**7**

OTHER WAVELET TOPICS

This chapter contains a variety of topics of a more advanced nature that relate to wavelets. Since these topics are more advanced, an overview will be presented with details left to the various references.

**7.1 COMPUTATIONAL COMPLEXITY**

**7.1.1 Wavelet Algorithm**

In this section we briefly discuss the number of operations required to compute the wavelet decomposition of a signal (the computational complexity of the wavelet decomposition algorithm).

To take a concrete setting, suppose *f* is a continuous signal defined on the unit interval 0 ≤ *t ≤* 1. Let

be a discrete sampling of ƒ. We wish to count the number of multiplications in the decomposition algorithm in terms of the number *N = 2 ^{n}.*

The decomposition algorithm stated in Eq. (5.12) or Eq. (5.17) computes the *a ^{j-i}* and

*b*1, using the formulas

^{j–i}, j = n,...,Note that there are half as many a^{j-i} coefficients as there are *a ^{j}* coefficients. So the index

*l*on the left runs from 0 to 2

^{j–}

This result is often summarized by stating that the wavelet decomposition algorithm requires *O(N)* multiplication operations where *N* is the number of data at the top level. Here, *O(N)* stands for a number that is proportional to *N.* The ...