Hilbert–Huang transform

The Hilbert–Huang transform (HHT) is a way to decompose a signal into so-called intrinsic mode functions (IMF) along with a trend, and obtain instantaneous frequency data.

It is the result of the empirical mode decomposition (EMD) and the Hilbert spectral analysis (HSA).

This is an important advantage of HHT since a real-world signal usually has multiple causes happening in different time intervals.

Using the EMD method, any complicated data set can be decomposed into a finite and often small number of components.

[7] Since the decomposition is based on the local characteristic time scale of the data, it can be applied to nonlinear and nonstationary processes.

By definition, an IMF is any function with the same number of extrema and zero crossings, whose envelopes are symmetric with respect to zero.

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 difference between the data and m1 is the first component h1: Ideally, h1 should satisfy the definition of an IMF, since the construction of h1 described above should have made it symmetric and having all maxima positive and all minima negative.

This procedure can be repeated for all the subsequent rj's, and the result is The sifting process finally stops when the residue, rn, becomes a monotonic function from which no more IMF can be extracted.

Flandrin et al. (2003) and Wu and Huang (2004) have shown that the EMD is equivalent to a dyadic filter bank.

[6][10] Having obtained the intrinsic mode function components, the instantaneous frequency can be computed using the Hilbert transform.

Datig and Schlurmann [2004] [29] conducted a comprehensive study on the performance and limitations of HHT with particular applications to irregular water waves.

Huang and Wu [2008] [30] 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".

[32] Source:[33] The masking method improves EMD by allowing for the separation of similar frequency components through the following steps: The optimal choice of amplitude depends on the frequencies Overall, the masking method enhances EMD by providing a means to prevent mode mixing, improving the accuracy and applicability of EMD in signal analysis Source:[34] EEMD adds finite amplitude white noise to the original signal.

The noise also enables the EMD method to be a truly dyadic filter bank for any data, which means that a signal of a similar scale in a noisy data set could be contained in one IMF component, significantly reducing the chance of mode mixing.

This approach preserves the physical uniqueness of decomposition and represents a major improvement over the EMD method.