# True Equalization of Polarization-Dependent Loss in Presence of Fast Rotation of SOP +
## 1 Introduction  
High-order modulation, polarization-division multiplexing (PDM), and coherent detection are very crucial techniques in optical communication systems for pursuing higher spectrum efficiency (SE). Furthermore, the integration of digital signal processing (DSP) in coherent receivers enables most of the impairments experienced by the optical signal, such as chromatic dispersion (CD), polarization mode dispersion (PMD), carrier frequency offset (CFO), and carrier phase noise (CPN), to be fully compensated in the electric domain [1]. Unfortunately, polarization-dependent loss (PDL), which induces polarization-dependent optical power fluctuations, remains the main limiting impairment and cannot be efficiently compensated by the state-of-art equalization algorithms at the receiver side [2]. PDL arises from some of the in-line components, such as isolators, multiplexers, amplifiers, and couplers. In long-haul links, we might have to face the problem of large PDL values of several dB. The PDL effect on PDM signals not only results in the inequality of signal power, most importantly, it causes the imbalance of optical signal-to-noise ratio (OSNR) between the two polarizations of a PDM signal. The OSNR asymmetry will cause quite different bit-error-rate (BER) degradations between two polarizations signals, and this may lead to significant penalties to the system performance. At present,lots of the equalization methods of PDL in the literature such as the multiple-input-multiple-output (MIMO)-based equalizer or 3D Stokes space method can only accomplish power imbalance equalization; therefore, they are not the true equalization of PDL [3; 4; 5; 6; 7; 8; 9]. In fact, the OSNR asymmetry in optical communication is similar to frequency-selective fading in multi-antenna wireless channels, which can be mitigated by space-time (ST) codes. The idea can be carried over and employed. In 2010, [PERSON] et al. first introduced these codes in PDM systems for PDL mitigation and renamed it to polarization-time code (PTC), which has been proven to be very effective to truly mitigate PDL [10; 11; 12; 13]. In their works, they supposed having perfect channel state information, or used a training sequence (TS) to estimate the channel matrix prior to implementation of the polarization-time (PT) decode at the receiver. However, there is no channel state information at the receiver. In addition, for the single-carrier systems, we prefer a blind equalizer adopted before the PT decoder at the receiver. Furthermore, in some extreme environments, like violent vibrations when trains pass by and lightning strikes on rainy days, an ultra-fast rotation of state of polarization (RSOP) would be induced with a rate of mega-radian per second [14; 15; 16; 17]. The fast RSOP in combination with large PDL will make the eigenmodes of the PDL randomly time-varying, putting a heavy burden on the DSP equalizer. The Kalman filter (KF) is an optimum adaptive filter algorithm, which has been widely studied for lots of kinds of impairment equalizations in optical communication in recent years. There has been a significant number of publications that have reported KF-based equalization methods covering phase noise tracking, carrier-frequency offset estimation [18; 19; 20], and nonlinear mitigation [21]. Particularly, thanks to the fast convergence ability of the KF, it achieved great success in ultra-fast RSOP tracking [22; 23; 24; 25; 26; 27; 28; 29]. Inspired by the previous works, in this paper, we propose a method for achieving the true equalization of PDL in the presence of ultra-fast RSOP by utilizing a polarization-time code combined with the Kalman filter. This scheme is proven to be very effective. With an approximate 1.2 dB OSNR penalty, it exhibits an equalization capacity with a PDL tolerance as large as 7 dB and a fast RSOP tolerance (up to 1 Mrad/s). The structure of this paper is as follows. In Section 2, we first elaborate the concept of PDL and its influence on PDM signals.