In this chapter, we review various applications of Kalman filtering for coherent optical communication systems. First, we briefly discuss the principles of Kalman filter and its variations including extended Kalman filter (EKF) and adaptive Kalman filter (AKF). Later on, we illustrate the applicability of Kalman filters for joint tracking of several optical transmission impairments, simultaneously, by formulating the state space model (SSM) and detailing the principles. A detailed methodology is presented for the joint tracking of linear and nonlinear phase noise along with amplitude noise using EKF. Also, approaches to enhance the performance obtained by EKF by combining with other existing digital signal processing (DSP) techniques are presented. Frequency and phase offset estimation using a two stage linear Kalman filter (LKF)/EKF is also discussed. A cascaded structure of LKF and EKF by splitting the SSM to jointly mitigate the effects of polarization, phase and amplitude noise is also presented. The numerical analysis concludes that the Kalman filter based approaches outperform the conventional methods with better tracking capability and faster convergence besides offering more feasibility for real-time implementations.
- optical communication systems
- coherent optical transmission
- digital signal processing
- nonlinear mitigation
- phase noise
- amplitude noise
In order to meet the yearning demands of bandwidth and capacity due to ever increasing data traffic, the contemporary research in the field of optical transmission, is focused on developing 400 Gbps and above, Ethernet transmission [1, 2, 3, 4, 5]. The achievable information rates using optical fiber as communication channel have been rapidly increased over the past few decades. Some of the technology breakthroughs behind this rapid increase in the transmission capacity, can be listed as the invention and development of the erbium doped fiber amplifiers (EDFA), wavelength division multiplexing (WDM) systems, coherent detection, digital signal processing (DSP) techniques and forward error correction (FEC) schemes ensuring reliable transmission. The advent of coherent detection along with subsequent DSP made it possible to deploy spectrally efficient higher order modulation formats and multiplexing techniques [6, 7]. Moreover, it has also made feasible to digitally equalize the optical fiber transmission impairments , which are the main hurdle to increase the bandwidth-distance product. The transmission capacity can be increased several times by employing complex modulation formats like m-ary quadrature amplitude modulation (with m = 4, 16, 64 and so on), and multiplexing techniques like polarization division multiplexing (PDM) and WDM. However, they are more vulnerable to the optical transmission impairments as well as to the carrier phase and frequency offset (FO). Hence, effective DSP algorithms for combatting with the channel impairments were under active research over the past decade [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Consequently, coherent optical receivers are well developed and employ digital filters that allow for effective equalization of fiber linear impairments like chromatic dispersion (CD) and polarization mode dispersion (PMD) in the electric domain . Typically, CD can be compensated by either frequency or time domain filters using finite impulse response (FIR) or infinite impulse response (IIR) design. Optical receivers exploiting polarization diversity should also compensate for the random fluctuations of the polarization state caused by the stochastic change of fiber birefringence. PMD compensation is widely performed using constant modulus algorithm (CMA)  or multi modulus algorithm (MMA) . Attributed to these well-developed linear equalization techniques, fiber nonlinearity still remains a bottleneck for increasing the capacity and transmission reach .
Although, multiple information bits being encoded in a single symbol significantly increase the spectral efficiency, the signal becomes more sensitive to the amplified spontaneous emission (ASE) noise that is added in the optical amplifiers along the transmission link. Therefore, a reliable transmission over long distance demands the signal to be launched into the optical fiber at higher power, to ensure a sufficiently high optical signal to noise ratio (OSNR) at the receiver. However, the maximum transmittable launch power per fiber span is constrained by the Kerr nonlinear effects, including self-phase modulation (SPM) and cross phase modulation (XPM) in case of WDM systems, which results in signal degradation . This degrading impact of Kerr nonlinearity is much more severe in multi-channel systems with increasing number of channels . Moreover, the nonlinear phase noise (NLPN) resulting from the signal and ASE noise interactions at high launch powers, deteriorates the signal quality further. On the other hand, signal transmission at low launch powers is limited by the ASE noise. Therefore, mitigation of fiber nonlinearity is vital to enhance the capacity ensuring reliable transmission. Consequently, several nonlinear mitigation techniques have been proposed in the recent era, of which, digital backward propagation (DBP) , maximum likelihood sequence estimation (MLSE) based nonlinearity mitigation , spectral inversion [18, 19, 20], phase conjugated twin waves [21, 22] and perturbation based approaches  gained considerable attention. However, the real time implementation of these algorithms is extremely challenging owing to either the high required computational effort or the higher bandwidth consumption. Although computationally complex, DBP has drawn significant attention owing to its capability of mitigating linear and nonlinear impairments simultaneously, provided the channel characteristics are known and the step size is sufficiently small when solving the inverse nonlinear Schrodinger equation (NLSE). Several strategies have been proposed in the literature to reduce the number of required DBP steps and enhancing the performance either by including temporal correlations [27, 28] or optimizing the parameters . Nevertheless, the performance of DBP significantly deteriorates in the presence of stochastic impairments like laser phase noise and NLPN [30, 31, 32]. Moreover, its applicability is limited to single channel systems [33, 34].
