Poles and Zeros: book recomendation

[Frank Miles]

What you don't know (and it took me awhile to realize myself, long ago) is that sheer
laziness has prevented the designer from saying that the pole is really at i(1 kHz),
on the imaginary axis [or -i(1 kHz) depending on your Fourier transform sign
convention].*

Of course, "a" (single) pole must be on the negative real axis, not imaginary.

Well, that of course is true in the Laplace transform, which is sort of an
intermediate between the +i and -i Fourier transforms. In that case,
though, you
can't see the real axis on the spectrum analyzer, and I have yet to hear
anyone say
"there's a pole at -1 kHz" or "the 3 dB points are +-i times six kiloradians per
second". ;-)

Use of radians vs. "cycles/sec" frequency doesn't depend on Fourier vs.
Laplace. As far as using the sqrt(-1) [i or j] in talking about frequency --
I don't see how Laplace or Fourier does anything about the convention of
omitting this detail, any more than our gibberish when we use the nonsensical
'conventional current' in talking about direction of current flow.

I haven't used a Laplace transform for anything in years and years,
mainly because
the two-sided Laplace is a renamed Fourier, and the one-sided Laplace
has such an
ugly inverse.

Perhaps you can educate me; but how can you have a purely imaginary single
pole with a one-pole circuit made from real components? Even with a Fourier
transform (ok, call me a Laplacian bigot -- I only occasionally use Fourier
for "real circuits"). Maybe you can give us the Fourier transform of
the transfer function of a 1-pole RC lowpass? Is its form something other
than k/(1+j*w/wo) ?

With the modern use of Heaviside unit step functions and

so forth,
which make Fourier transforms of one-sided functions well defined, there
doesn't seem
to be any point in using the (one-sided) Laplace any more.

Is there?

Using 2-sided Fourier involves singularities (delta functions) as well as
step (Heaviside) functions when you work with transient signals. These
messy details can trip you up. Most of the filters and controls books (for
continuous time real circuits) that I've used _still_ use Laplace. Of
course, if you're mostly working with sinusoids then Fourier makes more
sense.

Always looking to fill in my mental crevasses --

-frank
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