In mathematics, 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 wavelets vary continuously.

In simple terms, the Continuous Wavelet Transform is an analysis tool similar to the Fourier Transform, in that it takes a time-domain signal and returns the signal’s components in the frequency domain.

In this article, the continuous wavelet transform is introduced as a signal processing tool for investigating time-varying frequency spectrum characteristics of nonstationary signals.

In this article, we will first define the continuous wavelet transform and then the orthogonal wavelet transform based on a multiresolution analysis. Properties of both transforms will be discussed and illustrated by examples.

Definition of the Continuous Wavelet Transform Like the Fourier transform, the continuous wavelet transform (CWT) uses inner products to measure the similarity between a signal and an analyzing function.

Continuous Wavelet Transform. In the present (Hilbert space) setting, we can now easily define the continuous wavelet transform in terms of its signal basis set:

I Continuous Wavelet Transform W (s,⌧)= Z 1 1 f(t) ⇤ s,⌧ dt = hf(t), s,⌧ i I Transforms a continuous function of one variable into a continuous function of two variables : translation and scale I For a compact representation, we can choose a mother wavelet (t) that matches the signal shape I Inverse Wavelet Transform f(t)= Z 1 1 Z 1 1 W ...

2. The continuous wavelet transform The wavelet analysis described in the introduction is known as the continuous wavelet transform or CWT. More formally it is written as: (s, ) f (t) s (t)dt * γ τ=∫ψ,τ) 1 , (where * denotes complex conjugation. This equation shows how a function ƒ (t) is decomposed into a set of basis

wt = cwt(x) returns the continuous wavelet transform (CWT) of x. The CWT is obtained using the analytic Morse wavelet with the symmetry parameter, gamma (γ), equal to 3 and the time-bandwidth product equal to 60. cwt uses 10 voices per octave.

The Continuous Wavelet Transform (CWT) is an analog filterin g function and is similar to what is known as the Gabor spectrogram [13]. Similarly to the Discrete Wavelet Transform, it