njit-etd2003-081 - New Jersey Institute of Technology

68
This condition forces that the wavelet function is a band pass function and
decays at least as fast as |t|^1-ε in time where e is some decay constant. (In practice one
needs to have much faster decay of tit (t) , in order to have good time localization).
The admissibility condition assures that the "resolution of the identity" holds
[45]. This guarantees that any function f(t) E 1,2 (R^n) can be reconstructed from the
wavelet space as
where the wavelet coefficients were defined earlier in Eq. (3.63). Its mathematical proof
is given in the references [45][46]. Whenever ψ(t) is a real function, the integral limits
of Ch in Eq. (3.68) are changed from 0 to 00 .
Resolution of the identity ensures that the continuous wavelet transform (CWT)
is complete if W f (a,b) are known for all a and b. A continuous signal f(t) is
represented by a pass band function ψ(t) and its dilated and translated versions. The
dilation in time leads to different resolutions in frequency. ψ(t) is a prototype window
and is normally referred to as the mother wavelet. The digital implementation of the
CWT can be computed directly by convolving the signal with a scaled and dilated
version of the mother wavelet. For a reasonably efficient implementation, an FFT may
be applied to perform the convolution [46a].
In its discrete form, a = ao^j -=n 0-3where j • and n za
are integers. This is
referred to as the discrete wavelet transform (DWT), which is discussed in more detail in