Apart from fiber linear and nonlinear impairment compensation, digital carrier synchronization has also become an essential component of the coherent receivers, for synchronizing the phase and frequency offsets between the transmitter laser and the local oscillator (LO), eliminating the necessity of a phase locked loop. Several carrier phase estimation (CPE) techniques have been proposed for suppressing the laser phase noise [35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. CPE being a low complex technique is also under wide investigation to compensate the nonlinear phase shift owing to Kerr effect, besides laser phase noise [30, 44, 45, 46, 47]. Investigations were also carried on the combined performance of DBP and CPE, in order to reduce the number of DBP steps per span and there by its complexity [31, 48]. It was reported in [30, 45], that the considered CPE methods outperform the DBP technique implemented using asymmetric split step Fourier method (SSFM) with one step per span and without any parameter optimization. However, the accumulated ASE noise and the NLPN at high signal powers pose a challenging constraint on the conventional CPE limiting its nonlinear mitigation capability. Moreover, a phase unwrapping function  is typically required by CPE, which might increase the probability of cycle slips  and error propagation. Furthermore, the commonly employed CPE techniques have low tolerance towards the frequency offset (FO) between the transmitter laser and the LO. Therefore, a separate FO estimation module is necessary .
In the recent era, Kalman filtering has gained huge attention in the field of optical communication systems, owing to its potential capability to mitigate several optical transmission impairments simultaneously. The Kalman filter was first developed by R. E. Kalman in 1960. In , he presented a new approach to the linear filtering and prediction problems by introducing the state space notation, where the random processes/signals to be estimated are represented as the output of a linear dynamic system perturbed by uncorrelated noise. This approach facilitates recursive computation of the optimal solution and highly reduces the computational effort as compared to the conventional Wiener filter besides eliminating the memory problems. The so called Kalman filter computes the optimal solution recursively in the minimum mean square error (MMSE) sense. While the applicability of the Wiener filter is limited to stationary processes, the Kalman filters can be also applied to the non-stationary processes. An added advantage of the Kalman filter is its extended applicability also to the nonlinear systems through an approximate linearization and the so called filter is known as extended Kalman filter (EKF). This has attracted the Kalman filters for numerous real-time applications in the fields of navigation, radar, mobile communications, speech signal processing and so forth. Currently, EKF is under wide investigation in coherent optical communication systems for tracking and mitigating linear and nonlinear phase noise, amplitude noise, phase and frequency offsets as well as polarization de-multiplexing [12, 13, 31, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66]. Moreover, enhancing the performance obtained from EKF by incorporating with other existing techniques like DBP has also been studied in [31, 58, 59]. EKF requires only a few complex multiplications to recover one data symbol. Besides the advantage from low complexity, it further offers other benefits including faster convergence, joint tracking and compensation of fiber impairments. Therefore, it is worth discussing and reviewing the applications of Kalman filters for coherent optical communications in a nutshell.
This chapter is organized as follows: In Section 2, we discuss the principles of Kalman filter by describing the state space notation and the recursive equations. We further present some variations of the Kalman filter, namely, EKF and adaptive Kalman filter (AKF). Section 3 details the applications of EKF for coherent optical communications. We illustrate how to employ Kalman filtering for the joint tracking of several optical transmission impairments by formulating the state space model (SSM) and detailing the working principles. We also describe our numerical model and present the results to justify the theoretical findings. Finally, the chapter is concluded with a note on the key points, in Section 4.
