Digital Signal Processing Reference
In-Depth Information
The GAUSSian numerical plane which was introduced in Chapter 5 (see Illustration 97
ff.) provides a clear presentation of these possible states of digitally modulated signals. In
contrast to the symmetry of frequency presented in that chapter (any sinusoidal oscillation
consists of two frequencies +f and
−
f), it is in this case sufficient to use just one vector +f.
In the signal space the various different discrete states of the carrier oscillation are repre-
sented as vectors with regard to amplitude and phase. It is, however, common practice to
represent only the corners of the vectors. In addition, a cosine-shaped carrier oscillation
is usually chosen as carrier or reference carrier with 2-ASK and 2-PSK modulations so
that the ends are on the horizontal axis on the GAUSSian numerical plane.
In Illustration 262 the ends of 2-ASK and 2-PSK are on one line, i.e. from a mathematical
point of view in a
one
-dimensional space. So why is the representation on the GAUSSian
plane (
two
-dimensional) so important with regard to the signal space? Consider the
following aspects:
• Precisely one area can be determined in which the (discrete) state is to be located in a
disturbed channel to be clearly identifiable in the receiver.
•
This shows the advantage of 2-PSK where the ends have twice the
distance compared with 2-ASK (provided that the amplitude of the
carrier frequency is identical).
•
In a disturbed channel the ends will not always be in the same position
as in an undisturbed ideal state. If, for instance, the channel is noisy the
end is randomly distributed within a given range with each procedure
(see Illustration 262).
• The one-dimensional space could easily be enlarged to become a two-dimensional
space, if not only the phase angles 0 and 180 degrees, but other phase angles and
amplitudes were permitted, too.
•
The more different (discrete) states of a carrier oscillation (constant
frequency) are permitted the better the bandwidth utilization of the
transmission should be.
•
On the other hand: the shorter the distance between the different ends
of the signal space, the greater the interference proneness of the signal.
This could perhaps be avoided by linking the channel coding and
channel modulation.
• The most important question now is to what extent the bandwidth of the signal changes
when the number of discrete states (amplitude and phase) of the carrier oscillation
increase.
