Phase (waves) - Citizendia

The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0. Phase is a frequency domain or Fourier transform domain concept, and as such, can be readily understood in terms of simple harmonic motion. Frequency domain is a term used to describe the analysis of Mathematical functions or signals with respect to frequency This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. The same concept applies to wave motion, viewed either at a point in space over an interval of time or across an interval of space at a moment in time. Simple harmonic motion is a displacement that varies cyclically, as depicted below:

Image:Simple harmonic motion.png

and described by the formula:

$x(t) = A\cdot \sin( 2 \pi f t + \theta ),\,$

where A is the amplitude of oscillation, and f is the frequency. Amplitude is the magnitude of change in the oscillating variable with each Oscillation, within an oscillating system Frequency is a measure of the number of occurrences of a repeating event per unit Time. A motion with frequency f has period $T=\frac{1}{f}.$ $t\,$ is the elapsed time and $θ$ is the phase of the oscillation. Frequency is a measure of the number of occurrences of a repeating event per unit Time. It determines or is determined by the initial displacement at time t = 0.

Two potential ambiguities can be noted:

One is that the initial displacement of $\cos( 2 \pi f t + \theta )\,$ is different than the sine function, yet they appear to have the same "phase".
The time-variant angle $2 \pi f t + \theta,\,$ or its modulo $2π$ value, is also commonly referred to as "phase". Then it is not an initial condition, but rather a continuously-changing condition.

The term instantaneous phase is used to distinguish the time-variant angle from the initial condition. In Signal processing, the instantaneous phase (or "local phase" or simply "phase" of a complex-valued function x(t\ is the real-valued It also has a formal definition that is applicable to more general functions and unambiguously defines a function's initial phase at t=0. I. e. , sine and cosine inherently have different initial phases. When not explicitly stated otherwise, cosine should generally be inferred. (also see phasor)

1 Phase shift
2 Phase difference
- 2.1 In-phase and quadrature (I&Q) components
3 Phase coherence
4 See also
5 External links

Phase shift

Illustration of phase shift. In Physics and Engineering, a phase vector ("phasor" is a representation of a Sine wave whose amplitude ( A) phase ( θ) The horizontal axis represents an angle (phase) that is increasing with time.

$θ$ is sometimes referred to as a phase-shift, because it represents a "shift" from zero phase. But a change in $θ$ is also referred to as a phase-shift.

For infinitely long sinusoids, a change in $θ$ is the same as a shift in time, such as a time-delay. If $x(t)\,$ is delayed (time-shifted) by $\begin{matrix} \frac{1}{4} \end{matrix}\,$ of its cycle, it becomes:

$x(t - \begin{matrix} \frac{1}{4} \end{matrix}T) \,$	$= A\cdot \sin(2 \pi f (t - \begin{matrix} \frac{1}{4} \end{matrix}T) + \theta) \,$
	$= A\cdot \sin(2 \pi f t - \begin{matrix}\frac{\pi }{2} \end{matrix} + \theta ),\,$

whose "phase" is now $\theta - \begin{matrix}\frac{\pi }{2} \end{matrix}.$ It has been shifted by $\begin{matrix}\frac{\pi }{2} \end{matrix}$ .

Phase difference

In-phase waves

Out-of-phase waves

$Left: the real part of a plane wave moving from top to bottom. Right: the same wave after a central section underwent a phase shift, for example, by passing through a glass of different thickness than the other parts. (The illustration on the right ignores the effect of diffraction whose effect increases over large distances).$

Left: the real part of a plane wave moving from top to bottom. In Mathematics, the real part of a Complex number z is the first element of the Ordered pair of Real numbers representing z In the Physics of Wave propagation (especially Electromagnetic waves, a plane wave (also spelled planewave) is a constant-frequency wave whose Right: the same wave after a central section underwent a phase shift, for example, by passing through a glass of different thickness than the other parts. (The illustration on the right ignores the effect of diffraction whose effect increases over large distances). Diffraction is normally taken to refer to various phenomena which occur when a wave encounters an obstacle

