Radix Representation Research Articles

Lagrange was Wrong, Pascal was Right

Abstract In this paper we compare the efficiency of the decimal system to the efficiency of different mixed radix representations. We use as a starting point for our study the duodecimal systems suggested by Pascal and the Maya “Long Count” system. Using the quality index we experimentally show that two slight deviations from the duodecimal system are more efficient than the previous two systems and also than the decimal system.

Radix-Partition-Based Over-the-Air Aggregation and Low-Complexity State Estimation for IoT Systems Over Wireless Fading Channels

We consider remote state estimation for an internet-of-things (IoT) system, where a number of distributed IoT sensors monitor a dynamic plant and deliver the state measurements to a remote state estimator over an unreliable wireless network. We propose novel radix-partition-based over-the-air aggregation to coordinate the transmissions of the IoT sensors, which strongly enhances the spectral efficiency of radio resources and enables a smaller state estimation mean square error (MSE) at the remote state estimator, taking into consideration the quantization effect and radio resource allocation for the IoT sensors. We consider a fixed-filtering design at the remote state estimator, which significantly reduces the signal processing overhead at the remote state estimator compared to using the conventional Kalman-filtering-based state estimation approach. We show that the proposed low-complexity state estimation scheme enables state estimation stability via the linear matrix inequality (LMI) technique for on-off fading channels and via Lyapunov stability analysis for random fading channels. Based on this, we further propose efficient algorithms for the offline design of the filtering gain, connection topology and radix representation for IoT systems. Numerical results show that the proposed scheme has superior scalability performance and can achieve a smaller state estimation MSE compared to various state-of-the-art baselines.

Design and Analysis of Efficient Maximum/Minimum Circuits for Stochastic Computing

In stochastic computing (SC), a real-valued number is represented by a stochastic bit stream, encoding its value in the probability of obtaining a one. This leads to a significantly lower hardware effort for various functions and provides a higher tolerance to errors (e.g., bit flips) compared to binary radix representation. The implementation of a stochastic max/min function is important for many areas where SC has been successfully applied, such as image processing or machine learning (e.g., max pooling in neural networks). In this work, we propose a novel shift-register-based architecture for a stochastic max/min function. We show that the proposed circuit has significantly higher accuracy than state-of-the-art architectures for uncorrelated bit streams at comparable hardware costs. Moreover, we analytically proof the correctness of the proposed circuit and provide a new error analysis, based on the individual bits of the stochastic streams. Interestingly, the analysis reveals that for a certain practical bit stream length a finite optimal shift register length exists and it allows to determine the optimal length.

Fast division in the residue number system {2n + 1,2n, 2n-1} based on shortcut mixed radix conversion

Unlike the parallelism residue number systems (RNS) provide for addition and multiplication, RNS division is often performed via a straightforward RDF (i.e., reverse conversion, binary division, and forward-conversion) scheme. Given the lack of any faster-than-RDF RNS divider, we designed one for the popular moduli set {2n + 1, 2n,2n − 1}, via mixed radix representation of the operands and two parallel 2n-bit dividers (vs. the 3n-bit counterpart required by the RDF). However, the ensued RNS integer quotient could be off the correct result by ± 1, which is tolerated in many approximate arithmetic processes. Nevertheless, we obtain the accurate quotient via additional hardware (i.e., six parallel n-bit comparators). Our experimental results, for n = 8(16), as compared to RDF, show 33%(36%) less delay, 3%(12%) less energy, and 35%(43%) less energy-delay product in case of the minimal-error approximate divider and 14%(21%) less delay in case of the exact implementation, at the cost of additional area consumption and power dissipation.

