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(N³), 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 3 × 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 Nk frequency-domain filters, leading to O(N_k N²) complexity, associated with negligible performance loss. Finally, a simplified VSNE approach with invariant Kernel coefficients is proposed, delivering O(N_k N) 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.