New approaches to coding information using inverse scattering transform
Date/Time: 17:30 28-Jun-2017
Abstract:
Remarkable mathematical properties of the integrable nonlinear Schr\"{o}dinger equation (NLSE) can offer attractive new solutions for mitigation of the nonlinear distortions of signal in optical fibre links. Indeed, fundamental optical soliton, continuous and discrete eigenvalues of the scattering data have been already considered for transmission of information in fibre-optic channels.
The efficiency of numerical algorithms of encoding/decoding processes is decisive in practical applications of these concepts in the real world fibre-optic networks. Nowadays, the orthogonal frequency division multiplexing (OFDM) method has become one of the most widely used modulation technics in wireless communications. The success of the OFDM method is, in particular, due to its high spectral efficiency and the exceptional computational performance of the background fast Fourier transform (FFT).
Here we propose to use modulation of the kernel of the Gelfand-Levitan-Marchenko equations (GLME) to transmit information and demonstrate that the OFDM scheme can be applied in a much more stable and efficient way compared to the alternative parametric encoding approaches. To recover the encoded kernel at the end of the transmission line we use the efficient T\"{o}eplitz inner-bordering (TIB) numerical scheme of direct scattering transform, that was recently introduced by L.L. Frumin and co-authors [1]. The general idea to apply the OFDM scheme to the continuous spectrum scattering data was previously considered by S.K. Turitsyn and co-authors [2]. Here we make the next important step, namely, we study how to control signal parameters by GLME kernel modulations.
We demonstrate that in the case of discrete (solitonic) spectrum, the analogue of the OFDM scheme can be applied when the soliton amplitudes are identical, while the soliton frequencies are chosen to be equidistant. Then we propose to use the general analytical $N$-soliton solutions of the NLSE for simultaneous coding of $N$ symbols involving $4\times N$ coding parameters. In comparison with the traditional single-soliton transmission lines [3] such approach naturally accounts the interactions between solitons, thus, increasing the spectral efficiency.
Finally, we discuss the noise stability and spectral efficiency of the proposed approaches. The practical applications of the solitonic OFDM scheme requires further development of noise stable methods for solving the direct scattering problem. At the same time, the application of TIB to the continuous spectrum is robust with respect to significant noise amplitudes and provide high spectral efficiency. We present the simulations of data transmission by $N$-soliton as well as continuous spectrum signals in the optical transmission line with distributed signal amplification and discuss the further perspectives of the proposed approaches.
[1] L. L. Frumin, O. V. Belai, E. V. Podivilov, and D. A. Shapiro, JOSA B 32, 290 (2015).
[2] J. E. Prilepsky , S.A. Derevyanko, K.J. Blow, I. Gabitov and S.K. Turitsyn, PRL 113(1), 013901 (2014).
[3] Yushko, O. V., Redyuk, A. A., Fedoruk, M. P., & Turitsyn, S. K., JETP, 119(5), 787-794, 2014.
The efficiency of numerical algorithms of encoding/decoding processes is decisive in practical applications of these concepts in the real world fibre-optic networks. Nowadays, the orthogonal frequency division multiplexing (OFDM) method has become one of the most widely used modulation technics in wireless communications. The success of the OFDM method is, in particular, due to its high spectral efficiency and the exceptional computational performance of the background fast Fourier transform (FFT).
Here we propose to use modulation of the kernel of the Gelfand-Levitan-Marchenko equations (GLME) to transmit information and demonstrate that the OFDM scheme can be applied in a much more stable and efficient way compared to the alternative parametric encoding approaches. To recover the encoded kernel at the end of the transmission line we use the efficient T\"{o}eplitz inner-bordering (TIB) numerical scheme of direct scattering transform, that was recently introduced by L.L. Frumin and co-authors [1]. The general idea to apply the OFDM scheme to the continuous spectrum scattering data was previously considered by S.K. Turitsyn and co-authors [2]. Here we make the next important step, namely, we study how to control signal parameters by GLME kernel modulations.
We demonstrate that in the case of discrete (solitonic) spectrum, the analogue of the OFDM scheme can be applied when the soliton amplitudes are identical, while the soliton frequencies are chosen to be equidistant. Then we propose to use the general analytical $N$-soliton solutions of the NLSE for simultaneous coding of $N$ symbols involving $4\times N$ coding parameters. In comparison with the traditional single-soliton transmission lines [3] such approach naturally accounts the interactions between solitons, thus, increasing the spectral efficiency.
Finally, we discuss the noise stability and spectral efficiency of the proposed approaches. The practical applications of the solitonic OFDM scheme requires further development of noise stable methods for solving the direct scattering problem. At the same time, the application of TIB to the continuous spectrum is robust with respect to significant noise amplitudes and provide high spectral efficiency. We present the simulations of data transmission by $N$-soliton as well as continuous spectrum signals in the optical transmission line with distributed signal amplification and discuss the further perspectives of the proposed approaches.
[1] L. L. Frumin, O. V. Belai, E. V. Podivilov, and D. A. Shapiro, JOSA B 32, 290 (2015).
[2] J. E. Prilepsky , S.A. Derevyanko, K.J. Blow, I. Gabitov and S.K. Turitsyn, PRL 113(1), 013901 (2014).
[3] Yushko, O. V., Redyuk, A. A., Fedoruk, M. P., & Turitsyn, S. K., JETP, 119(5), 787-794, 2014.
Authors
Gelash Andrey A.
(Presenter)
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(Presenter)