2. The Kalman filter
A Kalman filter is an optimal recursive linear MMSE estimator that estimates the state of a linear dynamic perturbed by noise. Since the true state of the system is not observable, instead we obtain the measurements or observations that are corrupted by noise. Now, the goal of the Kalman filter is to obtain an optimal estimate of the unknown state from the noisy observations recursively. The stochastic process under estimation is modeled by a state space model (SSM) which facilitates the recursive nature of the Kalman filter. In the following, we present the general framework of the Kalman filtering and also discuss briefly the principles of the EKF and AKF.
2.1. Principles of Kalman filter
Consider a discrete-time, linear, time varying system in the state space notation, given by Eqs. (1) and (2). Eq. (1) describes how the true state of the system evolves over time and is known as the state or the process equation. Eq. (2) describes how the measurements are related to the states and is known as measurement or observation equation. Here,
The Kalman filter computes the optimal state recursively, following a predictor-corrector structure, where a prediction is computed prior to the availability of the observation at current time instant
During the update step, when the new observation at time
2.2. Extended Kalman filtering
In Section 2.1, we addressed the problem of estimating the unknown state of a linear dynamic system from noisy observations. Now, we consider the filtering problem for nonlinear system dynamics (either the process or observation model or both being nonlinear). The Kalman filter solution can be adopted for the nonlinear dynamic systems through an approximate linearization procedure and the resulting filter is known as EKF. Consider a nonlinear dynamic system described by the SSM given in Eqs. (12) and (13). Here,
The nonlinear system dynamics can be linearized through a first order Taylor approximation at each time instant, around the most recent state estimate. This forms the basic idea of EKF. Let,
2.3. Adaptive Kalman filtering
The Kalman filter computes the optimal solution, provided the process noise and measurement noise covariances,
Since the Kalman gain
In the case of EKF, the same procedure can be followed for adaptive estimation of noise covariances, by replacing the measurement matrix with the Jacobian matrix.
3. Kalman filtering for coherent optical communications
3.1. Kalman filtering for carrier phase and amplitude noise estimation (CPANE)
Digital carrier phase estimation (CPE) has become an essential component of coherent optical receivers to recover the carrier phase perturbed by laser phase noise arising from the transmitter laser or LO [35, 36, 37, 38, 39, 40, 41, 42, 43]. Several CPE techniques have been developed in the literature based on feedback [39, 40] or feed forward loops [35, 36, 37]. Depending on how the data phase is wiped off, they can be further classified into decision directed (DD) [35, 40, 42, 46] or non-decision directed (NDD) methods [39, 41, 43]. NDD methods like Viterbi-Viterbi  CPE has gained high attention due to its ease of implementation. However, it employs m-th power scheme to remove the data modulation and therefore, are only better suited for QPSK systems. However, for higher QAM systems, DD-CPE methods exhibit better performance compared to NDD CPE methods [35, 42].
Apart from tracking the carrier phase, CPE being a low complex technique, can also be employed for compensating the nonlinear phase shift stemming from the Kerr nonlinear effects [30, 44, 45, 46, 47]. However, the nonlinear mitigation performance of CPE is limited in the presence of ASE noise and at high launch powers. Moreover, a phase unwrapping function is typically required for CPE that might increase the probability of cycle slips [35, 49]. Addressing these problems, we have proposed a CPANE algorithm using EKF in [12, 53] for the joint mitigation of linear and nonlinear phase noise as well as ASE induced phase and amplitude distortions. Unlike CPE, EKF-CPANE estimates a complex quantity, and therefore, no argument function is required which eliminates the ambiguity associated with multiples of 2π and consequent cycle slips.
Kalman filter based CPE has been introduced and numerically verified in . From the numerical results, it was reported that the Kalman based phase estimation combined with DD equalizer in a feedback configuration outperforms the conventional CMA based approach . CPE based on EKF was demonstrated and verified experimentally for QPSK and 16-QAM systems in . In , EKF has been investigated for characterizing the laser phase and amplitude noise. EKF based carrier synchronization has also been verified experimentally, in combination with expectation maximization (EM). A carrier recovery scheme based on block estimation process with Kalman filter has been demonstrated in . This approach was verified experimentally for 16 and 64-QAM signals. However, these Kalman filter based approaches estimate an argument which involves sine and cosine functions, computation of the Jacobian matrix and also require matrix multiplications and inversions, which increases the computational complexity. The proposed method in [12, 53], estimates a complex quantity accounting also for the phase and amplitude distortions arising from the ASE noise in addition to the carrier phase. The variables in the SSM reduce to scalars and therefore, the vectors and matrices are reduced to scalars which will ease the computational effort. In the following, we first describe the general principles of CPE. Later, we explain our proposed CPANE algorithm illustrating its principles and implementation details using EKF.
