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26 Some Fundamental Numerical Methods
The error criterion will be the weighted least-squared-error (LSE) function
over the frequency samples 0, 1, ...,m:
where
m
E= L [Wdekl]z,
k=O
(2.48)
(2.49)
The complex numbers F(w k ) are the given data to be fitted, i.e., the target
function. The complex approximating function Z(wk) is given in (2.47). The
Wk values in (2.48) are the weighting values at each frequency wk' The
necessary condition for a minimum value of E (generally not zero) is that the
partial derivatives of E with respect to the coefficients ~,al"'" a p1 b l1
bz,"" b n
in (2.47) be equal to zero. A set of simultaneous nonlinear equations
will result if the formulation in (2.48) and (2.49) is used with independent
weights Wk' The equations are badly conditioned and extremely difficult to
solve. Gradient optimizers (Chapter Five) usually are not successful in finding
a solution (according to Jong and Shanmugam, 1977).
E. C. Levy's method will be described. It employs a weighted LSE objective
function similar to (2.48), except that the weights are dependent functions.
This produces a system of simultaneous linear equations that are readily
solved by the Gauss-Jordan program described in Section 2.2. The derivation
will be outlined, the matrix of linear equation coefficients will be tabulated,
and a brief BASIC language program will be furnished to calculate the four
kinds of matrix coefficients. An example will be provided here, and others will
be given in Section 6.7.
2.5.1. The Basis of Levy's Complex Curve-Fitting Method. The definition of
Z(s) in (2.47) is expanded, with s= jw, to produce a set of linear equations:
(a o -a Z
w z +a 4 U>4+ .. , )+jU>(al-a3wz+asw4+ ... )
Z(s) = Z 4 Z 4 ,(2.50)
(1- bzw + b 4 w + ... )+jW(b, - b,w + bsw + ... )
which is further defined by
Z(s) ~ a +jwf3 = N(w) (2.51)
O+jWT D(w)'
The real terms in the numerator and denominator of (2.50) are even functions
of frequency, and the imaginary terms are odd. Quantities in parentheses are
equated by relative position witb the variables appearing in (2.51), where the
numerator and denominator functions are also identified.
With these definitions, the unweighted error function in (2.49) becomes
N(w)
e(w)=F(w)- D(w)'
(2.52)