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Sensitivities 101
the corresponding impulse response time samples. Finally, the numerical
convolution integral can be evaluated Jar arbitrary time functions specified at
matching time samples. The memory requirements for practical problems
usually fit easily into desktop computers having 8-32 kilobytes of randomaccess
memory.
The fast Fourier transform (FFT) must be mentioned before leaving this
topic. It is clearly superior to the above and is available as standard software
from several desktop computer manufacturers. Serious users of the frequencyto-time-domain
transform should consider special programs built around this
technique, which are more efficient by at least an order of magnitude. It was
not described because of its computational complexity.
4.7. Sensitivities
Sensitivity quantifies the relative change in a response function (Z) with
respect to a relative change in anyone of several independent variables; i.e.,
(4.80)
for small changes in the klh variable x,. Often, Z is complex, and is evaluated
at some given frequency. In this case, the sensitivity is also a complex number.
For example, Z might be a ladder network input impedance, and x, might be
an inductance value in microhenrys. Alternatively, Z might be a time function
evaluated at some given time. Hopefully, a system being built will have
sensitivities with magnitudes less than unity, otherwise it might react badly to
component tolerances and to its environment.
For each response, there are as many sensitivity numbers at a frequency or
time value as there are variables in the problem. Applications include component
tolerances, optimization (Chapter Five), and large-change calculations,
e.g., network tuning. This section further defines real and complex sensitivities,
relates them to partial derivatives, shows ways to obtain partial derivatives
approximately by finite differences and exactly by Tellegen's theorem,
and provides several examples. Most of the discussion is limited to the
frequency domain, as justified in Section 4.7.4. Programs A2-1 and B4·1 will
he used for calculations.
4.7.1. Sensitivity Relationships. The partial derivative operator abbreviation
(4.81 )
will be used throughout. As the change in the variable, ~Xk'
(4.80) approaches the common sensitivity definition
Sz= x,~
)(k
zax k
'
approaches zero,
(4.82)