# Signal

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Varying physical quantity that conveys information

For other uses, see [Signal (disambiguation)](/source/Signal_(disambiguation)).

Not to be confused with [Cignal](/source/Cignal) or [Cygnal](/source/Cygnal).

In *The Signal* by [William Powell Frith](/source/William_Powell_Frith), a woman sends a signal by waving a white handkerchief.

A **signal** is both the process and the result of [transmission](/source/Signal_transmission) of [data](/source/Data) over some [media](/source/Transmission_media) accomplished by embedding some variation. Signals are important in multiple subject fields, including [signal processing](/source/Signal_processing), [information theory](/source/Information_theory) and [biology](/source/Biology).

In signal processing, a signal is a function that conveys [information](/source/Information) about a phenomenon.[1] Any quantity that can vary over space or time can be used as a signal to share messages between observers.[2] The *[IEEE Transactions on Signal Processing](/source/IEEE_Transactions_on_Signal_Processing)* includes [audio](/source/Audio_signal), [video](/source/Video), speech, [image](/source/Image), [sonar](/source/Sonar), and [radar](/source/Radar) as examples of signals.[3] A signal may also be defined as *any* observable change in a quantity over space or time (a [time series](/source/Time_series)), even if it does not carry information.[a]

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. Signaling occurs in all organisms, even at cellular levels, with [cell signaling](/source/Cell_signaling). [Signaling theory](/source/Signaling_theory), in [evolutionary biology](/source/Evolutionary_biology), proposes that a substantial driver for [evolution](/source/Evolution) is the ability of animals to communicate with each other by developing ways of signaling. In human engineering, signals are typically provided by a [sensor](/source/Sensor), and often the original form of a signal is converted to another form of energy using a [transducer](/source/Transducer). For example, a [microphone](/source/Microphone) converts an acoustic signal to a voltage waveform, and a [speaker](/source/Loudspeaker) does the reverse.[1]

Another important property of a signal is its [entropy](/source/Entropy_(information_theory)) or [information content](/source/Information_content). [Information theory](/source/Information_theory) serves as the formal study of signals and their content. The information of a signal is often accompanied by [noise](/source/Noise_(electronics)), which primarily refers to unwanted modifications of signals, but is often extended to include unwanted signals conflicting with desired signals ([crosstalk](/source/Crosstalk)). The reduction of noise is covered in part under the heading of [signal integrity](/source/Signal_integrity). The separation of desired signals from background noise is the field of [signal recovery](/source/Signal_recovery),[5] one branch of which is [estimation theory](/source/Estimation_theory), a probabilistic approach to suppressing random disturbances.

Engineering disciplines such as electrical engineering have advanced the design, study, and implementation of systems involving [transmission](/source/Data_transmission), [storage](/source/Data_storage_device), and manipulation of information. In the latter half of the 20th century, electrical engineering itself separated into several disciplines: [electronic engineering](/source/Electronic_engineering) and [computer engineering](/source/Computer_engineering) developed to specialize in the design and analysis of systems that manipulate physical signals, while [design engineering](/source/Industrial_design) developed to address the functional design of signals in [user–machine interfaces](/source/User_interface).

## Definitions

Definitions specific to sub-fields are common:

- In [electronics](/source/Electronics) and [telecommunications](/source/Telecommunications), *signal* refers to any time-varying [voltage](/source/Voltage), [current](/source/Electric_current), or [electromagnetic wave](/source/Electromagnetic_wave) that carries information.

- In [signal processing](/source/Signal_processing), signals are analog and digital representations of analog physical quantities.

- In [information theory](/source/Information_theory), a signal is a codified message, that is, the sequence of [states](/source/State_variable) in a [communication channel](/source/Communication_channel) that encodes a message.

- In a communication system, a *transmitter* encodes a *message* to create a signal, which is carried to a *receiver* by the communication channel. For example, the words "[Mary had a little lamb](/source/Mary_had_a_little_lamb)" might be the message spoken into a [telephone](/source/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](/source/Signaling_(telecommunications)), for example [common-channel signaling](/source/Common-channel_signaling), refers to phone number and other digital control information rather than the actual voice signal.

