A high-performance inner-product processor for real and complex numbers.

A novel, high-performance fixed-point inner-product processor based on a redundant binary number system is investigated in this dissertation. This scheme decreases the number of partial products to 50%, while achieving better speed and area performance, as well as providing pipeline extension opportunities. When modified Booth coding is used, partial products are reduced by almost 75%, thereby significantly reducing the multiplier addition depth. The design is applicable for digital signal and image processing applications that require real and/or complex numbers inner-product arithmetic, such as digital filters, correlation and convolution. This design is well suited for VLSI implementation and can also be embedded as an inner-product core inside a general purpose or DSP FPGA-based processor. Dynamic control of the computing structure permits different computations, such as a variety of inner-product real and complex number computations, parallel multiplication for real and complex numbers, and real and complex number division. The same structure can also be controlled to accept redundant binary number inputs for multiplication and inner-product computations. An improved 2's-complement to redundant binary converter is also presented

Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)

Oxford, UK, 26 August 200

Shift Radix Systems - A Survey

Let $d\ge 1$ be an integer and ${\bf r}=(r_0,\dots,r_{d-1}) \in \mathbf{R}^d$. The {\em shift radix system} $\tau_\mathbf{r}: \mathbb{Z}^d \to \mathbb{Z}^d$ is defined by $$\tau_{{\bf r}}({\bf z})=(z_1,\dots,z_{d-1},-\lfloor {\bf r} {\bf z}\rfloor)^t \qquad ({\bf z}=(z_0,\dots,z_{d-1})^t).$$ $\tau_\mathbf{r}$ has the {\em finiteness property} if each ${\bf z} \in \mathbb{Z}^d$ is eventually mapped to ${\bf 0}$ under iterations of $\tau_\mathbf{r}$. In the present survey we summarize results on these nearly linear mappings. We discuss how these mappings are related to well-known numeration systems, to rotations with round-offs, and to a conjecture on periodic expansions w.r.t.\ Salem numbers. Moreover, we review the behavior of the orbits of points under iterations of $\tau_\mathbf{r}$ with special emphasis on ultimately periodic orbits and on the finiteness property. We also describe a geometric theory related to shift radix systems.Comment: 45 pages, 16 figure

30 years of collaboration

We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptography School of Debrecen. However, we do not plan to be complete in any sense but give some interesting data and selected results that we find particularly nice. At the end we focus on two topics in more details, namely a problem that origins from a conjecture of Rényi and Erdős (on the number of terms of the square of a polynomial) and another one that origins from a question of Zelinsky (on the unit sum number problem). This paper evolved from a plenary invited talk that the authors gaveat the Joint Austrian-Hungarian Mathematical Conference 2015, August 25-27, 2015 in Győr (Hungary)

Deep Encoding: Where Genetic Algorithms and Numeral Systems Meet

It has been shown empirically that certain benefits can be gained by modelling genetic algorithm encoding as a numeral system and imple-menting mutation as a form of the numeral system’s arithmetic. It is the aim of this project to strengthen these findings. We will do this in three stages.Firstly, by creating meaningful classifications of numeral systems and formally proving crucial properties such as termination of standardisation and normalisation.Secondly, by developing a programming framework centered around these system classes. The framework is used on strings and can impose numeral systems on them. This allows the user to write code that can be run with a selection of systems to see different results. For example, taking the string ”10” and treating it as binary or decimal depending on what the user dictates.Thirdly, by writing a genetic algorithm and using the aforementioned framework to write an encoding method and mutation function that are based off of numeral system arithmetic. The mutation function adds a random unit value to the digit string and mutates the string by utilising arithmetic overflow