On the Subcarrier Averaged Channel Estimation for Polarization Mode Dispersion CO-OFDM Systems

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1 Vol. 1, No. 1, pp: 1-7, 2017 Published by Noble Academic Publisher URL: Open Access On the Subcarrier Averaged Channel Estimation for Polarization Mode Dispersion CO-OFDM Systems Mohamed B. El_Mashade a*, Ahmed M. Attiya b, B. Mounir a a* Electrical Engineering Dept., Faculty of Engineering, Al Azhar University, Nasr City, Cairo, Egypt b Electronics Research Institute, El-Tahreer St., Dokki, Giza, Egypt Abstract: A performance comparison among different channel estimation techniques for polarization division multiplexing coherent optical orthogonal frequency division multiplexing systems has been performed from the BER point of view. Our obtained results show that the subcarrier averaged over the maximum likelihood blind channel estimation method gives the best performance for low dispersed channels. However, for high dispersed channels, it cannot be applied directly to the blind estimated channel transfer function. So, we propose a new technique for it to be applied. The proposed algorithm makes the subcarrier averaged ML blind channel estimation to be superior in its performance. Keywords: PDM, CO-OFDM, Channel Estimation, ICA. 1. Introduction Coherent optical orthogonal frequency division multiplexing (CO-OFDM) attracts a lot of attention due to its ability to combat optical fiber channel impairments, such as chromatic dispersion (CD) and polarization mode dispersion (PMD), through electrical compensation without the need of optical compensation [1, 2]. As the channel estimation technique affects the performance of the system, a variety of channel estimation techniques has been proposed for the PDM CO-OFDM systems. They are divided into two categories: training symbol [3-5] based methods and blind methods [6, 7]. In order to enhance the performance of the channel estimation method, a pair of training symbols is transmitted in each polarization (x or y), multiple pairs of these symbols should be transmitted in time, and their average is going to be taken over time. On the other hand, the method presented in [4] is based on the previous one with the exception that only one training symbol can be used and the performance enhancement could be done by averaging over the adjacent subcarriers per OFDM symbol. The performance of this method is limited to the CD and PMD values in the optical fiber. Moreover, the method introduced in [5] offers the use of single training symbol in each polarization and channel estimation is going to be done in time domain before FFT processing. The performance of this technique is dependent upon the length of channel taps in such a way that as the number of channel taps increases, its performance becomes degraded in comparison with the time averaged method proposed in [3]. One drawback of using this method in PDM systems is its dependence on the existence of a left pseudo-inverse matrix. Generally, blind channel estimation techniques of PDM systems employ the concept of independent component analysis (ICA) [6]. Maximum likelihood (ML) method of BSS is used as an algorithm for channel estimation and leads it to have the best performance [8]. In this paper, we suggest a new channel estimation procedure which is based on frequency averaging of the blind estimated channel transfer function. For low dispersed channels, this method can be directly applied to the result of the blind channel estimation method. However, for high dispersed channels, applying the frequency averaging directly gives worst performance. So, by means of oversampling the channel transfer function before applying the subcarrier averaging method, the subcarrier averaged method performance can be enhanced. The rest of this paper is organized as follows. Section II is concerned with the system model. The channel estimation methods for PDM systems are 1

2 discussed in section III. Section IV is devoted to displaying the simulation results. Finally, our concluded remarks are outlined in section V. 2. System Model The discrete time OFDM signal with N subcarriers can be described as ( ) ( ) (1) Where n= 0, 1, 2,.., N-1 and S(k) is the information symbol at the k th subcarrier. For the sake of simplicity, we denote s(n) & S(k) by s n & S k, respectively, n & k vary as 0, 1, 2,.., N-1. On the other hand, the PDM system, in frequency domain, of each subcarrier can be represented by [ ] [ ] [ ] (2) ( ) (3) In the above expression, S kx & S ky denote the transmitted information symbols at k th subcarrier for x and y polarization respectively. Similarly, R kx & R ky symbolize their received versions at the same polarizations, and Z kx & Z ky represent the amplified spontaneous emission (ASE) noise for x and y polarizations. φ D (k) is the phase dispersion due to chromatic dispersion (CD), and H kpdm indicates the jones matrix of the optical fiber channel [7]. This matrix is given by the concatenation of M birefringence elements of short lengths, which are greater than the correlation length, and each element has its own PMD vector. The transmission matrix U for each section can be given by [ ] where, and denotes the differential group delay (DGD) value, symbolizes the angular frequency difference between adjacent subcarriers, and [r 1, r 2, r 3 ] is a unit stock vector points in the slower principal state of polarization (PSP) direction. Both the DGD value and the PSP vector formulate the PMD vector. Without the loss of generality, the polarization dependent loss (PDL) is neglected and the channel transfer function is assumed to be constant over the OFDM frame. 3. Channel Estimation Methods In this section, the previous and the proposed channel estimation techniques are summarized Time Averaging Method (T avg ) According to [3], using a pair of training symbols at two consecutive OFDM symbols as: Consequently, the received data would be (4) [ ] * + at t 1 (5a) [ ] [ ] at t 2 (5b) [ ] [ ] (6) Then, the estimated channel transfer function at subcarrier k is given by [ ] (7) 2