3.1.1. Principles of CPE
Consider an m-ary QAM received signal on single polarization, which is sampled and compensated for linear impairments. Assuming perfect linear equalization, the
3.1.2. Principles of CPANE
The effects of
3.1.3. EKF-CPANE for joint mitigation of phase and amplitude noise
As discussed earlier, CPANE algorithm can be employed for the joint mitigation of phase and amplitude noise. However, it requires a reliable tracking of the complex quantity
3.1.4. Numerical analysis of EKF-CPANE
The performance of EKF-CPANE algorithm for mitigating the laser phase noise, fiber nonlinearity besides the ASE induced phase and amplitude distortions has been verified through numerical simulations on single channel systems in [12, 53] and multi-channel systems in . Here, we briefly discuss the numerical model and present a few simulation results reproduced from , so that the flow of the readers is not interrupted. The numerical model of polarization multiplexed (PM) m-QAM coherent transmission system including a DSP module at the receiver, is depicted in Figure 3 . Here, we consider the PM-m-QAM transmitter with m = 16 and 64, operated at 28 and 18.667 GBaud, respectively. These signals are transmitted through a standard single mode fiber (SSMF) link at different launch powers. The SSMF has the following parameters: attenuation coefficient (
The bit error rate (BER) performance of the considered algorithms is evaluated and a Q-factor is computed as 20log10 erfcinv(2*BER). The Q-factor vs. launch power curves for 16-QAM and 64-QAM are depicted in Figure 4(a) and (b) , respectively. It can be seen that EKF-CPANE exhibits better performance compared to DD-CPE, DD-PLL and U-CPE in both linear and nonlinear regimes. This performance enhancement is better visible compared to the DD-CPE method. For PM-64-QAM, it can also be seen that the DD-CPE experiences cycle slips occurring through the error propagation of wrong decisions which can be seen in Figure 4(b) at launch powers ranging from −2 to 1 dBm . Since the performance of DD algorithms strongly depends on the pre-decisions made by the algorithm, we study the impact of ideal error free decisions on their performance by replacing the pre-decisions
3.2. EKF and DBP for fiber nonlinear mitigation
In Section 3.1, we have described how the EKF can be employed for the joint mitigation of phase and amplitude noise. From the numerical results discussed in Section 3.1.4, it can be concluded that the EKF-CAPNE algorithm shows promising results in mitigating the linear and nonlinear phase noise as well as amplitude noise simultaneously besides less computational effort. Although, EKF-CPANE outperforms several other considered CPE methods, the effectiveness of EKF-CPANE in mitigating fiber nonlinear effects can be further enhanced by reducing the number of errors in the pre-decisions
In , it was reported that the EKF-CPANE outperforms the asymmetric split step Fourier method (ASSFM) based one step per span (OSPS) DBP with optimized nonlinear co-efficient
For the case of multi-channel systems, also, a detailed analysis has been performed in , on the combined performance of DBP and EKF for mitigation of inter and intra channel nonlinearities besides phase and amplitude noise. Here, the DBP is employed by considering the temporal correlations between the neighboring signal samples and is termed as correlated DBP (CDBP) [27, 28]. This approach will improve the accuracy in computing the nonlinear phase shift and there by enhances the nonlinear mitigation performance. Since the optimization of nonlinear coefficient plays a vital role on the performance of DBP, we proposed an amplitude dependent optimization (AO)  of the nonlinear coefficient, according to the discrete amplitude levels present in the higher order modulation formats like 16-QAM. The combined performance of AO-CDBP and EKF-CPANE for WDM systems with varying number of channels has been investigated in . Analogous to the single channel systems, the combined performance of AO-CDBP and EKF yields an improved performance also for the WDM case. However, with increasing impact of the cross phase modulation (XPM) as the number of channels increase, the gain obtained from their combined performance starts vanishing which can be observed in Figure 5(b) .
