Editing Phase (waves) (section)

== Phase shift {{anchor|Phase shift|Shift|Phase difference|Difference}} ==
[[Image:Phase shift.svg|thumb|Illustration of phase shift. The horizontal axis represents an angle (phase) that is increasing with time.]]
[[File:Phase shifter using IQ modulator.gif|thumb|Phase shifter using [[In-phase and quadrature components|IQ modulator]]]]

The difference <math>\varphi(t) = \varphi_G(t) - \varphi_F(t)</math> between the phases of two periodic signals <math>F</math> and <math>G</math> is called the ''phase difference'' or ''phase shift'' of <math>G</math> relative to <math>F</math>.<ref name=Ballou2005/>  At values of <math>t</math> when the difference is zero, the two signals are said to be ''in phase;'' otherwise, they are ''out of phase'' with each other.

In the clock analogy, each signal is represented by a hand (or pointer) of the same clock, both turning at constant but possibly different speeds. The phase difference is then the angle between the two hands, measured clockwise.

The phase difference is particularly important when two signals are added together by a physical process, such as two periodic sound waves emitted by two sources and recorded together by a microphone. This is usually the case in [[linear algebra|linear]] systems, when the [[superposition principle]] holds.

For arguments <math>t</math> when the phase difference is zero, the two signals will have the same sign and will be reinforcing each other. One says that [[interference (wave propagation)|constructive interference]] is occurring.  At arguments <math>t</math> when the phases are different, the value of the sum depends on the waveform.

===For sinusoids===
For sinusoidal signals, when the phase difference <math>\varphi(t)</math> is 180° (<math>\pi</math> radians), one says that the phases are ''opposite'', and that the signals are ''in antiphase''.  Then the signals have opposite signs, and [[interference (wave propagation)|destructive interference]] occurs.  
{{anchor|Reversal}}Conversely, a ''phase reversal'' or ''phase inversion'' implies a 180-degree phase shift.<ref>{{Cite web|url=https://www.its.bldrdoc.gov/fs-1037/fs-1037c.htm|title = Federal Standard 1037C: Glossary of Telecommunications Terms}}</ref>

{{anchor|Quadrature}}When the phase difference <math>\varphi(t)</math> is a quarter of turn (a right angle, {{nowrap|1=+90° = π/2}} or {{nowrap|1=−90° = 270° = −π/2 = 3π/2}}), sinusoidal signals are sometimes said to be in ''quadrature'', e.g., [[in-phase and quadrature components]] of a composite signal or even different signals (e.g., voltage and current).

If the frequencies are different, the phase difference <math>\varphi(t)</math> increases linearly with the argument <math>t</math>.  The periodic changes from reinforcement and opposition cause a phenomenon called [[beat (acoustics)|beating]].

===For shifted signals===
The phase difference is especially important when comparing a periodic signal <math>F</math> with a shifted and possibly scaled version <math>G</math> of it. That is, suppose that <math>G(t) = \alpha\,F(t + \tau)</math> for some constants <math>\alpha,\tau</math> and all <math>t</math>.  Suppose also that the origin for computing the phase of <math>G</math> has been shifted too.  In that case, the phase difference <math>\varphi</math> is a constant (independent of <math>t</math>), called the 'phase shift' or 'phase offset' of <math>G</math> relative to <math>F</math>.   In the clock analogy, this situation corresponds to the two hands turning at the same speed, so that the angle between them is constant.

In this case, the phase shift is simply the argument shift <math>\tau</math>, expressed as a fraction of the common period <math>T</math> (in terms of the [[modulo operation]]) of the two signals and then scaled to a full turn:
<math display="block">\varphi = 2\pi \left[\!\!\left[ \frac{\tau}{T} \right]\!\!\right].</math>

If <math>F</math> is a "canonical" representative for a class of signals, like <math>\sin(t)</math> is for all sinusoidal signals, then the phase shift <math>\varphi</math> called simply the ''initial phase'' of <math>G</math>.

Therefore, when two periodic signals have the same frequency, they are always in phase, or always out of phase.  Physically, this situation commonly occurs, for many reasons.  For example, the two signals may be a periodic soundwave recorded by two microphones at separate locations. Or, conversely, they may be periodic soundwaves created by two separate speakers from the same electrical signal, and recorded by a single microphone.  They may be a [[radio]] signal that reaches the receiving antenna in a straight line, and a copy of it that was reflected off a large building nearby.

A well-known example of phase difference is the length of shadows seen at different points of Earth.  To a first approximation, if <math>F(t)</math> is the length seen at time <math>t</math> at one spot, and <math>G</math> is the length seen at the same time at a [[longitude]] 30° west of that point, then the phase difference between the two signals will be 30° (assuming that, in each signal, each period starts when the shadow is shortest).

===For sinusoids with same frequency===
For sinusoidal signals (and a few other waveforms, like square or symmetric triangular), a phase shift of 180° is equivalent to a phase shift of 0° with negation of the amplitude.  When two signals with these waveforms, same period, and opposite phases are added together, the sum <math>F+G</math> is either identically zero, or is a sinusoidal signal with the same period and phase, whose amplitude is the difference of the original amplitudes.

The phase shift of the co-sine function relative to the sine function is +90°.  It follows that, for two sinusoidal signals <math>F</math> and <math>G</math> with same frequency and amplitudes <math>A</math> and <math>B</math>, and <math>G</math> has phase shift +90° relative to <math>F</math>, the sum <math>F+G</math> is a sinusoidal signal with the same frequency, with amplitude <math>C</math> and phase shift <math>-90^\circ < \varphi < +90^\circ</math> from <math>F</math>, such that
<math display="block">C = \sqrt{A^2 + B^2} \quad\quad \text{ and } \quad\quad \sin(\varphi) = B/C.</math>

[[Image:Sine waves same phase.svg|thumb|In-phase signals]]
[[Image:Sine waves different phase.svg|thumb|Out-of-phase signals]]
[[File:Phase Comparison image two waves.gif|thumb|Representation of phase comparison.<ref name=phnist/>]]
[[Image:Phase-shift illustration.png|thumb|200px|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.]]

[[File:Out of phase AE.gif|thumb|Out of phase AE]] 
A real-world example of a sonic phase difference occurs in the [[Native American flute#The Warble|warble of a Native American flute]]. The amplitude of different [[Harmonics|harmonic components]] of same long-held note on the flute come into dominance at different points in the phase cycle.  The phase difference between the different harmonics can be observed on a [[spectrogram]] of the sound of a warbling flute.<ref>{{cite web |url=http://Flutopedia.com/warble.htm |title=The Warble |work=Flutopedia |author1=Clint Goss |author2=Barry Higgins |year=2013 |access-date=2013-03-06}}</ref>