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Time Response From Frequency Response
tor frequency stability calculations. Numerical differentiation will be discussed
in Section 4.7.2. It is the easiest way to compute time delay, but it trades
execution time for computer coding. Tellegen's theorem (Section 4.7.3) provides
a basis for computing exact time delay and its exact sensitivities, but this
particular application requires much more memory and coding. Therefore, it
is recommended that the angle be computed at a frequency perturbed (increased)
by 0.01% (a 1.0001 factor) and again at the desired frequency. The
difference between these two angles, in degrees, is used in the numerator of
the formula
Yl
(4.68)
which gives the delay in reciprocal frequency units. Suppose that the frequency
of interest-fa in (4.68)-is 50 MHz. According to Program B4-1 in
Section 4.1.4, the frequency units would be input as IE6; therefore, the time
delay would be in microseconds when calculated by (4.68). The only remaining
problem is the occasional 360-degree jump in the calculated angle that
might occur between the perturbed and desired frequencies. A simple program
test can prevent this.
4.6. TIme Response From Frequency Response
For most industrial engineers, there has been a gap between academic
concepts and applied design and analysis. Thi.s section uses a desktop computer
to close that gap for the Fourier and convolution integrals. A means for
rapid steady-state frequency analysis of ladder networks has been developed
that requires very little code and avoids most trivial calculations, such as
complex multiplication by zeros and ones. This makes practical a method of
numerical evaluation of the Fourier integral and, subsequently, numerical
evaluation of the convolution integral. This enables the conversion of a
system's band-limited frequency response to its impulse response in the time
domain. Then the convolution integral enables the response to any arbitrary
time excitation to be calculated in a reasonable amount of time, using a
desktop microcomputer. Complicated networks and dense time samples could
weaken this claim; the understanding of this process and its fundamental
simplicity may be reward enough for design engineers. After all, bigger and
faster computers are always available at some additional expense of time and
convenience.
This section begins with a review of the Fourier integral under the conditions
that the system impulse time response is a real function and causal, Le.,
cannot anticipate the excitation. Then Simpson's rule for numerical integration
will be applied, as previously discussed. Finally, the convolution integral
will also be evaluated by Simpson's rule according to a related general
I, formula. This material follows Ley (\970), and the program has been adapted
to BASIC language from the original FORTRAN.