A new universal class of discrete non-linear basis functions

Proceedings of the 2000 Third IEEE International Caracas Conference on Devices, Circuits and Systems (Cat. No.00TH8474) Pub Date : 2000-03-15 DOI:10.1109/ICCDCS.2000.869871

James J. Soltis

{"title":"A new universal class of discrete non-linear basis functions","authors":"James J. Soltis","doi":"10.1109/ICCDCS.2000.869871","DOIUrl":null,"url":null,"abstract":"The discrete Fourier transform is of fundamental importance in the digital processing of signals. By using the Jacobian elliptic functions sn(u,m) and cn(u,m) as basis functions in place of the trigonometric sine and cosine, one can obtain a generalized transform which includes the Fourier transform as a special case, viz. m=0, where m, the squared modulus of elliptic function theory, can have any positive value less than 1, and hence the new transform is extremely flexible. It is found that the associated inverse transform consists of basis functions whose appearance can be described as a set of dithered trigonometric functions. The dithering level increases in a monotonic fashion with the parameter m. The latter can be described as a new universal discrete non-linear basis set. The universality derives from the fact that, for the same precision of computer computation (e.g. 12 digits), identical values are obtained independent of the machine used. The new set has as selectable parameters both the number of samples per period and the squared modulus 'm' (0<m<1). This same generalization technique can also be applied to the fractional Fourier transform - itself a generalization of the Fourier transform with a real or complex parameter (alpha). This, in turn, means that the number of possible unique universal waveform patterns is immense. An estimate is /spl sim/2/sup 151/ different waveforms if one assumes data lengths up to 1024 and 'm' in double precision. Sample applications of this non-linear function set in signal processing are presented.","PeriodicalId":301003,"journal":{"name":"Proceedings of the 2000 Third IEEE International Caracas Conference on Devices, Circuits and Systems (Cat. No.00TH8474)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2000-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the 2000 Third IEEE International Caracas Conference on Devices, Circuits and Systems (Cat. No.00TH8474)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/ICCDCS.2000.869871","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 0

Abstract

The discrete Fourier transform is of fundamental importance in the digital processing of signals. By using the Jacobian elliptic functions sn(u,m) and cn(u,m) as basis functions in place of the trigonometric sine and cosine, one can obtain a generalized transform which includes the Fourier transform as a special case, viz. m=0, where m, the squared modulus of elliptic function theory, can have any positive value less than 1, and hence the new transform is extremely flexible. It is found that the associated inverse transform consists of basis functions whose appearance can be described as a set of dithered trigonometric functions. The dithering level increases in a monotonic fashion with the parameter m. The latter can be described as a new universal discrete non-linear basis set. The universality derives from the fact that, for the same precision of computer computation (e.g. 12 digits), identical values are obtained independent of the machine used. The new set has as selectable parameters both the number of samples per period and the squared modulus 'm' (0

查看原文本刊更多论文

一类新的通用离散非线性基函数

离散傅里叶变换在信号的数字处理中具有重要的基础意义。利用雅可比椭圆函数sn(u,m)和cn(u,m)作为基函数代替三角函数正弦和余弦，可以得到一个广义变换，其中包括作为特殊情况的傅里叶变换，即m=0，其中m，椭圆函数理论的平方模，可以是小于1的任何正值，因此新的变换是非常灵活的。发现关联逆变换由基函数组成，基函数的外观可以描述为一组抖动三角函数。抖动电平随参数m的增大呈单调增长。参数m可以描述为一种新的通用离散非线性基集。这种通用性源于这样一个事实，即对于相同的计算机计算精度(例如12位数字)，与所使用的机器无关，可以获得相同的值。新集合具有每个周期的样本数和平方模量m (0

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Proceedings of the 2000 Third IEEE International Caracas Conference on Devices, Circuits and Systems (Cat. No.00TH8474)

自引率

0.00%

发文量

京公网安备 11010802042870号

Book学术文献互助群
群号：604180095