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The Hilbert–Huang transform (HHT) is a way to decompose a signal into so-called intrinsic mode functions (IMF), and obtain instantaneous frequency data. It is designed to work well for data that is nonstationary and nonlinear. In contrast to other common transforms like the Fourier transform, the HHT is more like an algorithm (an empirical approach) that can be applied to a data set, rather than a theoretical tool.
The Hilbert–Huang transform (HHT), a NASA designated name, was proposed by Huang et al. (1996, 1998, 1999, 2003, 2012). It is the result of the empirical mode decomposition (EMD) and the Hilbert spectral analysis (HSA). The HHT uses the EMD method to decompose a signal into so-called intrinsic mode functions, and uses the HSA method to obtain instantaneous frequency data. The HHT provides a new method of analyzing nonstationary and nonlinear time series data.
The fundamental part of the HHT is the empirical mode decomposition (EMD) method. Using the EMD method, any complicated data set can be decomposed into a finite and often small number of components, which are intrinsic mode functions (IMF). An IMF represents a generally simple oscillatory mode as a counterpart to the simple harmonic function. By definition, an IMF is any function with the same number of extrema and zero crossings, whose envelopes are symmetric with respect to zero. This definition guarantees a well-behaved Hilbert transform of the IMF. This decomposition method operating in the time domain is adaptive and highly efficient. Since the decomposition is based on the local characteristic time scale of the data, it can be applied to nonlinear and nonstationary processes.
Hilbert spectral analysis (HSA) is a method for examining each IMF's instantaneous frequency as functions of time. The final result is a frequency-time distribution of signal amplitude (or energy), designated as the Hilbert spectrum, which permits the identification of localized features.
The EMD method is a necessary step to reduce any given data into a collection of intrinsic mode functions (IMF) to which the Hilbert spectral analysis can be applied. An IMF is defined as a function that satisfies the following requirements:
Therefore, an IMF represents a simple oscillatory mode as a counterpart to the simple harmonic function, but it is much more general: instead of constant amplitude and frequency in a simple harmonic component, an IMF can have variable amplitude and frequency along the time axis. The procedure of extracting an IMF is called sifting. The sifting process is as follows:
The upper and lower envelopes should cover all the data between them. Their mean is m_{1}. The difference between the data and m_{1} is the first component h_{1}:
Ideally, h_{1} should satisfy the definition of an IMF, since the construction of h_{1} described above should have made it symmetric and having all maxima positive and all minima negative. After the first round of sifting, a crest may become a local maximum. New extrema generated in this way actually reveal the proper modes lost in the initial examination. In the subsequent sifting process, h_{1} can only be treated as a proto-IMF. In the next step, h_{1} is treated as data: h_{1}-m_{11}=h_{11}.\,
After repeated sifting up to k times, h_{1} becomes an IMF, that is
Then, h_{1k} is designated as the first IMF component of the data:
The stoppage criterion determines the number of sifting steps to produce an IMF. Two different stoppage criteria have been used traditionally:
Once a stoppage criterion is selected, the first IMF, c_{1}, can be obtained. Overall, c_{1} should contain the finest scale or the shortest period component of the signal. We can, then, separate c_{1} from the rest of the data by X(t)-c_1=r_1.\, Since the residue, r_{1}, still contains longer period variations in the data, it is treated as the new data and subjected to the same sifting process as described above.
This procedure can be repeated for all the subsequent r_{j}'s, and the result is
The sifting process finally stops when the residue, r_{n}, becomes a monotonic function from which no more IMF can be extracted. From the above equations, we can induce that
Thus, a decomposition of the data into n-empirical modes is achieved. The components of the EMD are usually physically meaningful, for the characteristic scales are defined by the physical data. Flandrin et al. (2003) and Wu and Huang (2004) have shown that the EMD is equivalent to a dyadic filter bank.
Having obtained the intrinsic mode function components, the instantaneous frequency can be computed using the Hilbert Transform. After performing the Hilbert transform on each IMF component, the original data can be expressed as the real part, Real, in the following form:
Chen and Feng [2003] proposed a technique to improve the HHT procedure.^{[2]} The authors noted that the EMD is limited in distinguishing different components in narrow-band signals. The narrow band may contain either (a) components that have adjacent frequencies or (b) components that are not adjacent in frequency but for which one of the components has a much higher energy intensity than the other components. The improved technique is based on beating-phenomenon waves.
Datig and Schlurmann [2004] ^{[3]} conducted a comprehensive study on the performance and limitations of HHT with particular applications to irregular waves. The authors did extensive investigation into the spline interpolation. The authors discussed using additional points, both forward and backward, to determine better envelopes. They also performed a parametric study on the proposed improvement and showed significant improvement in the overall EMD computations. The authors noted that HHT is capable of differentiating between time-variant components from any given data. Their study also showed that HHT was able to distinguish between riding and carrier waves.
Huang and Zu [2008] ^{[4]} reviewed applications of the Hilbert–Huang transformation emphasizing that the HHT theoretical basis is purely empirical, and noting that "one of the main drawbacks of EMD is mode mixing". They also outline outstanding open problems with HHT, which include: End effects of the EMD, Spline problems, Best IMF selection and uniqueness. Although the ensemble EMD (EEMD) may help mitigate the latter.
In the US, where patents on algorithms are permitted, the HHT is heavily encumbered by patents in almost all of its domains of possible application .
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