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In electronics, acoustics, and related fields, the **waveform** of a signal is the shape of its graph as a function of time, independent of its time and magnitude scales and of any displacement in time.^{ [1] }^{ [2] }

**Electronics** comprises the physics, engineering, technology and applications that deal with the emission, flow and control of electrons in vacuum and matter. The identification of the electron in 1897, along with the invention of the vacuum tube, which could amplify and rectify small electrical signals, inaugurated the field of electronics and the electron age.

**Acoustics** is the branch of physics that deals with the study of all mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an **acoustician** while someone working in the field of acoustics technology may be called an acoustical engineer. The application of acoustics is present in almost all aspects of modern society with the most obvious being the audio and noise control industries.

In communication systems, signal processing, and electrical engineering, a **signal** is a function that "conveys information about the behavior or attributes of some phenomenon". In its most common usage, in electronics and telecommunication, this is a time varying voltage, current or electromagnetic wave used to carry information. A signal may also be defined as an "observable change in a quantifiable entity". In the physical world, any quantity exhibiting variation in time or variation in space is potentially a signal that might provide information on the status of a physical system, or convey a message between observers, among other possibilities. The *IEEE Transactions on Signal Processing* states that the term "signal" includes audio, video, speech, image, communication, geophysical, sonar, radar, medical and musical signals. In a later effort of redefining a signal, 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 electronics, the term is usually applied to periodically varying voltages, currents, or electromagnetic fields. In acoustics, it is usually applied to steady periodic sounds -- variations of pressure in air or other media. In these cases, the waveform is an attribute that is independent of the frequency, amplitude, or phase shift of the signal. The term can also be used for non-periodic signals, like chirps and pulses.

In mathematics, a **periodic function** is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of 2*π* radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called **aperiodic**.

**Voltage**, **electric potential difference**, **electric pressure **or **electric tension** is the difference in electric potential between two points. The difference in electric potential between two points in a static electric field is defined as the work needed per unit of charge to move a test charge between the two points. In the International System of Units, the derived unit for voltage is named *volt*. In SI units, work per unit charge is expressed as joules per coulomb, where 1 volt = 1 joule per 1 coulomb. The official SI definition for *volt* uses power and current, where 1 volt = 1 watt per 1 ampere. This definition is equivalent to the more commonly used 'joules per coulomb'. Voltage or electric potential difference is denoted symbolically by ∆*V*, but more often simply as *V*, for instance in the context of Ohm's or Kirchhoff's circuit laws.

An **electric current** is the rate of flow of electric charge past a point or region. An electric current is said to exist when there is a net flow of electric charge through a region. In electric circuits this charge is often carried by electrons moving through a wire. It can also be carried by ions in an electrolyte, or by both ions and electrons such as in an ionized gas (plasma).

The waveform of an electrical signal can be visualized in an oscilloscope or any other device that can capture and plot its value at various times, with a suitable scales in the time and value axes. The electrocardiograph is a medical device to record the waveform of the electric signals that are associated with the beating of the heart; that waveform has important diagnostic value. Waveform generators, that can output a periodic voltage or current with one of several waveforms, are a common tool in electronics laboratories and workshops.

An **oscilloscope**, previously called an **oscillograph**, and informally known as a **scope** or **o-scope**, **CRO**, or **DSO**, is a type of electronic test instrument that graphically displays varying signal voltages, usually as a two-dimensional plot of one or more signals as a function of time. Other signals can be converted to voltages and displayed.

The **scale ratio** of a model represents the proportional ratio of a linear dimension of the model to the same feature of the original. Examples include a 3-dimensional scale model of a building or the scale drawings of the elevations or plans of a building. In such cases the scale is dimensionless and **exact** throughout the model or drawing.

**Electrocardiography** is the process of producing an **electrocardiogram**, a recording - a graph of voltage versus time - of the electrical activity of the heart using electrodes placed on the skin. These electrodes detect the small electrical changes that are a consequence of cardiac muscle depolarization followed by repolarization during each cardiac cycle (heartbeat). Changes in the normal ECG pattern occur in numerous cardiac abnormalities, including cardiac rhythm disturbances, inadequate coronary artery blood flow, and electrolyte disturbances.

