuous quantity is a representation of a sequence of discrete values which can only take on one of a finite number of values.^{[6]}^{[7]}

The term *analog signal* usually refers to electrical signals; however, analog signals may use other mediums such as mechanical, pneumatic or hydraulic. An analog signal uses some property of the medium to convey the signal's information. For example, an aneroid barometer uses rotary position as the signal to convey pressure information. In an electrical signal, the voltage, current, or frequency of the signal may be varied to represent the information.

Any information may be conveyed by an analog signal; often such a signal is a measured response to changes in physical phenomena, such as sound, light, temperature, position, or pressure. The physical variable is converted to an analog signal by a transducer. For example, in sound recording, fluctuations in air pressure (that is to say, sound) strike the diaphragm of a microphone which induces corresponding electrical fluctuations. The voltage or the current is said to be an *analog* of the sound.

A digital signal is a signal that is constructed from a discrete set of waveforms of a physical quantity so as to represent a sequence of discrete values.^{[8]}^{[9]}^{[10]} A *logic signal* is a digital signal with only two possible values,^{[11]}^{[12]} and describes an arbitrary bit stream. Other types of digital signals can represent three-valued logic or higher valued logics.

Alternatively, a digital signal may be considered to be the sequence of codes represented by such a physical quantity.^{[13]} The physical quantity may be a variable electric current or voltage, the intensity, phase or polarization of an optical or other electromagnetic field, acoustic pressure, the magnetization of a magnetic storage media, etc. Digital signals are present in all digital electronics, notably computing equipment and data transmission.

With digital signals, system noise, provided it is not too great, will not affect system operation whereas noise always degrades the operation of analog signals to some degree.

Digital signals often arise via sampling of analog signals, for example, a continually fluctuating voltage on a line that can be digitized by an analog-to-digital converter circuit, wherein the circuit will read the voltage level on the line, say, every 50 microseconds and represent each reading with a fixed number of bits. The resulting stream of numbers is stored as digital data on a discrete-time and quantized-amplitude signal. Computers and other digital devices are restricted to discrete time.

According to the strengths of signals, practical signals can be classified into two categories: energy signals and power signals.

Energy signals: Those signals' energy are equal to a finite positive value, but their average powers are 0;

Power signals: Those signals' average power are equal to a finite positive value, but their energy are infinite.

Deterministic signals are those whose values at any time are deterministic and predictable, and it can be calculated by a mathematical equation.

Random signals are signals that take on random values at any given time instant and must be modeled probabilistically.^{[15]}

Even signal satisfies the condition

or equivalently if the following equation holds for all and in the domain of :

Odd signal satisfies the condition

or equivalently if the following equation holds for all and in the domain of :

A signal is said to be periodic if it satisfies the condition

or

Where

= fundamental time period,

= fundamental frequency.

A periodic signal will repeat for every period.

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; most often it is time.

If for a signal, the quantities are defined only on a discrete set of times, we call it a discrete-time signal. A simple source for a discrete time signal is the sampling of a continuous, approximating the signal by a sequence of its values at particular time instants.

A discrete-time 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 continuous-time real (or complex) signal is any real-valued (or complex-valued) function which is defined at every time *t* in an interval, most commonly an infinite interval.

If a signal is to be represented as a sequence of numbers, it is impossible to maintain exact precision - each number in the sequence must have a finite number of digits. As a result, the values of such a signal belong to a finite set; in other words, it is quantized. Quantization is the process of converting a continuous analog audio signal to a digital signal with discrete numerical values.

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.^{[16]}For example, radar can provide an electromagnetic signal for following aircraft motion. A motion signal is one-dimensional (time), and the range is generally three-dimensional. Position is thus a 3-vector signal; position and orientation of a rigid body is a 6-vector signal. Orientation signals can be generated using a gyroscope.^{[17]}*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 2-vector 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.^{[18]}*Images*. A picture or image consists of a brightness or color signal, a function of a two-dimensional 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 charge-coupled 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 vector-valued with dimension three.*Videos*. A video signal is a sequence of images. A point in a video is identified by its two-dimensional position and by the time at which it occurs, so a video signal has a three-dimensional domain. Analog video has one continuous domain dimension (across a scan line) and two discrete dimensions (frame and line).- Biological
*membrane potentials*. The value of the signal is an electric potential ("voltage"). The domain is more difficult to establish. Some cells or organelles have the same membrane potential throughout; 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.

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]}

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.

