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Performance Analysis of Antipodal Binary signals with Correlation

Module by: Behnaam Aazhang

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

Figure4-32.png
Figure 1
The bit-error probability for a correlation receiver with an antipodal signal set (Figure 1) can be found as follows:
P e =Pr m ̂ ≠m=Pr b ̂ ≠b= π 0 Pr r 1 <γ|m=1+ π 1 Pr r 1 ≥γ|m=2= π 0 ∫-∞γf r 1 s 1 trdr+ π 1 ∫γ∞f r 1 s 2 trdr P e m ̂ m b ̂ b π 0 m 1 r 1 γ π 1 m 2 r 1 γ π 0 r γ f r 1 s 1 t r π 1 r γ f r 1 s 2 t r (1)
if π 0 = π 1 =1/2 π 0 π 1 12 , then the optimum threshold is γ=0 γ 0 .
f r 1 | s 1 tr= E s N 0 2 f | r 1 s 1 t r E s N 0 2 (2)
f r 1 | s 2 tr=- E s N 0 2 f | r 1 s 2 t r E s N 0 2 (3)
If the two symbols are equally likely to be transmitted then π 0 = π 1 =1/2 π 0 π 1 12 and if the threshold is set to zero, then
P e =1/2∫-∞012π N 0 2ⅇ-|r- E s |2 N 0 dr+1/2∫0∞12π N 0 2ⅇ-|r+ E s |2 N 0 dr P e 12 r 0 1 2 π N 0 2 r E s 2 N 0 12 r 0 1 2 π N 0 2 r E s 2 N 0 (4)
P e =1/2∫-∞-2 E s N 0 12πⅇ-| r ′ |22d r ′ +1/2∫2 E s N 0 ∞12πⅇ-| r ″ |22d r ″ P e 12 r ′ 2 E s N 0 1 2 π r ′ 2 2 12 r ″ 2 E s N 0 1 2 π r ″ 2 2 (5)
with r ′ =r- E s N 0 2 r ′ r E s N 0 2 and r ″ =r+ E s N 0 2 r ″ r E s N 0 2
P e =12Q2 E s N 0 +12Q2 E s N 0 =Q2 E s N 0 P e 1 2 Q 2 E s N 0 1 2 Q 2 E s N 0 Q 2 E s N 0 (6)
where Qb=∫b∞12πⅇ-x22dx Q b x b 1 2 π x 2 2 .
Note that
Figure4-34_1.png
Figure 2
P e =Q d 1 2 2 N 0 P e Q d 1 2 2 N 0 (7)
where d 1 2 =2 E s =∥s1-s2∥2 d 1 2 2 E s s 1 s 2 2 is the Euclidean distance between the two constellation points (Figure 2).
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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