Two oscillators that have the same frequency and different phases have a phase difference, and the oscillators are said to be out of phase with each other. The amount by which such oscillators are out of step with each other can be expressed in degrees from 0° to 360°, or in radians from 0 to 2π. This article describes the unit of angle For other meanings see Degree. The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 If the phase difference is 180 degrees (π radians), then the two oscillators are said to be in antiphase. If two interacting waves meet at a point where they are in antiphase, then destructive interference will occur. A wave is a disturbance that propagates through Space and Time, usually with transference of Energy. In physics interference is the addition ( superposition) of two or more Waves that result in a new wave pattern It is common for waves of electromagnetic (light, RF), acoustic (sound) or other energy to become superimposed in their transmission medium. When that happens, the phase difference determines whether they reinforce or weaken each other. Complete cancellation is possible for waves with equal amplitudes.

Time is sometimes used (instead of angle) to express position within the cycle of an oscillation.

A phase difference is analogous to two athletes running around a race track at the same speed and direction but starting at different positions on the track. They pass a point at different instants in time. But the time difference (phase difference) between them is a constant - same for every pass since they are at the same speed and in the same direction. If they were at different speeds (different frequencies), the phase difference would only reflect different starting positions.
We measure the rotation of the earth in hours, instead of radians. And therefore time zones are an example of phase differences.

In-phase and quadrature (I&Q) components

The term in-phase is also found in the context of communication signals:

$\begin{align} A(t)\cdot \sin[2\pi ft + \phi(t)] &= I(t)\cdot \sin(2\pi ft) + Q(t)\cdot \cos(2\pi ft) \\ &=I(t)\cdot \sin(2\pi ft) + Q(t)\cdot \sin(2\pi ft + \begin{matrix} \frac{\pi}{2} \end{matrix})\end{align}$

and:

$\begin{align} A(t)\cdot \cos[2\pi ft + \phi(t)] &= I(t)\cdot \cos(2\pi ft) - Q(t)\cdot \sin(2\pi ft) \\ &= I(t)\cdot \cos(2\pi ft) + Q(t)\cdot \cos(2\pi ft + \begin{matrix} \frac{\pi}{2} \end{matrix}), \end{align}$

where $f\,$ represents a carrier frequency, and

$I(t)\ \stackrel{\mathrm{def}}{=}\ A(t)\cdot \cos[\phi(t)], \,$

$Q(t)\ \stackrel{\mathrm{def}}{=}\ A(t)\cdot \sin[\phi(t)].\,$

$A(t)\,$ and $\phi(t)\,$ represent possible modulation of a pure carrier wave, e. In Telecommunications, modulation is the process of varying a periodic Waveform, i g. : $\sin(2\pi ft).\,$ The modulation alters the original $\sin\,$ component of the carrier, and creates a (new) $\cos\,$ component, as shown above. The component that is in phase with the original carrier is referred to as the in-phase component. The other component, which is always 90° ( $\begin{matrix} \frac{\pi}{2} \end{matrix}$ radians) "out of phase", is referred to as the quadrature component. Communication signals often have the form': A(t\cdot \sin ft + \phi(t which is called envelope-and-phase form

Phase coherence

Coherence is the quality of a wave to display well defined phase relationship in different regions of its domain of definition. In Physics, coherence is a property of waves that enables stationary (i

In physics, quantum mechanics ascribes waves to physical objects. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons The wave function is complex and since its square modulus is associated with the probability of observing the object, the complex character of the wave function is associated to the phase. A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system Since the complex algebra is responsible for the striking interference effect of quantum mechanics, phase of particles is therefore ultimately related to their quantum behavior.

External links

Relationship of phase difference and time-delay

In Signal processing, the instantaneous phase (or "local phase" or simply "phase" of a complex-valued function x(t\ is the real-valued In physics interference is the addition ( superposition) of two or more Waves that result in a new wave pattern Polarization ( ''Brit'' polarisation) is a property of Waves that describes the orientation of their oscillations

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Contents

Phase shift

Phase difference

In-phase and quadrature (I&Q) components

Phase coherence

See also

External links