3.3. EKF for mitigation of nonlinearities in dispersion managed links
Since the advent of coherent detection and DSP for coherent optical receivers, CD can be effectively compensated by digital equalization in the electric domain ad thereby, eliminating the need for dispersion compensating fibers (DCF). However, nonlinear mitigation in the dispersion managed (DM) links is also vital in order to upgrade existing links. Although, the computational complexity of DBP is quite high, for DM links, the DBP algorithm can be simplified by assuming that the nonlinear behavior repeats itself every span and therefore, the total nonlinearity after
The numerical model of PM-16-QAM coherent transmission system over DM link  is depicted in Figure 6 . Here, a fully compensated periodical DM link with several spans has been considered. Each span consists of 80 km of SSMF and 17 km of dispersion compensating fiber (DCF). The SSMF has the following parameters:
3.4. Kalman filtering for polarization de-multiplexing
An effective way to double the transmission capacity is to employ PDM which allows the transmission of two information signals simultaneously on the orthogonal polarization states of the same optical carrier wave. However, due to fiber birefringence, the state of polarization is not preserved during the propagation on the fiber that leads to crosstalk upon the receipt of the signal. In coherent receivers, CMA  or MMA  is commonly employed in order to align the polarization states and recover the transmitted signal fully. However, CMA or MMA suffer from the drawbacks of low convergence speed and singularity problem . Moreover, a separate phase estimation scheme is required to track the laser phase noise. Since the Kalman filter allows simultaneous tracking of several state variables provided a precise SSM, the Kalman filter and its variations including radius directed linear Kalman filter (RD-LKF), EKF and UKF are widely investigated for tracking the complex elements of the Jones matrix along with the carrier phase [61, 62, 63].
3.4.1. RD-LKF, EKF and UKF for joint tracking polarization state and phase noise
An EKF has been proposed in  for joint tracking of the polarization and phase noise. It has also been reported that the EKF shows faster convergence than the conventional approach based on CMA and VV-CPE . However, the variables in the state vector are restricted to real values, which would lead to singularity problems or divergence of the filter , besides increasing the dimensions of the vectors and matrices in the Kalman recursive equations. A polarization state tracking scheme using Kalman filter, which is immune to phase/frequency offset, has been introduced in , and is termed as RD-LKF. Although, it shows faster convergence compared to CMA, this method needs significant modifications for applying to higher order QAM. Moreover, it is not possible to track the carrier phase simultaneously with the polarization state. The joint tracking of polarization state and carrier phase using EKF has been experimentally verified in . A reduced SSM using UKF has been introduced in , which facilitates the joint tracking of polarization state and phase noise. Here, the variables of the state vector are considered to be complex valued. This approach exhibits better performance compared to EKF at high OSNRs at the expense of additional computational effort.
3.4.2. Adaptive cascaded Kalman filtering (A-CKF) for polarization de-multiplexing with simultaneous tracking of phase and amplitude distortions
A cascaded Kalman filtering (CKF), a series of EKF and linear Kalman filtering (LKF) for joint tracking of phase and amplitude distortions besides polarization state, has been proposed in . By splitting up the conventional SSM into linear and nonlinear SSM, the inaccuracies in the linearization of the SSM as a whole can be reduced and thus CKF exhibits enhanced performance besides no increased computational cost compared to the approaches like UKF  and radius directed (RD) LKF . Since the optimal performance of the Kalman filter depends on the noise covariances, we proposed an adaptive CKF (A-CKF)  to adapt the process noise covariance recursively using the covariance matching method as described in Section 2.3.
220.127.116.11. Principles of A-CKF
The transmitted and received signal in the presence of phase noise and polarization rotation can be related as given in Eq. (36). Here,
Numerical investigations on both back-to-back (BTB) and transmission scenarios, have been carried out in , on the variations of the Kalman filter including EKF, UKF, CKF and A-CKF, for tracking the polarization state and phase noise jointly and are compared to the conventional MMA algorithm. Since the MMA can track only the polarization state, it is accompanied by a DD-CPE algorithm for the phase noise mitigation. It can be concluded from  that the CKF and A-CKF outperform the rest of the considered algorithms with a better tolerance towards polarization rotations, phase and amplitude noise. This can be attributed to the decrement in the inaccuracies through the linearization of the whole SSM in CKF/A-CKF, compared to EKF and UKF. The benefit from the adaptive computation of process noise covariance compared to the CKF can be observed at rotation angular frequencies of 400 Mrad/s and higher in the BTB case and at higher launch powers of 5 dBm in the transmission case .
