New Approaches to Coding Information using Inverse Scattering Transform

Remarkable mathematical properties of the integrable nonlinear Schrödinger equation (NLSE) can offer advanced solutions for the mitigation of nonlinear signal distortions in optical fiber links. Fundamental optical soliton, continuous, and discrete eigenvalues of the nonlinear spectrum have already been considered for the transmission of information in fiber-optic channels. Here, we propose to apply signal modulation to the kernel of the Gelfand-Levitan-Marchenko equations that offers the advantage of a relatively simple decoder design. First, we describe an approach based on exploiting the general $N$-soliton solution of the NLSE for simultaneous coding of $N$ symbols involving $4\times N$ coding parameters. As a specific elegant subclass of the general schemes, we introduce a soliton orthogonal frequency division multiplexing (SOFDM) method. This method is based on the choice of identical imaginary parts of the $N$-soliton solution eigenvalues, corresponding to equidistant soliton frequencies, making it similar to the conventional OFDM scheme, thus, allowing for the use of the efficient fast Fourier transform algorithm to recover the data. Then, we demonstrate how to use this new approach to control signal parameters in the case of the continuous spectrum.

Figure 1

Left: The modulus of the 6-soliton solution at the beginning ($q_0$, red dashed line) and at the end ($q_1$, blue solid line) of the NLSE governed transmission line. Right: 6-soliton normalized kernel of the GLME equations at the beginning of the transmission line $|\mathrm{\Omega }(0,t)\exp (-At)|$. Blue solid line—the exact normalized kernel encoded by the SOFDM method with QPSK; red dashed line—the same kernel restored by the use of direct TIB method.

Figure 2

Left: 16 encoded kernel harmonics (grey, solid line) and the window function (black, dashed line). Parameters of the window transformation function (12) are the following: $\tilde{A}=15$, $\mathrm{\Gamma }=20$. Right: The double kernel (blue, solid line) obtained by multiplying the encoded harmonics by the window function and corresponding signal, obtained by the use of the inverse TIB algorithm (red, dashed line). The inset picture shows the absolute value of signal Fourier spectrum; the frequency index $n$ is defined in (8).

Figure 3

Dependence of the signal from the parameters of the kernel window transformation function (12).

Figure 4

Left: Propagation of the burst-mode signal in the model NLSE channel, with $L=1000\mathrm{km}$ and $\mathrm{SNR}=19.7\mathrm{dB}$. Blue solid lines correspond to the signal at the beginning of the line; red dashed lines show the signal at the end of the line. Parameters of the window transformation function are the same as in Fig. 2. The bottom picture corresponds to the encoded (blue, solid lines) and decoded, using the direct TIB algorithm (red, dashed lines) kernel harmonics for the central burst interval. Right: Constellation diagram for the central burst interval (statistics on a $10^3$ randomly encoded initial signals).