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Performance Analysis of Binary Antipodal Signals with Matched Filters

Module by: Behnaam Aazhang

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

Matched Filter receiver

Recall r t = s m t+ N t r t s m t N t where m=1 m 1 or m=2 m 2 and s 1 t=- s 2 t s 1 t s 2 t (see Figure 1).
Figure4-34_2.png
Figure 1
Y 1 T= E s + ν 1 Y 1 T E s ν 1 (1)
Y 2 T=- E s + ν 2 Y 2 T E s ν 2 (2)
since s 1 t=- s 2 t s 1 t s 2 t then ν 1 ν 1 is 0 N 0 2 E s 0 N 0 2 E s . Furthermore ν 2 =- ν 1 ν 2 ν 1 . Given ν 1 ν 1 then ν 2 ν 2 is deterministic and equals - ν 1 ν 1 . Then Y 2 T=- Y 1 T Y 2 T Y 1 T if s 1 t s 1 t is transmitted.
If s 2 T s 2 T is transmitted
Y 1 T=- E s + ν 1 Y 1 T E s ν 1 (3)
Y 2 T= E s + ν 2 Y 2 T E s ν 2 (4)
ν 1 ν 1 is 0 N 0 2 E s 0 N 0 2 E s and ν 2 =- ν 1 ν 2 ν 1 .
The receiver can be simplified to (see Figure 2)
Figure4-35.png
Figure 2
If s 1 t s 1 t is transmitted Y 1 T= E s + ν 1 Y 1 T E s ν 1 .
If s 2 t s 2 t is transmitted Y 1 T=- E s + ν 1 Y 1 T E s ν 1 .
P e =1/2Pr Y 1 T<0| s 1 t+1/2Pr Y 1 T≥0| s 2 t=1/2∫-∞012π N 0 2 E s ⅇ-|y- E s |2 N 0 E s dy+1/2∫0∞12π N 0 2 E s ⅇ-|y+ E s |2 N 0 E s dy=Q E s N 0 2 E s =Q2 E s N 0 P e 12 s 1 t Y 1 T 0 12 s 2 t Y 1 T 0 12 y 0 1 2 N 0 2 E s y E s 2 N 0 E s 12 y 0 1 2 N 0 2 E s y E s 2 N 0 E s Q E s N 0 2 E s Q 2 E s N 0 (5)
This is the exact bit-error rate of a correlation receiver. For a bit-error analysis for orthogonal signals using a matched filter receiver, refer here.

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