Instantaneous phase and frequency

Formulations

Instantaneous angular frequency is defined as:

${\displaystyle \omega (t)=\frac{d\phi (t)}{dt},}$

and instantaneous (ordinary) frequency is defined as:

${\displaystyle f(t)=\frac{1}{2\pi }\omega (t)=\frac{1}{2\pi }\frac{d\phi (t)}{dt}}$

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:

${\displaystyle \begin{array}{cc}\phi (t)& ={\int }_{-\mathrm{\infty }}^t\omega (\tau )d\tau =2\pi {\int }_{-\mathrm{\infty }}^{t}f(\tau )d\tau \\ & ={\int }_{-\mathrm{\infty }}^0\omega (\tau )d\tau +{\int }_0^t\omega (\tau )d\tau \\ & =\phi (0)+{\int }_0^t\omega (\tau )d\tau .\end{array}}$

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.

${\displaystyle \begin{array}{cc}\phi (t)& =\arg \{s_{\mathrm{a}}(t)\}\\ & =\mathrm{atan2}(\mathcal{I}\mathcal{m}[s_{\mathrm{a}}(t)],\mathcal{R}\mathcal{e}[s_{\mathrm{a}}(t)])+2m_1\pi \\ & =\arctan \left(\frac{\mathcal{I}\mathcal{m}[s_{\mathrm{a}}(t)]}{\mathcal{R}\mathcal{e}[s_{\mathrm{a}}(t)]}\right)+m_2\pi \end{array}}$

2m₁π and m₂π 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

${\displaystyle \begin{array}{cc}\omega (t)=\frac{d\phi (t)}{dt}& =\frac{d}{dt}\arctan \left(\frac{\mathcal{I}\mathcal{m}[s_{\mathrm{a}}(t)]}{\mathcal{R}\mathcal{e}[s_{\mathrm{a}}(t)]}\right)\\ & =\frac{1}{1+{\left(\frac{\mathcal{I}\mathcal{m}[s_{\mathrm{a}}(t)]}{\mathcal{R}\mathcal{e}[s_{\mathrm{a}}(t)]}\right)}^2}\frac{d}{dt}\left(\frac{\mathcal{I}\mathcal{m}[s_{\mathrm{a}}(t)]}{\mathcal{R}\mathcal{e}[s_{\mathrm{a}}(t)]}\right)\\ & =\frac{\mathcal{R}\mathcal{e}[s_{\mathrm{a}}(t)]\frac{d\mathcal{I}\mathcal{m}[s_{\mathrm{a}}(t)]}{dt}-\mathcal{I}\mathcal{m}[s_{\mathrm{a}}(t)]\frac{d\mathcal{R}\mathcal{e}[s_{\mathrm{a}}(t)]}{dt}}{(\mathcal{R}\mathcal{e}[s_{\mathrm{a}}(t)]{)}^2+(\mathcal{I}\mathcal{m}[s_{\mathrm{a}}(t)]{)}^2}\\ & =\frac{1}{|s_{\mathrm{a}}(t){|}^2}\left(\mathcal{R}\mathcal{e}[s_{\mathrm{a}}(t)]\frac{d\mathcal{I}\mathcal{m}[s_{\mathrm{a}}(t)]}{dt}-\mathcal{I}\mathcal{m}[s_{\mathrm{a}}(t)]\frac{d\mathcal{R}\mathcal{e}[s_{\mathrm{a}}(t)]}{dt}\right)\\ & =\frac{1}{(s(t){)}^2+{\left(\hat{s}(t)\right)}^2}\left(s(t)\frac{d\hat{s}(t)}{dt}-\hat{s}(t)\frac{ds(t)}{dt}\right)\end{array}}$

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

${\displaystyle \begin{array}{cc}\phi [n]& =\phi [n-1]+\omega [n]\\ & =\phi [n-1]+\underset{\mathrm{\Delta }\phi [n]}{\underbrace{\arg \{s_{\mathrm{a}}[n]\}-\arg \{s_{\mathrm{a}}[n-1]\}}}\\ & =\phi [n-1]+\arg \left\{\frac{s_{\mathrm{a}}[n]}{s_{\mathrm{a}}[n-1]}\right\}\end{array}}$

Discontinuities can then be removed by adding 2π whenever Δφ[n] ≤ −π, and subtracting 2π 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 2π operation with a complex multiplication is:

${\displaystyle \phi [n]=\phi [n-1]+\arg \{s_{\mathrm{a}}[n]s_{\mathrm{a}}^{\ast }[n-1]\},}$

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

${\displaystyle \omega [n]=\arg \{s_{\mathrm{a}}[n]s_{\mathrm{a}}^{\ast }[n-1]\}.}$

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:

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

This representation is similar to the wrapped phase representation in that it does not distinguish between multiples of 2π 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.