Cascade Algorithm - Successive Approximation

Successive Approximation

The iterative algorithm generate 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

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

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

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

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

Plots of the function at each iteration is shown in Figure 1.

Successive approximation can also be derived in the frequency domain.

Read more about this topic:  Cascade Algorithm

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