**6. Conclusion**

Any original data or signal can be represented in terms of a wavelet expansion. The best representation of a data using a wavelet depends on the best or close wavelet of what we are choosing. There are many numbers of wavelets available as per the literature. Some of the examples of the wavelets are Haar and Daubechies [3]; under Gaussian-based wavelets, we have Mexican hat wavelet and Morlet wavelet; under polynomial-based wavelets, we have Battle-lemarie, Coiflet and Spline-based wavelets; and under Sinc wavelets, we have Meyer wavelet and Shannon

From the previous understanding, it is clear that CWT is a redundant transform, which means that the translation parameter 'b' and scaling parameter 'a' seem to be infinite making them difficult in terms of implementation. It is always seems to be CWT that is computable but not implementable. The solution for the implementation of wavelet transform arises from discrete wavelet transform (DWT). Sampling in the time-frequency plane on a dyadic (octave) grid is happening in DWT that makes them efficient in terms of implementation. The scaling parameter 'a' is replaced by 2−j and 'b' is made proportional to 'a', i.e., b = k 2−j. Here 'j' is called as scaling parameter and 'k' is the proportionality constant taking

> *<sup>ψ</sup>*(*a*, *b*) = \_\_\_1 √ \_\_\_ <sup>|</sup>*a*<sup>|</sup> <sup>∫</sup> *t*

In multiresolution analysis, the signal can be viewed as the sum of a smooth ("coarse") part—reflects main features of the signal (approximation signal) and a detailed ("fine") part—faster fluctuations represent the details of the signal [1]. The separation of the signal

2*<sup>j</sup> ψ*(2*<sup>j</sup> t* − *k*) *j*, *k* ∈ *Z*

; *b* = 2<sup>−</sup>*<sup>j</sup>*

*<sup>x</sup>*(*t*)*ψ*(\_\_\_ *<sup>t</sup>* <sup>−</sup> *<sup>b</sup>*

*<sup>a</sup>* )*dt* 

*k* (j and k are integers) in

(6)

wavelet.

10 Wavelet Theory and Its Applications

Eq. (5), we get Eq. (6).

**5. Discrete wavelet transform (DWT)**

the role of shifting parameter in DWT. Substituting *a* = 2<sup>−</sup>*<sup>j</sup>*

(*t*) = √

**Figure 9.** Filter bank implementation of DWT (courtesy by Robi polikar).

*<sup>ψ</sup>*(*a*, *b*) = Ψ*<sup>x</sup>*

\_\_

*CWTx*

*ψj*,*k*

I hope that this chapter gives a definite and thoughtful introduction to all the beginners who are new to wavelets. As there are different number of wavelets available with different signal processing properties like compact support, symmetry, regularity and vanishing moments make them suitable in the field of signal de-noising, detecting discontinuities and breakdown points in a signal, compressing images, identifying pure frequencies, seismic and geophysical signal processing, video compression, acoustic data analysis, nuclear engineering, neurophysiology, music, magnetic resonance imaging, speech discrimination, optics, fractals, turbulence, earthquake-prediction, radar, human vision, etc. Some of applications from the perspective scientists and researchers are discussed in the forthcoming chapters.

#### **Author details**

Sudhakar Radhakrishnan

Address all correspondence to: sudha\_radha2000@yahoo.co.in

Department of Electronics and Communication Engineering, Dr. Mahalingam College of Engineering and Technology, Pollachi, India