## Classification

Signals can be categorized in various ways. The most common[*[verification needed](https://en.wikipedia.org/wiki/Wikipedia:Verifiability)*] distinction is between discrete and continuous spaces that the functions are defined over, for example, discrete and continuous-time domains. [Discrete-time signals](/source/Discrete-time_signal) are often referred to as *[time series](/source/Time_series)* in other fields. [Continuous-time signals](/source/Continuous-time_signal) are often referred to as *continuous signals*.

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

Signals may also be categorized by their spatial distributions as either point source signals (PSSs) or distributed source signals (DSSs).[2]

In Signals and Systems, signals can be classified according to many criteria, mainly: according to the different feature of values, classified into [analog signals](/source/Analog_signal) and [digital signals](/source/Digital_signal); according to the determinacy of signals, classified into deterministic signals and random signals; according to the [strength of signals](/source/Signal_strength_in_telecommunications), classified into energy signals and power signals.

### Analog and digital signals

A digital signal has two or more distinguishable waveforms, in this example, high voltage and low voltages, each of which can be mapped onto a digit. Characteristically, noise can be removed from digital signals provided it is not too extreme.

Two main types of signals encountered in practice are *[analog](/source/Analog_signal)* and *[digital](/source/Digital_signal)*. The figure shows a digital signal that results from approximating an analog signal by its values at particular time instants. Digital signals are *[quantized](/source/Quantization_(signal_processing))*, while analog signals are continuous.

#### Analog signal

Main article: [Analog signal](/source/Analog_signal)

An analog signal is any [continuous signal](/source/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](/source/Audio_signal), the instantaneous [voltage](/source/Voltage) of the signal varies continuously with the [sound pressure](/source/Sound_pressure). It differs from a [digital signal](/source/Digital_signal_(signal_processing)), in which the continuous quantity is a representation of a sequence of [discrete values](/source/Discrete_value) which can only take on one of a finite number of values.[6][7]

The term *analog signal* usually refers to [electrical signals](/source/Electrical_signal); however, analog signals may use other mediums such as [mechanical](/source/Classical_mechanics), [pneumatic](/source/Pneumatic) or [hydraulic](/source/Hydraulic). An analog signal uses some property of the medium to convey the signal's information. For example, an [aneroid barometer](/source/Aneroid_barometer) uses rotary position as the signal to convey pressure information. In an electrical signal, the [voltage](/source/Voltage), [current](/source/Electric_current), or [frequency](/source/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](/source/Sound), [light](/source/Light), [temperature](/source/Temperature), position, or [pressure](/source/Pressure). The physical variable is converted to an analog signal by a [transducer](/source/Transducer). For example, in sound recording, fluctuations in air pressure (that is to say, [sound](/source/Sound)) strike the diaphragm of a [microphone](/source/Microphone), which induces corresponding electrical fluctuations. The voltage or the current is said to be an *analog* of the sound.

#### Digital signal

Main article: [Digital signal](/source/Digital_signal)

A binary signal, also known as a logic signal, is a digital signal with two distinguishable levels.

A digital signal is a signal that is constructed from a discrete set of [waveforms](/source/Waveform) of a physical quantity so as to represent a sequence of [discrete](/source/Discrete_space) values.[8][9][10] A *logic signal* is a digital signal with only two possible values,[11][12] and describes an arbitrary [bit stream](/source/Bit_stream). Other types of digital signals can represent [three-valued logic](/source/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](/source/Polarization_(waves)) of an [optical](/source/Optical) or other [electromagnetic field](/source/Electromagnetism), acoustic pressure, the [magnetization](/source/Magnetization) of a [magnetic storage](/source/Magnetic_storage) media, etc. Digital signals are present in all [digital electronics](/source/Digital_electronics), notably computing equipment and [data transmission](/source/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](/source/Analog_signals) to some degree.

Digital signals often arise via [sampling](/source/Sampling_(signal_processing)) of analog signals, for example, a continually fluctuating voltage on a line that can be digitized by an [analog-to-digital converter](/source/Analog-to-digital_converter) circuit, wherein the circuit will read the voltage level on the line, say, every 50 [microseconds](/source/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](/source/Computer) and other [digital](/source/Digital_data) devices are restricted to discrete time.

