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Waveform
Waveform
Waveform
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Sine, square, triangle, and sawtooth waveforms.
A sine, square, and sawtooth wave at 440 Hz
A composite waveform that is shaped like a teardrop.
A waveform generated by a synthesizer

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] Periodic waveforms repeat regularly at a constant period. The term can also be used for non-periodic or aperiodic signals, like chirps and pulses.[3]

In electronics, the term is usually applied to time-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 waveform of an electrical signal can be visualized with an oscilloscope or any other device that can capture and plot its value at various times, with 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, which can output a periodic voltage or current with one of several waveforms, are a common tool in electronics laboratories and workshops.

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

Common periodic waveforms

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

  • 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.

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.

See also

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References

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Further reading

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Waveform

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A waveform is a graphical representation of the shape of a wave that indicates its characteristics, such as and . In physics and , it depicts the variation in magnitude of a signal—such as voltage, current, , or strength—over time or another independent variable, typically plotted as a on a graph. Waveforms are classified as periodic if they repeat at regular intervals or aperiodic if they do not; common periodic types include the sinusoidal waveform, which is smooth and undulating, the square waveform with abrupt transitions between high and low levels, the triangular waveform with linear rises and falls, and the sawtooth waveform featuring a slow rise followed by a rapid fall. Key properties of a waveform include , the maximum extent of deviation from the equilibrium value; period, the duration of one complete cycle; , the number of cycles per unit time (measured in hertz); and phase, the offset of the waveform relative to a reference point. These elements determine the waveform's behavior and suitability for specific uses. Waveforms underpin numerous scientific and technological domains, including electronics for circuit design and testing with oscilloscopes, acoustics for representing sound signals, optics for light pulses, and seismology for ground motion analysis. In radar systems, engineered waveforms optimize signal transmission, detection, and interference resistance, while in power quality monitoring, they enable assessment of voltage and current distortions to ensure system reliability. Additionally, in and , waveforms model bioelectric signals like electrocardiograms (ECGs) and electroencephalograms (EEGs) to diagnose physiological conditions.

Definition and Characteristics

Definition

A waveform is the graphical representation of the variation of a , such as voltage, , or displacement, with respect to another variable, typically time or position. This depiction illustrates the and characteristics of the wave, independent of specific scales in time or magnitude. Common examples include the displacement of particles in mechanical waves, the strength in electromagnetic waves, and the sound level in . The term "waveform" originated in the 19th century, with the earliest known use recorded in within reports of the British Association for the Advancement of Science. Its adoption accelerated with the invention of early oscilloscopes in the 1890s and pivotal experiments on electromagnetic waves conducted by in 1887, which demonstrated the propagation of radio waves and highlighted wave patterns visually. In contrast to a signal, which denotes the actual physical entity carrying information—such as a varying electrical voltage or current generated by a circuit—the waveform emphasizes the geometric shape and temporal pattern of that signal's changes. This distinction underscores how waveforms provide a visual tool for analyzing signal behavior without altering the underlying information content. Waveforms may be periodic, repeating cyclically, or aperiodic, varying irregularly.

Key Characteristics

Waveforms are characterized by several fundamental properties that quantify their behavior and structure, including , , phase, , and additional parameters such as and for specific types. refers to the maximum displacement of the waveform from its equilibrium position, representing the peak value or the total range of variation, such as peak-to-peak amplitude in electrical signals measured in volts. In physical contexts, amplitude is typically expressed in meters for displacement waves. Frequency denotes the number of complete cycles occurring per unit time, measured in hertz (Hz), where one hertz equals one cycle per second. It is inversely related to the period TT, the time for one full cycle, by the equation T=1/fT = 1/f, with period in seconds. Phase describes the position of a point within the waveform cycle relative to a reference point, often expressed in radians or degrees, indicating any shift in the starting point of the oscillation. For sinusoidal waves, phase is a key parameter alongside and . For traveling waveforms, is the spatial distance between consecutive corresponding points, such as crests, measured in meters, and related to and wave speed vv by λ=v/f\lambda = v/f. In non-sinusoidal waveforms like pulses, is the duration for the signal to transition from 10% to 90% of its on the rising edge, measured in seconds. , relevant for pulsed or rectangular waveforms, is the fraction of the period during which the signal is active (high), expressed as a . These properties are measured using standard SI units: amplitude in volts for electrical waveforms or meters for mechanical ones, in hertz, phase in radians or degrees, in meters, and period in seconds, and as a unitless or .

Mathematical Representation

General Mathematical Forms

Waveforms are commonly represented in the time domain as functions of time y(t)y(t), capturing variations in amplitude over time. For periodic waveforms, the general mathematical form is y(t)=A⋅p(2πft+ϕ)y(t) = A \cdot p(2\pi f t + \phi), where AA denotes the amplitude (maximum deviation from the mean value), ff is the frequency (cycles per unit time), ϕ\phi is the phase shift (initial offset in radians), and p(θ)p(\theta) is a periodic function with period 2π2\pi that defines the shape of the waveform. This form encapsulates the repetitive nature of periodic signals by scaling and shifting a base periodic function, allowing description of diverse shapes like sines or more complex patterns while preserving key parameters such as amplitude and frequency. In contexts involving , such as mechanical or electromagnetic waves, waveforms are described in both space and time using a two-dimensional function y(x,t)y(x, t). A fundamental representation for a traveling sinusoidal wave is given by y(x,t)=Asin⁡(kx−ωt+ϕ),y(x, t) = A \sin(kx - \omega t + \phi), where k=2π/λk = 2\pi / \lambda is the (with λ\lambda as the ), ω=2πf\omega = 2\pi f is the , xx is the position, and tt is time. This equation models a wave propagating in the positive xx-direction at speed v=ω/k=fλv = \omega / k = f \lambda, with the argument kx−ωt+ϕkx - \omega t + \phi ensuring constant phase along characteristics of the wave./Book%3A_University_Physics_I_-Mechanics_Sound_Oscillations_and_Waves(OpenStax)/16%3A_Waves/16.03%3A_Mathematics_of_Waves) Variations account for direction, such as +ωt+ \omega t for negative , but the core structure highlights spatiotemporal . Complex notation simplifies analysis of sinusoidal components through phasors, representing waves as y~(t)=Aej(ωt+ϕ)\tilde{y}(t) = A e^{j(\omega t + \phi)}, where j=−1j = \sqrt{-1}
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