Coherent Optical OFDM with Index Modulation for Long-Haul Transmission in Optical Communication Systems

In this paper, it is proposed to enhance coherent optical orthogonal frequency-division multiplexing (OFDM) system by using index modulation in terms of bit error rate (BER) and spectral efficiency for different levels of the fiber nonlinearity. In the coherent optical OFDM with index modulation system (CO-OFDM-IM), the information is conveyed not only through activated subcarriers but also by indices of the active subcarrier that are dynamically updated at the transmitted symbol rates. Unlike the conventional coherent optical OFDM (CO-OFDM) system, adjusting the number of the active subcarriers in the CO-OFDM-IM system provides improved BER performance and reduction the signal peak-to-average-ratios, when the fiber nonlinearity becomes significant. The simulation results illustrate that the proposed CO-OFDM-IM system has better BER performance when the proposed CO-OFDM-IM system and the CO-OFDM system have similar spectral efficiency. Also, the CO-OFDM-IM system can provide better spectral efficiency than the CO-OFDM system at the low-to-medium order modulation such as BPSK, QPSK, and 16QAM.

about what are the best suitable waveforms. Various modulation techniques, such as the filter bank based multicarrier (FBMC) [5], windowed-OFDM (W-OFDM) [6], filtered-OFDM (F-OFDM) [7] and generalized frequency division multiplexing (GFDM) [8], have been proposed for both the obtaining of high spectral efficiency and the using of them in low power applications. Among possible candidates, orthogonal frequency-division multiplexing (OFDM) is already used in most modern communication systems, including 4G, 5G, and Wi-Fi. The main benefits of OFDM are a very efficient hardware implementation and single-tap equalization at the receiver. Also, OFDM has been widely used in both wireless and optical communication systems.
Index modulation (IM) is a technique inspired by the idea of a spatial modulation scheme that is capable of achieving high spectral efficiency by way of allowing transmitting additional bits compared with the conventional modulation techniques [9]. The IM has been intensively investigated in different communication systems such as multi-carrier systems [9], MIMO systems [10], molecular communication [11], millimeter-wave transmission [12], visible light com-munication, [13]- [15] and polar codes [16].
In 2013, Başar et al. proposed firstly OFDM with index modulation (OFDM-IM) scheme [9]. OFDM-IM both provides innovative ways [8], [17]- [22] to convey information compared to conventional OFDM and offers advantages in terms of spectral and energy efficiency [23]. In OFDM-IM, the information is conveyed not only by the active subcarriers, which are activated to carry constellation symbols but also by the indices of the active subcarriers, which are used for the transmission of the additional binary information. So OFDM-IM offers a trade-off between spectral efficiency and system performance by the adjustment of the number of active subcarriers [9].
Higher launch power levels are desirable to extend the reach of transmission in long-haul links. However, in longhaul CO-OFDM, significant distortions are due to the fiber nonlinearity. Higher input power levels result in performance degradation due to the effect of the fiber nonlinearity [24]. To improve spectral efficiency and use the advantages of OFDM-IM compared to OFDM, OFDM-IM can be employed in coherent optical communication systems.
The effect of fiber nonlinearity causes nonlinear interference between subcarriers caused by adjacent subcarriers in OFDM. Therefore, in this paper, it is aimed to reduce the number of active subcarriers at the same spectral efficiency in order to mitigate the fiber nonlinearity effect. While index modulation provides the possibility to transmit additional information bits using subcarrier indices in OFDM, some subcarriers become inactive. So, index modulation is proposed to reduce the number of nonlinear interfering subcarriers by using inactive subcarriers and to enhance BER performance at the same spectral efficiency for coherent optical OFDM systems.
Simulation results demonstrate that the proposed coherent optical OFDM-IM can improve the spectral efficiency, and the BER performance can be enhanced at the same spectral efficiency in comparison with conventional coherent optical OFDM when the fiber nonlinearity is considered.