### Energy and power

According to the strengths of signals, practical signals can be classified into two categories: energy signals and power signals.[14]

Energy signals: Those signals' [energy](/source/Energy) are equal to a finite positive value, but their average powers are 0;

0 < E = ∫ − ∞ ∞ s 2 ( t ) d t < ∞ {\displaystyle 0<E=\int _{-\infty }^{\infty }s^{2}(t)dt<\infty }

Power signals: Those signals' average [power](/source/Power_(physics)) are equal to a finite [positive](/source/Sign_(mathematics)) value, but their energy are [infinite](/source/Infinity).

P = lim T → ∞ 1 T ∫ − T / 2 T / 2 s 2 ( t ) d t {\displaystyle P=\lim _{T\rightarrow \infty }{\frac {1}{T}}\int _{-T/2}^{T/2}s^{2}(t)dt}

### Deterministic and random

Deterministic signals are those whose values at any time are predictable and 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 [stochastically](/source/Stochastic).[15]

### Even and odd

Even and odd signals

        f
        (
        x
        )
        =

          x

            2

    {\displaystyle f(x)=x^{2}}

 is an example of an even signal.

        f
        (
        x
        )
        =

          x

            3

    {\displaystyle f(x)=x^{3}}

 is an example of an odd signal.

An [even signal](/source/Even_and_odd_functions) satisfies the condition x ( t ) = x ( − t ) {\displaystyle x(t)=x(-t)}

or equivalently if the following equation holds for all t {\displaystyle t} and − t {\displaystyle -t} in the domain of x {\displaystyle x} :

- x ( t ) − x ( − t ) = 0. {\displaystyle x(t)-x(-t)=0.}

An odd signal satisfies the condition x ( t ) = − x ( − t ) {\displaystyle x(t)=-x(-t)}

or equivalently if the following equation holds for all t {\displaystyle t} and − t {\displaystyle -t} in the domain of x {\displaystyle x} :

- x ( t ) + x ( − t ) = 0. {\displaystyle x(t)+x(-t)=0.}

### Periodic

A signal is said to be [periodic](/source/Periodic_function) if it satisfies the condition:

x ( t ) = x ( t + T ) ∀ t ∈ [ t 0 , t m a x ] {\displaystyle x(t)=x(t+T)\quad \forall t\in [t_{0},t_{max}]} or x ( n ) = x ( n + N ) ∀ n ∈ [ n 0 , n m a x ] {\displaystyle x(n)=x(n+N)\quad \forall n\in [n_{0},n_{max}]}

Where:

T {\displaystyle T} = fundamental time [period](/source/Period_(physics)),

1 / T = f {\displaystyle 1/T=f} = fundamental [frequency](/source/Frequency).

The same can be applied to N {\displaystyle N} . A periodic signal will repeat for every period.

#### Time discretization

Discrete-time signal created from a continuous signal by [sampling](/source/Sampling_(signal_processing))

Signals can be classified as [continuous](/source/Continuous_signal) or [discrete time](/source/Discrete_time). In the mathematical abstraction, the domain of a continuous-time 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](/source/Integer) (or other subsets of real numbers). What these integers represent depends on the nature of the signal; most often, it is time.

A continuous-time signal is any [function](/source/Mathematical_function) which is defined at every time *t* in an interval, most commonly an infinite interval. A simple source for a discrete-time signal is the [sampling](/source/Sampling_(signal_processing)) of a continuous signal, approximating the signal by a sequence of its values at particular time instants.

### Amplitude quantization

If a signal is to be represented as a sequence of digital data, 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 must be [quantized](/source/Quantization_(signal_processing)) into a [finite set](/source/Finite_set) for practical representation. Quantization is the process of converting a continuous analog audio signal to a digital signal with discrete numerical values of integers.

