I am doing a complete review of Fourier and how it relates to my understanding of frequency decomposition of information. I designed an RF analyzer for $AGENCY at one time, long ago, and it was supposedly more elegant than any signal processor that existed in the world, however that remains to be proved as it is not open to observation for a few reasons.

I am learning SciLab and attempting to simulate the functions in my program in openGL as I observe them in MatLab, Mathematica, SciLab, and Octave.

ADDED: Now things are getting real ℜ hinky , as they say. Below is the DTFT (Discrete Time Fourier Transfer) and it includes the imaginary. Here is where things get interesting as I think I have a way that resolves all this. Symmetry is the way that I have heard discussion of why it is real and as such I don't disagree, however I believe the problem lies very deep and Q or quantity can be manipulated in a lot of strange ways, however when you add that second point as 0(Zero) or any other reference point, then you have defined dimensionality itself and you are dealing with a vector, whether you recognize that fact or not. It is all very much fun, whatever the outcome.

This is not the way I solve it and I will put up my account of how I deal with periodic signals, phase, decomposition, recomposition, identification, eigenvalues, and amplitude, etc. I think that my way of representing and utilizing a periodic is more effective and more straight forward. The DTFT is very elegant and fun to play with, but it may not be the best way. Time will tell if I am being skeptical of something I am bound to accept for its utility and application, IDK.

And here is the La^{T}ex, just to be consistent.

S_T(f) = \sum_{k=-\infty}^{\infty} S\left(f - \frac{k}{T}\right) \equiv \sum_{n=-\infty}^{\infty} \underbrace{T\cdot s(nT)}_{s[n]} \cdot e^{-i 2\pi f n T},

I see math in the same way that I see Python functions or Objective C++, there are methods which can be associated with objects and some objects cannot have some methods assigned without getting indeterminate results. Lisp and Forth and to some extent other languages deal with this hierarchy of function in such a way that are easier, but I don't like to sacrifice the complex utility for that level of control. It needs more complexity, and I think that it is like XML, if it doesn't work, add more XML or in this case, more complexity.

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