Efficient division in the binary representation of complex numbers

We introduce a novel method for complex number representation. The proposed, redundant complex binary number system (RCBNS) is developed by combining a redundant binary number and a complex number in base (-1+j). Donald (1960) and Walter Penny (1964, 1965) represented complex numbers using base -j and (-1+j) in the classified algorithmic models. A redundant complex binary number system consists of both real and imaginary-radix number systems that form a redundant integer digit set. This system is formed by using complex radix of (-1+j) and a digit set of /spl alpha/=3, where /spl alpha/ assumes a value of -3, -2, -1, 0, 1, 2, 3. The arithmetic operations of complex numbers with this system treat the real and imaginary parts as one unit. The carry-free addition has the advantage of redundancy in number representation in the arithmetic operations. Results of the arithmetic operations are in the RCBNS form. The two methods for conversion from the RCBNS form to the standard binary number form have been presented. The RCBNS reduces the number of steps required to perform complex number arithmetic operations, thus enhancing the speed.

Complex numbers play a very important role in various applications of electrical and computer engineering. These days, arithmetic operations dealing with these numbers rely on a divide-and-conquer technique wherein a complex number is broken into its real and imaginary parts and then, after representing each part in binary number system, operation is carried out on each part as if part of the real arithmetic. Thus, addition of two complex numbers requires two separate additions and their multiplication requires four individual multiplications, one subtraction and one addition. In an effort to reduce the number of arithmetic operations within the realm of complex arithmetic, binary number system with base (-1+j), called complex binary number system, has been proposed in the literature which allows a complex number to be represented as single-unit instead of two separate units as in the base-2 binary number system. In this paper, the effects of shift-left operations on complex binary numbers have been examined

A quantum theory representations of real (R) and complex (C) numbers is given that is based on states of single, finite strings of qukits for any base k > 1. Both unary representations and the possibility that qukits with k a prime number are elementary and the rest composite are discussed. Cauchy sequences of qukit string states are defined from the arithmetic properties. The representations of R and C, as equivalence classes of these sequences, differ from classical kit string state representations in two ways: the freedom of choice of basis states, and the fact that each quantum theory representation is part of a mathematical structure that is itself based on the real and complex numbers. These aspects enable the description of 3 dimensional frame fields labeled by different k values, different basis or gauge choices, and different iteration stages. The reference frames in the field are based on each R and C representation where each frame contains representations of all physical theories as mathematical structures based on the R and C representation. Approaches to integrating this with physics are described. It is observed that R and C values of physical quantities, matrix elements, etc. which are viewed in a frame as elementary and featureless, are seen in a parent frame as equivalence classes of Cauchy sequences of qukit string states.

We argue that a compact right-associated binary number representation gives simpler operators and better efficiency than the leftassociated binary number representation proposed by den Hoed and investigated by Goldberg. This representation is then generalised to higher number-bases and it is argued that bases between 3 and 5can give higher efficiency than binary representation.

For many years, vector space models have been used in information retrieval and computational linguistics to represent terms, queries, and documents, using vector addition as a simple operator to model semantic composition. Though surprisingly successful, many aspects of meaning including word order, typed relationships, and nested structures are not captured by this modelling process.

A complex number is formed by a real part and an imaginary part, the latter being composed of a real number multiplied by the imaginary unit “i,” defined as the square root of . For this reason, imaginary numbers cannot be placed on the real number line. The geometric representation of complex numbers is done in the so-called complex plane, composed of a real axis and an imaginary axis, orthogonal to the real line. However, from a philosophical point of view, the fact that the imaginary axis is not identified with any axis in three-dimensional Euclidean space raises ontological questions that transcend the well-established mathematical formalism of the set of complex numbers: are imaginary numbers just a sophism, an arbitrary invention, or a true mathematical discovery? Are imaginary numbers a glimpse of a parallel reality that is beyond our perception? In this article, I seek to demonstrate that the imaginary unit can be interpreted as an algebraic ambiguity that arises when representing certain vectors in the real plane as one-dimensional variables. Consequently, we will see how it is possible to express complex numbers in terms of real numbers.

