In mathematics, the Morlet wavelet (or Gabor wavelet) is a wavelet composed of a complex exponential (carrier) multiplied by a Gaussian window (envelope). This wavelet is closely related to human perception, both hearing and vision.
Wavelet#History
In 1946, physicist Dennis Gabor, applying ideas from quantum physics, introduced the use of Gaussian-windowed sinusoids for time-frequency decomposition, which he referred to as atoms, and which provide the best trade-off between spatial and frequency resolution. These are used in the Gabor transform, a type of short-time Fourier transform. In 1984, Jean Morlet introduced Gabor's work to the seismology community and, with Goupillaud and Grossmann, modified it to keep the same wavelet shape over equal octave intervals, resulting in the first formalization of the continuous wavelet transform.
The wavelet is defined as a constant subtracted from a plane wave and then localised by a Gaussian window:
where is defined by the admissibility criterion,
and the normalisation constant is:
The Fourier transform of the Morlet wavelet is:
The "central frequency" is the position of the global maximum of which, in this case, is given by the positive solution to:
which can be solved by a fixed-point iteration starting at (the fixed-point iterations converge to the unique positive solution for any initial ).
The parameter in the Morlet wavelet allows trade between time and frequency resolutions. Conventionally, the restriction is used to avoid problems with the Morlet wavelet at low (high temporal resolution).
For signals containing only slowly varying frequency and amplitude modulations (audio, for example) it is not necessary to use small values of . In this case, becomes very small (e.g. ) and is, therefore, often neglected. Under the restriction , the frequency of the Morlet wavelet is conventionally taken to be .
The wavelet exists as a complex version or a purely real-valued version. Some distinguish between the "real Morlet" vs the "complex Morlet".
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
We discuss a set of topics that are important for the understanding of modern data science but that are typically not taught in an introductory ML course. In particular we discuss fundamental ideas an
Machine learning and data analysis are becoming increasingly central in sciences including physics. In this course, fundamental principles and methods of machine learning will be introduced and practi
Study of advanced image processing; mathematical imaging. Development of image-processing software and prototyping in Jupyter Notebooks; application to real-world examples in industrial vision and bio
Related units (7)
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the number and direction of its pulses. Wavelets are imbued with specific properties that make them useful for signal processing. For example, a wavelet could be created to have a frequency of Middle C and a short duration of roughly one tenth of a second.
Fault diagnosis plays an essential role in reducing the maintenance costs of rotating machinery manufacturing systems. In many real applications of fault detection and diagnosis, data tend to be imbalanced, meaning that the number of samples for some fault ...
In this paper, the recommended implementation of the post-quantum key exchange SIKE for Cortex-M4 is attacked through power analysis with a single trace by clustering with the k-means algorithm the power samples of all the invocations of the elliptic curve ...
The usual explanation of the efficacy of wavelet-based methods hinges on the sparsity of many real-world objects in the wavelet domain. Yet, standard wavelet-shrinkage techniques for sparse reconstruction are not competitive in practice, one reason being t ...