Cascade algorithm

The iterative algorithm generates successive approximations to ψ(t) or φ(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

${\displaystyle \varphi ^{(k+1)}(t)=\sum _{n=0}^{N-1}h[n]{\sqrt {2}}\varphi ^{(k)}(2t-n)}$

For the kth iteration, where an initial φ⁽⁰⁾(t) must be given.

The frequency domain estimates of the basic scaling function is given by

${\displaystyle \Phi ^{(k+1)}(\omega )={\frac {1}{\sqrt {2}}}H\left({\frac {\omega }{2}}\right)\Phi ^{(k)}\left({\frac {\omega }{2}}\right)}$

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

${\displaystyle \Phi ^{(\infty )}(\omega )=\prod _{k=1}^{\infty }{\frac {1}{\sqrt {2}}}H\left({\frac {\omega }{2^{k}}}\right)\Phi ^{(\infty )}(0).}$

If such a limit exists, the spectrum of the scaling function is

${\displaystyle \Phi (\omega )=\prod _{k=1}^{\infty }{\frac {1}{\sqrt {2}}}H\left({\frac {\omega }{2^{k}}}\right)\Phi ^{(\infty )}(0)}$

The limit does not depends on the initial shape assume for φ⁽⁰⁾(t). This algorithm converges reliably to φ(t), even if it is discontinuous.

From this scaling function, the wavelet can be generated from

${\displaystyle \psi (t)=\sum _{n=-\infty }^{\infty }g[n]{\sqrt {2}}\varphi ^{(k)}(2t-n).}$

Successive approximation can also be derived in the frequency domain.

• C.S. Burrus, R.A. Gopinath, H. Guo, Introduction to Wavelets and Wavelet Transforms: A Primer, Prentice-Hall, 1988, ISBN 0-13-489600-9.
• http://cnx.org/content/m10486/latest/
• https://web.archive.org/web/20070615055323/http://cm.bell-labs.com/cm/ms/who/wim/cascade/index.html