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Performance Analysis of Binary Orthogonal Signals with Correlation

Module by: Behnaam Aazhang. E-mail the author

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)

Figure 1
Figure 1 (Figure4-36.png)

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 =1/2⁢Prr∈ R 2 | s 1 ⁢t  transmitted +1/2⁢Prr∈ R 1 | s 2 ⁢t  transmitted =1/2⁢∫ R 2 ∫f r s 1 ⁢t rd r 1 d r 2 +1/2⁢∫ R 1 ∫f r s 2 ⁢t rd r 1 d r 2 =1/2⁢∫ R 2 ∫12⁢π⁢ N 0 2⁢e−| 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 ∫12⁢π⁢ N 0 2⁢e−| 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 ∞12⁢π⁢ N 0 ⁢e− η ′ 22⁢ N 0 d η ′ +1/2⁢Pr 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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