A. TRANSMITTER
The block diagram of the proposed coherent optical OFDM with index modulation (CO-OFDM-IM) system is depicted in Fig. 1. For the transmission of each OFDM symbol, a variable number of the information bits enter the transmitter of the CO-OFDM-IM system. These bits are firstly split into OFDM subblocks. G is the total number of the OFDM subblocks. Each OFDM subblock contains subcarriers of length n = N F F T /G , where N F F T is the size of the fast Fourier transform (FFT). p = p 1 + p 2 bits is mapped to an OFDM subblock of length n. The mapping operation is not only performed by means of the modulated symbols, but also by the indices of the subcarriers. The indices of k active subcarriers are determined by a selection procedure based on the first p 1 bits of the incoming p-bit sequence for transmitting additional information bits. The number of additional information bits can be given by where . is the floor operation and C (·) is the combinatorial operation.
For each g th OFDM subblock (g = 1, 2, . . . , G), p 2 bits of the incoming p-bit sequence are mapped onto the M -ary signal constellation, where M is the constellation size, in order to determine the data symbols that are transmitted over the k active subcarriers. The number of the remaining p 2 bits can be given by p 1 bits are sent to an index selector to generate indices which indicate to k active subcarriers. The index selection is implemented by using combinatorial method. The combinational number system of degree k provides a one-toone mapping relationship with the set of k active subcarrier indices [9]. For any n and k values, the natural number Z ∈ [0, C (n, k) + 1] can be presented by a unique descending sequence J = {j k , . . . , j 1 }, which takes elements from the set according to the following equation: where j k > . . . > j 1 ≥ 0 and j k , . . . , j 1 ∈ {0, . . . , n − 1} [9]. The obtained J indices sequence from input p 1 additional information bits can be explained by Algorithm 1 [25].
Algorithm 1: Obtain J indices sequence from input p 1 bits 1: Input : The binary sequence of p 1 bits, the number of subcarriers n, the number of active subcarriers k 2: Initialization : Convert p 1 bits into a decimal number Z, jj ← n 3: for i = k : −1 : 1 do 4: repeat 5: jj ← jj − 1 6: ComCoef ← C(jj, i) 7: until ComCoef ≤ Z 8: j i ← jj 9: Z ← Z− ComCoef 10: end for 11: return array j i To make easy understanding, as an example of how active or inactive subcarriers are determined, for n = 4 and k = 2, the number of p 1 additional information bits is calculated using (1) that p 1 = log 2 (C(4, 2)) = 2. The p 1 bits are first converted to a decimal number Z. Then the combinational number system provides a one-to-one mapping between natural number Z and k-combinations (see Algorithm 1 to find the k-combination for a given number Z).
After J indices sequence is obtained from input p 1 additional information bits, the symbol in the inactive subcarriers is setted to zero. In Table 1, it is shown the subblocks obtained for different p 1 additional information bits. s a and s b are transmitted constellation symbols on active subcarriers.
The g th OFDM subblock can be represented as where s g (i) are independently obtained from a M -ary signal constellation. After concatenating X g , the equivalent timedomain signal can be calculated by using the inverse Fourier transform as follows where f g,i is the i th subcarrier frequency in the g th OFDM subblock and T is the time duration of a OFDM-IM symbol. Then parallel to serial conversion and Cyclic prefix (CP) insertion are followed. The electrical signal obtained after the inverse conversion process is converted into a transmitted optical signal with two optical IQ modulators (Mach-Zehnder Modulator -MZM) [26].

B. OPTICAL CHANNEL
The optical channel includes many Standart Single Mode Fiber (SSMF) lengths. For SSMF, within each span, the nonlinear Schrodinger equation (NLSE) models loss, chromatic dispersion (CD), and nonlinear effects as where E is the complex light envelope of the received signal after a fiber distance of z km. D is the linear part for linear effects (CD and attenuation) and N is the nonlinear part, which is given by: where α is the attenuation factor, β 2 is the group velocity dispersion parameter, γ is the nonlinearity coefficient, n 2 is the nonlinear-index coefficient, λ c is the center wavelength and A ef f is the effective core area [27]. At the end of each span, the optical signal is amplified to compensate for fiber losses by an EDFA.