The waveform of a steady periodic sound affects its timbre. Synthesizers and modern keyboards can generate sounds with many complicated waveforms.^{ [1] }

In music, **timbre** is the perceived sound quality of a musical note, sound or tone. Timbre distinguishes different types of sound production, such as choir voices and musical instruments, such as string instruments, wind instruments, and percussion instruments. It also enables listeners to distinguish different instruments in the same category.

A **synthesizer** or **synthesiser** is an electronic musical instrument that generates audio signals that may be converted to sound. Synthesizers may imitate traditional musical instruments such as piano, flute, vocals, or natural sounds such as ocean waves; or generate novel electronic timbres. They are often played with a musical keyboard, but they can be controlled via a variety of other devices, including music sequencers, instrument controllers, fingerboards, guitar synthesizers, wind controllers, and electronic drums. Synthesizers without built-in controllers are often called *sound modules*, and are controlled via USB, MIDI or CV/gate using a controller device, often a MIDI keyboard or other controller.

Simple examples of **periodic waveforms** include the following, where is time, is wavelength, is amplitude and is phase:

**Time** is the indefinite continued progress of existence and events that occur in apparently irreversible succession from the past, through the present, to the future. Time is a component quantity of various measurements used to sequence events, to compare the duration of events or the intervals between them, and to quantify rates of change of quantities in material reality or in the conscious experience. Time is often referred to as a fourth dimension, along with three spatial dimensions.

In physics, the **wavelength** is the **spatial period** of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is commonly designated by the Greek letter *lambda* (λ). The term *wavelength* is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

The **amplitude** of a periodic variable is a measure of its change over a single period. There are various definitions of amplitude, which are all functions of the magnitude of the difference between the variable's extreme values. In older texts the phase is sometimes called the amplitude.

- Sine wave . The amplitude of the waveform follows a trigonometric sine function with respect to time.
- Square wave . This waveform is commonly used to represent digital information. A square wave of constant period contains odd harmonics that decrease at −6 dB/octave.
- Triangle wave . It contains odd harmonics that decrease at −12 dB/octave.
- Sawtooth wave . This looks like the teeth of a saw. Found often in time bases for display scanning. It is used as the starting point for subtractive synthesis, as a sawtooth wave of constant period contains odd and even harmonics that decrease at −6 dB/octave.

A **sine wave** or **sinusoid** is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph. It occurs often in pure and applied mathematics, as well as physics, engineering, signal processing and many other fields. Its most basic form as a function of time (*t*) is:

A **square wave** is a non-sinusoidal periodic waveform in which the amplitude alternates at a steady frequency between fixed minimum and maximum values, with the same duration at minimum and maximum. Although not realizable in physical systems, the transition between minimum and maximum is instantaneous for an ideal square wave.

**Frequency** is the number of occurrences of a repeating event per unit of time. It is also referred to as **temporal frequency**, which emphasizes the contrast to spatial frequency and angular frequency. The

The Fourier series describes the decomposition of periodic waveforms, such that any periodic waveform can be formed by the sum of a (possibly infinite) set of fundamental and harmonic components. Finite-energy non-periodic waveforms can be analyzed into sinusoids by the Fourier transform.

Other periodic waveforms are often called composite waveforms and can often be described as a combination of a number of sinusoidal waves or other basis functions added together.

In physics and mathematics, the **phase** of a periodic function of some real variable is the relative value of that variable within the span of each full period.

In antenna theory, a **phased array** usually means an **electronically scanned array**, a computer-controlled array of antennas which creates a beam of radio waves that can be electronically steered to point in different directions without moving the antennas. In an array antenna, the radio frequency current from the transmitter is fed to the individual antennas with the correct phase relationship so that the radio waves from the separate antennas add together to increase the radiation in a desired direction, while cancelling to suppress radiation in undesired directions. In a phased array, the power from the transmitter is fed to the antennas through devices called *phase shifters*, controlled by a computer system, which can alter the phase electronically, thus steering the beam of radio waves to a different direction. Since the array must consist of many small antennas to achieve high gain, phased arrays are mainly practical at the high frequency end of the radio spectrum, in the UHF and microwave bands, in which the antenna elements are conveniently small.

