Hello.

To introduce myself, I'm a blind music composer, I play the piano and the flute,
and I'm also very interested in lots of topics concerning acoustics and
digitized sound. Unfortunately, the last time I had any regular math lessons was
when I was about 16 and since then I had to look things up very
non-systematically on the web. But I hope I'm finally beginning to understand
some of the mathematical topics which are so crucial in the field of audio DSP.
To make sure I'm not terribly wrong in some of my conclusions, I'd like to ask a
couple of questions. Hope you don't mind if I ask them using just "the words I
can think of" -- sometimes I realize other people use slightly different
terminology.

- #1. I was looking for an algorithm for frequency demodulation somewhere on the
web. Unfortunately, the one I found required me to first get a 90-degree
phase-shifted copy of the modulated signal before I could apply the procedure.
But if if the signal contains both positive and negative frequencies (which many
FM signals do) and I try to phase-shift it via convolution with a particular
static impulse, then the positive ones are phase-shifted in a different
direction than the negative ones. So my question now is:

A) Is there an algorithm for frequency demodulation which doesn't require having
a phase-shifted copy of the original?

B) And if not, how can I phase-shift both the positive and negative frequencies
in the same direction?

- #2. I'm slowly beginning to understand complex numbers and currently I'm
getting familiar with things like complex exponentials or some such. What I'd
like to know is this:

A) Am I right in assuming that "i^z = e^(i*z*pi/2)" for any z?

B) And if not, is it true when z is purely real?

Thank you very much for any comments.

Petr, Czech Rep.