3 To enhance the performance of this method, multiple training symbols should be employed and the averaging process is done over Eq.(7) Intra-Symbol Frequency Domain Averaging (ISFA) Based on Eq.(7), rather than applying average over multiple training symbols, averaging is done over adjacent m subcarriers left and right, that results in a totally averaging across (2m+1) subcarriers. Therefore, the ISFA estimated channel transfer function of subcarrier k is given by (8) 3.3. Time Domain Maximum Likelihood (TDML) The received OFDM symbol in time domain over a channel has a 2L+1 taps with a sufficient cyclic prefix and suffix to transform the linear convolution into a circular one that can be represented in two forms: In this representation, r=[r 0 r 1 r 2 r N-1 ] T, where the superscript T denotes transpose, is the received symbol, T is a circulated matrix of dimension NxN whose first line is given by [h L h L-1.. h h 2L h 2L-1.. h L+1 ], where h i symbolizes i th channel tap, and s represents the transmitted OFDM symbol s=[s 0 s 1 s 2 s N-1 ] T. In the second form, A is N x (2L+1) matrix whose elements can be formulated as: ([ ] ) (10) On the other hand, is an 2L+1 length channel impulse response (CIR) vector, and z indicates the ASE noise vector of length N. For the PDM system, by using a single training symbol for each polarization, the corresponding received symbol is given by (9) [ ] [ ] [ ] [ ] (11a) [ ] [ ] [ ] (11b) [ ] According to Eq.(11b), the ML of the CIR can be specified as: [( ) ] [ ] (12a) With [ ] [ ] (12b) This method depends on the existence of the left pseudo-inverse to the matrix of dimensions 2Nx(8L+4) which in turn requires (2L+1) N/2. Fourier transform is applied to ( ), given by Eq.(12), then frequency domain equalization can be processed. The estimated channel transfer function for the k th subcarrier can be evaluated with the aid of: [ ( ) ( ) ( ) ( ) ] (13) 3

4 It is evident that the performance of this method depends on the memory length of the channel in such a way that as the length increases, as the performance closes to the T avg method Blind Complex Maximum Likelihood (CML) Using ICA, we can estimate the inverse of the channel transfer function for each subcarrier separately, taking into account the fact that the transmitted information has a non-gaussian distribution. Dropping the noise term in Eq.(2), for sake of simplicity, we can rewrite it as: [ ] [ ] [ ] (14) Using the CML estimator, the de-mixing matrix W defined as: [ ] [ ] [ ] (15) Can be estimated. In other words, the k th subcarrier de-mixing matrix W can be calculated, according to the CML algorithm, using the recurrence relation: [ ( )( ) ] Where "i" is the iteration number and (16) ( ) ( ( )) ( ( )) (17) Appling the de-correlation property to the de-mixing matrix yields: [ ( ) ] (18) Repeating Eqs.(16 & 18) till the convergence between two consecutive iterations attains, the channel transfer function, for subcarrier k, will be estimated and the result is given by (19) The blind channel estimation suffers from two problems: the scaling and ordering. A pilot symbol will be used to resolve these two problems. However, the performance of blind estimators can be enhanced if the averaging process is taken over a larger numbers of OFDM symbols, where a better statistical averaging will be achieved Subcarrier Averaging Technique (proposed) Averaging across adjacent subcarriers reduces the noise level and thus improving the system performance. However, averaging is limited to the coherence bandwidth of the channel. Low dispersed channel has a large coherence bandwidth whilst high dispersed channel results in small coherence bandwidth. For low dispersed channel, subcarrier averaging is applied directly. Appling subcarrier averaging to the 2m+1 adjacent subcarriers, the CML blind channel estimator can be computed through the following formula: (20) 4

