Fast Numerical Nonlinear Fourier Transforms

作     者：Wahls, Sander Poor, H. Vincent 

作者机构：Princeton Univ Dept Elect Engn Princeton NJ 08544 USA 

出 版 物：《IEEE TRANSACTIONS ON INFORMATION THEORY》 (IEEE Trans. Inf. Theory)

年 卷 期：2015年第61卷第12期

页      面：6957-6974页

核心收录：

学科分类：0808[工学-电气工程] 08[工学] 0701[理学-数学] 0812[工学-计算机科学与技术（可授工学、理学学位）] 

基　　金：German Research Foundation (DFG) [WA 3139/1-1] U.S. National Science Foundation [CCF-1420575] 

主　　题：Nonlinear Fourier transform forward scattering transform nonlinear Schrodinger equation fast algorithms 

摘      要：The nonlinear Fourier transform, which is also known as the forward scattering transform, decomposes a periodic signal into nonlinearly interacting waves. In contrast to the common Fourier transform, these waves no longer have to be sinusoidal. Physically relevant waveforms are often available for the analysis instead. The details of the transform depend on the waveforms underlying the analysis, which in turn are specified through the implicit assumption that the signal is governed by a certain evolution equation. For example, water waves generated by the Korteweg-de Vries equation can be expressed in terms of cnoidal waves. Light waves in optical fiber governed by the nonlinear Schrodinger equation (NSE) are another example. Nonlinear analogs of classic problems such as spectral analysis and filtering arise in many applications, with information transmission in optical fiber, as proposed by Yousefi and Kschischang, being a very recent one. The nonlinear Fourier transform is eminently suited to address them-at least from a theoretical point of view. Although numerical algorithms are available for computing the transform, a fast nonlinear Fourier transform that is similarly effective as the fast Fourier transform is for computing the common Fourier transform has not been available so far. The goal of this paper is to address this problem. Two fast numerical methods for computing the nonlinear Fourier transform with respect to the NSE are presented. The first method achieves a runtime of O(D-2) floating point operations, where D is the number of sample points. The second method applies only to the case where the NSE is defocusing, but it achieves an O(D log(2) D) runtime. Extensions of the results to other evolution equations are discussed as well.