View source for Instantaneous phase and frequency

{{Short description|Electrical engineering concept}}
{{Technical|date=July 2022}}

'''Instantaneous phase and frequency''' are important concepts in [[signal processing]] that occur in the context of the representation and analysis of time-varying functions.<ref>{{Cite journal|last1=Sejdic|first1=E.|last2=Djurovic|first2=I.|last3=Stankovic|first3=L.|date=August 2008|title=Quantitative Performance Analysis of Scalogram as Instantaneous Frequency Estimator|journal=IEEE Transactions on Signal Processing|volume=56|issue=8|pages=3837–3845|doi=10.1109/TSP.2008.924856|bibcode=2008ITSP...56.3837S |s2cid=16396084 |issn=1053-587X}}</ref> The '''instantaneous phase''' (also known as '''local phase''' or simply '''phase''') of a ''complex-valued'' function ''s''(''t''), is the real-valued function:
:<math>\varphi(t) = \arg\{s(t)\},</math>
where '''arg''' is the [[Argument (complex analysis)|complex argument function]].
The '''instantaneous frequency''' is the [[temporal rate of change]] of the instantaneous phase.

And for a ''real-valued'' function ''s''(''t''), it is determined from the function's [[analytic signal|analytic representation]], ''s''<sub>a</sub>(''t''):<ref>{{cite book|last=Blackledge|first=Jonathan M.|title=Digital Signal Processing: Mathematical and Computational Methods, Software Development and Applications|year=2006|publisher=Woodhead Publishing|isbn=1904275265|page=134|edition=2}}</ref>
:<math>\begin{align}
  \varphi(t) &= \arg\{s_\mathrm{a}(t)\} \\[4pt]
             &= \arg\{s(t) + j \hat{s}(t)\},
\end{align}</math>
where <math>\hat{s}(t)</math> represents the [[Hilbert transform]] of ''s''(''t'').

When ''φ''(''t'') is constrained to its [[principal value]], either the interval {{open-closed|−''π'', ''π''}} or {{closed-open|0, 2''π''}}, it is called '''''wrapped phase'''''. Otherwise it is called '''''unwrapped phase''''', which is a continuous function of argument ''t'', assuming ''s''<sub>a</sub>(''t'') is a continuous function of ''t''. Unless otherwise indicated, the continuous form should be inferred.

[[File:Phase vs Time, wrapped and unwrapped.jpg|thumb|400px|Instantaneous phase vs time. The function has two true discontinuities of 180° at times 21 and 59, indicative of amplitude zero-crossings. The 360° "discontinuities" at times 19, 37, and 91 are artifacts of phase wrapping.]]

[[File:Instantaneous (wrapped) phase; one 360° plot stacked 3 times vertically.jpg|thumb|400px|Instantaneous phase of a frequency-modulated waveform: MSK (minimum shift keying). A 360° "wrapped" plot is simply replicated vertically two more times, creating the illusion of an unwrapped plot, but using only 3x360° of the vertical axis.]]

==Examples==
===Example 1===
:<math>s(t) = A \cos(\omega t + \theta),</math>
where ''ω'' > 0.
:<math>\begin{align}
  s_\mathrm{a}(t) &= A e^{j(\omega t + \theta)}, \\
       \varphi(t) &= \omega t + \theta.
\end{align}</math>
In this simple sinusoidal example, the constant ''θ'' is also commonly referred to as ''phase'' or ''phase offset''. ''φ''(''t'') is a function of time; ''θ'' is not. In the next example, we also see that the phase offset of a real-valued sinusoid is ambiguous unless a reference (sin or cos) is specified. ''φ''(''t'') is unambiguously defined.

===Example 2===
:<math>s(t) = A \sin(\omega t) = A \cos\left(\omega t - \frac{\pi}{2}\right),</math>
where ''ω'' > 0.
:<math>\begin{align}
  s_\mathrm{a}(t) &= A e^{j \left(\omega t - \frac{\pi}{2}\right)}, \\
       \varphi(t) &= \omega t - \frac{\pi}{2}.
\end{align}</math>
In both examples the local maxima of ''s''(''t'') correspond to ''φ''(''t'') =&nbsp;2{{pi}}''N'' for integer values of&nbsp;''N''. This has applications in the field of computer vision.

==Formulations==
'''Instantaneous angular frequency''' is defined as:
:<math>\omega(t) = \frac{d\varphi(t)}{dt},</math>
and '''instantaneous (ordinary) frequency''' is defined as:
:<math>f(t) = \frac{1}{2\pi} \omega(t) = \frac{1}{2\pi} \frac{d\varphi(t)}{dt}</math>
where ''φ''(''t'') must be the '''unwrapped phase'''; otherwise, if ''φ''(''t'') is wrapped, discontinuities in ''φ''(''t'') will result in [[Dirac delta]] impulses in ''f''(''t'').

The inverse operation, which always unwraps phase, is:
:<math>\begin{align}
  \varphi(t) &= \int_{-\infty}^t \omega(\tau)\, d\tau = 2 \pi \int_{-\infty}^t f(\tau)\, d\tau\\[5pt]
             &= \int_{-\infty}^0 \omega(\tau)\, d\tau + \int_0^t \omega(\tau)\, d\tau\\[5pt]
             &= \varphi(0) + \int_0^t \omega(\tau)\, d\tau.
\end{align}</math>

This instantaneous frequency, ''ω''(''t''), can be derived directly from the [[Complex number|real and imaginary parts]] of ''s''<sub>a</sub>(''t''), instead of the [[Argument (complex analysis)|complex arg]] without concern of phase unwrapping.

