  Efficient nonlinearity compensation in fiber-optic communication systems is
considered a key element to go beyond the "capacity crunch''. One guiding
principle for previous work on the design of practical nonlinearity
compensation schemes is that fewer steps lead to better systems. In this paper,
we challenge this assumption and show how to carefully design multi-step
approaches that provide better performance--complexity trade-offs than their
few-step counterparts. We consider the recently proposed learned digital
backpropagation (LDBP) approach, where the linear steps in the split-step
method are re-interpreted as general linear functions, similar to the weight
matrices in a deep neural network. Our main contribution lies in an
experimental demonstration of this approach for a 25 Gbaud single-channel
optical transmission system. It is shown how LDBP can be integrated into a
coherent receiver DSP chain and successfully trained in the presence of various
hardware impairments. Our results show that LDBP with limited complexity can
achieve better performance than standard DBP by using very short, but jointly
optimized, finite-impulse response filters in each step. This paper also
provides an overview of recently proposed extensions of LDBP and we comment on
potentially interesting avenues for future work.