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.^{[19]}

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.^{[20]}

- Current loop - a signaling system in widespread use for process control
- Impulse function
- Signal noise
- Signal to noise ratio
- Signal processing
- Signal strength
- Image processing

- ^
^{a}^{b}^{c}Roland Priemer (1991).*Introductory Signal Processing*. World Scientific. p. 1. ISBN 978-9971509194. Archived from the original on 2013-06-02. - ^
^{a}^{b}^{c}Pragnan Chakravorty, "What Is a Signal? [Lecture Notes],"IEEE Signal Processing Magazine, vol. 35, no. 5, pp. 175-177, Sept. 2018. https://doi.org/10.1109/MSP.2018.2832195 **^**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. ISBN 978-0387755809. Archived from the original on 2013-06-02.To put it very generally, a signal is any time-varying physical quantity.

**^**"Aims and scope".*IEEE Transactions on Signal Processing*. IEEE. Archived from the original on 2012-04-17.**^**T. H. Wilmshurst (1990).*Signal Recovery from Noise in Electronic Instrumentation*(2nd ed.). CRC Press. pp. 11*ff*. ISBN 978-0750300582. Archived from the original on 2015-03-19.**^**"Digital signals".*www.st-andrews.ac.uk*. Archived from the original on 2017-03-02. Retrieved .**^**"Analog vs. Digital - learn.sparkfun.com".*learn.sparkfun.com*. Archived from the original on 2017-07-05. Retrieved .**^**Robert K. Dueck.*Digital Design with CPLD Applications and VHDL*. Archived from the original on 2017-12-17.A digital representation can have only specific discrete values

**^**Proakis, John G.; Manolakis, Dimitris G. (2007-01-01).*Digital Signal Processing*. Pearson Prentice Hall. ISBN 9780131873742. Archived from the original on 2016-05-20.**^***Analogue and Digital Communication Techniques*. Archived from the original on 2017-12-17.A digital signal is a complex waveform and can be defined as a discrete waveform having a finite set of levels

**^**"Digital Signal". Retrieved .**^**Paul Horowitz; Winfield Hill (2015).*The Art of Electronics*. Cambridge University Press. ISBN 9780521809269.**^**Vinod Kumar Khanna (2009).*Digital Signal Processing*. p. 3. ISBN 9788121930956.A digital signal is a special form of discrete-time signal which is discrete in both time and amplitude, obtained by permitting each value (sample) of a discrete-time signal to acquire a finite set of values (quantization), assigning it a numerical symbol according to a code ... A digital signal is a sequence or list of numbers drawn from a finite set.

**^**Sklar, Bernard, 1927- (2001).*Digital communications : fundamentals and applications*(2nd ed.). Upper Saddle River, N.J.: Prentice-Hall PTR. ISBN 0130847887. OCLC 45823120.CS1 maint: multiple names: authors list (link)**^**Ziemer, Rodger E.,.*Principles of communication : systems, modulation, and noise*. Tranter, William H., (Seventh ed.). Hoboken, New Jersey. ISBN 9781118078914. OCLC 856647730.CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link)**^**For an example from robotics, see K Nishio & T Yasuda (2011). "Analog-digital circuit for motion detection based on vertebrate retina and its application to mobile robot". In Bao-Liang Lu; Liqing Zhang & James Kwok (eds.).*Neural Information Processing: 18th International Conference, Iconip 2011, Shanghai, China, November 13-17, 2011*. Springer. pp. 506*ff*. ISBN 978-3642249648. Archived from the original on 2013-06-02.**^**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. ISBN 978-3642154935. Archived from the original on 2013-06-02.**^**The optical reading process is described by Mark L. Chambers (2004).*CD & DVD Recording for Dummies*(2nd ed.). John Wiley & Sons. p. 13. ISBN 978-0764559563. Archived from the original on 2013-06-02.**^**David McMahon (2007).*Signals & Systems Demystified*. New York: McGraw Hill. ISBN 978-0-07-147578-5.**^**M.J. Roberts (2011).*Signals and Systems: Analysis Using Transform Methods & MATLAB*. New York: McGraw Hill. ISBN 978-0073380681.

- Hsu, P. H.
*Schaum's Theory and Problems: Signals and Systems*, McGraw-Hill 1995, ISBN 0-07-030641-9 - Lathi, B.P.,
*Signal Processing & Linear Systems*, Berkeley-Cambridge Press, 1998, ISBN 0-941413-35-7 - 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.

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