Instantaneous phase and frequency

Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions.^[1] The instantaneous phase (also known as local phase or simply phase) of a complex-valued function s(t), is the real-valued function:

\varphi (t)=\arg\{s(t)\},

where arg is the 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 representation, s_a(t):^[2]

{\begin{aligned}\varphi (t)&=\arg\{s_{\mathrm {a} }(t)\}\\[4pt]&=\arg\{s(t)+j{\hat {s}}(t)\},\end{aligned}}

where ${\hat {s}}(t)$ represents the Hilbert transform of s(t).

When φ(t) is constrained to its principal value, either the interval $(- π, π]$ or $[0, 2 π)$ , it is called wrapped phase. Otherwise it is called unwrapped phase, which is a continuous function of argument t, assuming s_a(t) is a continuous function of t. Unless otherwise indicated, the continuous form should be inferred.

Examples

Example 1

s(t)=A\cos(\omega t+\theta ),

where ω > 0.

{\begin{aligned}s_{\mathrm {a} }(t)&=Ae^{j(\omega t+\theta )},\\\varphi (t)&=\omega t+\theta .\end{aligned}}

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

s(t)=A\sin(\omega t)=A\cos \left(\omega t-{\frac {\pi }{2}}\right),

where ω > 0.

{\begin{aligned}s_{\mathrm {a} }(t)&=Ae^{j\left(\omega t-{\frac {\pi }{2}}\right)},\\\varphi (t)&=\omega t-{\frac {\pi }{2}}.\end{aligned}}

In both examples the local maxima of s(t) correspond to φ(t) = 2 $π$ N for integer values of N. This has applications in the field of computer vision.

Formulations

Instantaneous angular frequency is defined as:

\omega (t)={\frac {d\varphi (t)}{dt}},

and instantaneous (ordinary) frequency is defined as:

f(t)={\frac {1}{2\pi }}\omega (t)={\frac {1}{2\pi }}{\frac {d\varphi (t)}{dt}}

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:

{\begin{aligned}\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{aligned}}

This instantaneous frequency, ω(t), can be derived directly from the real and imaginary parts of s_a(t), instead of the complex arg without concern of phase unwrapping.

{\begin{aligned}\varphi (t)&=\arg\{s_{\mathrm {a} }(t)\}\\[4pt]&=\operatorname {atan2} ({\mathcal {Im}}[s_{\mathrm {a} }(t)],{\mathcal {Re}}[s_{\mathrm {a} }(t)])+2m_{1}\pi \\[4pt]&=\arctan \left({\frac {{\mathcal {Im}}[s_{\mathrm {a} }(t)]}{{\mathcal {Re}}[s_{\mathrm {a} }(t)]}}\right)+m_{2}\pi \end{aligned}}

2m₁ $π$ and m₂ $π$ are the integer multiples of $π$ necessary to add to unwrap the phase. At values of time, t, where there is no change to integer m₂, the derivative of φ(t) is

{\begin{aligned}\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{aligned}}

For discrete-time functions, this can be written as a recursion:

{\begin{aligned}\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{aligned}}

Discontinuities can then be removed by adding 2 $π$ whenever Δφ[n] ≤ − $π$ , and subtracting 2 $π$ whenever Δφ[n] > $π$ . That allows φ[n] to accumulate without limit and produces an unwrapped instantaneous phase. An equivalent formulation that replaces the modulo 2 $π$ operation with a complex multiplication is:

\varphi [n]=\varphi [n-1]+\arg\{s_{\mathrm {a} }[n]\,s_{\mathrm {a} }^{*}[n-1]\},

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

\omega [n]=\arg\{s_{\mathrm {a} }[n]\,s_{\mathrm {a} }^{*}[n-1]\}.

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:^[3]

e^{i\varphi (t)}={\frac {s_{\mathrm {a} }(t)}{|s_{\mathrm {a} }(t)|}}=\cos(\varphi (t))+i\sin(\varphi (t)).

This representation is similar to the wrapped phase representation in that it does not distinguish between multiples of 2 $π$ in the phase, but similar to the unwrapped phase representation since it is continuous. A vector-average phase can be obtained as the arg of the sum of the complex numbers without concern about wrap-around.

References

^ Sejdic, E.; Djurovic, I.; Stankovic, L. (August 2008). "Quantitative Performance Analysis of Scalogram as Instantaneous Frequency Estimator". IEEE Transactions on Signal Processing. 56 (8): 3837–3845. Bibcode:2008ITSP...56.3837S. doi:10.1109/TSP.2008.924856. ISSN 1053-587X. S2CID 16396084.
^ Blackledge, Jonathan M. (2006). Digital Signal Processing: Mathematical and Computational Methods, Software Development and Applications (2 ed.). Woodhead Publishing. p. 134. ISBN 1904275265.
^ Wang, S. (2014). "An Improved Quality Guided Phase Unwrapping Method and Its Applications to MRI". Progress in Electromagnetics Research. 145: 273–286. doi:10.2528/PIER14021005.