Energy-Efficient VLSI Realization of Binary64 Division With Redundant Number Systems

VLSI realizations of digit-recurrence binary division usually use redundant representation of partial remainders and quotient digits. The former allows for fast carry-free computation of the next partial remainder, and the latter leads to less number of the required divisor multiples. In studying the previous relevant works, we have noted that the binary carry-save (CS) number system is prevalent in the representation of partial remainders, and redundant high radix representation of quotient digits is popular in order to reduce the cycle count. In this paper, we explore a design space containing four division architectures. These are based on binary CS or radix-16 signed digit (SD) representations of partial remainders. On the other hand, they use full or partial precomputation of divisor multiples. The latter uses smaller multiplexer at the cost two extra adders, where one of the operands is constant within all cycles. The quotient digits are represented by radix-16 [−9, 9] SDs. Our synthesis-based evaluation of VLSI realizations of the best previous relevant work and the four proposed designs show reduced power and energy figures in the proposed designs at the cost of more silicon area and delay measures. However, our energy-delay product is 26%–35% less than that of the reference work.

Radix Representation of Triangular Discrete Grid System

Discrete Global Grid Systems (DGGSs) are spatial references that use a hierarchical tessellation of cells to partition and address the entire globe. It provides an organizational structure that permits fast integration between multiple sources of large and variable geospatial data. Although many endeavors have been done to describe certain discrete grid systems, there still lack of a uniform mathematical framework for them. This paper simplifies the planar class I aperture 4 triangular discrete grid system into a hierarchical lattice model which is proved to be a radix system in the complex number plane. Mathematical properties of the radix system reveal the discrete grid system is equivalent to the set of complex numbers with special form. The conclusion provides a potential way to build a uniform mathematical framework of DGGS and can be used to design efficient encoding and spatial operation scheme for DGGS.

Joint spectral radius, dilation equations, and asymptotic behavior of radix-rational sequences

Radix-rational sequences are solutions of systems of recurrence equations based on the radix representation of the index. For each radix-rational sequence with complex values we provide an asymptotic expansion, essentially in the scale NαlogℓN. The precision of the asymptotic expansion depends on the joint spectral radius of the linear representation of the sequence of first-order differences. The coefficients are Hölderian functions obtained through some dilation equations, which are usual in the domains of wavelets and refinement schemes. The proofs are ultimately based on elementary linear algebra.

$\epsilon$-shift radix systems and radix representations with shifted digit sets

$\epsilon$-shift radix systems and radix representations with shifted digit sets

Generalized radix representations and dynamical systems. IV

For r = ( r 1,…, r d ) ∈ ℝ d the mapping τ r :ℤ d →ℤ d given by τ r ( a 1,…, a d ) = ( a 2, …, a d ,−⌊ r 1 a 1+…+ r d a d ⌋) where ⌊·⌋ denotes the floor function, is called a shift radix system if for each a ∈ ℤ d there exists an integer k > 0 with τ r k ( a) = 0. As shown in Part I of this series of papers, shift radix systems are intimately related to certain well-known notions of number systems like β-expansibns and canonical number systems. After characterization results on shift radix systems in Part II of this series of papers and the thorough investigation of the relations between shift radix systems and canonical number systems in Part III, the present part is devoted to further structural relationships between shift radix systems and β-expansions. In particular we establish the distribution of Pisot polynomials with and without the finiteness property (F).

Generalized radix representations and dynamical systems III

Generalized radix representations and dynamical systems III

Finding Trott Constants

A Trott constant is a number whose continued fraction representation (simple or not) is the same as the digits of its radix representation. For instance, in base 10, 0.1084101... = 0 + 1 / (1 + 1 / (0 + 1 / (8 + 1 / (4 + 1 / (1 + 1 / (0 + 1 / (1 ...))))))). This Corner implements a search method for such numbers and lists its results. The best number we found has more than 400 digits in common between its nonsimple, zero-free continued fraction representation and its radix representation.

Radix representations, self-affine tiles, and multivariable wavelets

We investigate the connection between radix representations for Z and self-affine tilings of R n . We apply our results to show that Haar-like multivariable wavelets exist for all dilation matrices that are sufficiently large.

Generalized radix representations and dynamical systems II

We are concerned with families of dynamical systems which are related to generalized radix representations. The properties of these dynamical systems lead to new results on the characterization of bases of Pisot number systems as well as canonical number systems.

Bases of canonical number systems in quartic algebraic number fields

Canonical number systems can be viewed as natural generalizations of radix representations of ordinary integers to algebraic integers. A slightly modified version of an algorithm of B. Kovács and A. Pethő is presented here for the determination of canonical number systems in orders of algebraic number fields. Using this algorithm canonical number systems of some quartic fields are computed.

