and ψj,k(t) are scaled and translated versions of φ and ψ. As equation (3) indicates, the scale
or dilation factor is 2j, whereas the translation or location parameter is 2jk. As j gets larger,
so does the scale factor 2j, and the functions φj,k(t) and ψj,k(t) get shorter and more spread
out. In other words, 2j is a measure of the width of the functions φj,k(t) and ψj,k(t). Likewise,
as j increases, the translation step gets correspondingly larger in order to match the scale
parameter 2j.
Applications of wavelet analysis commonly make use of a discrete wavelet
transform (DWT). The DWT calculates the coefficients of the approximation in (2) for a
discrete signal of final extent, f1, f2,.., fn. That is, it maps the vector f=(f1, f2,...,fn)' to a
vector ω of n wavelet coefficients that contains sJ,k and dj,k, j=1,2,..., J. The sJ,k are called
the smooth coefficients and the dj,k are called the detail coefficients. Intuitively, the smooth
coefficients represent the underlying smooth behavior of the data at the coarse scale 2J,
whereas the detail coefficients provide the coarse scale deviations from it.
When the length of the data n is divisible by 2J, there are n/2 coefficients d1,k at the
finest scale 21=2. At the next finest scale, there are n/22 coefficients d2,k. Similarly, at the
coarsest scale, there are n/2J dJ,k coefficients and n/2J sJ,k coefficients. Altogether, there are
J
n ∑ ■
— 2i 2j
∖^ i=1 y y
= n coefficients. The number of coefficients at a given scale is related to the
width of the wavelet function. For instance, at the finest scale, it takes n/2 terms for the
functions ψ1,k(t) to cover the interval 1≤t≤n.
The wavelet coefficients are ordered from coarse scales to fine scales in the vector
ω. If n is divisible by 2J, ω will be given by