Optical expressions2/18/2023 ![]() When σ 1 = σ 0, I D = ( I 1 + I 0)/2, which corresponds to setting the decision threshold in the middle. The last term in this equation is negligible in most cases of practical interest, and I D is approximately obtained from The minimum occurs when I D is chosen such that In practice, I D is optimized to minimize the BER. This equation shows that the BER depends on the decision threshold I D. Where erfc stands for the complementary error function, defined asīy substituting these two equations in the BER equation above, the BER is given by If σ 1 2 and σ 0 2 are the corresponding variances, the conditional probabilities are given by However, both the average and the variance are different for 1 and 0 bits since I p equals I 1 or I 0, depending on the bit received. Since the sum of two Gaussian random variables is also a Gaussian random variable, the sampled value I has a Gaussian probability density function with variance σ 2 = σ s 2 + σ T 2. A common approximation treats i s as a Gaussian random variable for both p-i-n and APD receivers but with different σ s 2 given by two equations, respectively. The statistics of shot-noise contribution i s is also approximately Gaussian for p-i-n receivers although that is not the case for APDs. Thermal noise i T is well described by Gaussian statistics with zero mean and variance σ T 2. The functional form of p(I) depends on the statistics of noise sources responsible for current fluctuations. Since 1 and 0 bits are equally likely to occur, p(1) = p(0) = 1/2, and the BER becomesįigure (b) above shows how P(0|1) and P(1|0) depend on the probability density function p(I) of the sampled value I. Where p(1) and p(0) are the probabilities of receiving bit 1 and 0, respectively, P(0|1) is the probability of deciding 0 when 1 is received, and P(1|0) is the probability of deciding 1 when 0 is received. Both sources of errors can be included by defining the error probability as The decision circuit compares the sampled value with a threshold value I D and calls it bit 1 if I > I D or bit 0 if I I D for bit 0. The sampled valued I fluctuates from bit to bit around an average value I 1 or I 0, depending on whether the bit corresponds to 1 or 0 in the bit stream. Bit-Error Rateįigure (a) above shows schematically the fluctuating signal received by the decision circuit, which samples it at the decision instant t D determined through clock recovery. Since depends on the BER, let us begin by calculating the BER. The receiver sensitivity is then defined as the minimum average received power required by the receiver to operate at a BER of 1 0 -9. A commonly used criterion for digital optical receivers requires the BER to be below 1 x 1 0 -9. Hence, a BER of 2 x 10 -6 corresponds to on average 2 errors per million bits. The performance criterion for digital receivers is governed by the bit-error rate (BER), defined as the probability of incorrect identification of a bit by the decision circuit of the receiver. Among a group of optical receivers, a receiver is said to be more sensitive if it achieves the same performance with less optical power incident on it.
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