C. RECEIVER
At the receiver, the received optical signal after transmission through the optical channel is converted to an electrical signal by a local oscillator laser (LO) by passing through a 90 • optical hybrid. The optical signal interferes with the local oscillator laser, which usually has the same frequency as the transmitter laser in the 90 • optical hybrid. In order to detect both real and imaginary parts, the input signal is mixed with the real part of the local oscillator laser in one branch and the imaginary part in the other branch through the 90 • phase shift between the signal and the local oscillator laser. At the CO-OFDM-IM receiver after removing CP and implementing the FFT operation, it is followed by a channel estimation operation. To be able to equalize the received signal in the frequency domain, it is important to estimate the channel frequency response.X g (i) is the received vector for the g th OFDM subblock after FFT operation, which is given bŷ where H g (i) and W g (i) denote respectively the channel frequency response and the zero-mean complex additive white VOLUME 4, 2016 Gaussian noise (AWGN) with variance N 0 on the i th subcarrier of the g th OFDM subblock. PN sequence block, which is suitable for the fiber optical channel with slow variation, is used for the channel estimation, which is given bŷ , i = 1, . . . , n (11) whereX P N g (i) and X P N g (i) are the PN sequence signal on the i th subcarrier of the g th OFDM subblock at the receiver and the transmitter respectively.
It needs to detect both the indices of active subcarriers and the corresponding additional information bits. The index detector is used to detect indices of active subcarriers from each OFDM subblock. The maximum likelihood (ML) detector of the CO-OFDM-IM system searches all combinations of subcarrier indices and the constellation points in order to a joint decision on the indices of active subcarriers and the transmitted symbols for each OFDM subblock.
The searching space of the ML detection per bit is given by the order of O C(n, k) + M k . Since the complexity increases exponentially with increasing k values, the optimal ML detector has high complexity. To achieve near optimal performance, the low-complex posteriori probability detection (PPD) method have been proposed [8].
The PPD detector of CO-OFDM-IM gives the logarithm of the ratio of a posteriori probabilities of frequency domain signals by considering the case that the values are either non-zero or zero [9]. For the g th OFDM subblock, the PPD detector provide the probability of the active status of the corresponding index i with i = 1, 2, . . . , n given by; where Q j is the element of a M -ary signal constellation. The larger the value of λ g (i), the higher the probability that the corresponding i is an active subcarrier. Using Bayes formula, (12) can be expressed as After obtaining all a posteriori probabilities based on (13), for each OFDM subblock, the receiver decides on k active indices (Ĵ) having maximum posteriori probabilities. After determining the k active indices, the additional information bits and the data symbols can be easily decided.
It takes the active indices sequences from the index detector and performs a summation of the k-combination coefficients (Algorithm 2) for each OFDM subblock. After calculating decimal numberẐ, the additional information bits can be decided with a decimal-to-binary converter. Transmitted symbols at sets of M -ary signal constellation (Q) on k active subcarriers are estimated aŝ Algorithm 2: Obtain decimal numberẐ from the indices sequenceĴ 1: Input : The indices sequenceĴ = ĵ k , . . . ,ĵ 1 , the number of subcarriers n, the number of active subcarriers k 2:Ẑ ← 0 3: for i = 1 : k do 4: ComCoef ← C ( i , i) 5:Ẑ ←Ẑ + ComCoef 6: end for 7: returnẐ

III. SPECTRAL EFFICIENCY
Spectral efficiency (SE) is an important performance metric in OFDM systems. It is defined as the bit rate that can be achieved within a unit bandwidth B. For the conventional CO-OFDM systems, the spectral efficiency can be given by, where N u is the number of active subcarriers for data transmission in the CO-OFDM systems, and T CP is the time of the cyclic prefix. For the CO-OFDM-IM systems, the spectral efficiency can be expressed as, Assuming N u = N F F T = 256 and N CP = 16, the CO-OFDM system can achieve a SE value 0.9412 bits/s/Hz for BPSK modulation. However, the CO-OFDM-IM system can achieve a SE value 0.9412 bits/s/Hz for n = 4, k = 2 and a SE value 1.1765 bits/s/Hz for n = 4, k = 3, respectively, for BPSK modulation. Hence, in the CO-OFDM-IM system, the SE value can be achieved higher than that in the CO-OFDM system according to the selected n and k values. So, the proper selection of n and k values can help enhance the SE.
The SE as a function of active subcarrier k is shown under different number of the OFDM subblocks for M = 2, N F F T = 256, N CP = 16 and BPSK modulation in Fig.  2. As shown in Fig. 2, when the proper number of the active subcarrier is chosen, the CO-OFDM-IM system can improve SE over the CO-OFDM system.
The CO-OFDM-IM can provide better spectral efficiency than the CO-OFDM at the low-to-medium order modulation such as BPSK, QPSK, and 16QAM as shown in Fig. 3. However, the spectral efficiency of the CO-OFDM-IM cannot compete with that of the CO-OFDM at the high order modulation.  The maximization of SE is almost synonymous with maximizing the number of bits per symbol. The sequence of binomial coefficients C(n, k) is a logarithmically concave sequence. Therefore, it is analytically possible to detect the optimal value of the active subcarrier which maximizes p bits [28].
Assuming dropping the floor function in (17) and then taking its derivative of p with respect to k, where the k th harmonic number Γ k is the sum of the reciprocals of the first k positive integers. Since the limit Γ k − log k goes to γ which is the Euler-Mascheroni constant, the approximate value of Γ k is expressed as, By substituting the approximate value of Γ k in (18) and making (18) equal to zero, it is obtained the optimal number of the active subcarrier,

