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4.11 Performance analysis of antipodal binary signals with correlation

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Bit-error analysis for an antipodal signal set by using a correlator-type receiver.

The bit-error probability for a correlation receiver with an antipodal signal set ( ) can be found as follows:

P e m m b b 0 m 1 r 1 1 m 2 r 1 0 r ∞ f r 1 s 1 t r 1 r ∞ f r 1 s 2 t r
if 0 1 1 2 , then the optimum threshold is 0 .
f r 1 s 1 t r E s N 0 2
f r 1 s 2 t r E s N 0 2
If the two symbols are equally likely to be transmitted then 0 1 1 2 and if the threshold is set to zero, then
P e 1 2 r ∞ 0 1 2 N 0 2 r E s 2 N 0 1 2 r 0 ∞ 1 2 N 0 2 r E s 2 N 0
P e 1 2 r ∞ 2 E s N 0 1 2 r 2 2 1 2 r 2 E s N 0 ∞ 1 2 r 2 2
with r r E s N 0 2 and r r E s N 0 2
P e 1 2 Q 2 E s N 0 1 2 Q 2 E s N 0 Q 2 E s N 0
where Q b x b ∞ 1 2 x 2 2 .

Note that

P e Q d 1 2 2 N 0
where d 1 2 2 E s s 1 s 2 2 is the Euclidean distance between the two constellation points ( ).

This is exactly the same bit-error probability as for the matched filter case.

A similar bit-error analysis for matched filters can be found here . For the bit-error analysis for correlation receivers with an orthogonal signal set, refer here .
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Source:  OpenStax, Digital communication systems. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10134/1.3
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