Hilbert-Huang transform

returns the Hilbert spectrum `hs`

= hht(`imf`

)`hs`

of the signal specified by
intrinsic mode functions `imf`

. `hs`

is
useful for analyzing signals that comprise a mixture of signals whose spectral
content changes in time. Use `hht`

to perform Hilbert
spectral analysis on signals to identify localized features.

`[___] = hht(___,`

estimates Hilbert spectrum parameters with additional options specified by one
or more `Name,Value`

)`Name,Value`

pair arguments.

`hht(___)`

with no output arguments plots the
Hilbert spectrum in the current figure window. You can use this syntax with any
of the input arguments in previous syntaxes.

`hht(___,`

plots the Hilbert spectrum with the optional `freqlocation`

)`freqlocation`

argument to specify the location of the frequency axis. Frequency is represented
on the *y*-axis by default.

The Hilbert-Huang transform is useful for performing time-frequency analysis of nonstationary and nonlinear data. The Hilbert-Huang procedure consists of the following steps:

`emd`

decomposes the data set*x*into a finite number of intrinsic mode functions.For each intrinsic mode function,

*x*, the function_{i}`hht`

:Uses

`hilbert`

to compute the analytic signal, $${z}_{i}(t)={x}_{i}(t)+jH\{{x}_{i}(t)\}$$, where*H*{*x*} is the Hilbert transform of_{i}*x*._{i}Expresses

*z*as $${z}_{i}(t)={a}_{i}(t)\text{\hspace{0.17em}}{e}^{j{\theta}_{i}(t)}$$, where_{i}*a*(_{i}*t*) is the instantaneous amplitude and $${\theta}_{i}(t)$$ is the instantaneous phase.Computes the instantaneous energy, $$|{a}_{i}(t){|}^{2}$$, and the instantaneous frequency, $${\omega}_{i}(t)\equiv d{\theta}_{i}(t)/dt$$. If given a sample rate,

`hht`

converts $${\omega}_{i}(t)$$ to a frequency in Hz.Outputs the instantaneous energy in

`imfinse`

and the instantaneous frequency in`imfinsf`

.

When called with no output arguments,

`hht`

plots the energy of the signal as a function of time and frequency, with color proportional to amplitude.

[1] Huang, Norden E, and Samuel S
P Shen. *Hilbert–Huang Transform and Its Applications*. 2nd ed.
Vol. 16. Interdisciplinary Mathematical Sciences. WORLD SCIENTIFIC, 2014.
https://doi.org/10.1142/8804.

[2] Huang, Norden E., Zhaohua Wu,
Steven R. Long, Kenneth C. Arnold, Xianyao Chen, and Karin Blank. “ON INSTANTANEOUS
FREQUENCY.” *Advances in Adaptive Data Analysis* 01, no. 02 (April
2009): 177–229. https://doi.org/10.1142/S1793536909000096.