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

10.8 Equivalent Signal Sets 569
and 1.04 MHz for modulated pulses (assuming raised-cosine pulses). Also,
Therefore,
6E b log2 16 ]
_ _
PeM - 4 X 10 6 _
- 2 ( M -- - 1 ) Q [
M N (M 2 -1)
For M = 16, this yields E b = 0.499 x 10- 5 . If the M-ary pulse rate is RM , then
S; = E p
MRM = Eb log2 M · RM
= 0.499 X 10- 5 X 4 X (0.52 X 10 6 ) = 9.34 W
(c) 16-ary PSK: We need transmit only RM = 0.52 x 10 6 pulses per second. For
baseband pulses, this will require a bandwidth of 520 kHz. But PSK is a modulated signal,
and the required bandwidth is 2(0.52 x 10 6 ) = 1.04 MHz. Also,
This yields E b = 137.8 x 10- 8 and
S; = Eb log2 16RM
= (137.8 X 10- 8 ) X 4 X (0.52 X 10 6 ) = 2.86W
10.8 EQUIVALENT SIGNAL SETS
The computation of error probabilities is greatly facilitated by the translation and rotation of
coordinate axes. We now show that such operations are permissible.
Consider a signal set with its corresponding decision regions, as shown in Fig. 10.30a.
The conditional probability P( C1m1) is the probability that the noise vector drawn from s1 lies
within R1. Note that this probability does not depend on the origin of the coordinate system.
We may translate the coordinate system any way we wish. This is equivalent to translating the
signal set and the corresponding decision regions. Thus, the P( Clm;) for the translated system
shown in Fig. 10.30b is identical to that of the system in Fig. 10.30a.
In the case of Gaussian noise, we make another important observation. The rotation of the
coordinate system does not affect the error probability because the noise-vector probability
density has spherical symmetry. To show this, we shall consider Fig. 10.30c, which shows the
signal set in Fig. 10.30a translated and rotated. Note that a rotation of the coordinate system
is equivalent to a rotation of the signal set in the opposite sense. Here for convenience we
rotate the signal set instead of the coordinate system. It can be seen that the probability that the
noise vector n drawn from s1 lies in R1 is the same in Fig. 10.30a and c, since this probability