IV. AVERAGE BIT ERROR PROBABILITY
Let X g →X g denote the pairwise error event that in g th subblock is incorrectly detected. Since the pairwise error events within different subblocks are identical and independent, the pairwise error event within a single subblock is sufficient to determine the OFDM block performance.
In the presence of channel estimation errors in which a mismatched ML detector is used for data detection, the conditional pairwise error probability (CPEP) can be expressed by [9] Pr where σ 2 e is the estimation error variance of the vector of the channel frequency response H g , (·) H , · and Q (·) denote the Hermitian transposition, the Frobenius norm and the Gaussian Q-function, respectively.
So, the CPEP can be expressed again as where EĤ g {·} denotes the expectation operation with respect to the estimated channel frequency response. To obtain the closed-form expression of the UPEP, the approximate Gaussian Q-function can be applied. To enhance the accuracy, the approximate Gaussian Q-function is proposed in [29], given by To the approximate UPEP, (26) is applied to (25), where q = n 2σ 2 e X g 2 + 2nN 0 . Using (27), the expectation can be calculated according to the spectral theorem in [9] and the UPEP upper bound can be calculated as where I n denotes n-dimensional identity matrix.K n = E Ĥ gĤ H g = K n + σ 2 e I n is the correlation matrix ofĤ g , Then, based on the UPEP derived above, an upper bound on the average bit error probability (ABEP) can be obtained as, Pr X g →X g e X g ,X g (29) where e X g ,X g represents the number of bit errors for the corresponding pairwise error event and n Xg = 2 p is the number of the possible realizations of X g .

