Hi Tim, Since this thread is still going can I ask a question?
(Well first a statement.) I learned Laplace transforms, back as an EE. mostly (I think) to help in solving some differential equations. Then I became more of a physicist, and don't use them anymore.
So now the question. Do you use the Z-transform as a way to understand the problem? Or is it more of a tool to help you write the software correctly?
(Or something else.)
George H.
The z-transform, like the Laplace transform, can be used as a tool to describe the behavior of a system (via the system transfer function). So in that sense it's a way to understand the problem.
It can also (like the Laplace transform) be used as a tool to help you to synthesize a description of how you want a system to work, and then as a tool to help you make your final system match your target system behavior.
And, when you're done, you can use it as a tool to help write the software correctly, and to verify its correctness.
There's a 1:1 relationship between the Z-transform and the band-limited Fourier transform. The unit delay 1/z equals exp(-2 pi f T), where T is the sampling interval. (Or exp(+2 pi f T) if you're using the physicists' sign convention.) ;)
This relationship is an example of a conformal map, which is familiar from undergraduate complex variables calculus. It maps the two-sided Nyquist interval of the real frequency axis into the unit circle, and the stable strip into the interior of the circle.
The approximations come in when you take continuous-time, non-band-limited things and express them in the difference-equation terms that Z-transforms require.
Cheers
Phil Hobbs
(Who usually sticks with Fourier transforms)
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