:<math>\begin{align}
  \varphi(t) &= \arg\{s_\mathrm{a}(t)\} \\[4pt]
             &= \operatorname{atan2}(\mathcal{Im}[s_\mathrm{a}(t)],\mathcal{Re}[s_\mathrm{a}(t)]) + 2 m_1 \pi \\[4pt]
             &= \arctan\left( \frac{\mathcal{Im}[s_\mathrm{a}(t)]}{\mathcal{Re}[s_\mathrm{a}(t)]} \right) + m_2 \pi
\end{align}</math>

2''m''<sub>1</sub>{{pi}} and ''m''<sub>2</sub>{{pi}} are the integer multiples of {{pi}} necessary to add to unwrap the phase. At values of time, ''t'', where there is no change to integer ''m''<sub>2</sub>, the derivative of ''φ''(''t'') is

:<math>\begin{align}
  \omega(t) = \frac{d\varphi(t)}{dt}
            &= \frac{d}{dt} \arctan\left( \frac{\mathcal{Im}[s_\mathrm{a}(t)]}{\mathcal{Re}[s_\mathrm{a}(t)]} \right) \\[3pt]
            &= \frac{1}{1 + \left(\frac{\mathcal{Im}[s_\mathrm{a}(t)]}{\mathcal{Re}[s_\mathrm{a}(t)]} \right)^2} \frac{d}{dt} \left( \frac{\mathcal{Im}[s_\mathrm{a}(t)]}{\mathcal{Re}[s_\mathrm{a}(t)]} \right) \\[3pt]
            &= \frac{\mathcal{Re}[s_\mathrm{a}(t)] \frac{d\mathcal{Im}[s_\mathrm{a}(t)]}{dt} - \mathcal{Im}[s_\mathrm{a}(t)] \frac{d\mathcal{Re}[s_\mathrm{a}(t)]}{dt} }{(\mathcal{Re}[s_\mathrm{a}(t)])^2 + (\mathcal{Im}[s_\mathrm{a}(t)])^2 } \\[3pt]
            &= \frac{1}{|s_\mathrm{a}(t)|^2} \left(\mathcal{Re}[s_\mathrm{a}(t)] \frac{d\mathcal{Im}[s_\mathrm{a}(t)]}{dt} - \mathcal{Im}[s_\mathrm{a}(t)] \frac{d\mathcal{Re}[s_\mathrm{a}(t)]}{dt} \right) \\[3pt]
            &= \frac{1}{(s(t))^2 + \left(\hat{s}(t)\right)^2} \left(s(t) \frac{d\hat{s}(t)}{dt} - \hat{s}(t) \frac{ds(t)}{dt} \right)
\end{align}</math>

For discrete-time functions, this can be written as a recursion:
:<math>\begin{align}
  \varphi[n] &= \varphi[n - 1] + \omega[n] \\
             &= \varphi[n - 1] + \underbrace{\arg\{s_\mathrm{a}[n]\} - \arg\{s_\mathrm{a}[n - 1]\}}_{\Delta \varphi[n]} \\
             &= \varphi[n - 1] + \arg\left\{\frac{s_\mathrm{a}[n]}{s_\mathrm{a}[n - 1]}\right\} \\
\end{align}</math>

Discontinuities can then be removed by adding 2{{pi}} whenever Δ''φ''[''n''] ≤ −{{pi}}, and subtracting 2{{pi}} whenever Δ''φ''[''n'']&nbsp;>&nbsp;{{pi}}. That allows ''φ''[''n''] to accumulate without limit and produces an unwrapped instantaneous phase. An equivalent formulation that replaces the modulo 2{{pi}} operation with a complex multiplication is:
:<math>\varphi[n] = \varphi[n - 1] + \arg\{s_\mathrm{a}[n] \, s_\mathrm{a}^*[n - 1]\},</math>
where the asterisk denotes complex conjugate. The discrete-time instantaneous frequency (in units of radians per sample) is simply the advancement of phase for that sample
:<math>\omega[n] = \arg\{s_\mathrm{a}[n] \, s_\mathrm{a}^*[n - 1]\}.</math>

==Complex representation==
In some applications, such as averaging the values of phase at several moments of time, it may be useful to convert each value to a complex number, or vector representation:<ref>{{cite journal|last1=Wang|first1=S.|title=An Improved Quality Guided Phase Unwrapping Method and Its Applications to MRI|journal=Progress in Electromagnetics Research|date=2014|volume=145|pages=273–286|doi=10.2528/PIER14021005|doi-access=free}}</ref>
:<math>
e^{i\varphi(t)}
= \frac{s_\mathrm{a}(t)}{|s_\mathrm{a}(t)|}
= \cos(\varphi(t)) + i \sin(\varphi(t)).
</math>

This representation is similar to the wrapped phase representation in that it does not distinguish between multiples of 2{{pi}} in the phase, but similar to the unwrapped phase representation since it is continuous. A vector-average phase can be obtained as the [[Argument (complex analysis)|arg]] of the sum of the complex numbers without concern about wrap-around.

==See also==
*[[Angular displacement]]
*[[Analytic signal]]
*[[Frequency modulation]]
*[[Group delay]]
*[[Instantaneous amplitude]]
*[[Negative frequency]]

==References==
{{reflist}}

==Further reading==
*{{cite book |first=Leon |last=Cohen |title=Time-Frequency Analysis |publisher=Prentice Hall |year=1995 }}
*{{cite book |last1=Granlund |last2=Knutsson |title=Signal Processing for Computer Vision |publisher=Kluwer Academic Publishers |year=1995 }}

[[Category:Signal processing]]
[[Category:Digital signal processing]]
[[Category:Time–frequency analysis]]
[[Category:Fourier analysis]]
[[Category:Electrical engineering]]
[[Category:Audio engineering]]