njit-etd2003-081 - New Jersey Institute of Technology

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3.7.1 Continuous Wavelet Transform
The continuous wavelet transform maps a function f (t) onto time-scale space as
This operation can be expressed in a simpler inner product notation as
ψab (t) represents a family of functions obtained from a single wavelet function ψ (t) and
the dilation and translation parameters a and b as
where a and b are continuous.
The wavelet function ψ(t) is a band-pass function. It is desired that this function
have a good time and frequency localization so that f (t) is decomposed into elementary
building blocks which are jointly well localized in time and frequency. The wavelet
function has to satisfy the "admissibility" condition that makes it an isometry (up to a
constant) of I! (R) onto 1,2 (R x R) . This requirement limits the wavelet functions which
must satisfy [45][46]
where ψ(Ω) is the Fourier transform of the wavelet function ψ(t). The admissibility
condition is directly related to the decay of the wavelet function ψ(t) which is required
to have good localization. The admissibility condition for a continuous ψ(Ω) is
equivalent to a zero-mean wavelet function in time: