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

and assume $h\left(n\right)\ne 0$ for $0\le n\le N-1$ .

This is the refinement matrix illustrated in [link] for $N=6$ which we write in matrix form as

In other words, the vector of $\phi \left(k\right)$ is the eigenvector of ${\mathbf{M}}_0$ for an eigenvalue of unity. The simple sum of ${\sum }_{n}h\left(n\right)=\sqrt{2}$ in [link] does not guarantee that $M_0$ always has such an eigenvalue, but ${\sum }_{n}h\left(2n\right)={\sum }_{n}h(2n+1)$ in [link] does guarantee a unity eigenvalue. This means that if [link] is not satisfied, $\phi \left(t\right)$ is not defined on the dyadic rationals and is, therefore, probably not a very nice signal.

Our problem is to now find that eigenvector. Note from [link] that $\phi \left(0\right)=\phi (N-1)=0$ or $h\left(0\right)=h(N-1)=1/\sqrt{2}$ . For the Haar wavelet system, the second is true but for longer systems, this would mean all the other $h\left(n\right)$ would have to be zero because of [link] and that is not only not interesting, it produces a very poorly behaved $\phi \left(t\right)$ . Therefore, the scaling function with $N>2$ and compact support will always be zero on the extremes of the support. Thismeans that we can look for the eigenvector of the smaller 4 by 4 matrix obtained by eliminating the first and last rows and columns of $M_0$ .

From [link] we form $[M_0-I]\underline{\phi }=\underline{0}$ which shows that $[M_0-I]$ is singular, meaning its rows are not independent. We remove the last row and assume the remaining rows are nowindependent. If that is not true, we remove another row. We next replace that row with a row of ones in order to implement the normalizing equation

This augmented matrix, $[M_0-I]$ with a row replaced by a row of ones, when multiplied by $\underline{\phi }$ gives a vector of all zeros except for a one in the position of the replaced row. This equationshould not be singular and is solved for $\underline{\phi }$ which gives $\phi \left(k\right)$ , the scaling function evaluated at the integers.

From these values of $\phi \left(t\right)$ on the integers, we can find the values at the half integers using the recursive equation [link] or a modified form

This is illustrated with the matrix equation [link] as

Here, the first and last columns and last row are not needed (because ${\phi }_0={\phi }_5={\phi }_{11/2}=0$ ) and can be eliminated to save some arithmetic.

The procedure described here can be repeated to find a matrix that when multiplied by a vector of the scaling function evaluated at the oddintegers divided by $k$ will give the values at the odd integers divided by $2k$ . This modified matrix corresponds to convolving the samples of $\phi \left(t\right)$ by an up-sampled $h\left(n\right)$ . Again, convolution combined with up- and down-sampling is the basis of wavelet calculations. It is also thebasis of digital filter bank theory. [link] illustrates the dyadic expansion calculation of a Daubechies scaling function for $N=4$ at each iteration of this method.

Not only does this dyadic expansion give an explicit method for finding the exact values of $\phi \left(t\right)$ of the dyadic rationals ( $t=k/2^j$ ), but it shows how the eigenvalues of $\mathbf{M}$ say something about the $\phi \left(t\right)$ . Clearly, if $\phi \left(t\right)$ is continuous, it says everything.

Matlab programs are included in Appendix C to implement the successive approximation and dyadic expansion approaches to evaluating the scaling function from the scaling coefficients. They were used to generate thefigures in this section. It is very illuminating to experiment with different $h\left(n\right)$ and observe the effects on $\phi \left(t\right)$ and $\psi \left(t\right)$ .