Coherent receiver structure for polarization multiplex transmission. 

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... detection and digital signal processing (DSP) allows employment of advanced transmission techniques like quadrature phase-shift keying (QPSK), multilevel quadrature amplitude modulation (M-QAM), and orthogonal frequency-division multiplexing (OFDM), in combination with polarization multiplex (PM) for optical transmission systems with 112-Gb/s data rate or above. The DSP also permits electronic compensation of intermediate frequency, phase noise [ 1], and fiber impairments that lead to polarization crosstalk, chromatic dispersion (CD), and polarization-mode dispersion (PMD) [2]. Several real-time experiments using such algorithms for QPSK transmission have already been demonstrated [3], [4]. Even single-chip implementations [5] and modularized implementations with high-speed analog-to-digital converters (ADCs) and a Complementary metal–oxide–semiconductor DSP chip [6] have been presented. On the other hand, research is also still going on with offline experiments that allow the investigation of new algorithms without the need for extensive hardware development [7]. Transition from QPSK to QAM formats would usually focus on square constellation formats because they have the optimum distance for additive noise. The disadvantage of these formats like B three-ring [ 16-QAM is that they require a considerable computation effort for phase estimation in order to achieve low bit error ratio (BER) values [8]. Beside additive optical noise, phase noise has to be considered, especially if standard distributed feedback (DFB) lasers with a sum linewidth of 1 MHz or above are employed. The dominance of phase noise may cause a better BER performance to be reached with alternative QAM constellations. Therefore, we investigate a phase-estimation algorithm that is suitable for certain star constellation formats. The advantage of this algorithm is the simplicity regarding realtime implementations. By the term regular star constellation (RSC), we refer to any QAM format where all constellation points are located on intercepts of circles around the origin ( B rings [ ) and R lines from the origin with equal angular spacing ( B rays [ ). Binary phase-shift keying (BPSK), QPSK, and other phase-shift keying (PSK) formats like 8-PSK are examples of single-ring RSCs. Their combination with amplitude modulation leads to multiring formats with a higher information content per symbol. Such modulation formats are most efficiently decoded in polar coordinates, and the amplitude modulation can be neglected for angle-based phase estimators like ours. Fig. 1 shows two possible constellations for 16-QAM: (left) the usual square constellation with point locations on three rings and (right) an alternative star constellation with only two rings. Note that, in the square constellation, middle-ring points do not have an equal angular spacing, which makes phase estimation more difficult. The star constellation can be described as 8-PSK combined with a 1-bit amplitude modulation. For all modulation formats including OFDM, the coherent receiver with PM and DSP can have the same structure (see Fig. 2): The receiver consists of an optical front end including optical 90 hybrids, optical/electrical conversion, ADCs, and a DSP unit (DSPU). ADCs and DSPU can either be integrated in a single chip to ease interfacing and reduce the footprint [4] or in a modular approach for optimized performance, where ADCs and DSPU can be developed in different technologies for maximum bandwidth and high integration, respectively [5]. In both topologies, the implemented algorithms must allow parallel processing to match system (transmission) standards and technology constraints [9]. The DSPU cannot operate at the sampling clock frequency of the ADC but requires demultiplexing to process the data in m parallel stages at reduced clock frequencies that are usable for the required signal processing. Therefore, the algorithms should allow parallel processing with an unlimited number m of demultiplexed channels, which implies that the output of one module is independent of the outputs of the other parallel modules. Another constraint for algorithms used for real-time DSP is hardware efficiency, which also originates from the parallel processing in the DSPU. For example, data recovery and phase estimation usually require coordinate transformations with lookup tables (LUTs). These have to be implemented for each module separately, and if a single coordinate conversion can be avoided, the hardware effort is remarkably reduced. For any QAM transmission without PM, the received symbols after optoelectronic conversion and discretization can be described as a time sequence of complex numbers Z ð k Þ with the argument ð k Þ and time index k . For PM, the received signal from four ADCs can be seen as a pair of two complex numbers Z 1 ð k Þ , Z 2 ð k Þ , which has (as a vector) to be multiplied by a compensation matrix in order to separate the transmission channels again [9]. The result is another complex pair X 1 ð k Þ , X 2 ð k Þ with arguments 1 ð k Þ , 2 ð k Þ . We will not go into details on PM and polarization control within this paper, but it should be noted that these angle pair time sequences could also be employed for the described phase-estimation algorithm and that a common phase estimation for both PM channels is advantageous in terms of performance and hardware efficiency. Because coherent transmission requires a second laser as a local oscillator (LO) on the receiver side, the transmitter (TX) and LO laser can differ in both frequency and phase. This fact leads to an intradyne receiver signal [10] and makes estimation of the intradyne frequency and the phase necessary. For frequency estimation and phase estimation of a regular star QAM transmission without nonlinear phase noise, only the argument of the received symbol is necessary, whereas the magnitudes of the complex numbers only contain the amplitude modulation and fluctuations due to noise. Nevertheless, the usual way to estimate the phase for QPSK proposed by Viterbi and Viterbi (V&V) involves complex calculations [11]. An estimated frequency sufficient for analog LO laser can be obtained from averaged phase increments [12]. Data recovery (also addressed as demodulation in this paper) for an RSC is done most convenient in polar coordinates. Because of the extraordinary high- speed and parallelization requirements, it is advantageous to convert the received signal into polar coordinates with LUTs instead of calculation algorithms for trigonometric functions like the coordinate rotation digital computer [13]. Amplitude modulation data bits can be recovered independent from and in parallel to phase-modulation data bits which require phase estimation and differential encoding. The decider threshold values for amplitude demodulation can be optimized independent from the phase estimation and demodulation with a slow decision-directed (DD) control structure. For a synchronous phase demodulator, an estimated phase angle ’ ^ ð k Þ is subtracted from each received signal angle ð k Þ in order to compensate for phase noise and residual frequency mismatch. Best results for high-phase-noise requirements can be achieved with estimators that perform a full phase tracking. The estimated phase is usually limited to a certain quadrant; therefore, it is wrapped. Phase unwrapping based on a maximum-likelihood decision is feasible in real time [14], but it is easier to detect the deviations between the physical and wrapped estimated phase, encode it into quadrant jump numbers, and to consider it in the differential decoder that recovers the original data. The phase-estimation algorithm of V&V, similar to our approach, employs a polar coordinate representation of the received signal [11]. It can be employed for any RSC with small adaptations, but in order to simplify notation, we concentrate on QPSK or, more generally, RSCs with four rays; in other words, R 1⁄4 4. The received symbol Z ð k Þ consists of the sent QPSK symbols c ð k Þ 1⁄4 Æ 1 Æ j , a time-variant phasor exp ð j ’ ð k ÞÞ and additional noise n ð k Þ ...