Cascade algorithm

An interesting numeric method for calculating the basic scaling function or wavelets uses an iterative algorithm, which computes wavelet coefficients at one scale from those at another. Because it applies the same operation over and over to the output of the previous application, it is known as the cascade algorithm.

Successive approximation

The iterative algorithm generate successive approximations to $\psi (t)$ or $\phi (t)$ from {h} and {g} filter coefficients. If the algorithm converges to a fixed point, then that fixed point is the basic scaling function or wavelet.
The iterations are defined by
$\phi ^{(k+1)}(t)=\sum _{n=0}^{N-1}h[n]{\sqrt {2}}\phi ^{(k)}(2t-n)$
For the $k^{th}$ iteration, where an initial $\phi ^{(0)}(t)$ must be given.
The frequency domain estimates of the basic scaling function is given by
$\Phi ^{(k+1)}(\omega )={\frac {1}{\sqrt {2}}}H({\frac {\omega }{2}}).\Phi ^{(k)}({\frac {\omega }{2}})$ and the limit can be viewed as an infinite product in the form
$\Phi ^{(\infty )}(\omega )=\prod _{k=1}^{\infty }{\frac {1}{\sqrt {2}}}H({\frac {\omega }{2^{k}}}).\Phi ^{(\infty )}(0)$
If such a limit exists, the spectrum of the scaling function is $\Phi (\omega )=\prod _{k=1}^{\infty }{\frac {1}{\sqrt {2}}}H({\frac {\omega }{2^{k}}}).\Phi ^{(\infty )}(0)$
The limit does not depends on the initial shape assume for $\phi ^{(0)}(t)$ . This algorithm converges reliably to $\phi (t)$ , even if it is discontinuous.
From this scaling function, the wavelet can be generated from $\psi (t)=\sum _{-\infty }^{\infty }g[n]{\sqrt {2}}\phi ^{(k)}(2t-n)$
Plots of the function at each iteration is shown in Figure 1.
Successive approximation can also be derived in the frequency domain.
Figure 1. Iterations of the sucessive approximations of Daubechies wavelet and scal function by the cascade algorithm.