## Examples

Naturally occurring signals can be converted to electronic signals by various [sensors](/source/Sensor). Examples include:

- *[Motion](/source/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](/source/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; the position and orientation of a rigid body is a 6-vector signal. Orientation signals can be generated using a [gyroscope](/source/Gyroscope).[17]

- *[Sound](/source/Sound)*. Since a sound is a [vibration](/source/Oscillation) of a medium (such as air), a sound signal associates a [pressure](/source/Pressure) value to every value of time and possibly three space coordinates indicating the direction of travel. A sound signal is converted to an electrical signal by a [microphone](/source/Microphone), generating a [voltage](/source/Voltage) signal as an analog of the sound signal. Sound signals can be [sampled](/source/Sampling_(signal_processing)) at a discrete set of time points; for example, [compact discs](/source/Compact_disc) (CDs) contain discrete signals representing sound, recorded at [44,100 Hz](/source/44%2C100_Hz); since CDs are recorded in [stereo](/source/Stereo), each sample contains data for a left and right channel, which may be considered to be a 2-vector signal. The CD encoding is converted to an electrical signal by reading the information with a [laser](/source/Laser), converting the sound signal to an optical signal.[18]

- *[Images](/source/Image)*. 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 emitted or reflected [light](/source/Light), an electromagnetic signal. It can be converted to voltage or current waveforms using devices such as the [charge-coupled device](/source/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 [digital image](/source/Digital_image). Color images are typically represented as a combination of monochrome images in three [primary colors](/source/Primary_colors).

- *[Videos](/source/Video)*. A video signal is a sequence of images. A point in a video is identified by its two-dimensional position in the image 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](/source/Scan_line)) and two discrete dimensions (frame and line).

- *[Temperature](/source/Temperature)*, conveyed for example by a [thermocouple](/source/Thermocouple) output.[1]

- *[pH](/source/PH)* or *acidity*, provided by a [pH meter](/source/PH_meter).[1]

- *[Biosignals](/source/Biosignal)*. A biosignal is any signal in a living organism that can be continually measured. The term biosignal is often used to refer to bioelectrical signals, but it may refer to both electrical and non-electrical signals. The usual understanding is to refer only to time-varying signals, although spatial parameter variations (e.g., the nucleotide sequence determining the [genetic code](/source/Genetic_code)) are sometimes subsumed as well. - *[Membrane potential](/source/Membrane_potential)*. This biosignal is the [voltage](/source/Voltage) across a [biological membrane](/source/Biological_membrane). The domain is more difficult to establish. Some [cells](/source/Cell_(biology)) or [organelles](/source/Organelle) have the same membrane potential throughout; [neurons](/source/Neuron) 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 [electrophysiology](/source/Electrophysiology) techniques.

## Signal processing

Main article: [Signal processing](/source/Signal_processing)

Signal transmission using electronic signals

Signal processing is the analysis and manipulation of signals. A common example is signal transmission between different locations. The embodiment of a signal in electrical form is made by a [transducer](/source/Transducer) that converts the signal from its original form to a [waveform](/source/Waveform) expressed as a [current](/source/Electric_current) or a [voltage](/source/Voltage), or [electromagnetic radiation](/source/Electromagnetic_radiation), for example, an [optical signal](/source/Free-space_optical_communication) or [radio transmission](/source/Radio_transmission). Once expressed as an electronic signal, the signal is available for further processing by electrical devices such as [electronic amplifiers](/source/Electronic_amplifier) and [filters](/source/Electronic_filters), and can be transmitted to a remote location by a [transmitter](/source/Transmitter) and received using [radio receivers](/source/Radio_receiver).

## Signals and systems

In [electrical engineering](/source/Electrical_engineering) (EE) programs, signals are covered in a class and field of study known as *signals and systems*. 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](/source/Linear_algebra) and [differential equation](/source/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 can also be transformed between these domains, like signals, with continuous to *s* and discrete to *z*.

Signals and systems is a subset of the field of [mathematical modeling](/source/Mathematical_model). It involves circuit analysis and design via mathematical modeling and some numerical methods, and was updated several decades ago with [dynamical systems](/source/Dynamical_system) tools, including differential equations, and recently, [Lagrangians](/source/Lagrangian_mechanics). Students are expected to understand the modeling tools as well as the mathematics, physics, circuit analysis, and transformations between the 8 domains.

