Author details

6. Wavelet transform design procedure using filter banks

diagram in Figure 10 are necessary.

Wavelet Transform and Complexity

non-zero coefficient.

time and frequency domains.

wavelets.

of the chart.

7. Conclusion

effective design.

30

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

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

ii. Paraunitary condition which guarantees the generation of orthonormal

i. Compact support which guarantees that the wavelet is characterized by finite

iii. Flatness/k-regularity which guarantees the smoothness of the wavelet in both

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

In this chapter, we have presented an analysis of the design of wavelets using filter bank technique. The chapter looked at the two major components of a filter bank which the analysis and the synthesis components. The properties of filter banks which are desirable in the design of wavelets were also investigated, alongside the mathematical description of these properties. The chapter also gave a brief mathematical description of the role the analysis and the synthesis filter banks play in the design of wavelets. Finally, the required general procedure for the design of wavelets was presented, showing the necessary steps to take in order to achieve an

The major contribution of this chapter is the provision of a step by step analysis

and procedure for the design of filter banks in a precise and concise manner.

Peter Yusuf Dibal<sup>1</sup> \*, Elizabeth Onwuka<sup>1</sup> , James Agajo<sup>2</sup> and Caroline Alenoghena<sup>1</sup>

1 Telecommunication Engineering Department, Federal University of Technology Minna, Minna, Nigeria

2 Computer Engineering Department, Federal University of Technology Minna, Minna, Nigeria

\*Address all correspondence to: yoksa77@gmail

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
