Fast and Reliable Detection of Significant Solitons in Signals With Large Time-Bandwidth Products

Abstract

We present a fast method to calculate the significantly large solitonic components of signals with large time-bandwidth products governed by the nonlinear Schrödinger equation, for which the computation typically becomes prohibitively expensive and/or numerically unstable. We partition the full signal in both frequency and time to obtain short signals with a constant number of samples, independent of the size of the full signal. The solitons within each short signal are computed using a conventional nonlinear Fourier transform (NFT) algorithm. The partitioning in general leads to spurious solitons not present in the full signal. We therefore design an acceptance scheme that removes spurious solitons. The remaining solitons are attributed to the full signal. Solitons that are too wide to fit into the short signals cannot be detected by this approach, but since wide solitons must be of low amplitude, the significant solitons will be found. This approach only requires <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {O}(N)$</tex-math></inline-formula> floating point operations, with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> the number of signal samples. It can furthermore be applied to signals with large time-bandwidth products for which conventional NFT algorithms become unreliable or even fail. When applying our proposed method to a signal of 15,000 samples, the significant solitonic components were computed 14 times faster than when considering the whole signal, for which the conventional algorithm furthermore provided wrong results. We found that time-partitioning yields accurate results, while frequency-partitioning causes a small loss in accuracy. Combined frequency-time partitioning leads to the fastest computation, but also suffers from the same loss in accuracy as with frequency-partitioning. As time-partitioning yields a significant speed-up at nearly no loss in accuracy, we regard this as the method of choice in most practical scenarios.