Generalized Rudin-Shapiro Systems

Benke, George

doi:10.1007/s00041-001-4004-9

Cited by 20 publications

(20 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An elegant construction achieving all combinations of sign changes is found in Benke [4] (Byrnes [8,10] gives a similar construction). In short,…”

Section: The Rudin-shapiro Transformmentioning

confidence: 94%

See 1 more Smart Citation

On the Rudin–Shapiro transform

Cour-Harbo

2008

Applied and Computational Harmonic Analysis

View full text Add to dashboard Cite

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.

show abstract

“…An elegant construction achieving all combinations of sign changes is found in Benke [4] (Byrnes [8,10] gives a similar construction). In short,…”

Section: The Rudin-shapiro Transformmentioning

confidence: 94%

“…Consequently, the matrices are called the Rudin-Shapiro transform (RST). It is shown in Benke [4] that this construction can be generalized in various ways. The individual entries in the Rudin-Shapiro transform can be found by the following equation, where…”

Section: The Rudin-shapiro Transformmentioning

confidence: 97%

On the Rudin–Shapiro transform

Cour-Harbo

2008

Applied and Computational Harmonic Analysis

View full text Add to dashboard Cite

show abstract

“…One interesting characteristic of a GRS sequence is that a uniform upper bound for the PAPR, which is independent of the length sequence [10][11][12][13][14][15][16], is considered as a flat polynomial having the properties of a spread spectrum sequence [16]. Additionally this polynomial is orthogonal and its amplitude is on the unit circle [10][11][12][13][14][15][16].…”

Section: B Short Preamblementioning

confidence: 99%

Synchronization algorithms based on weighted CAZAC preambles for OFDM systems

Silva

Harris

Dolecek

2013

2013 13th International Symposium on Communications and Information Technologies (ISCIT)

View full text Add to dashboard Cite

In this paper, we study the performances of preambles with a weighted structure for estimate the Timing Offset and Frequency Offset in Orthogonal Frequency Division Multiplexing (OFDM) systems. We have reviewed the configuration in short and long preambles of weighted preambles, based on Constant Amplitude Zero Auto Correlation (CAZAC) sequences weighted by a Pseudo Noise (PN) sequences. In this paper we have designed short and long preambles with a CAZAC sequence weighted by a Golay Rudin Shapiro sequence and their respective timing metrics. By simulation in MATLAB, we get the Mean Square Error (MSE) of the estimation, the Bit Error Rate (BER) of transmitted data over Rayleigh fading channel with Additive White Gaussian Noise (AWGN), coarse timing and frequency offset.

show abstract

“…In fact, arbitrarily interchanges of the signs in each recursive step does not affect the properties of the constructed sequences. An elegant construction achieving all combinations of sign changes is found in Benke [6] (Byrnes [7,8] gives a similar construction). In short,…”

Section: The Rudin-shapiro Transformmentioning

confidence: 99%

“…The 2 × 2 matrix in (10) aside, it is not immediately obvious neither how this definition is linked to (6), nor that it defines a symmetric transform. However, a fairly straightforward proof shows that the rows of P ( J ) are the coefficients of the polynomials defined in (6), see la Cour-Harbo [9], and thus that the defined transform does indeed possess the properties derived above for the Rudin-Shapiro transform.…”

Section: Definition 1 Define the Mappingmentioning

confidence: 99%

The symmetric rudin-shapiro transform - an easy, stable, and fast construction of multiple orthogonal spread spectrum signals

Cour-Harbo

3rd International Symposium on Image and Signal Processing and Analysis, 2003. ISPA 2003. Proceedings of The

View full text Add to dashboard Cite

A method for constructing spread spectrum sequences is presented. The method is based on a linear, orthogonal, symmetric transform, the Rudin-Shapiro transform (RST), which is in many respects quite similar to the Haar wavelet packet transform. The RST provides the means for generating large sets of spread spectrum signals. This presentation provides a simple definition of the symmetric RST that leads to a fast N log(N) and numerically stable implementation of the transform.

show abstract

Generalized Rudin-Shapiro Systems

Cited by 20 publications

References 8 publications

On the Rudin–Shapiro transform

On the Rudin–Shapiro transform

Synchronization algorithms based on weighted CAZAC preambles for OFDM systems

The symmetric rudin-shapiro transform - an easy, stable, and fast construction of multiple orthogonal spread spectrum signals

Contact Info

Product

Resources

About