7.1 Error Probabilities for Binary Signaling Results for gaussian noise Then the two conditionalpdes are (n0-s01)2/(2 e (7.12) 2 丌O f(ro s,) -sa2)(2o)(7 2兀00 e. Using equally likely source statistics, the ber becomes: P=P(S1sen)∫f(vos1)dh+P(S2sen)∫∫(ro|S2)d f(ro l sy)dro d+-∫f(b|s2)dh e(0-s01)206) (-s02)2 2兀oo 16
16 7.1 Error Probabilities for Binary Signaling Results for Gaussian Noise • Then the two conditional PDFs are ( ) (7.12) 2 1 ( | ) /(2 ) 0 0 1 2 0 2 0 0 1 r s f r s e − − = ( ) (7.13) 2 1 ( | ) /(2 ) 0 0 2 2 0 2 0 0 2 r s f r s e − − = • Using equally likely source statistics, the BER becomes: ( ) ( ) = + = + = + − − − − − − − − − − T T T T T T V r s V r s V V V V e e d r e d r f r s d r f r s d r P P s sent f r s d r P s sent f r s d r 0 /(2 ) 0 0 /(2 ) 0 0 1 0 0 2 0 1 0 1 0 2 0 2 0 2 0 2 0 0 2 2 0 2 0 0 1 2 1 2 1 2 1 2 1 ( | ) 2 1 ( | ) 2 1 ( ) ( | ) ( ) ( | )
7.1 Error Probabilities for Binary Signaling Results for gaussian noise thus -x212d 22/2 e (-s01yo0√ 2(r-52)/a√2丌 S S To find the vr that minimizes p we need to solve dPe/dVT=0 dP 1 1 e-(o-)2 e -(0-s0,)21(203)=0 2√2兀σ 2√2丌00 S+s (best)=- 2 Pe(min)=g 01 4 0 17
17 7.1 Error Probabilities for Binary Signaling Results for Gaussian Noise • thus ( ) ( ) − + − + = = + − − − − − 0 01 0 01 / / 2 / 2 2 1 2 1 2 1 2 1 2 1 2 1 0 2 0 2 0 / 0 1 2 V s Q V s Q P e d e d T T V s e T VT s • To find the VT that minimizes Pe we need to solve dPe /dVT=0 ( ) ( ) 0 2 1 2 1 2 1 2 1 /(2 ) 0 /(2 ) 0 2 0 2 0 0 2 2 0 2 0 0 1 = − = − − − − r s r s T e e e d V d P = − = + = 2 0 2 0 2 01 02 01 02 4 4 ( ) (min) 2 ( ) d e T E Q s s P Q s s V best
7.1 Error Probabilities for Binary Signaling Results for Gaussian Noise and Matched-Filter Reception If the receiving filter is optimized the ber can be reduced. To minimize p. we need to maximize the argument of Q, thus we need to find the linear filter that maximizes: sn(t)-s2(t0)2S(t0) 010 0 e. sa(tol is the instantaneous power of the difference output signal at t=t 0 Transmitter Processing r(r)=s()+n(r) ro(r) ro(4)7今eve at fo 18 Figure 7-1 General binary communication system
18 7.1 Error Probabilities for Binary Signaling Results for Gaussian Noise and Matched-Filter Reception • If the receiving filter is optimized the BER can be reduced. To minimize Pe , we need to maximize the argument of Q, thus we need to find the linear filter that maximizes: ( ( ) ( )) ( ) 2 0 2 0 2 0 2 01 0 02 0 s t s t s t d = − • [sd (t0 )]2 is the instantaneous power of the difference output signal at t=t0