October 07, 2011
Complex Frequency Visualizers
by Marco Vitanza

When analyzing continuous-time linear systems, the Laplace transform is used to work with signals in the frequency domain. The Laplace transform can be thought of as the Fourier transform generalized to complex frequencies.

X(s) = L{x(t)} = integral[-inf, inf] exp(-st) x(t) dt

The transfer function H(s) of an LTI system represents how the system transforms the input signal exp(st). But what does such an input signal look like, and what is the meaning of a complex frequency?

Wonder no further. Play around with this complex frequency visualizer (continuous).

But what about discrete-time signals? We can sample signals at a period T and use the Z-transform to examine them in the frequency domain.

X(z) = Z{x(n)} = sum[-inf,inf] z^-n x(n)

So what's the relationship between the s-domain and the z-domain? Check out this complex frequency visualizer (discrete).

Comment by Candie on January 27, 2012 at 06:22pm
Well put, sir, well put. I'll creaitnly make note of that.
Comment by Rubik on June 08, 2013 at 10:24am
Very cool. It'd be nice to have a small display of how far one is zoomed in or out, because it is too easy to loose track and have no frame of reference!! Kind of what happens when someone turns a cube around in their hands to look for a piece, without holding in place a top, and a front face...

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