
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the continuous wavelet transform (CWT) is a formal (i.e., non-numerical) tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the
wavelet
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 n ...
s vary continuously.
Definition
The continuous wavelet transform of a function
at a scale
and translational value
is expressed by the following integral
:
where
is a continuous function in both the time domain and the frequency domain called the mother wavelet and the overline represents operation of
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - ...
. The main purpose of the mother wavelet is to provide a source function to generate the daughter wavelets which are simply the translated and scaled versions of the mother wavelet. To recover the original signal
, the first inverse continuous wavelet transform can be exploited.
:
is the
dual function of
and
:
is admissible constant, where hat means Fourier transform operator. Sometimes,
, then the admissible constant becomes
:
Traditionally, this constant is called wavelet admissible constant. A wavelet whose admissible constant satisfies
: