John von Neumann’s Time-Frequency Orthogonal Transforms

Abstract

John von Neumann (JvN) was one of the greatest scientists and minds of the 20th century. His research encompassed a large variety of topics (especially from mathematics), and the results he obtained essentially contributed to the progress of science and technology. Within this article, one function that JvN defined long time ago, namely the cardinal sinus (sinc), was employed to define transforms to be applied on 1D signals, either in continuous or discrete time. The main characteristics of JvN Transforms (JvNTs) are founded on a theory described at length in the article. Two properties are of particular interest: orthogonality and invertibility. Both are important in the context of data compression. After building the theoretical foundation of JvNTs, the corresponding numerical algorithms were designed, implemented and tested on artificial and real signals. The last part of the article is devoted to simulations with such algorithms by using 1D signals. An extensive analysis on JvNTs effectiveness is performed as well, based on simulation results. In conclusion, JvNTs prove to be useful tools in signal processing.

Keywords

• numerable bases in Lebesgue-Hilbert space
• orthogonal transforms
• time-frequency dictionary
• Fourier transforms
• analysis and synthesis of finite energy signals