malcolm (09/06/82)
#N:pur-ee:6600005:000:1109
pur-ee!malcolm Sep 5 14:08:00 1982
I had excellent luck the last time I submitted one of my research problems
to the network, so I am going to try again. Part of the problem is that
I am not sure the best way to pose the problem....please bear with me.
If anybody has seen a problem of this type or knows of any references.
I am doing work on Ultrasonic Diffraction Tomagraphy and we have to
interpolate complex numbers. The actual interpolation is easy and
is well described by a number of books. The problem is do we represent
the complex numbers in rectangular form or in polar form. Consider the
numbers 0+j1 and 0-j1. In rectangular form this averages to 0, while
in polar form they become mag=1, phase=90 and mag=1, phase=-90 and they
average to +1. The results are quite different.
Since we are using the interpolation to supply numbers for an FFT algorithm
my inclination is to say that the rectangular form should be used. I
can't even begin to justify my choice though.
How about it net-land??? Does anybody have any ideas??
Malcolm Slaney
Purdue EE Dept.
{decvax,harpo,ihnss,ucbvax}
!pur-ee!malcolmmalcolm (09/13/82)
#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