Wavelets

Since w(x) may be complex valued, \w(x)\2 is used instead of w(x)2. The square of L2 norm, ||re||2, is called the energy of the function w. We may assume ||re|| = 1 by normalizing w by

The mother wavelet w satisfies the following condition, called the compatibility condition

where w is the Fourier transform of w. The compatibility con­ dition implies

if the integral exists. The sine function does not satisfy these conditions and sin(a;)

does not have finite L2 norm. Since sin2(a;) has period tt and

r% . 9/ , , r% 1 ^ cos 2a: , 7r sm (x)dx = dx

Jo Jo 2

it follows / OO sin2 (a:) dx = oo

The wavelets are generated from the analyzing wavelet by translation and dilation,

Wab(x) = A=W(~-----) \ /a a

where a is a positive real number and b is a real number. The factor is multiplied to preserve the L2 norm, that is, each wah has norm 1 if in has norm 1 ,

For example, w2,o is the dilation of w by the factor of 2, wi# is a translation of w by 3, and w2 , 3 is the dilation of w by the factor of 2 then translation by 3, The order of dilation then translation is important. If w(x) is translated by b and then dilated by a, it would be

w { - - b) a

which is equivalent to

dilation by a then translation by ab. A certain (sufficiently large) class of functions can be rep­

resented as linear combinations of the wavelets. That is, the functions are expressed as a finite linear combination of the di­ lation and translation of a single function re or as a limit of such finite linear combinations. Wavelet representations are more ef­ ficient than the traditional Fourier series representation in many situations, where the signal is nonstationarv, which means the

signal changes its behaviour (frequency) with time (or space) or has local singularities. Also, a wavelet is more flexible since we can choose the analyzing wavelet that is suitable for the signal being analyzed whereas Fourier representation has a fixed basis, namely the sine and cosine functions,

Let f ( x ) be any real valued function defined on the real line R, A new function f ab{x ) defined by

fab(x ) = f (~— -) Cl

where a ^ 0, is scaled (or dilated) by the factor of o, then shifted (translated) by b version of / , For example, the graph of f 2 ,o(x ) is the graph of f ( x ) horizontally stretched by a factor of 2, and the graph of fi , 2 (x ) is the graph of f ( x ) shifted horizontally to the right by 2,

Here the familar sine function is used to show the scaling and shifting. Let f ( t ) = sin t be a function of the independent variable t (say, time), and the graphs of f (x t ) = sin7rt, /(2-7rf) = An 2;:/. and /(3-7rf) = sin3-7rf are shown in Figure 2,1,

A function / is called periodic if there is a positive number T with the property that

f ( t + T) = f( t ) , V t e R ,

The smallest positive number T with this property is called the period of / ,

For example, sin t is of period 2tt and sin2-7rf is of period 1, In general, if f ( t ) is of period T then the dilation f a(t) = f ( t /a) is of period oT, Obviously, the period is invariant under translation,

sin ------ is of period 2o7r a

Figure 2,1: sin(7r;r), sin(27r;r), and sin(37r.i:).