Английская Википедия:Instantaneous phase and frequency

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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:

<math>\varphi(t) = \arg\{s(t)\},</math>

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):

<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 Шаблон:Open-closed or Шаблон: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.

Файл:Phase vs Time, wrapped and unwrapped.jpg

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.

Файл:Instantaneous (wrapped) phase; one 360° plot stacked 3 times vertically.jpg

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) = 2Шаблон:PiN for integer values of 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 real and imaginary parts of s_a(t), instead of the 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>

2m₁Шаблон:Pi and m₂Шаблон:Pi are the integer multiples of Шаблон: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

<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 2Шаблон:Pi whenever Δφ[n] ≤ −Шаблон:Pi, and subtracting 2Шаблон:Pi whenever Δφ[n] > Шаблон:Pi. That allows φ[n] to accumulate without limit and produces an unwrapped instantaneous phase. An equivalent formulation that replaces the modulo 2Шаблон: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:

<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 2Шаблон: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

• Шаблон:Cite book
• Шаблон:Cite book