## 4 Simulation Model of DQPSK Modulated Data Transmission
In order to evaluate the BER performance for variation values of the required OSNR or SNR per bit (Eb/No) with a relatively short simulation run-time (i.e., compared to the Monte Carlo simulation, which requires a very long extended run-time), we decided to use a semi-analytical BER estimation method based on the Karhunen-Loeve series expansion (KLSE) in our BER performance simulation. The term \"semi-analytical\" means that some portions of this method are based on analytical derivations, while others are based on numerical computation. The BER performance simulation using the KLSE method was initially proposed by [PERSON] [37] to calculate the BER value in an optically pre-amplified on-off keying (OOK) modulated system. An extension to the phase-modulated system (DBPSK) was proposed in [38] by using the Fourier transform of the input electric field. The stream of the simulated input data signal is modeled to be a periodic repetition of a binary sequence such that the input data signal electric field can be expanded in a Fourier series. Using a Karhunen-Loeve expansion, the ASE noise generated by optical amplifiers can be expanded in a Fourier series. After expressing both the data signal and the ASE noise in a Fourier series, the decision sample can be readily formulated and will comprise three terms: data signal, signal-ASE beat noise, and ASE-ASE beat noise. By writing these three terms in a matrix form and using matrix transformation techniques, both types of noises can be combined into one term. This combined noise term can be expressed as a non-central quadratic form of Gaussian random variables and has a non-central chi-square distribution. Therefore, the moment generating function (MGF) of the decision sample can be derived. The probability density function (PDF) of the photocurrent can be obtained  
Figure 7: Unrepeated ultra-long span optical transmission system over 390 km length consisting of a bidirectional distributed Raman amplifier, a three-segment hybrid fiber effective core area, chromatic dispersion compensation fiber (DCF), and erbium-doped fiber amplifier (EDFA). The blue and red arrows indicate the optical signal and Raman pump, respectively. from the MGF by using the inverse Laplace transform and saddle-point integration approximation. The calculated PDF reflects the exact statistics of the photocurrent appearing at the decision gate and shows the overall estimated error probability for different required OSNR values [31; 37; 38; 39]. In a phase-modulated DBPSK receiver, the KLSE method is used to obtain the MGF of the combined decision sample by multiplying the MGFs of the two photocurrent branches of a balanced photodiodes [38]. The probability of an error occurring at the sample time upon the data sequence can be calculated after setting the decision threshold value. Therefore, the BER can be estimated over a range of required OSNR values. In computing the BER for a DQPSK system, we need to compute the BER for the in-phase (I) and quadrature (Q) signals separately and then average the BER values of I and Q. In conclusion, with knowledge of the exact statistics of the photocurrents appearing at the decision gates, it becomes possible to estimate the BER performance by integrating over the PDF of the photocurrents means over a different OSNR or SNR per bit (Eb/No), without needing to simulate a very large number of bits, as is necessary in Monte Carlo simulations. The bit sequence used in the simulation is a De Bruijn bit sequence (DBBS) of length \(2^{12}\), which is equivalent to a pseudo-random binary sequence (PRBS) \(2^{12}-1\) with an additional \"0\" bit inserted into the longest chain of zeros. In the DBBS-generated bit pattern, every possible distinct bit combination occurs exactly once, thus correctly representing the average distribution of bit combinations generated by the statistically independent data generator [40].