Because mechanical engineering (ME) 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 repellers 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 signals and systems 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]

## Gallery

		- A signalman sends a semaphore message from a Pearl Harbor Control Tower, c. 1960.

		- A Finnish distant signal at the western approach to [Muhos](/source/Muhos) station is displaying *Expect Stop*.

		- A woman hailing a cab is sending a signal of availability to be picked up.

		- A [flare](/source/Flare) is a common means to signal during dark or smoke-filled conditions.

## See also

Wikibooks has a book on the topic of: ***[Signals and Systems](https://en.wikibooks.org/wiki/Signals_and_Systems)***

- [A Mathematical Theory of Communication](/source/A_Mathematical_Theory_of_Communication) – 1948 scholarly article by Claude Shannon

- [Beacon](/source/Beacon)

- [Current loop](/source/Current_loop) – a signaling system in widespread use for process control

- [Signal-to-noise ratio](/source/Signal-to-noise_ratio)

## Notes

1. **[^](#cite_ref-5)** Some authors do not emphasize the role of information in the definition of a signal.[4]

## References

1. ^ [***a***](#cite_ref-Priemer_1-0) [***b***](#cite_ref-Priemer_1-1) [***c***](#cite_ref-Priemer_1-2) [***d***](#cite_ref-Priemer_1-3) Roland Priemer (1991). [*Introductory Signal Processing*](https://books.google.com/books?id=QBT7nP7zTLgC&pg=PA1). World Scientific. p. 1. [ISBN](/source/ISBN_(identifier)) [978-9971509194](https://en.wikipedia.org/wiki/Special:BookSources/978-9971509194). [Archived](https://web.archive.org/web/20130602161539/http://books.google.com/books?id=QBT7nP7zTLgC&pg=PA1) from the original on 2013-06-02. A signal is a function that conveys information about the behavior of a system or attributes of some phenomenon.

1. ^ [***a***](#cite_ref-PC_2-0) [***b***](#cite_ref-PC_2-1) Chakravorty, Pragnan (2018). "What Is a Signal? [Lecture Notes]". *IEEE Signal Processing Magazine*. **35** (5): 175–177. [Bibcode](/source/Bibcode_(identifier)):[2018ISPM...35e.175C](https://ui.adsabs.harvard.edu/abs/2018ISPM...35e.175C). [doi](/source/Doi_(identifier)):[10.1109/MSP.2018.2832195](https://doi.org/10.1109%2FMSP.2018.2832195). [S2CID](/source/S2CID_(identifier)) [52164353](https://api.semanticscholar.org/CorpusID:52164353). Consequently, a signal, represented as a function of one or more variables, may be defined as an observable change in a quantifiable entity.

1. **[^](#cite_ref-IEEE_3-0)** ["Aims and scope"](https://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=78). *IEEE Transactions on Signal Processing*. [IEEE](/source/IEEE). [Archived](https://web.archive.org/web/20120417233106/http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=78) from the original on 2012-04-17.

1. **[^](#cite_ref-Sinha_4-0)** Priyabrata Sinha (2009). [*Speech processing in embedded systems*](https://books.google.com/books?id=mrOHsYChY-oC&pg=PA9). Springer. p. 9. [ISBN](/source/ISBN_(identifier)) [978-0387755809](https://en.wikipedia.org/wiki/Special:BookSources/978-0387755809). [Archived](https://web.archive.org/web/20130602161013/http://books.google.com/books?id=mrOHsYChY-oC&pg=PA9) from the original on 2013-06-02. To put it very generally, a signal is any time-varying physical quantity.

1. **[^](#cite_ref-Wilmshurst_6-0)** T. H. Wilmshurst (1990). [*Signal Recovery from Noise in Electronic Instrumentation*](https://books.google.com/books?id=49hfsIPpGwYC&pg=PP11) (2nd ed.). CRC Press. pp. 11 *ff*. [ISBN](/source/ISBN_(identifier)) [978-0750300582](https://en.wikipedia.org/wiki/Special:BookSources/978-0750300582). [Archived](https://web.archive.org/web/20150319012552/http://books.google.com/books?id=49hfsIPpGwYC&pg=PP11) from the original on 2015-03-19.

