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Performance Analysis of Binary Orthogonal Signals with Correlation

Module by: Behnaam Aazhang

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

Orthogonal signals with equally likely bits, r t = s m t+ N t r t s m t N t for 0≤t≤T 0 t T , m=1 m 1 , m=2 m 2 , and < s 1 , s 2 >=0 s 1 s 2 0 .

Correlation (correlator-type) receiver

r t ⇒r= r 1 r 2 T=sm+η r t r r 1 r 2 s m η (see Figure 1)
Figure4-36.png
Figure 1
Decide s 1 t s 1 t was transmitted if r 1 ≥ r 2 r 1 r 2 .
P e =Pr m ̂ ≠m=Pr b ̂ ≠b P e m ̂ m b ̂ b (1)
P e =1/2Prr∈ R 2 | s 1 t  transmitted +1/2Prr∈ R 1 | s 2 t  transmitted =1/2∫ R 2 ∫fr s 1 trd r 1 d r 2 +1/2∫ R 1 ∫fr s 2 trd r 1 d r 2 =1/2∫ R 2 ∫12π N 0 2ⅇ-| r 1 - E s |2 N 0 1π N 0 ⅇ-| r 2 |2 N 0 d r 1 d r 2 +1/2∫ R 1 ∫12π N 0 2ⅇ-| r 1 |2 N 0 1π N 0 ⅇ-| r 2 - E s |2 N 0 d r 1 d r 2 P e 12 s 1 t  transmitted r R 2 12 s 2 t  transmitted r R 1 12 r 2 R 2 r 1 f r s 1 t r 12 r 2 R 1 r 1 f r s 2 t r 12 r 2 R 2 r 1 1 2 N 0 2 r 1 E s 2 N 0 1 N 0 r 2 2 N 0 12 r 2 R 1 r 1 1 2 N 0 2 r 1 2 N 0 1 N 0 r 2 E s 2 N 0 (2)
Alternatively, if s 1 t s 1 t is transmitted we decide on the wrong signal if r 2 > r 1 r 2 r 1 or η 2 > η 1 + E s η 2 η 1 E s or when η 2 - η 1 > E s η 2 η 1 E s .
P e =1/2∫ E s ∞12π N 0 ⅇ- η ′ 22 N 0 d η ′ +1/2Pr r 1 ≥ r 2 | s 2 t  transmitted =Q E s N 0 P e 12 η ′ E s 1 2 N 0 η ′ 2 2 N 0 12 s 2 t  transmitted r 1 r 2 Q E s N 0 (3)
Note that the distance between s 1 s 1 and s 2 s 2 is d 1 2 =2 E s d 1 2 2 E s . The average bit error probability P e =Q d 1 2 2 N 0 P e Q d 1 2 2 N 0 as we had for the antipodal case. Note also that the bit-error probability is the same as for the matched filter receiver.

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