5 For high dispersed channel, subcarrier averaging cannot be applied directly due to the small coherence bandwidth. With the help of oversampling method, the coherence bandwidth can be occupied with a sufficient number of pseudo-subcarriers and consequently it becomes suitable for applying subcarrier averaging technique. After the processing of average, the down-sampling is applied to the net oversampled channel transfer function to return it to its natural state. At first, we transform the estimated channel transfer function into time domain, then oversampling at level O is done by padding {(O-1)N} zeros at the middle of the CIR as Eq.(21) demonstrates. Then, the oversampled channel transfer function can be obtained by transforming it back to the frequency domain. ( ) { ( ( ) ) (21) The application of subcarrier average over the oversampled channel transfer function, will lead to: (22) The net estimated averaged transfer function is accomplished after down-sampling Eq.(22) at the order O and the result becomes: ( ) ( ) (23) 4. Simulation and Results In this section, our numerical simulation results are provided to investigate the performance of the proposed channel estimation method against the previous discussed methods. Simulation is done using MATLAB. An original data rate of 100 Gbps is transmitted over a 1000 km SSMF, using 16-QAM modulation format. For high dispersion scenario, we use a GVD value of and the process of equalization is achieved through the using of 256 OFDM subcarriers. For low dispersion scenario, on the other hand, a GVD value of and the procedure of equalization is carried out by using 128 OFDM subcarriers. The overall DGD value is fixed to in both cases. Sufficient cyclic prefix and suffix in each case is employed. The OFDM frame is taken to be of 100 OFDM symbol. (Fig.1) shows the BER performance for the different channel estimation techniques under low dispersed channel. Using two training symbols for each polarization, time averaging (T avg ) method has the poorest performance, whilst the ML blind estimator gives better performance than T avg by about 2dB. TDML, with only one training symbol for each polarization, has a performance which lies in the mid-way of the above mentioned methods. With the aid of subcarrier averaging over T avg method accompanied by ISFA with m=2, the second level performance is accomplished. Due to the fact that, ISFA is the subcarrier averaging version of the T avg method and the ML blind method has a better performance than T avg, subcarrier averaging over the ML blind method introduces a better performance than ISFA method. The confirmation of this conclusion is demonstrated throughout the displayed results in (Fig.1), where the ML blind with subcarrier averaging using m=2 has the highest performance than the other methods. For high dispersed channel, the TDML could not be applicable owing to the channel memory length which is greater than half of the number of subcarriers. Therefore, the left pseudo-inverse of the channel matrix given in Eq.(12), is no longer exist. So, the BER performance comparison, as shown in (Fig.2) is limited to T avg and ML blind methods along with their subcarrier averaged versions. It is showed that both T avg and ML blind methods have a stable performance for low and high dispersed channels. ISFA with 5

6 m=2 is out of competition due to its worst performance for high SNR values. It is clear that ISFA with m=1 reaches saturation at 10-3 BER level. Applying oversampling with O=4 to ISFA for m=2 method gives a better performance than T avg with about 1dB but still less than the ML Blind method. As in low dispersed channel, the best performance belongs to subcarrier averaged ML blind method, taking into account that oversampled has to be firstly performed with O=4 and m=2. 5. Conclusions In this paper, we compare the BER performance of different channel estimation techniques along with our proposed one. The displayed results demonstrate that the proposed subcarrier averaging over ML blind method gives the best performance in low dispersed channel. Additionally, we introduce an oversampling technique as a solution to enable the subcarrier averaging process in high dispersed channel. The application of this process showed that, for high dispersed channel, the modified subcarrier averaged ML blind method incorporated the processing of oversampling still has the best BER performance. Fig 1. BER performance comparison among different channel estimation techniques for low dispersed channel. Fig 2. BER performance comparison among different channel estimation techniques for high dispersed channel References [1] W. Shieh and C. Athaudage, "Coherent optical orthogonal frequency division multiplexing," Electronics Letters, vol. 42, pp , [2] W. Shieh, "PMD-supported coherent optical OFDM systems," IEEE Photonics Technology Letters, vol. 19, pp , [3] S. L. Jansen and I. Morita, "Polarization-division-multiplexed coherent optical OFDM transmission enabled by MIMO processing, high spectral density optical communication technologies, optical and fiber communications reports," vol. 6, pp ,

7 [4] X. Liu and F. Buchali, "Intra-symbol frequency-domain averaging based channel estimation for coherent optical OFDM," Opt. Express, vol. 16, pp , [5] X. Fang, C. Yang, and F. Zhang, "Time-domain maximum-likelihood channel estimation for PDM CO-OFDM systems," IEEE Photonics Technology Letters, vol. 25, pp , [6] A. Hyvarinen and O. Erkki, "Independent component analysis: Algorithms and applications," Neural Networks, vol. 13, pp , [7] J. Gordon and H. Kogelnik, "PMD fundamentals: Polarization mode dispersion in optical fibers," in Proceedings of the National Academy of Sciences, 2000, pp [8] X. Li, W. Zhong, A. Alphones, and C. Yu, "Channel equalization using independent component analysis in PDM-CO-OFDM," IEEE Photonics Technology Letters, vol. 26, pp ,