From: utzoo!decvax!pur-ee!malcolm
Newsgroups: net.math
Title: Re: Interpolation of Complex Numbers - (nf)
Article-I.D.: pur-ee.542
Posted: Sun Sep 12 22:27:26 1982
Received: Mon Sep 13 02:15:44 1982

#R:pur-ee:6600005:pur-ee:6600007:000:2026
pur-ee!malcolm    Sep 12 20:15:00 1982

I sort of blew it.  In my original letter of 9/5 I described the process
of interpolating complex numbers by showing the "average" of two numbers
in both rectangular and polar forms.  A large number of people (cbosgd!djb,
psuvax!sibley, houti!kdh, burdvax!puder, utcsrgv!wessels, tekcad!franka)
were good enough to point out that my calculations of the average of two
complex numbers when expressed in polar forms was incorrect.  The key to
this (as pointed out by houti!kdh) is that a vector operation ALWAYS has
the same physical result, no matter what the representation.

Tim Grogan (pur-ee!grogan) suggested a better example to illustrate my
problem.  Consider a simple exponential function of the form
			j*PI*t
		y(t) = e
If this function is known for integer values of t then it makes perfect
sense to consider this function to have magnitude equal to 1 and a phase
of (j*PI*t).  This was the type of function that I was alluding to in
my previous letter.  If the value of the function is known at times 0,
y(0) = 1+j0, and t=1, y(1) = -1+j0, then the best approximation to y(.5) 
is not found from simply averaging the two vectors in rectangular form
but from considering the magnitude and phase seperately.   In this case
the magnitude is always equal to 1, but the phase is a linear function 
of time.  The "polar interpolate" at t=.5 is therefore magnitude=1 and
phase=PI/2, or 0+j1.

So my problem still remains.  It is not simply a matter of vector arithmetic
as I first thought, but a matter of choosing an appropiate interpolating
function.  

Does anybody have any ideas on how I can decide whether I should use
a rectangular coordinate system or a polar form when I do the interpolation?
My data represents complex amplitude of a waveform and I need to interpolate
the measured values onto a uniform grid so that I can do further computations
on the matrix.  So if anybody has any ideas, please mail me a note.

					Malcolm Slaney
					Purdue EE Dept.
					{decvax,harpo,ihnss,ucbvax,gsp86}!
						malcolm