Instantaneous phase and frequency

 ( Signal Processing ) 

In 1922, according to Nahin, John Renshaw Carson defined the instantaneous frequency of a signal "as the time derivative of the signal's phase angle." In frequency modulation, instantaneous frequency describes the frequency varying above and below the carrier frequency, at the audio tone 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:

$\backslash varphi(t)\; =\; \backslash arg\backslash \{s(t)\backslash \},$

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 signal, s_a( t):

$\backslash begin\{align\}$

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

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

When φ( t) is constrained to its principal value, either the interval or , 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.

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Examples

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Example 1

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

where ω > 0.

$\backslash begin\{align\}$

 s_\mathrm{a}(t) &= A e^{j(\omega t + \theta)}, \\
      \varphi(t) &= \omega t + \theta.
     

\end{align} 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.

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Example 2

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

where ω > 0.

$\backslash begin\{align\}$

 s_\mathrm{a}(t) &= A e^{j \left(\omega t - \frac{\pi}{2}\right)}, \\
      \varphi(t) &= \omega t - \frac{\pi}{2}.
     

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

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Formulations Instantaneous angular frequency is defined as:

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

and instantaneous (ordinary) frequency is defined as:

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

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:

$\backslash begin\{align\}$

 \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}

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

$\backslash begin\{align\}$

 \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}

2 m₁ and m₂ are the integer multiples of 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

$\backslash begin\{align\}$

 \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}
&= \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}
     

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

     
$\backslash begin\{align\}$
     
     
 \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}
     

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

     
$\backslash varphin\; =\; \backslash varphin\; +\; \backslash arg\backslash \{s\_\backslash mathrm\{a\}n\; \backslash ,\; s\_\backslash mathrm\{a\}^*n\backslash \},$
     
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
     

     
$\backslash omegan\; =\; \backslash arg\backslash \{s\_\backslash mathrm\{a\}n\; \backslash ,\; s\_\backslash mathrm\{a\}^*n\backslash \}.$
     
     

 

 
 
  

  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\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 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
 
 

 
 
 
Categories: Signal Processing, Digital Signal Processing, Time–frequency Analysis, Fourier Analysis, Electrical Engineering, Audio Engineering

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