1. **[^](#cite_ref-7)** ["Digital signals"](https://www.st-andrews.ac.uk/~www_pa/Scots_Guide/info/signals/digital/digital.htm). *www.st-andrews.ac.uk*. [Archived](https://web.archive.org/web/20170302055200/http://www.st-andrews.ac.uk/~www_pa/Scots_Guide/info/signals/digital/digital.htm) from the original on 2017-03-02. Retrieved 2017-12-17.

1. **[^](#cite_ref-8)** ["Analog vs. Digital - learn.sparkfun.com"](https://learn.sparkfun.com/tutorials/analog-vs-digital/digital-signals). *learn.sparkfun.com*. [Archived](https://web.archive.org/web/20170705100418/https://learn.sparkfun.com/tutorials/analog-vs-digital/digital-signals) from the original on 2017-07-05. Retrieved 2017-12-17.

1. **[^](#cite_ref-9)** Robert K. Dueck (2005). [*Digital Design with CPLD Applications and VHDL*](https://web.archive.org/web/20171217234501/https://books.google.com/books?id=1eO7kLWUmYIC&lpg=PA49&ots=EBhxEwk7bY&dq=digital%20signal%20logic&pg=PA4). Thomson/Delmar Learning. [ISBN](/source/ISBN_(identifier)) [1401840302](https://en.wikipedia.org/wiki/Special:BookSources/1401840302). Archived from [the original](https://books.google.com/books?id=1eO7kLWUmYIC&q=digital%20signal%20logic&pg=PA4) on 2017-12-17. A digital representation can have only specific discrete values

1. **[^](#cite_ref-10)** Proakis, John G.; [Manolakis, Dimitris G.](/source/Dimitris_Manolakis) (2007-01-01). [*Digital Signal Processing*](https://books.google.com/books?id=H_5SAAAAMAAJ). Pearson Prentice Hall. [ISBN](/source/ISBN_(identifier)) [9780131873742](https://en.wikipedia.org/wiki/Special:BookSources/9780131873742). [Archived](https://web.archive.org/web/20160520061336/https://books.google.com/books?id=H_5SAAAAMAAJ) from the original on 2016-05-20.

1. **[^](#cite_ref-waveform_11-0)** Smillie, Grahame (1999-04-02). [*Analogue and Digital Communication Techniques*](https://web.archive.org/web/20171217234501/https://books.google.com/books?id=WJ9qkNpScFAC&pg=PA3&dq=%22a+digital+signal+is%22+communication&hl=sv&sa=X&ved=0CFoQ6AEwB2oVChMI-MD09ZjjxwIVBAssCh1wnQu4). Elsevier. [ISBN](/source/ISBN_(identifier)) [9780080527147](https://en.wikipedia.org/wiki/Special:BookSources/9780080527147). Archived from [the original](https://books.google.com/books?id=WJ9qkNpScFAC&q=%22a+digital+signal+is%22+communication&pg=PA3) 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

1. **[^](#cite_ref-12)** ["Digital Signal"](http://www.chegg.com/homework-help/definitions/digital-signal-4). [Archived](https://web.archive.org/web/20190402155546/https://www.chegg.com/homework-help/definitions/digital-signal-4) from the original on 2019-04-02. Retrieved 2016-08-13.

1. **[^](#cite_ref-13)** Paul Horowitz; Winfield Hill (2015). *The Art of Electronics*. Cambridge University Press. [ISBN](/source/ISBN_(identifier)) [9780521809269](https://en.wikipedia.org/wiki/Special:BookSources/9780521809269).

1. **[^](#cite_ref-Khanna_14-0)** Vinod Kumar Khanna (2009). [*Digital Signal Processing*](https://books.google.com/books?id=Vf2qXAbn58oC). S. Chand. p. 3. [ISBN](/source/ISBN_(identifier)) [9788121930956](https://en.wikipedia.org/wiki/Special:BookSources/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.

