Yeah, I'm not getting much joy from that either :/
I'm still working on it....I've had some luck at improving it, but I've managed to show there's some pesky effects from the windowed short time fourier transform business. The time I made it work for some force control signals in the lab, I used the fourier transform of the whole signal.
I'm somewhat familiar with the principle I think, at least in practice as a "phase shifter" for removing sidebands with a quadrature mixer etc, but even though I understand Hilbert is fundamental there's a big hole in my maths in that subject, around about the point I fell asleep in engineering maths lectures.
We all fell asleep in engineering math lectures. I just make it up as I go along. (My brother looks at me sideways as I do this, all the time)
Can you give a description, in words rather than equations if possible please, how Hilbert arises and why its central to complex signals. Maybe with another example of why you would use the transform for a practical purpose.
I don't really have a text on this subject any more....so the details are a little fuzzy where the hilbert transform comes from.....I can tell you this, it's a funky integral equation with some kernel, that has do to with the eigenvectors/eigenvalues of some linear operator...but then, that's generally what all transforms are about....I'll have to look it up again, just to make sure I have the right idea.
for signal analysis, the whole idea is this: we can take any signal , that has any number of frequencies, and transform it into a joint amplitude/frequency modulation of a single frequency described by amplitude and phase separately as functions of time.
In the lab, I was working on using this for analyzing force control data, which showed a kind of damped resonance. But it was a natural signal so there was no clearly defined resonance. Instead there were several damped resonances of different frequencies. When you plot the hilbert tranform in the complex plane, you get a "phase plane" (is that what it's called?) portrait of the signal. There were decaying slow frequencies of high amplitude, with some smaller higher frequencies added in, which showed up as epicycles in the graph. Ultimately, we scrapped the analysis, since it didn't seem like I could make any significant conclusions.
One thing that confuses me right now is how you can have a phase signal? Relative to what? That is why I said phase comparator, it kinda doesn;t make sense to me to have absolute phase, it must be relative to something.
To me, the phase is absolute. It describes a relationship, at a certain point in time, between the signal's value and it's rate of change.
........ for improvement, maybe I should try doing it on the whole signal....This sounds like a job for the script function. I haven't tried it yet, but this would be a good chance to learn. I could make a little script with octave that would take the hilbert transform of the whole signal, and smoothly match the phase at the end of the file to the beginning and return it....then the signal could be loaded as an array, and could try looping it to see if it works. sounds a little ambitious at the moment.... 's getting late tonight
Have a good weekend! Chuck