Modern digital communication systems rely heavily on baseband signal processing for in-phase and quadrature (I-Q) channels, and complex number processing in low-voltage CMOS has become a necessity for channel equalization, timing recovery, modulation, and demodulation. In this work, redundant binary (RB) arithmetic is applied to complex number multiplication for the first time so that an N-bit parallel complex number multiplier can be reduced to two RE multiplications (i.e., an addition of N RB partial products) corresponding to real and imaginary parts, respectively. This efficient RE encoding scheme proposed can generate RB partial products with no additional hardware and delay overheads. A prototype 8-bit complex number multiplier containing 11.5 K transistors is integrated on 1.05/spl times/1.33 mm/sup 2/ using 0.8 /spl mu/m CMOS. The chip consumes 90 mW with 2.5 V supply when clocked at 200 MHz.

In this study, dimensions, mass and shape characteristics of five commercial loquat cultivars (‘Akko XIII’, ‘Champagne de Grasse’, ‘Guzelyurt 6’, ‘Hafifcukurgobek’ and ‘Sayda’) and two genotypes (‘KKTC3’ and ‘KKTC4’) were determined. Physical properties such as dimension and shape were determined both by digital caliper and image processing to compare each other. Among the cultivars and genotypes tested, ‘KKTC4’ (58.88 mm) had the highest length while ‘KKTC3’ had the highest major diameter, minor diameter, mass, geometric mean diameter, surface area and volume, with a mean of 46.31 mm, 43.61 mm, 60.12 g, 48.69 mm, 74.82 cm2, and 61.25 cm3, respectively. The greatest value of elongation and the smallest value of sphericity were obtained from ‘Akko XII’ with 1.33, and ‘Akko XII’ was more oblong than the other cultivars and genotypes. The smallest value of elongation and the greatest value of sphericity were obtained from ‘Sayda’ with 1.12 and 91.16%, respectively. The results of comparison between the digital caliper and image processing showed that using the image processing method could potentially be rapid and alternative to the conventional measurement method. Key words: Eriobotrya japonica, image processing, loquat, physical properties.

The digital image acquisition and processing technology already mature also in domains and so on visual communication obtains day by day[1]. However, the image acquisition and processing technology had many kinds, for example white balance processing and image denoising methods. The traditional image acquisition and processing systems are implemented using software by the PC or DSP chip[2], and sometimes software in series can not meet the requirement of high speed image processing. this article has given FPGA in the image acquisition and processing simple application, applied FPGA in the image acquisition and processing to be possible to enhance the image processing the speed, reduced takes the logical unit. The system designed by above-mentioned way achieves a satisfying effect by experimental verification.

10 - COMPLEX NUMBERS

Chapter 2 - Complex Numbers and Exponentials

CHAPTER 25 - THE ARGAND DIAGRAM

Rotations and Lorentz transformations are compared geometrically and algebraically for two-, three-, and four-dimensional space and space-time. The close analogy between the two types of transformation is revealed by means of orthogonal and pseudo-orthogonal matrices, complex numbers and double numbers, and quaternions and tetrons. The most natural and elegant forms are shown to be the complex and hypercomplex number representations. The relationship is essentially based on the direct correspondence between the circular functions sin θ and cos θ, and the hyperbolic functions sinh φ and cosh φ.

In this work, we apply quantum cellular automata (QCA) to study pattern formation and image processing in quantum-diffusion Schrödinger metamedia with generalized complex diffusion coefficients. Generalized complex numbers have the real part and imaginary part with the imaginary unit 2 i  1 (classical case), 2 i  1 (double numbers) and 2 i  0 (dual numbers). They form three 2-D complex algebras. Discretization of the Schrödinger equation gives the quantum Schrödinger cellular automaton with various complex–valued physical parameters. The process of excitation in these media is described by the Schrödinger equations with the wave functions that have values in algebras of the generalized complex numbers. This medium can be used for creation of the eye-prosthesis (so called the ”silicon eye”). The medium suggested can serve as the prosthesis prototype for perception of the bichromatic images.

In this paper, we present multipliers using a modified binary tree of the modulo m signed-digit (SD) number residue adders where m=2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> -1, 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> +1. New additions rules are used for generating the intermediate sum and carry with a binary number representation. The sums and carries are directly inputted into the next stage of adders, so that the modulo m multiplier using binary modulo m adder tree proposed in can be improved. Moreover residue multipliers using the SD residue adders are also designed with inputs/outputs in binary number representation. The design and simulation results of the proposed residue arithmetic circuits show that high speed arithmetic circuits can be obtained.