A Scheme for Teaching Wavelets at the Introductory Level

Wavelet analysis and synthesis are becoming highly important in the area of signal processing. Therefore, it is felt that undergraduate students should be exposed to the rudiments of this treatment before they graduate. A scheme for an elementary coverage of this topic with required minimum amount of mathematics is discussed in this paper. 1.Introduction One of the major Signal Processing tools is the Fourier Analysis, which has different forms like Fourier Series, Fourier Transform, Discrete Fourier Transform, and so on. Fourier analysis methods use orthogonal sets of functions in order to expand a given periodic function into an infinite series. While a large number of orthogonal sets exists, the Fourier Analysis form which is commonly utilized makes use of the cosine and sine functions. However, for non-periodic time-varying signals, the above mentioned method has several inconveniences. To a certain extent, these could be overcome by the application of different windowing techniques leading to the Short-Time-Fourier Transforms. But these have also their own limitations. An alternative method is to use wavelets. The Wavelet Analysis differs from the Fourier Analysis in several aspects. Naturally their use requires different skills. So it is the opinion of the authors that it is important to expose students to the concept of wavelets and to their uses in signal analysis early and gradually in the curriculum. It is possible to teach the undergraduate students the basis of wavelets analysis without all the mathematical considerations and proofs which are usually found in texts written for graduate students and researchers. This paper proposes a simple approach in order to introduce wavelet analysis to electrical engineering students at the introductory level. Wavelets can be considered from different points of views [1-9], one of which is the filterbanks. This is possibly one of the easiest and most 'natural' approaches to wavelets for these students. The paper is arranged as follows : Sections 2 to 4 contain the key ideas which in our opinion should be conveyed to the students. Section 2 consists of a short theoretical introduction to the concept of continuous wavelet transform which can be interpreted as a generalization of the Short time Fourier Transform. Sections 3 and 4 link this concept with filter-banks along with decimation for analysis and interpolation for synthesis. The link is mostly heuristic in its approach in order to keep the mathematical treatment to a minimum. Section 5 contains curriculum considerations. Section 6 contains the summary of the paper. 2. Mathematical background A general transform equation can be written as : X a b x t t dt a b ( , ) ( ) ( ) , = −∞ ∞ ∫ ψ (1) The signal to be processed is given as x(t). Depending on the function ψa,b(t) chosen, one gets different transforms. In order to evaluate the integral, certain conditions have to be imposed on this function. In this paper, we shall consider only three such transforms and these are the Fourier Transform, Short-Time Fourier Transform and the Wavelet Transform. a) Fourier Transform : The function ψa,b(t) is given by : ψ π a b j f t t e , ( ) = − 2 (2) where f is the frequency, and the length of the window is infinite since the basis functions are the sine and the cosine functions. b) Short-Time Fourier Transform : The function ψa,b(t) is given by : ψ ψ π a b j at t e t b , ( ) ( ) = − − 2 (3) Where a = f is the frequency , ψ(τ) is a window of finite width, and b is the window translation parameter. This means that ψ(t-b) is a time-shifted version of ψ(t) and is of finite duration. c) Wavelet Transform : The function ψa,b(t) is given by : ψ ψ a b t a t b a , ( ) = −     1 (4) where ψa,b(t) is a window of finite length. The quantity b is the window translation parameter and is a real number; a is the dilation or contraction parameter and is a positive real number. This can be interpreted as follows : ψ(t-b) by itself is a time-shifted window and is of finite duration. By introducing the factor a as shown by (4) , the resulting window can be made larger or smaller than the starting window. In the special case where a = 1 , it will be the same as the starting window. Because of the large number of possibilities, ψa,b(t) as defined in Eq. (4) is called the mother wavelet. When (4) is replaced in (1), the product ( ) x t t b a ( ) ( ) / ψ − appears and when it is integrated , we get the continuous wavelet transform which can be viewed as the area under the curve representing this product. Thus the continuous wavelet transform (CWT) can be written as: X a b x t t dt x t t b a dt W a b ( , ) ( ) ( ) ( ) , = = −     −∞ ∞