**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 *k*th 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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### Famous quotes containing the word successive:

“I think, for the rest of my life, I shall refrain from looking up things. It is the most ravenous time-snatcher I know. You pull one book from the shelf, which carries a hint or a reference that sends you posthaste to another book, and that to *successive* others. It is incredible, the number of books you hopefully open and disappointedly close, only to take down another with the same result.”

—Carolyn Wells (1862–1942)