A new universal class of discrete non-linear basis functions

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 ~2<sup>151</sup> 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