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Joint moment of the time-frequency distribution of a signal

Time-frequency moments provide an efficient way to characterize signals whose frequencies change in time (that is, are nonstationary). Such signals can arise from machinery with degraded or failed hardware. Classical Fourier analysis cannot capture the time-varying frequency behavior. Time-frequency distribution generated by short-time Fourier transform (STFT) or other time-frequency analysis techniques can capture the time-varying behavior, but directly treating these distributions as features carries a high computational burden, and potentially introduces unrelated and undesirable feature characteristics. In contrast, distilling the time-frequency distribution results into low-dimension time-frequency moments provides a method for capturing the essential features of the signal in a much smaller data package. Using these moments significantly reduces the computational burden for feature extraction and comparison — a key benefit for real-time operation [1], [2].

The Predictive Maintenance Toolbox™ implements the three branches of time-frequency moment:

`momentJ = tfmoment(xt,order)`

`momentJ = tfmoment(x,fs,order)`

`momentJ = tfmoment(x,ts,order) `

`momentJ = tfmoment(p,fp,tp,order) `

`momentJ = tfmoment(___,Name,Value)`

returns the Joint Time-Frequency Moments of
`momentJ`

= tfmoment(`xt`

,`order`

)`timetable`

`xt`

as a vector with one or more components. Each
`momentJ`

scalar element represents the joint moment for
one of the orders you specify in `order`

. The data in
`xt`

can be nonuniformly sampled.

returns the joint time-frequency moment of time-series vector
`momentJ`

= tfmoment(`x`

,`fs`

,`order`

)`x`

, sampled at rate `Fs`

. The moment
is returned as a vector, in which each scalar element represents the joint
moment corresponding to one of the orders you specify in
`order`

. With this syntax, `x`

must be
uniformly sampled.

returns
the joint time-frequency moment of `momentJ`

= tfmoment(`x`

,`ts`

,`order`

) `x`

sampled at the time
instants specified by `ts`

in seconds.

If

`ts`

is a scalar`duration`

, then`tfmoment`

applies it uniformly to all samples.If

`ts`

is a vector, then`tfmoment`

applies each element to the corresponding sample in`x`

. Use this syntax for nonuniform sampling.

returns
the joint time-frequency moment of a signal whose power spectrogram is
`momentJ`

= tfmoment(`p`

,`fp`

,`tp`

,`order`

) `p`

. `fp`

contains the frequencies
corresponding to the spectral estimate contained in `p`

.
`tp`

contains the vector of time instants corresponding
to the centers of the windowed segments used to compute short-time power
spectrum estimates. Use this syntax when:

You already have the power spectrogram you want to use.

You want to customize the options for

`pspectrum`

, rather than accept the default`pspectrum`

options that`tfmoment`

applies. Use`pspectrum`

first with the options you want, and then use the output`p`

as input for`tfmoment`

. This approach also allows you to plot the power spectrogram.

specifies additional properties using name-value pair arguments. Options include
moment centralization, frequency-limit specification, and time-limit
specification.`momentJ`

= tfmoment(___,`Name,Value`

)

You can use `Name,Value`

with any of the input-argument
combinations in previous syntaxes.

[1] Loughlin, P. J. "What Are the
Time-Frequency Moments of a Signal?" *Advanced Signal Processing Algorithms,
Architectures, and Implementations XI, SPIE Proceedings*. Vol. 4474,
November 2001.

[2] Loughlin, P., F. Cakrak, and
L. Cohen. "Conditional Moment Analysis of Transients with Application to Helicopter
Fault Data." *Mechanical Systems and Signal Processing*. Vol 14,
Issue 4, 2000, pp. 511–522.