# Tech Resources

Astrapi® Brochure. Learn more about Astrapi’s spiral-based signal modulation with this two-page Technology Summary.

Instantaneous Spectral Analysis (ISA). A peer-reviewed article in the Journal of Communication and Information Systems (JCIS) that presents the theory and benefits of ISA and contrasts the technique to the standard Fourier Transform (FT).

Technical Introduction to Spiral Modulation. A unified introduction to spiral modulation, including its underlying mathematics and range of applications.

The Astrapi Technical Value Proposition. Astrapi is the pioneer of spiral-based signal modulation, a generalization of the circle-based signal modulation that is the basis for standard techniques such as Quadrature Amplitude Modulation (QAM) and Phase-Shift Keying (PSK). Spiral-based signal modulation is a wide-open field for innovation. This technical report summarizes the benefits of this new technology and reviews the classical channel capacity literature, showing that the Shannon-Hartley law rests on an implicit assumption of signal periodicity that does not apply to spiral-based signal modulation.

The Shannon Law for Non-Periodic Channels. Making use of new mathematical tools arising from a generalization of Euler’s formula, this technical report removes the periodic assumption implicit in the derivation of the Shannon-Hartley law and proves that non-periodic spiral-based signal modulation is in principle capable of spectral efficiencies dramatically higher than the Shannon limit.

Euler’s Formula for Fractional Powers of i. This paper introduces a new generalization of Euler’s formula to describe spirals in the complex plane, rather than the circles which arise from the classical Euler’s formula. The paper defines an infinite family of new functions that contains and combines the traditional exponential and sinusoidal functions, and describes their geometric, analytic, and algebraic properties. It is shown that any Taylor polynomial can be easily decomposed into this new class of functions. While focused purely on mathematical topics, this paper later became the theoretical basis for Astrapi’s non-periodic spiral-based signal modulation.