Simplified Volterra Series Nonlinear Equalizer for Polarization-Multiplexed Coherent Optical Systems

Starting from a previously proposed frequency-domain Volterra series nonlinear equalizer (VSNE), whose complexity evolves as O(N-3), with N being the frequency-domain block length, we derive a symmetric VSNE filter array formulation for polarization-multiplexed (PM) signals, whose full VSNE equivalent is up to 3x more computationally efficient, with zero performance penalty. By gradually reconstructing the third-order kernel from its column/diagonal components, the full VSNE can be reduced to a restrict set of N-k frequency-domain filters, leading to O(NkN2) complexity, associated with negligible performance loss. Finally, a simplified VSNE approach with invariant Kernel coefficients is proposed, delivering O(NkN) complexity at the expense of controlled performance penalty. The proposed array of symmetric VSNE filters significantly increases the scalability of the previous matrix-based VSNE, providing a more flexible balance between performance and complexity, which can be freely adjusted to match the available computational resources. Performing a direct comparison between the simplified VSNE and the widely used split-step Fourier method in a long-haul 224 Gb/s PM-16QAM transmission system, we demonstrate a reduction of over 60% in terms of computational effort and 90% in terms of equalization latency.