1. **[^](#cite_ref-15)** Sklar, Bernard (2001). *Digital communications : fundamentals and applications* (2nd ed.). Upper Saddle River, N.J.: Prentice-Hall PTR. [ISBN](/source/ISBN_(identifier)) [0130847887](https://en.wikipedia.org/wiki/Special:BookSources/0130847887). [OCLC](/source/OCLC_(identifier)) [45823120](https://search.worldcat.org/oclc/45823120).

1. **[^](#cite_ref-16)** Ziemer, Rodger E.; Tranter, William H. (2014-03-17). *Principles of communication: systems, modulation, and noise* (Seventh ed.). Hoboken, New Jersey: Wiley. [ISBN](/source/ISBN_(identifier)) [9781118078914](https://en.wikipedia.org/wiki/Special:BookSources/9781118078914). [OCLC](/source/OCLC_(identifier)) [856647730](https://search.worldcat.org/oclc/856647730).

1. **[^](#cite_ref-Lu_17-0)** 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"](https://books.google.com/books?id=cU4skcyGRFUC&pg=PA506). 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](/source/ISBN_(identifier)) [978-3642249648](https://en.wikipedia.org/wiki/Special:BookSources/978-3642249648). [Archived](https://web.archive.org/web/20130602151534/http://books.google.com/books?id=cU4skcyGRFUC&pg=PA506) from the original on 2013-06-02.

1. **[^](#cite_ref-gyro_18-0)** For example, see M. N. Armenise; Caterina Ciminelli; Francesco Dell'Olio; Vittorio Passaro (2010). ["§4.3 Optical gyros based on a fiber ring laser"](https://books.google.com/books?id=lJUiyigJRBgC&pg=PA47). *Advances in Gyroscope Technologies*. Springer. p. 47. [ISBN](/source/ISBN_(identifier)) [978-3642154935](https://en.wikipedia.org/wiki/Special:BookSources/978-3642154935). [Archived](https://web.archive.org/web/20130602165007/http://books.google.com/books?id=lJUiyigJRBgC&pg=PA47) from the original on 2013-06-02.

1. **[^](#cite_ref-Chambers_19-0)** The optical reading process is described by Mark L. Chambers (2004). [*CD & DVD Recording for Dummies*](https://books.google.com/books?id=VKsAroUnIxsC&pg=PA13) (2nd ed.). John Wiley & Sons. p. 13. [ISBN](/source/ISBN_(identifier)) [978-0764559563](https://en.wikipedia.org/wiki/Special:BookSources/978-0764559563). [Archived](https://web.archive.org/web/20130602154240/http://books.google.com/books?id=VKsAroUnIxsC&pg=PA13) from the original on 2013-06-02.

1. **[^](#cite_ref-McMahon_s_and_s_20-0)** David McMahon (2007). [*Signals & Systems Demystified*](https://www.amazon.com/gp/product/0071475788?selectObb=new). New York: McGraw Hill. [ISBN](/source/ISBN_(identifier)) [978-0-07-147578-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-147578-5). [Archived](https://web.archive.org/web/20200122152613/https://www.amazon.com/gp/product/0071475788?selectObb=new) from the original on 2020-01-22. Retrieved 2017-09-11.

1. **[^](#cite_ref-Signals_and_systems_21-0)** M.J. Roberts (2011). *Signals and Systems: Analysis Using Transform Methods & MATLAB*. New York: McGraw Hill. [ISBN](/source/ISBN_(identifier)) [978-0073380681](https://en.wikipedia.org/wiki/Special:BookSources/978-0073380681).

## Further reading

- Hsu, P. H. (1995). *Schaum's Theory and Problems: Signals and Systems*. McGraw-Hill. [ISBN](/source/ISBN_(identifier)) [0-07-030641-9](https://en.wikipedia.org/wiki/Special:BookSources/0-07-030641-9).

- Lathi, B.P. (1998). *Signal Processing & Linear Systems*. Berkeley-Cambridge Press. [ISBN](/source/ISBN_(identifier)) [0-941413-35-7](https://en.wikipedia.org/wiki/Special:BookSources/0-941413-35-7).

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Adapted from the Wikipedia article [Signal](https://en.wikipedia.org/wiki/Signal) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Signal?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.