B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

570 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
Fi g ure 10.30
Translation and
rotation of
coordinate axes.
'P 2
(a)
'P1 -
s, ..
- I - -
I
t
'P 2
(b)
'P1 -
t
'P 2
S/._
2
/
\
\
\
(c)
'P1 -
is given by the integral of the noise probability density Pn (n) over the region R1 . Because
Pn (n) has a spherical symmetry for Gaussian noise, the probability will remain unaffected by a
rotation of the region R1. Clearly, for additive Gaussian channel noise, translation and rotation
of the coordinate system (or translation and rotation of the signal set) do not affect the error
probability. Note that when we rotate or translate a set of signals, the resulting set represents
an entirely different set of signals. Yet the error probabilities of the two sets are identical. Such
sets are called equivalent sets.
The following example demonstrates the utility of translation and rotation of a signal set
in the computation of error probability.
Example l 0.6 A quaternary PSK (QPSK) signal set is shown in Fig. 10.31a:
S ] = -S2 = v'£ <{) 1
S3 = -S4 = v'£ <{)2