Hilbert spectrum

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The Hilbert spectrum (sometimes referred to as the Hilbert amplitude spectrum), named after

With the Hilbert transform, the singular vectors give instantaneous frequencies that are functions of time, so that the result is an energy distribution over time and frequency.

The result is an ability to capture time-frequency localization to make the concept of instantaneous frequency and time relevant (the concept of instantaneous frequency is otherwise abstract or difficult to define for all but monocomponent signals).

Definition

For a given signal ${\displaystyle x(t)}$ decomposed (with for example Empirical Mode Decomposition) to

${\displaystyle x(t)=r(t)+\sum _{j=1}^{k}c_{j}(t)}$

where ${\displaystyle k}$ is the number of intrinsic mode functions that ${\displaystyle x(t)}$ consists of and

${\displaystyle c_{j}(t)=\mathbb {R} {\big \{}a_{j}(t)e^{-i\theta _{j}(t)}{\big \}}=a_{j}(t)\cos {\big (}\theta _{j}(t){\big )}}$

The

${\displaystyle \omega _{j}(t)={\frac {d\theta _{j}(t)}{dt}}}$

From this, we can define the Hilbert Spectrum^[1] for ${\displaystyle c_{j}(t)}$ as

${\displaystyle H_{j}(\omega ,t)={\begin{cases}a_{j}(t),&\omega =\omega _{j}(t)\\0,&{\text{otherwise}}\end{cases}}}$

The Hilbert Spectrum of ${\displaystyle x(t)}$ is then given by

${\displaystyle H(\omega ,t)=\sum _{j=1}^{k}H_{j}(\omega ,t)}$

Marginal Hilbert Spectrum

A two dimensional representation of a Hilbert Spectrum, called Marginal Hilbert Spectrum, is defined as

${\displaystyle h(\omega )={\frac {1}{T}}\int _{0}^{T}H(\omega ,t)dt}$

where ${\displaystyle T}$ is the length of the sampled signal ${\displaystyle x(t)}$. The Marginal Hilbert Spectrum show the total energy that each frequency value contribute with.^[1]

Applications

The Hilbert spectrum has many practical applications. One example application pioneered by Professor

See also

• Hilbert–Huang transform

References

• Huang, et al., "The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis" Proc. R. Soc. Lond. (A) 1998
• Huang, N.E.; et al. (2016). "On Holo-Hilbert spectral analysis: a full informational spectral representation for nonlinear and non-stationary data". Phil. Trans. R. Soc. Lond. A. 374: 20150206.
  PMID 26953180
  .