Paradoxes on Instantaneous Frequency

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Paradoxes on Instantaneous Frequency

• a la Leon Cohen
• Time-Frequency Analysis, Prentice Hall, 1995
• Chapter 2 Instantaneous Frequency, P. 40

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The Five Paradoxes

• 1. Instantaneous frequency of a signal may not be
  one of the frequencies in the spectrum.
• 2. For a signal with a line spectrum consisting
  of only a few sharp frequencies, the
  instantaneous frequency may be continuous and
  range over an infinite number of values.
• 3. Although the spectrum of analytic signal is
  zero for negative frequencies, the instantaneous
  frequency may be negative
• 4. For the band limited signal the instantaneous
  frequency may be outside the band.
• 5. The value of the Instantaneous frequency
  should depend only on the present time, but the
  analytic signal, from which the instantaneous
  frequency is computed, depends on the signal
  values for the whole time space.

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Observations I

• By spectrum, Cohn is limiting the term to
  Fourier spectrum.
• By instantaneous Frequency, Cohn is limiting
  the terms to be the IF obtained through Hilbert
  Transform. In fact, as we see, IF could be
  determined through many other methods.

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Observations II

• 1. Paradoxes 1, 2 and 4 are essentially the
  same Instantaneous Frequency values may be
  different from the frequency in the spectrum.
• 2. The negative frequency in analytic signal
  seems to violate Gabors construction.
• 3. The analytic function, or the Hilbert
  Transform, involves the functional values over
  the whole time domain therefore, it is not
  local.

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Resolution for paradoxes 1, 2 and 4

• Two Examples

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The First Example

• Sin A cSin B

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Data Sin (pt/360) Sin (pt/320) t023040
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Hilbert Spectrum X
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Spectrogram X
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Morlet Wavelet X
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Instantaneous frequency X
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Instantaneous frequency X Details
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Marginal Spectra X
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Data Sin (pt/360) 0.8 Sin (pt/320)
t023040
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Hilbert Spectrum X08
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Marginal Spectra X08
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Two ways to view modulated wave
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New developments

• G. RILLING, P. FLANDRIN, 2008  "One or Two
  Frequencies? The Empirical Mode Decomposition
  Answers, IEEE Trans. on Signal Proc., Vol. 56,
  No. 1, pp. 85-95.
• .close tones are no longer perceived as such
  by the human ear but are rather considered as a
  whole, one can wonder whether a decomposition
  into tones is a good answer if the aim is to get
  a representation matched to physics (and/or
  perception) rather than to mathematics.

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Example
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General case
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Derivatives of HF and LF components
Af lt 1
Af2 gt 1
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Numerical experiments
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Numerical Experiments C
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Numerical Experiments C
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One or two-frequency?

• Mathematically, if we select strict Fourier
  basis, it is two-frequency signal.
• Physically, it is a modulated one frequency
  signal.
• Using EMD, we could separate the signal, if the
  amplitude-frequency combination satisfies certain
  condition, the condition coincides with physical
  perception.

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Example 2

• Duffings Pendulum

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Duffing Pendulum
x
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Duffing Type Wave Data x cos(wt0.3 sin2wt)
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Duffing Type Wave Perturbation Expansion
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Duffing Type Wave Wavelet Spectrum
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Duffing Type Wave Hilbert Spectrum
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Duffing Type Wave Marginal Spectra
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Duffing Equation
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Duffing Equation Data
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Duffing Equation IMFs
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Duffing Equation IMFs
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Duffing Equation Hilbert Spectrum
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Duffing Equation Detailed Hilbert Spectrum
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Duffing Equation Wavelet Spectrum
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Duffing Equation Hilbert Wavelet Spectra
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Duffing Equation Marginal Hilbert Spectrum
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Rössler Equation
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Rössler Equation Data
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Rössler Equation 3D Phase
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Rössler Equation 2D Phase
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Rössler Equation IMF Strips
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Rössler Equation IMF
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Rössler Equation Hilbert Spectrum
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Rössler Equation Hilbert Spectrum Data Details
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Rössler Equation Wavelet Spectrum
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Rössler Equation Hilbert Wavelet Spectra
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Rössler Equation Marginal Spectra
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Rössler Equation Marginal Spectra
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Resolution for Paradox 3

• Negative Frequency

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Examples of Negative Frequency 1

• Different references

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Hilbert Transform a cos ? b Data
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Hilbert Transform a cos ? b Phase Diagram
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Hilbert Transform a cos ? b Phase Angle
Details
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Hilbert Transform a cos ? b Frequency
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The Empirical Mode Decomposition Method and
Hilbert Spectral AnalysisSifting
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Examples of Negative Frequency 2

• FM and AM Frequencies
• a sin ? t b sin f t

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sin ? t 0.4 sin 4 ? t
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Hilbert sin ? t 0.4 sin 4 ? t
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sin ? t sin 4 ? t
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Hilbert sin ? t sin 4 ? t
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a sin ?t b sin ft

• The data need to be sifted first.
• Whenever Hilbert Transform has a loop away from
  the original (negative maximum or positive
  minimum), there will be negative frequency.
• Whenever the Hilbert pass through the original
  (both real and imaginary parts are zero), there
  will be a frequency singularity.
• Hilbert Transform is local to a degree of 1/t.

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IMF sin ? t 0.2 sin 4 ?
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IMF sin ? t 0.4 sin 4 ?
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IMF sin ? t sin 4 ?
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Negative Frequency

• Negative instantaneous frequency values are
  mostly due to riding waves.
• IMF is a necessary (but not a sufficient)
  condition for having non-negative frequency.
• There are occasion when abrupt amplitude change
  in an IMF (but no riding waves) can also generate
  negative frequency. The amplitude induced problem
  is covered by Bedrosian theorem normalized HHT
  will take care of it.
• Physically, the abrupt amplitude change also
  shows the non-local characteristics of the
  Hilbert Transform.

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Resolution for Paradox 5

• Non-local influence does exist, they may come
  from Gibbs Phenomenon, end effects, and the
  limitation of the 1/t window in the Hilbert
  Transform. But most of the problems can be
  rectified through the Normalized HHT.
• In fact, the non-local property of Hilbert
  transform is fully resolved by Quadrature method,
  though the solution is no longer a Hilbert
  Spectrum.

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Data with magnitude jump Signal
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Data with magnitude jump Signal
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Hilbert Spectrum
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Spectrogram
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Morlet Wavelet
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Data with magnitude jump
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Data with magnitude jump Details
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Normalized Hilbert Spectrum
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Amplitude Effects on Marginal Hilbert Fourier
Spectra
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Instantaneous frequency
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Instantaneous frequency Details
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Summary The so called paradoxes are really not
problems, once some misconceptions are clarified

• Instantaneous Frequency (IF) has very different
  meaning than the Fourier frequency.
• IF for special mono-component functions only
  IMFs a necessary but not a sufficient condition.
  
• Even for IMFs, there are still problems
  associated with IF through Hilbert Transform. We
  can rectify most of them with the Normalized HHT.
• The better solution is through quadrature.