6. Wavelet transform design procedure using filter banks

In the design of a wavelet system using filter banks, it is of utmost importance that the filters which will execute the filter bank system as shown in Figure 1, possess the properties discussed in Section 4. Owing to the fact that in a filter bank, all the filters can be derived from an initial filter H<sup>0</sup> as described in Eq. (13), then this initial filter must be designed in such a manner that the relationships in Sections 5.1 and 5.2 are realized. To this end, the following steps as shown in the state diagram in Figure 10 are necessary.

In the first state in Figure 10, the design problem formulation which can be achieved using trigonometric polynomial, takes the following into consideration [14]:


The second state which involves conditioning the problem as a tractable problem involves, if necessary, transforming a non-linear formulation of the problem to a linear formulation, and then optimizing the problem using techniques like convex optimization. The generation of the filter coefficients using solvers in the third state of the machine involves techniques like spectral factorization. Through simulation in the fourth state of the chart, the generated coefficients can be verified whether or not they meet the design constraints. Using the QMF or CQF relationships in Eqs. (13) and (14), the other filters in the filter bank are generated in the fifth state of the chart.