Generalized radix representations and dynamical systems. I

We are concerned with families of dynamical systems which are related to generalized radix representations. The properties of these dynamical systems lead to new results on the characterization of bases of Pisot number systems as well as canonical number systems.

Disk-like Tiles Derived from Complex Bases

For each positive integer k, the radix representation of the complex numbers in the base –k+i gives rise to a lattice self-affine tile Tk in the plane, which consists of all the complex numbers that can be expressed in the form ∑j≥1dj(–k+i)–j, where dj∈{0, 1, 2, ...,k2}. We prove that Tk is homeomorphic to the closed unit disk {z∈C:∣z∣ ≤ 1} if and only if k ≠ 2.

Real functions computable by finite automata using affine representations

This paper tries to classify the functions of type I n → I (for some interval I⊆ R ) that can be exactly realized by a finite transducer. Such a class of functions strongly depends on the choice of real number representation. This paper considers only the so-called affine representations where numbers are represented by infinite compositions of affine contracting functions on I. Affine representations include the radix (e.g. decimal, signed binary) representations. The first result is that all piecewise affine functions of n variables with rational coefficients are computable by a finite transducer which uses the signed binary representation. The second and main result is that any function computable by a finite transducer using an affine representation is affine on any open connected subset of I n on which it is continuously differentiable. This limitation theorem shows that the set of finitely computable functions is very restricted.

A generalized reverse jacket transform

Generalization of the well-known Walsh-Hadamard transform (WHT), namely center-weighted Hadamard transform (CWHT) and complex reverse-jacket transform (CRJT) have been proposed and their fast implementation and simple index generation algorithms have recently been reported. These transforms are of size 2/sup r//spl times/2/sup r/ for integral values or r, and defined in terms of binary radix representation of integers. In this paper, using appropriate mixed-radix representation of integers, we present a generalized transform called general reverse jacket transform (GRJT) that unifies all the three classes of transforms, WHT, CWHT, and CRJT, and is also applicable for any even length vectors, that is of size 2/sup r//spl times/2/sup r/. A subclass of GRJT which includes CRJT (but not CWHT) is applicable for finite fields and useful for constructing error control codes.

Guest editors' introduction - Special issue on computer arithmetic

—————————— ✦ —————————— DVANCES in VLSI technology and the availability of sophisticated circuit design and testing tools have made it feasible to implement very complex algorithms directly in hardware. Consequently, we are now witnessing the design and fabrication of special-purpose capabilities beyond those previously realized only in software. Arithmetic calculations play a crucial role in many applications for which speed is essential, e.g., digital signal processing, cryptography, and telecommunications. The accelerating pursuit of higher speed must be realized in tandem with satisfying user demands for higher numeric precision and standardization of the results of arithmetic operations. This necessitates algorithm design which exploits parallelism at all levels and meshes with the underlying physical technology. The explosion of research and implementation proceeding in this manner is nowhere more evident than in the advanced arithmetic capabilities provided in today's contemporary microprocessors. Computer arithmetic draws upon mathematics, computer science, and electrical engineering both in creating a theoretical framework and in improving the specific implementation technology. While the basic principles and inherent limitations of arithmetic on standard radix representations have been known for some decades, the variations and fine tunings appropriate to the currently available implementation technologies, and today's more aggressive utilization of parallelism, provide fertile areas for fundamental research and innovative design. The objective of this special issue is to present some of the state-of-the-art developments in computer arithmetic. The papers included in this special issue were selected from

Redundant radix representations of rings

This paper presents an analysis of radix representations of elements from general rings; in particular, we study the questions of redundancy and completeness in such representations. Mappings into radix representations, as well as conversions between such, are discussed, in particular where the target system is redundant. Results are shown valid for normed rings containing only a finite number of elements with a bounded distance from zero, essentially assuring that the ring is "discrete." With only brief references to the more usual representations of integers, the emphasis is on various complex number systems, including the "classical" complex number systems for the Gaussian integers, as well as the Eisenstein integers, concluding with a summary on properties of some low-radix representations of such systems.