3.2 MATHEMATICAL APPROXIMATIONS

The ideal responses discussed in Section 3.1 are unrealizable, hence cannot be achieved in practice. They are presented to illustrate the expected tradeoffs as these responses are approximated by realizable functions. Each ideal function has specific properties that we attempt to preserve by the approximation. These desired qualities usually dictate the "best" approximation technique.

Approximation theory is a specialty, the subject of numerous theoretical and practical studies. Because the derived equations often do not allow explicit solutions to be obtained, this task is performed on a digital computer by suitable search algorithms. Many of the sophisticated filter responses could not have been realized without the digital computer. It offers so many computational benefits that it is now a necessary ingredient of most approximation procedures.

Here we briefly examine three useful and popular approximations-the Taylor, the Chebyshev, and the least squares. The complexity of the resulting equations verifies that the digital computer is indispensable for carrying out the necessary computations, except for the low-order approximations.

The function to be approximated is represented by g( x), the approximating function is represented by f( x), and the interval of interest is ? x = x ₂ - x ₁. The error function e( x) is the difference of these functions,

(3.2-1)

The usual procedure is to form an error criterion in terms of e( x) and then adjust the parameters of f( x) to satisfy this criterion.