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Performance Analysis of Antipodal Binary signals with Correlation

Module by: Behnaam Aazhang. E-mail the author

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

Figure 1
Figure 1 (Figure4-32.png)

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 t rd r + π 1 γf r 1 s 2 t rd r 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 t r=𝒩 E s N 0 2 f | r 1 s 1 t r E s N 0 2
(2)
f r 1 | s 2 t r=𝒩 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 =1/2012π N 0 2e|r E s |2 N 0 d r +1/2012π N 0 2e|r+ E s |2 N 0 d r 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 =1/22 E s N 0 12πe| r |22d r +1/22 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=b12πex22d x Q b x b 1 2 π x 2 2 .

Note that

Figure 2
Figure 2 (Figure4-34_1.png)
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 =s1s22 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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