Selected Topics in Applied Signal Processing - Digital Signal Processing: An Introduction with MATLAB and Applications - page 135

Fig. 3.2 Gaussian pdf. Note
that the shaded area repre-
sents Pr(x [ E)

p ( x )

x

0 E

SNR ¼ E

g ;

The probability of error P e in correctly detecting a data symbol is often used as the
basis for performance evaluation of communication systems. Although the prob-
ability of error P e in (3.2) was obtained under the assumption that an ''off'' was
transmitted, the same kind of result would have obtained if an ''on'' was trans-
mitted. Hence, the average probability of error is given by:

P e av ¼ P e :

Figure 3.3 shows the general shape of P e versus SNR.

3.2.2 Binary Transmission Using Antipodal Signals

Two signals s 0 (t) and s 1 (t) are said to be antipodal if s 0 (t) =-s 1 (t) = s(t). One
possible antipodal configuration is the use of ± V as the two different signals. An
alternative configuration is depicted in Fig. 3.4 .
If one uses orthogonal signals for transmission, normally a bank of two matched
filters are needed for optimum reception. However, if one uses two antipodal
signals, only one matched filter is needed. The received signal is r(t) =

± s(t) ? n(t), and the matched filter is matched to s(t).
Assume that two antipodal signals are used for transmission in a binary base-
band communication system. Following the same kind of analysis as was used for
orthogonal signals, the following result can be obtained (see [ 2 ]):

Output of the matched filter ¼ E þ n a ;

where n a ¼ R T

0 n ð t Þ s ð t Þ dt ; and the variance of n a ¼ 2 E (i.e., the same as that of
n 0 and n 1 for orthogonal transmission). The decision process used for antipodal
transmission is as follows: if r [ 0, then s(t) was assumed to be transmitted (which
represents the ''on'' symbol), otherwise -s(t) is assumed to have been transmitted.