science

Hilbert–Huang transform

Hilbert–Huang transform is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Hilbert–Huang transform rather than just read about it. In short: 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 designed to work well for data that is nonstationary and nonlinear.

Key takeaways

Hilbert–Huang transform belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hilbert–Huang transform to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hilbert–Huang transform from memory before moving on to harder problems.

Reference excerpt

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 designed to work well for data that is nonstationary and nonlinear. The Hilbert–Huang transform (HHT), a NASA designated name, was proposed by Norden E. Huang. 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 (IMF) with a trend, and applies the HSA method to the IMFs to obtain instantaneous frequency data. Since the signal is decomposed in time domain and the length of the IMFs is the same as the original signal, HHT preserves the characteristics of the varying frequency. This is an important advantage of HHT since a real-world signal usually has multiple causes happening in different time intervals. The HHT provides a new method of analyzing nonstationary and nonlinear time series data.

Definition

Empirical mode decomposition The fundamental part of the HHT is the empirical mode decomposition (EMD) method. Breaking down signals into various components, EMD can be compared with other analysis methods such as Fourier transform and Wavelet transform. Using the EMD method, any complicated data set can be decomposed into a finite and often small number of components. These components form a complete and nearly orthogonal basis for the original signal. In addition, they can be described as intrinsic mode functions (IMF). Because the first IMF usually carries the most oscillating (high-frequency) components, it can be rejected to remove high-frequency components (e.g., random noise). EMD based smoothing algorithms have been widely used in seismic data processing, where high-quality seismic records are highly demanded. Without leaving the time domain, EMD 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.

Intrinsic mode functions An intrinsic mode function (IMF) is defined as a function that satisfies the following requirements:

In the whole data set, the number of extrema and the number of zero-crossings must either be equal or differ at most by one. At any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero. It 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.

Hilbert spectral analysis 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.

Techniques The Intrinsic Mode Function (IMF) amplitude and frequency can vary with time and it must satisfy the rule below:

The number of extremes(local maximums & local minimums) and the number of zero-crossings must either equal or differ at most by one. At any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is near zero.

Empirical mode decomposition

The empirical mode decomposition (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. 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:

Identify all the local extrema in the test data. Connect all the local maxima by a cubic spline line as the upper envelope. Repeat the procedure for the local minima to produce the lower envelope. The upper and lower envelopes should cover all the data between them. Their mean is m1. The difference between the data and m1 is the first component h1:

X ( t ) − m 1 = h 1 . {\displaystyle X(t)-m_{1}=h_{1}.\,}

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. 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, h1 can only be treated as a proto-IMF. In the next step, h1 is treated as data:

h 1 − m 11 = h 11 . {\displaystyle h_{1}-m_{11}=h_{11}.\,}

After repeated sifting up to k times, h1 becomes an IMF, that is

h 1 ( k − 1 ) − m 1 k = h 1 k . {\displaystyle h_{1(k-1)}-m_{1k}=h_{1k}.\,}

Then, h1k is designated as the first IMF component of the data:

c 1 = h 1 k . {\displaystyle c_{1}=h_{1k}.\,}

Stoppage criteria of the sifting process The stoppage criterion determines the number of sifting steps to produce an IMF. Following are the four existing stoppage criterion:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Hilbert–Huang transform

Start with the simplest possible case. Write down what Hilbert–Huang transform claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hilbert–Huang transform before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hilbert–Huang transform ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hilbert–Huang transform

In research: Hilbert–Huang transform appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Hilbert–Huang transform in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Hilbert–Huang transform is common in secondary-school and first-year university syllabi. It links to neighbouring topics Signal processing, Telecommunication theory, so understanding it makes those chapters shorter.
In everyday life: Look for Hilbert–Huang transform outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Hilbert–Huang transform in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hilbert–Huang transform means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hilbert–Huang transform out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hilbert–Huang transform in simple terms?

Why does Hilbert–Huang transform matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hilbert–Huang transform?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hilbert–Huang transform.