0.5 The scaling function and scaling coefficients, wavelet and  (Page 11/13)

Matlab programs that calculate $h\left(n\right)$ for $N=2,4,6$ are furnished in Appendix C . They calculate $h\left(n\right)$ from $\alpha$ and $\beta$ according to [link] , [link] , and [link] . They also work backwards to calculate $\alpha$ and $\beta$ from allowable $h\left(n\right)$ using [link] . A program is also included that calculates the Daubechies coefficients for anylength using the spectral factorization techniques in [link] and Chapter: Regularity, Moments, and Wavelet System Design of this book.

Longer $h\left(n\right)$ sequences are more difficult to parameterize but can be done with the techniques of Pollen [link] and Wells [link] or the lattice factorization by Vaidyanathan [link] developed in Chapter: Filter Banks and Transmultiplexers . Selesnick derived explicit formulas for $N=8$ using the symbolic software system, Maple, and set up the formulation forlonger lengths [link] . It is over the space of these independent parameters that one can findoptimal wavelets for a particular problem or class of signals [link] , [link] .

Calculating the basic scaling function and wavelet

Although one never explicitly uses the scaling function or wavelet (one uses the scaling and wavelet coefficients) in most practicalapplications, it is enlightening to consider methods to calculate $\phi \left(t\right)$ and $\psi \left(t\right)$ . There are two approaches that we will discuss. The first is a form of successive approximations that is used theoreticallyto prove existence and uniqueness of $\phi \left(t\right)$ and can also be used to actually calculate them. This can be done in the time domain to find $\phi \left(t\right)$ or in the frequency domain to find the Fourier transform of $\phi \left(t\right)$ which is denoted $\text{Φ}\left(\omega \right)$ . The second method solves for the exact values of $\phi \left(t\right)$ on the integers by solving a set of simultaneous equations. From these values, it is possible to then exactlycalculate values at the half integers, then at the quarter integers and so on, giving values of $\phi \left(t\right)$ on what are called the dyadic rationals.

Successive approximations or the cascade algorithm

In order to solve the basic recursion equation [link] , we propose an iterative algorithm that will generate successive approximations to $\phi \left(t\right)$ . If the algorithm converges to a fixed point, then that fixed point is a solution to [link] . The iterations are defined by

for the $k^{th}$ iteration where an initial ${\phi }^{\left(0\right)}\left(t\right)$ must be given. Because this can be viewed as applying the same operation overand over to the output of the previous application, it is sometimes called the cascade algorithm .

Using definitions [link] and [link] , the frequency domain form becomes

and the limit can be written as an infinite product in the form

If this limit exists, the Fourier transform of the scaling function is

The limit does not depend on the shape of the initial ${\phi }^{\left(0\right)}\left(t\right)$ , but only on ${\text{Φ}}^{\left(k\right)}\left(0\right)=\int {\phi }^{\left(k\right)}\left(t\right)dt=A_0$ , which is invariant over the iterations. This only makes sense if the limit of $\text{Φ}\left(\omega \right)$ is well-defined as when it is continuous at $\omega =0$ .

The Matlab program in Appendix C implements the algorithm in [link] which converges reliably to $\phi \left(t\right)$ , even when it is very discontinuous. From this scaling function, the wavelet can be generated from [link] . It is interesting to try this algorithm, plotting the function at each iteration, on both admissible $h\left(n\right)$ that satisfy [link] and [link] and on inadmissible $h\left(n\right)$ . The calculation of a scaling function for $N=4$ is shown at each iteration in [link] .