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. The instantaneous phase (also known as local phase or simply phase) of a complex-valued function s(t), is the real-valued function:

$\phi (t)=\arg \{s(t)\},$

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):

$\begin{array}{cc}\phi (t)& =\arg \{s_{\mathrm{a}}(t)\}\\ & =\arg \{s(t)+j\hat{s}(t)\},\end{array}$

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

When φ(t) is constrained to its principal value, either the interval Template:Open-closed or Template:Closed-open, 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.

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.

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

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

where ω > 0.

$\begin{array}{cc}{s}_{\mathrm{a}}(t)& =A{e}^{j(\omega t+\theta )},\\ \phi (t)& =\omega t+\theta .\end{array}$

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

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

where ω > 0.

$\begin{array}{cc}{s}_{\mathrm{a}}(t)& =A{e}^{j(\omega t-\frac{\pi }{2})},\\ \phi (t)& =\omega t-\frac{\pi }{2}.\end{array}$

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

Formulations

Instantaneous angular frequency is defined as:

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

and instantaneous (ordinary) frequency is defined as:

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

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:

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

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.

$\begin{array}{cc}\phi (t)& =\arg \{s_{\mathrm{a}}(t)\}\\ & =\mathrm{atan2}(ℐ𝓂[s_{\mathrm{a}}(t)],ℛℯ[s_{\mathrm{a}}(t)])+2m_1\pi \\ & =\arctan \left(\frac{ℐ𝓂[s_{\mathrm{a}}(t)]}{ℛℯ[s_{\mathrm{a}}(t)]}\right)+m_2\pi \end{array}$

2m₁Template:Pi and m₂Template:Pi are the integer multiples of Template:Pi 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

$\begin{array}{cc}\omega (t)=\frac{d\phi (t)}{dt}& =\frac{d}{dt}\arctan \left(\frac{ℐ𝓂[s_{\mathrm{a}}(t)]}{ℛℯ[s_{\mathrm{a}}(t)]}\right)\\ & =\frac{1}{1+{\left(\frac{ℐ𝓂[s_{\mathrm{a}}(t)]}{ℛℯ[s_{\mathrm{a}}(t)]}\right)}^2}\frac{d}{dt}\left(\frac{ℐ𝓂[s_{\mathrm{a}}(t)]}{ℛℯ[s_{\mathrm{a}}(t)]}\right)\\ & =\frac{ℛℯ[s_{\mathrm{a}}(t)]\frac{dℐ𝓂[s_{\mathrm{a}}(t)]}{dt}-ℐ𝓂[s_{\mathrm{a}}(t)]\frac{dℛℯ[s_{\mathrm{a}}(t)]}{dt}}{(ℛℯ[s_{\mathrm{a}}(t)]{)}^2+(ℐ𝓂[s_{\mathrm{a}}(t)]{)}^2}\\ & =\frac{1}{|s_{\mathrm{a}}(t){|}^2}(ℛℯ[s_{\mathrm{a}}(t)]\frac{dℐ𝓂[s_{\mathrm{a}}(t)]}{dt}-ℐ𝓂[s_{\mathrm{a}}(t)]\frac{dℛℯ[s_{\mathrm{a}}(t)]}{dt})\\ & =\frac{1}{(s(t){)}^2+{(\hat{s}(t))}^2}(s(t)\frac{d\hat{s}(t)}{dt}-\hat{s}(t)\frac{ds(t)}{dt})\end{array}$

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

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

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

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

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:

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

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

See also

• Angular displacement
• Analytic signal
• Frequency modulation
• Group delay
• Instantaneous amplitude
• Negative frequency

Further reading

• Template:Cite book
• Template:Cite book