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Network Objective Functions 151
and frequency. The second integrand might represent the difference between a
response function (R) and the goal function (G).
Since integration on digital computers is discrete anyhow, the measure of
goodness of fit can be a process of frequency sampling. The Euclidean norm
(inner product) mentioned in (5.82) applies here as well:
2 2 1/2
IIEII=(el+e~+.. ·+eN) . (5.87)
This might correspond to sampling at the ith frequency, where ej is the
difference between the response and the goal. The next section combines these
concepts in a form convenient for optimizing network response functions
sampled at several frequencies or times.
5.5.2. Discrete Objective Functions. A typical discrete objective function for
network response is shown in Figure 5.25, as described mathematically by
M
E(x,w)= 2: W;(R,-Gj)P, (5.88)
i=l
where P is an even integer (the Pth difference), R, is the response, G j is the
goal, and Wj is the weight factor at the ith frequency. None of these quantities
are complex. For example, if a network is to be adjusted so that an impedance
approximates some given impedance values at various frequencies, then an
approximate response might be SWR, according to (4.59) and (4.54). Compare
(5.88), with P=2 and Wj= I, to (5.87). Also, (5.88) may be generalized to
account for more than one kind of response, R;k' by adding a second, nested
summation on k. Two responses might then be'SWR and voltage, where the
weights W, k must equalize the scales for the two different kinds of responses.
In practic~, only very few kinds of responses are successfully considered
simultaneously, and there is a good chance for a standoff (over constraint), so
that optimization is ineffective.
A "satisfied-when-exceeded" feature can be included in a program for
(5.88), so that W,=O is employed whenever R,>G,. This feature is useful
Least Pth:
Response
Figure 5.25.
Least·Pth error function with weighted frequency samples.
w