7.1 Error Probabilities for Binary Signaling - The ro is a random variable, basically from the e to noise e. Errors occur when ro falls in the wrong areas di randomness of n(t)in r(t. The distribution could be as following: f(rols, and f(rols2) are generated from n(t) by detect- processing Vr is a proper-defined threshold f(rols2 sent) f(rols, sent) P(error 5, sent) P(error s2 sent) Figure 7-2 Error probability for binary signaling
11 7.1 Error Probabilities for Binary Signaling • Errors occur when r0 falls in the wrong areas due to noise. • The r0 is a random variable, basically from the randomness of n(t) in r(t). The distribution could be as following: • f(r0 |s1 ) and f(r0 |s2 ) are generated from n(t) by detectprocessing – VT is a proper-defined threshold
7.1 Error Probabilities for Binary Signaling When signal plus noise is present at the receiver input, Error can occur in two ways, an error occurs when <VT if a binary l is sent, and an error occurs when rO?VT if a binary 0 is sent P(error|S, sent) =f(ro s, )dr (7.5) P(error s, sent)=f(ro I, )dro (7.6 一 f(rolsz sent) fcrolsisent P(error I s, sent) P(error I s? sent) Figure 7-2 Error probability for binary signaling
12 7.1 Error Probabilities for Binary Signaling • When signal plus noise is present at the receiver input,Error can occur in two ways, an error occurs when r0<VT if a binary 1 is sent, and an error occurs when r0>VT if a binary 0 is sent ( | ) ( | ) (7.6) ( | ) ( | ) 7.5 2 0 2 0 1 0 1 0 = = − − T T V V P error s sent f r s d r P error s sent f r s d r （ ）
7.1 Error Probabilities for Binary Signaling P(S, sent)and P(s, sent) are source statistics and are known before transmission, so called prior statistics; P(S, sent) and P(s, sent) are normally considered to be equally likely. Pe is a standard measure of error rate of the system. It is on a bit-basis, so called as(Average)bit error rate For given n(t), good processing and opt-Vr yield good · The ber is Pe= P(error s, sent)P(S, sent)+P(error l s2 sent)P(s2 sent)(7.7) .=P(s sent) I f(o l 1 )dro+ P(S2 sent) I f(o I s2 )dro We assume that the source statistics are equally likel 13
13 7.1 Error Probabilities for Binary Signaling • The BER is = + = + − − VT VT e P s sent f r s d r P s sent f r s d r P P error s sent P s sent P error s sent P s sent 1 0 1 0 2 0 2 0 1 1 2 2 ( ) ( | ) ( ) ( | ) ( | ) ( ) ( | ) ( )（ 7.7） – P(s1 sent) and P(s2 sent) are source statistics and are known before transmission, so called prior statistics; P(s1 sent) and P(s2 sent) are normally considered to be equally likely. • Pe is a standard measure of error rate of the system. – It is on a bit-basis, so called as (Average) bit error rate • For given n(t), good processing and opt-VT yield good Pe • We assume that the source statistics are equally likely
7.1 Error Probabilities for Binary Signaling Results for gaussian noise Two assumptions make things easy I. linear detection except for the threshold device 2. The AWGn channel) white and Gaussian n(t) .. notith zero mean and edl n(t2=002 For baseband signaling, the processing circuits consisting of linear filter with some gain. For bandpass signaling, a superheterodyne circuit consisting of a mixer, IF stage and product detector being also a linear circuit If automatic gain control(AGC)or limiters or a nonlinear detector such as envelope detector is used the results of this section will not be applicable
14 7.1 Error Probabilities for Binary Signaling Results for Gaussian Noise • Two assumptions make things easy: – 1. linear detection except for the threshold device – 2. The AWGN channel → white and Gaussian n(t) with zero mean and E{| n(t) |2 }=σ0 2 • Notes: • For baseband signaling, the processing circuits consisting of linear filter with some gain. • For bandpass signaling, a superheterodyne circuit consisting of a mixer, IF stage, and product detector being also a linear circuit. • If automatic gain control (AGC) or limiters or a nonlinear detector such as envelope detector is used, the results of this section will not be applicable
7.1 Error Probabilities for Binary Signaling Results for gaussian noise For the case of a linear-processing receiver circuit with a binary signal plus noise at the input, the sampled output is 70=S0+no (7.10) So is a constant that depends on the signal being sent for a binary l sent 0 (7.11) for a binary o sent Since the output noise no is a zero-mean Gaussian random variable, the total output sample ro is a gaussian random e variable with a mean value of either so or so2, depending on whether a binary l or a binary 0 was sent. That is to say for a binary l sent So2 for a binary Sent 15
15 7.1 Error Probabilities for Binary Signaling Results for Gaussian Noise • For the case of a linear-processing receiver circuit with a binary signal plus noise at the input, the sampled output is: (7.10) 0 0 n0 r = s + • s0 is a constant that depends on the signal being sent (7.11) for a binary 0 sent for a binary 1sent 02 01 0 = s s s • Since the output noise n0 is a zero-mean Gaussian random variable, the total output sample r0 is a Gaussian random variable with a mean value of either s01 or s02, depending on whether a binary 1 or a binary 0 was sent. That is to say: for a binary 0 sent for a binary 1sent 02 01 0 = s s mr