278
7 Optical Receiver Operation
P 1 (v) =
v
−∞
p(y|1 )dy
(7.6)
which is the probability that the equalizer output voltage is less than υ when a
logical 1 pulse is sent, and
P 0 (v) =
∞
v
p(y|0 )dy
(7.7)
which is the probability that the output voltage exceeds υ when a logical 0 is transmitted. Note that the different shapes of the two probability distributions in Fig. 7.7
indicate that the noise power for a logical 0 is usually not the same as that for a
logical 1. This occurs in optical systems because of signal distortion from transmission impairments (e.g., dispersion, optical amplifier noise, and distortion from
nonlinear effects) and from noise and ISI contributions at the receiver. The functions
p(y|l) and p(y|0) are the conditional probability distribution functions; that is, p(y|x)
is the probability that the output voltage is y, given that an x was transmitted.
If the threshold voltage is υ th then the error probability P e is defined as
P e = a P 1 (υ th ) + bP 0 (υ th )
(7.8)
The weighting factors a and b are determined by the a priori distribution of the
data. That is, a and b are the probabilities that either a 1 or a 0 occurs, respectively.
For unbiased data with equal probability of 1 and 0 occurrences, a = b = 0.5. The
problem to be solved now is to select the decision threshold at that point where P e
is minimum.
To calculate the error probability it is necessary to know the mean-square noise
voltage
υ
2
N
, which is superimposed on the signal voltage at the decision time.
The statistics of the output voltage at the sampling time are quite complex, so that
an exact calculation is rather tedious to perform. Therefore a number of different
approximations have been used to calculate the performance of a binary optical
fiber receiver. In applying these approximations, one has to make a tradeoff between
computational simplicity and accuracy of the results. The simplest method is based
on a Gaussian approximation. In this method, it is assumed that, when the sequence
of optical input pulses is known, the equalizer output voltage υ out (t) is a Gaussian
random variable. Thus, to calculate the error probability, one only needs to know the
mean and standard deviation of υ out (t).
Thus, assume that a signal s (which can be either a noise disturbance or a desired
information-bearing signal) has a Gaussian probability distribution function with a
mean value m. If the signal voltage level s(t) is sampled at any arbitrary time t 1 , the
probability that the measured sample s(t 1 ) falls in the range s to s + ds is given by
7 Optical Receiver Operation
P 1 (v) =
v
−∞
p(y|1 )dy
(7.6)
which is the probability that the equalizer output voltage is less than υ when a
logical 1 pulse is sent, and
P 0 (v) =
∞
v
p(y|0 )dy
(7.7)
which is the probability that the output voltage exceeds υ when a logical 0 is transmitted. Note that the different shapes of the two probability distributions in Fig. 7.7
indicate that the noise power for a logical 0 is usually not the same as that for a
logical 1. This occurs in optical systems because of signal distortion from transmission impairments (e.g., dispersion, optical amplifier noise, and distortion from
nonlinear effects) and from noise and ISI contributions at the receiver. The functions
p(y|l) and p(y|0) are the conditional probability distribution functions; that is, p(y|x)
is the probability that the output voltage is y, given that an x was transmitted.
If the threshold voltage is υ th then the error probability P e is defined as
P e = a P 1 (υ th ) + bP 0 (υ th )
(7.8)
The weighting factors a and b are determined by the a priori distribution of the
data. That is, a and b are the probabilities that either a 1 or a 0 occurs, respectively.
For unbiased data with equal probability of 1 and 0 occurrences, a = b = 0.5. The
problem to be solved now is to select the decision threshold at that point where P e
is minimum.
To calculate the error probability it is necessary to know the mean-square noise
voltage
υ
2
N
, which is superimposed on the signal voltage at the decision time.
The statistics of the output voltage at the sampling time are quite complex, so that
an exact calculation is rather tedious to perform. Therefore a number of different
approximations have been used to calculate the performance of a binary optical
fiber receiver. In applying these approximations, one has to make a tradeoff between
computational simplicity and accuracy of the results. The simplest method is based
on a Gaussian approximation. In this method, it is assumed that, when the sequence
of optical input pulses is known, the equalizer output voltage υ out (t) is a Gaussian
random variable. Thus, to calculate the error probability, one only needs to know the
mean and standard deviation of υ out (t).
Thus, assume that a signal s (which can be either a noise disturbance or a desired
information-bearing signal) has a Gaussian probability distribution function with a
mean value m. If the signal voltage level s(t) is sampled at any arbitrary time t 1 , the
probability that the measured sample s(t 1 ) falls in the range s to s + ds is given by