On the Rudin–Shapiro transform

Abstract

The Rudin-Shapiro transform (RST) is a linear transform derived from the remarkable Rudin-Shapiro polynomials discovered in 1951. The transform has the notable property of forming a spread spectrum basis for R N , i.e. the basis vectors are sequences with a nearly flat power spectrum. It is also orthogonal and Hadamard, and it can be made symmetric. This presentation is partly a tutorial on the RST, partly some new results on the symmetric RST that makes the transform interesting from an applicational point-of-view. In particular, it is shown how to make a very simple O(N log N) implementation, which is quite similar to the Haar wavelet packet transform.The Rudin-Shapiro transform was originally conceived as a series of coefficient sequences from a set of trigonometric polynomials discovered by Shapiro and Rudin in the 1950s. Since then the transform has come to exist in its own right. This is because the transform has a series of nice properties among which the spread spectrum property of the basis elements is the most noticeable one. The transform proves useful for designing signals in low-cost hardware, not least due to the existence of a fast and numerically robust implementation.The Rudin-Shapiro polynomials are categorized as flat polynomials. This refers to the fact that the amplitude of the complex polynomials are, on the unit circle, bounded by a constant times the energy of the polynomial. There exists many other examples of flat polynomials besides the Rudin-Shapiro polynomials, and the history of the development in the field of flat polynomials is quite interesting. This is in no small part due to the fact that a number of seemingly simple questions within the field have remained unanswered for several decades.Flat polynomials are also interesting for applications. The author has demonstrated, see la Cour-Harbo [23], that a spread spectrum transform has a role to play in the attempt to increase the robustness of active sensors. In that context the aim of this presentation is to show the mathematical background for properties exploited in real applications.The paper is divided into four parts (sections). Section 1 reviews some of the important definitions and notions in the field of flat polynomials. This is followed by a short historical background listing some of the major contributions in this field. In Section 2 the classical Rudin-Shapiro polynomials are presented along with some previously known results on the crest factor and auto and cross correlation properties of RS polynomials.

Coding Metasurfaces for Diffuse Scattering: Scaling Laws, Bounds, and Suboptimal Design

“…Another very interesting example is provided by the Golay‐Rudin‐Shapiro (GRS) polynomials, which have found interesting applications ranging from antenna arrays to spread‐spectrum communications . The P ν and Q ν GRS polynomials can be recursively defined via two intertwined formulas with It can be verified that these polynomials belong to the general class in Equation , with N = 2 ν .…”

Electrical Impedance Spectroscopy Using a Wide-Band Signal Based on the Rudin-Shapiro Polynomials

“…Most of the applications of the RS polynomials have been mainly in antenna design for communication systems, 15 in signal processing, 16 and in information theory, 17 while more work has been done on their mathematical characteristics 18 and geometrical features. 19 To the best of our knowledge no attempt to use their wide-band frequency spectra property has been reported in electrochemical impedance spectroscopy measurements.…”