# Gaussian approximation for bit error rate calculation

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## Effect of impairments on system performance

The effects of impairments, discussed in chapter 2, on system performance will be considered in this chapter. An additional impairment, extinction ratio, will also be included. An important performance parameter for a digital communication system is bit-error-rate or error probability. The error probability calculation in this chapter will be approximated by the Guassian approximation, which will be discussed first. Other performance parameters for optical communication systems are the minimum received power and the power penalty. The minimum received power in this chapter will be expressed by the average number of photons per bit at the receiver since the received power is proportional to the number of incident photons. The formula for these parameters will be given. Also, some plots of these parameters will be shown and discussed.

### Gaussian approximation for the error probability calculation

In a binary communications system, one important parameter indicating the system performance is the error probability. To compute the error probability, the probabilities of sending bit 0 and bit 1, and the probability density function of the signal at the receiver have to be known. The error probability, in general, is given by Pe  = PrPr[sˆ = 1 | s = 0] + PrPr[sˆ = 0 | s = 1]    (3.1.1) where  Pr and Pr are the probabilities of sending bit 0 and bit 1 from the transmitter, respectively sˆ is the estimated bit from the decision circuit s is the bit that was transmitted.
In an optical communications system, the optical signal at the receiver is converted to a current by the photodetector, and then is passed through electrical components for electrical processing; for example, amplification and filtering. Finally, the signal current is sent to the decision circuit to estimate the data bit. Many types of noise are added to the signal as mentioned in chapter 2. The more noise in the system, the more fluctuation on the input current to the decision circuit. That is, the spread of the probability density function of the signal will increase. In many cases, a good approximation of the probability density function for the input current to the decision circuit is given by the Gaussian distribution. The Gaussian probability density function is given by.
From equation (3.1.2), only two parameters are needed to define the Gaussian probability density function; that is, mean and variance. The probability density function of the input current to the decision circuit when bit 1 and bit 0 are transmitted can be approximated using equation (3.1.2). The probability that the estimate is bit 1 when bit 0 was transmitted and the probability that the estimate is bit 0 when bit 1 was transmitted are given by.
Equation (3.1.5) allows us to find the error probability if values of all parameters are known. To get a minimum error probability, the current threshold should be chosen such that the probability density functions of bit 0 and bit 1 are the same. This follows by differentiating equation (3.1.5) with respect to the current threshold (Ith). However, setting Pr[sˆ = 1 | s = 0] = Pr[sˆ = 0 | s = 1] is generally a good approximation of choosing the optimum threshold. That is, setting.

#### Effect of impairments for zero extinction ratio

The input current to the decision circuit combines the signal and the various sources of noise, which are added to the signal along the transmission path. The combined noise is assumed to be a zero mean Gaussian process. This assumption is good as long as the system is not in the shot noise limit. The input current to the decision circuit can be written as.
The variance of input current is needed to calculate the minimum error probability. The variance of input current for each bit (bit 0 or bit 1) is just the sum of variances of the currents induced by each source of noise since the various noise sources are assumed to be independent. Therefore, the variance of input current is given by.
The minimum error probability can be calculated by using k in equation (3.2.14) with equation (3.1.9). Figure 3.1 shows the minimum error probability versus the average number of photons per bit for different ratios of [ f/B]. The effective capacitance is 1 pF and the quantum efficiency is 0.8. As the number of photons increases, the minimum error probability decreases. As the ratio of the receiver bandwidth to bit rate, [ f/B], increases, the minimum error probability for a given number of photons per bit increases. For example, at N p =10,000 photons, the minimum error probability for [ f/B] of 0.7, 0.9, and 1.1 are 5.2*10-9, 4.2*10-6, and 1.3*10-4, respectively. The reason is when the ratio [ f/B] increases, the effective noise bandwidth of the receiver increases; thus, there is more noise in the input current to the decision circuit. Consequently, the minimum error probability with a high [ f/B] is poorer than the minimum error probability with a low [ f/B].
Comparing equation (3.2.21) with equation (3.2.15), it is seen that the effect of relative intensity noise is to multiply N p by the first factor in equation (3.2.21).
The minimum error probability versus the average number of photons per bit for different values of RIN(0) is plotted in Figure 3.2. In this Figure, the effective capacitance is 1 pF, the quantum efficiency is 0.8, and the bit rate is 200 Mbps. As the value of RIN(0) increases, the minimum error probability increases. For example, at the average number of photons per bit of 10,000, the minimum error probabilities for zero RIN,-97dB/Hz RIN,and– 94dB/Hz RIN are 4.4*10 -9,        1.4*10-4, and 1.6*10-3 respectively. Furthermore, there is a minimum error probability (error floor) resulting from the RIN.
Using equation (3.3.5), the plot of the error probability versus the average number of photons per bit is shown in Figure 3.5. The effective capacitance is 1 pF and the quantum efficiency is 0.8. There are four values of extinction ratios; that is, 0, 0.1, 0.2, and 0.4. The error probability decreases as the average number of photons per bit increases. The best error probability is for the case of zero-extinction ratio. From the figure, it is seen that as the extinction ratio increases, the error probability increases; for example, at N p =10,000 photons/bit, the error probabilities for the extinction ratios of 0, 0.1, 0.2, and 0.4 are 4.4*10-9, 1.3*10-6, 6.3*10-5, and 6.9*10-3, respectively. However, if the extinction ratio is kept less than 1.0, the effect of nonzero-extinction ratio will not introduce an error floor as the effect of RIN does. This means that the error probability can be reduced if the average number of photons per bit increases; thus, more power is required.
The additional required power can be interpreted in terms of power penalty, which was shown in equation (3.3.9). In Figure 3.6, the power penalty due to the extinction ratio with k=6 (i.e., Pe=10-9) is plotted. The power penalty increases as the extinction ratio increases. It is seen that the power penalty is lower than 1 dB if the extinction ratio is less than 0.12. From this figure, it is also seen that the degradation due to the extinction ratio does not give an error floor as long as the extinction ratio less than 1, which is reasonable in practice.

Effect of relative intensity noise (RIN)

In this section, the effect of RIN is included to the calculation with a nonzero-extinction ratio. As mentioned previously, the average input current for bit 0 is not zero because of a nonzero-extinction ratio; thus, RIN will affect the signal at bit 0 also.

1.0 Introduction
2.0 Impairments in fiber optic systems
2.1 Thermal noise
2.2 Shot noise
2.3 Signal-dependent noise
2.4 Intersymbol interference (ISI)
3.0 Effect of impairments on system performance
3.1 Gaussian approximation for bit error rate calculation
3.2 Effect of impairments for zero extinction ratio
3.4 Effect of vertical offset from the decision threshold
4.0 Effect of system bandwidth on intersymbol interference
4.1 Modeling fiber optic systems using Butterworth filters
4.2 Intersymbol interference simulation
4.3 Power penalty due to IS
5.0 Conclusions and suggestions for future research
Appendix
References

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Modeling and Analysis of the Effects of Impairments in Fiber Optic Links