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30 Some Fundamental Numerical Methods
2.53. Summary 01 Compkx Curve Filting by RalionJJl Polynomials. Levy's
method for fitting complex data at sampled frequencies in the weighted
least-squared-error sense is straightforward. The weighting versus frequency is
inversely proportional to the value of the rational polynomial thus found.
Since the polynomial should roughly correspond to data values to be of any
use, nonuniform samples versus frequency should produce emphasis on the
frequencies where the magnitude of the complex data is least. For example, if
a lowpass function is to be fitted over several decades, then the fit to the
passband (lower) frequency data may be poor unless samples are spaced more
closely in this frequency range. Proposed iterative schemes have been based
on a sequence of solutions similar to those presented here; they tend to
converge to a situation equivalent to uniform weighting (see Jong and
Shanmugam, 1977, and Sanathanan and Koerner, 1963). However, equal
weighting may still require some experimenting. Thus the built-in inverse
weighting does not seem too severe a limitation.
The method requires the user to input real and imaginary data parts, with
the associated frequency, into Program 82-4 to obtain coefficients for a
system of linear equations. The system's matrix elements are partitioned from
a general matrix format (Figure 2.10) according to the approximating rational
polynomial's numerator and denominator degrees. The system of linear equations
is then solved by Gauss-Jordan Program 82-1 or by any other program
that solves linear systems of real equations. This method will play an important
role in Carlin's broadband impedance-matching technique in Section 6.7.
Problems
2.1. If
Z-Z
p(Z)= Z+Z: ' ,
where Z,=2+j3, find p when Z=3-j5.
2.2. Given that V= V, + jV; and I = I, + jI;, show that
2.3. Show that IZj2 = ZZ'.
2.4. Show that 2Re(Z)=Z+Z'.
2.5. If
a= V+ZI b= V-Z'I and Z=R+jX,
2.fR ' 2.fR'
show that lal'-Ib['= Re(IV').