The **sawtooth wave** is a kind of non-sinusoidal waveform. It is so named based on its resemblance to the teeth of a plain-toothed saw with a zero rake angle.

A **triangle wave** is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function.

A **chirp** is a signal in which the frequency increases (*up-chirp*) or decreases (*down-chirp*) with time. In some sources, the term *chirp* is used interchangeably with **sweep signal**. It is commonly used in sonar, radar, and laser, but has other applications, such as in spread-spectrum communications.

The **great-circle distance** or **orthodromic distance** is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere. The distance between two points in Euclidean space is the length of a straight line between them, but on the sphere there are no straight lines. In spaces with curvature, straight lines are replaced by geodesics. Geodesics on the sphere are circles on the sphere whose centers coincide with the center of the sphere, and are called *great circles*.

The **Josephson effect** is the phenomenon of supercurrent, a current that flows indefinitely long without any voltage applied, across a device known as a **Josephson junction** (JJ), which consists of two or more superconductors coupled by a weak link. The weak link can consist of a thin insulating barrier, a short section of non-superconducting metal (S-N-S), or a physical constriction that weakens the superconductivity at the point of contact (S-s-S).

In physics and engineering, a **phasor**, is a complex number representing a sinusoidal function whose amplitude (*A*), angular frequency (*ω*), and initial phase (*θ*) are time-invariant. It is related to a more general concept called analytic representation, which decomposes a sinusoid into the product of a complex constant and a factor that encapsulates the frequency and time dependence. The complex constant, which encapsulates amplitude and phase dependence, is known as **phasor**, **complex amplitude**, and **sinor** or even **complexor**.

The **Cassini projection** is a map projection described by César-François Cassini de Thury in 1745. It is the transverse aspect of the equirectangular projection, in that the globe is first rotated so the central meridian becomes the "equator", and then the normal equirectangular projection is applied. Considering the earth as a sphere, the projection is composed of the operations:

Diffraction processes affecting waves are amenable to quantitative description and analysis. Such treatments are applied to a wave passing through one or more slits whose width is specified as a proportion of the wavelength. Numerical approximations may be used, including the Fresnel and Fraunhofer approximations.

In statistical signal processing, the goal of **spectral density estimation** (**SDE**) is to estimate the spectral density of a random signal from a sequence of time samples of the signal. Intuitively speaking, the spectral density characterizes the frequency content of the signal. One purpose of estimating the spectral density is to detect any periodicities in the data, by observing peaks at the frequencies corresponding to these periodicities.

The **contrast transfer function** (CTF) mathematically describes how aberrations in a transmission electron microscope (TEM) modify the image of a sample. This contrast transfer function (CTF) sets the resolution of high-resolution transmission electron microscopy (HRTEM), also known as phase contrast TEM.

In physics and engineering, the **envelope** of an oscillating signal is a smooth curve outlining its extremes. The envelope thus generalizes the concept of a constant amplitude. The figure illustrates a modulated sine wave varying between an upper and a lower envelope. The envelope function may be a function of time, space, angle, or indeed of any variable.

In optics, the **Fraunhofer diffraction equation** is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens.

A **damped sine wave** is a sinusoidal function whose amplitude approaches zero as time increases.

A **phase detector characteristic** is a function of phase difference describing the output of the phase detector.

The spectrum of a chirp pulse describes its characteristics in terms of its frequency components. This frequency-domain representation is an alternative to the more familiar time-domain waveform, and the two versions are mathematically related by the Fourier transform.

The spectrum is of particular interest when pulses are subject to signal processing. For example, when a chirp pulse is compressed by its matched filter, the resulting waveform contains not only a main narrow pulse but, also, a variety of unwanted artifacts many of which are directly attributable to features in the chirp's spectral characteristics.

The simplest way to derive the spectrum of a chirp, now that computers are widely available, is to sample the time-domain waveform at a frequency well above the Nyquist limit and call up an FFT algorithm to obtain the desired result. As this approach was not an option for the early designers, they resorted to analytic analysis, where possible, or to graphical or approximation methods, otherwise. These early methods still remain helpful, however, as they give additional insight into the behavior and properties of chirps.

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*Electronics*, 2nd ed., ISBN 0748770364, CRC Press, 2002, p. 62

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- Collection of single cycle waveforms sampled from various sources

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