Morlet wavelet

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. 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. 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. (See also Wavelet history) The wavelet is defined as a constant κ σ {displaystyle kappa _{sigma }} subtracted from a plane wave and then localised by a Gaussian window: where κ σ = e − 1 2 σ 2 {displaystyle kappa _{sigma }=e^{-{frac {1}{2}}sigma ^{2}}} is defined by the admissibility criterion,and the normalisation constant c σ {displaystyle c_{sigma }} is: The Fourier transform of the Morlet wavelet is: The 'central frequency' ω Ψ {displaystyle omega _{Psi }} is the position of the global maximum of Ψ ^ σ ( ω ) {displaystyle {hat {Psi }}_{sigma }(omega )} which, in this case, is given by the positive solution to: which can be solved by a fixed-point iteration starting at ω Ψ = σ {displaystyle omega _{Psi }=sigma } (the fixed point iterations converge to the unique positive solution for any initial ω Ψ > 0 {displaystyle omega _{Psi }>0} ). The parameter σ {displaystyle sigma } in the Morlet wavelet allows trade betweentime and frequency resolutions. Conventionally, the restriction σ > 5 {displaystyle sigma >5} is used to avoid problems with the Morlet wavelet at low σ {displaystyle sigma } (high temporal resolution). For signals containing only slowly varying frequency and amplitudemodulations (audio, for example) it is not necessary to use smallvalues of σ {displaystyle sigma } . In this case, κ σ {displaystyle kappa _{sigma }} becomes very small (e.g. σ > 5 ⇒ κ σ < 10 − 5 {displaystyle sigma >5quad Rightarrow quad kappa _{sigma }<10^{-5},} ) and is, therefore, often neglected. Under the restriction σ > 5 {displaystyle sigma >5} , the frequency of the Morlet wavelet is conventionally taken to be ω Ψ ≃ σ {displaystyle omega _{Psi }simeq sigma } .