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Everything about Hilbert transform

Integral transform and linear operator

In mathematics and signal processing, the Hilbert transform is a specific singular integral that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). The Hilbert transform is given by the Cauchy principal value of the convolution with the function (see § Definition). The Hilbert transform has a particularly simple representation in the frequency domain: It imparts a phase shift of ±90° (π/2 radians) to every frequency component of a function, the sign of the shift depending on the sign of the frequency (see § Relationship with the Fourier transform). The Hilbert transform is important in signal processing, where it is a component of the analytic representation of a real-valued signal u(t). The Hilbert transform was first introduced by David Hilbert in this setting, to solve a special case of the Riemann–Hilbert problem for analytic functions. Wikipedia →

🧭 Start here: a 6-step reading path

  1. Read the overview: Hilbert transformA 5-minute orientation
  2. Skim a recent paper: Improved Hilbert–Huang transform with soft sifting stopping criterion and its application to fault diagnosis of wheelset bearingsInfluential work since 2022
  3. Pick up a classic book: Théorie des opérations linéairesThe most-reprinted book on the subject
  4. Read the landmark paper: Hamiltonian Systems and Transformation in Hilbert SpaceCited 2,180 times
  5. Go sideways: Fourier transformA closely related topic
  6. Go sideways: Singular integral operators of convolution typeA closely related topic

📖 Definitions

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🗂️ The subject at a glance

  1. Definition
  2. History
  3. Relationship with the Fourier transform
  4. Table of selected Hilbert transforms
  5. Domain of definition
  6. Properties
  7. Extending the domain of definition
  8. Conjugate functions
  9. Hilbert transform on the circle
  10. Hilbert transform in signal processing
  11. Discrete Hilbert transform
  12. Number-theoretic Hilbert transform

🏆 Landmark papers (most cited)

🆕 Influential since 2022

📚 Books worth reading

Ranked by number of editions on Open Library, a rough measure of how lasting a book is.

🔗 Related topics