njit-etd2003-081 - New Jersey Institute of Technology

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section 3.7.3. If one chooses a 0≈1 and n≈
0 , one is close to the continuous case. The
choice of a = 2' and r = n • is the most common choice; this is referred to as a dyadic
wavelet basis. It allows much greater computational efficiency than the CWT; fast
algorithms have been developed that are 0(n) . The dyadic form also simplifies the
constraints on the mother wavelet to achieve another desirable property: orthogonality of
the analysis windows, which subtends an efficient representation. The dyadic DWT is
almost always the chosen implementation because of its computational efficiency and
the mutual orthogonality of its analysis windows (or subbands), and it will heretofore be
referred to as simply the wavelet transform (WT) unless otherwise specified.
Because the mother wavelet may take many forms, the term scale is often
preferred to frequency when using the WT. An analogy between scale and frequency is
useful however, at this point, to illustrate the difference between the STFT and the WT.
Consider a mother wavelet chosen to be
where g(t) is the STFT analysis window, and f0 is the fundamental frequency in the
STFT analysis. This implies that the frequency analogous to the WT scale is
f = af0 = 2 1 f0 . As the scale is increased, however, the frequency resolution (or
equivalently, the bandwidth) of the orthogonal WT analysis window is proportional to
its center frequency: