Since the wire diagrams are generally composed of elliptical loops and
some transitions, we have chosen to consider whether the use of conformal maps
will simplify the search for an effective ``geometric deconvolution
mapping." We will discuss the steps required
to project a deformed reconstruction of a convolved MSK signal onto
the original MSK signal. To begin, we looked more closely
at the time delay reconstruction of an MSK signal. The MSK signal is
defined for a data stream 
, 
, 
, 
by splitting the
bit stream into odd-numbered bits, 
and
even-numbered bits, 
. The MSK signal
is then:
where 
is the length of time that one bit is transmitted, and 
is the carrier frequency. Note that the system is defined so the 
terms
switch on every odd time unit and the 
terms switch on every even
time unit. When the 
and 
bits have the same sign, the lower
frequency is generated, with the higher frequency occurring when the
signs differ. Since the bit changes occur out of synch, there are
eight possible types of switches, each resulting in the
signal changing from one frequency to the other.
In the figure below, you can see periods when the 
and 
bits are
the same, as well as when they differ. In between these states the signal
is in transition from one frequency to the other. Also shown
is the time delay reconstruction
of the signal. In this reconstruction two ellipses can be seen, oriented
with their major axes along the two main diagonals of the figure. Each of
the ellipses represents one of the main frequencies in the MSK signal. Motion
along these ellipses occurs when the MSK signal is in either the high- or
low-frequency state. All of the other traces represent transitions from
one state to the other. By carefully tracing the curves, it can be seen
that there are eight transitions
between the two ellipses representing the eight different switches in
the 
's and 
's.
In examining the reconstruction in 2 dimensions, it is difficult to determine when intersections of the lines are the result of the projection and when they represent true transitions. In order to be able to map a deformed reconstruction onto the original, we need a reconstruction that removes the false intersections resulting from the 2-d projection. One way to achieve this is to form the reconstruction in 3-dimensions so the ellipses no longer intersect. Ideally we want to find a third dimension such that the two ellipses lie in two unique, constant planes. Once we can plot the ellipses in this way, we should be able to use a conformal mapping to project the deformed ellipses onto the originals.
Since we know the two ellipses represent the two frequencies, we examined
the integral of the signal over one period of the higher frequency, and we call this interval T.
When we are on the higher frequency, the integral over one period is zero.
When we are on the lower frequency or there is a transition involved, the
integral over an interval T will be changing. The following figure shows the
reconstruction in 3-d, where the third dimension is the result of the
integration. Notice that the ellipses still
appear to intersect.
When we rotate this figure, we can see the ellipses lie in separate planes. However, these planes are intersecting (as seen below).
The ellipses project to straight lines in the xz-plane.
This reconstruction makes it possible to calculate the slope of
these planes, which we then use to find a third dimension which will
separate the ellipses. If the slope is zero, the trajectory is in
Plane 1, so we choose z=0 as the third dimension.
If the slope is 
, and we set the
third component of the reconstruction to z=1. We also let the
transitions be linear in tz-space,
so in the transition regions, the z-component increases linearly
from z=0 to z=1. The results of the reconstruction using this
slope estimation are shown below. Notice that we now have the two
ellipses in unique, constant planes in the third dimension.
The pure ellipses have been highlighted to be seen more easily.
Notice that there are still only eight transition curves between the two
ellipses.
The next stage in the process is to apply the same reconstruction techniques to an MSK signal that has passed through a channel, which we simulate by convolving the MSK signal with a response function that is one symbol in width. We apply the same reconstruction to the convolved signal, and the resulting three-dimensional reconstruction is shown below. Notice there are still two ellipses (which have been highlighted) representing the two frequencies of the signal.
In order to map the convolved signal onto the original signal, we examined the
ellipses in the planes z=0 and z=1. Using a Least-Squares approximation
to find the equations for the ellipses, we can map the deformed ellipse to
the pure ellipse in the plane z=0. We can call this map 
. We can
then map the deformed ellipse to the pure ellipse in the plane z=1, call
this 
. We can now use the map:
where 
is the coordinate at any point. Notice, when z=1, we
use only the map 
and, when z=0, we use the map 
. The results of
this mapping are shown below. The two figures show the results in three
dimensions. The mapped signal is the solid line while the original signal is
dotted. The pure ellipses have been highlighted. Notice that the general
shape of the transitions is mimiced in the deformed transitions. This can be
seen more clearly in the second figure. The ellipses have been mapped onto the target ellipses, as desired, and in the transition regions the
trajectories are close to the original transitions, but they are not yet mapped perfectly.