3.5. Kalman filtering for joint compensation of phase and frequency offset
Apart from digital equalization, carrier synchronization is also vital to mitigate the phase and frequency offsets between the transmitter laser and free running LO. Since the CPE methods have low tolerance towards FO, which may go as high as ±5 GHz, a separate FO estimation (FOE) is required. Consequently, several FOE algorithms have been proposed in the literature that are either based on the phase increments between adjacent symbols  or spectrum based methods . These methods are either not accurate for higher order QAM systems or computationally complex.
3.5.1. LKF and EKF for FO estimation
A novel FOE algorithm using Kalman filtering have been proposed and numerically verified in . The simulation results in  concludes that the Kalman filter can achieve faster convergence and outperforms the conventional FO estimation at low OSNR. In , FOE schemes based on blind and training data, using LKF and EKF have been proposed for QPSK systems. These Kalman based FOE algorithms are evaluated both numerically and experimentally, and are compared to FFT based FOE methods. The investigations in  report that the training data based Kalman FOE methods show better accuracy in estimating the FO in case of fewer symbols and high OSNR, compared to FFT based methods. However, a separate phase estimation has to be carried out after FO compensation.
3.5.2. Two stage EKF for joint compensation of FO, phase and amplitude noise
The Kalman based FOE algorithms proposed in [60, 64] can compensate only for the FO and therefore, the carrier phase has to be recovered separately after FO compensation. In , a two stage EKF method based on training data has been proposed for joint tracking of FO, phase and amplitude noise. In the first stage, a coarse estimate of FO is obtained using a set of training data symbols following the training data scheme proposed in . In the second stage, CPANE algorithm has been employed to jointly compensate for the residual FO, phase and amplitude noise.
18.104.22.168. Principles of two stage EKF
After linear equalization, the received signal on single polarization, with frequency and phase offset can be represented as given in Eq. (39). Here,
The BER vs OSNR curves for LKF and EKF after the 2nd stage, using 200 and 500 training data symbols, for a FO of 1 GHz, are depicted in Figure 9 . It can be concluded from , that both LKF and EKF show faster convergence irrespective of the number of training data symbols utilized in the first stage. However, since the EKF estimates a complex quantity, it facilitates in compensating also for the amplitude noise and therefore, outperforms LKF. Moreover, as discussed earlier, EKF does not require any angle operations unlike LKF, and thereby the additional few computations required by the EKF compared to LKF can be sought to be compensated with the additional benefit of better tracking capability.
This two stage EKF model has been extended in  to compensate also for the fiber nonlinearity in addition to FO, phase and amplitude noise. The first stage is similar and compensates FO coarsely, as discussed earlier. In the second stage, the total phase noise to be estimated comprises of both laser phase noise and fiber nonlinearities. The EKF-CPANE algorithm is employed for tracking the residual FO and the total phase noise in addition to amplitude noise. From the numerical analysis, it was reported in  that compared to LKF, the maximum possible transmission reach can be increased by an additional 500 km using EKF, at a BER of 2.4*10−2.
We have discussed in detail on how to exploit the potential of Kalman filters for the joint mitigation of several fiber optical transmission impairments in coherent optical transmission systems. Various Kalman based approaches for tracking carrier phase and frequency offset, polarization state have been reviewed. The CPANE algorithm and its implementation details using EKF for joint mitigation of linear and nonlinear phase noise as well as amplitude noise have been illustrated in detail. It is also verified that the combination of DBP and EKF enhances the nonlinear mitigation performance, at the expense of few DBP steps. A cascaded structure using LKF and EKF is illustrated for tracking the polarization state and carrier phase besides amplitude noise, simultaneously. A two stage EKF model for simultaneous tracking of FO, phase and amplitude noise is also discussed. From the discussed numerical verifications, it can be concluded that the Kalman filter based approaches for tracking the optical transmission impairments outperforms the conventional methods in coherent optical communication systems, with faster convergence, better tracking ability and more tolerance towards the optical transmission impairments. Since the Kalman filter is an optimal recursive MMSE estimator, with its attractive properties of hardware efficient implementation feasibility, less computational effort as well as memory requirements, it seems to be an essential component of future coherent optical receivers.
The authors gratefully acknowledge the support of the Erlangen Graduate School in Advanced Optical Technologies (SAOT) funded by the German National Science Foundation (DFG) in the framework of the excellence initiative.