V. SIMULATION AND DISCUSSION
To investigate the channel impairments of the long-haul coherent optical communication system, the system was installed in the receiver and transmitter shown in Fig. 1 and  simulated on the MATLAB platform. The channel block was simulated with the Split-Step Fourier Method (SSFM) for solving NLSE. This system can be made use of by many applications to simulate impairment such as dispersion effects and fiber nonlinearity in optical communication systems [30].
Optical fiber parameters and OFDM parameters are presented in Table 2 and Table 3 respectively. It is assumed in the simulation that a perfectly synchronized system compensates for carrier frequency offset. The simulations were performed over 500 independent channels. On the transmitter side, the data bits were generated using the Pseudorandom Binary Sequence (PRBS) generator, and then data mapped with the QPSK and 16QAM modulations.
It is well known that the OFDM signal is a high PAPR waveform that requires a large dynamic range of nonlinear devices such as digital-to-analog converters (DACs), power amplifiers, and external optical modulators (Mach-Zehnder Modulator). Also, the high PAPR leads to the increase of nonlinearity of optical fiber. However, the PAPR is ideally required to stay within the linear operating range of nonlinear devices at the transmitter [31]. It has been aimed that the OFDM signal has a low PAPR by using PAPR reduction techniques.
Here, taking into consideration the splitting of a N F F T subcarriers into G subcarrier blocks with k active subcarrier each, the PAPR behavior of the proposed system was analyzed. The PAPR is defined for a transmit signal v(t) as where E {·} represents the expectation or mean value over a sufficient long time period. When the performance for PAPR reduction techniques needs to be evaluated, the complementary cumulative distribution function (CCDF), which is the probability that PAPR exceeds a predefined threshold value P AP R 0 , is used as [32] CCDF (P AP R dB ) = P rob {P AP R dB > P AP R 0 } (31) Fig. 4 shows the CCDFs of PAPR reduction performance of the proposed CO-OFDM-IM with different numbers of active subcarriers and the conventional CO-OFDM. For example, the proposed CO-OFDM-IM with n = 4 and k = 3 provides about 0.5 dB reduction in PAPR than that of the conventional CO-OFDM at a CCDF value of 10 −4 for QPSK modulation. At the same CCDF value, the proposed CO-OFDM-IM with BPSK and 16QAM modulation, respectively, achieves about 0.2 dB and 0.4 dB reduction in PAPR. As a result, the partial selection of the active subcarriers in the subcarrier blocks of the proposed CO-OFDM-IM results in a reduction of the PAPR.  In Fig. 5 and Fig. 6, it is compared the BER performance of the proposed CO-OFDM-IM with the CO-OFDM at the different launch power values for QPSK and 16QAM modulation. In Fig. 5 and Fig. 6, both the effect of the launch power and, the effect of the transmission distance are observed, because the effect of the fiber nonlinearity depends on the instantaneous power of the signal and the transmission distance according to NLSE. To evaluate the effect of the fiber nonlinearity only, the laser is assumed to have no phase noise.
At a low launch power, the optical fiber channel can be modeled as a linear dispersive channel impaired by the amplified spontaneous emission (ASE) noise. However, the proposed CO-OFDM-IM performance is lower than the CO-OFDM in low launch power, because the proposed CO-OFDM-IM performance is decreased by the bit errors caused by incorrectly recovered active subcarrier indices. The minimum point of the BER curve is called the optimum launch power. Optimum launch power varies depending on modu-lation type, transmission distance, and different parameters. The right side of the optimum launch power shows the region where the effect of the fiber nonlinearity is effective. That is, the fiber nonlinearity becomes significant when the launch power increases to allow long-haul transmission. Because the fiber nonlinearity causes randomly a spreading around the central point of each subcarrier leading to adjacent subcarriers interfering with each other.
As shown in Fig. 5 and Fig. 6, the proposed CO-OFDM-IM can provide better BER performance than the CO-OFDM for mid-to-high launch power values where the fiber nonlinearity is effective. The better BER performance of the proposed CO-OFDM-IM can be explained by additional binary bits transmitted by the active subcarrier indices and the higher robustness against inter-carrier interference as a result of the number and the location of the inactive subcarriers. In other words, the CO-OFDM-IM successfully mitigates the effect of the fiber nonlinearity and supports a long-haul transmission.   Fig. 8 show the BER performance of the proposed CO-OFDM-IM with the CO-OFDM for QPSK and 16QAM modulation at SE levels of 2 bit/s/Hz and 3 bit/s/Hz, respectively. The number of both the subcarriers n and the active subcarriers k in each subblock were chosen to achieve the SE levels as close as possible. As shown in Fig. 7 and Fig.  8, the proposed CO-OFDM-IM outperforms the CO-OFDM at high launch power. For example, for the spectral efficiency value of 1.8824 bits/s/Hz, the proposed CO-OFDM-IM can provide significant BER performance improvement.
Also, the proposed CO-OFDM-IM (8,7) attains 0.12 bit/s/Hz spectral efficiency gain, although it has a similar BER performance as CO-OFDM in the right side of the optimum launch power as shown in Fig. 7. Considering these achievements, it can be said that the CO-OFDM-IM is capable of taking a good tradeoff between BER performance and spectral efficiency. Error vector magnitude (EVM) is a useful system-level metric to quantify the impact of the impairments in a system. For coherent optical communication systems, the EVM is commonly used as a measure of the received signal's quality after the channel equalizer. The EVM describes the effective distance of the received complex symbol from its ideal position in the constellation diagram [33].
The performance of average EVM is obtained for launch power = 4dB, 6dB and 8dB to quantify the impact of the linear impairment on subcarriers at a low launch power for 16QAM modulation after 500 km transmission distance. Average EVM is calculated at the receiver for the CO-OFDM and CO-OFDM-IM symbols over 1000 channel realization after the channel equalizer. At each realization, the EVM value on a given subcarrier is the square root of the ratio of the power of the error vector to the average reference constellation power. The performance of the average EVM is shown in Fig. 9 for the CO-OFDM and CO-OFDM-IM. As a result, the average EVMs of CO-OFDM and CO-OFDM-IM are similar in terms of EVM variations over subcarriers at low launch power levels as shown in Fig. 9. The performance of the average EVM is obtained for launch power = 10dB, 12dB and 14dB to quantify the impact of the nonlinear impairment on subcarriers at mid-to-high launch power values where the fiber nonlinearity is effective. The performance of the average EVM is shown in Fig. 10 for the CO-OFDM and CO-OFDM-IM.
Comparing CO-OFDM-IM with CO-OFDM, in CO-OFDM-IM system larger average EVM has been observed on the subcarriers as shown in Fig. 10. Since inactive subcarriers donâȂŹt cause nonlinear interference with adjacent subcarriers, the effect of the fiber nonlinearity on active subcarriers is mitigated.

VI. CONCLUSION
In this paper, it has been introduced the coherent optical OFDM-IM for the long-haul optical communication systems suffering from especially the fiber nonlinearity. The coherent optical OFDM-IM provides an interesting trade-off between BER and spectral efficiency. Since the number of active subcarriers can be adjusted accordingly to reach the desired spectral efficiency or BER performance, the coherent optical OFDM-IM system can bring important spectral efficiency improvement. Also, the coherent optical OFDM-IM system can provide better BER performance than the coherent optical OFDM systems for mid-to-high launch power values when the proposed CO-OFDM-IM system and the CO-OFDM system have similar spectral efficiency.
The variation of the number and the location of inactive subcarriers directly affects PAPR reduction and is robust to inter-carrier interference.