Representation Of Complex Numbers Research Articles

Matrix representation of octonions and generalizations

We define a special matrix multiplication among a special subset of 2N×2N matrices, and study the resulting (nonassociative) algebras and their subalgebras. We derive the conditions under which these algebras become alternative nonassociative, and when they become associative. In particular, these algebras yield special matrix representations of octonions and complex numbers; they naturally lead to the Cayley–Dickson doubling process. Our matrix representation of octonions also yields elegant insights into Dirac’s equation for a free particle. A few other results and remarks arise as byproducts.

Polygonal radix representations of complex numbers

Complex numbers can be represented in positional notation using certain digit sets. In this paper, we present the polygonal representation which uses zero and the n-roots of unity as digits. We give conditions on the base in order that every complex number be representable in such a system. We totally characterize complete polygonal numeration systems in imaginary quadratic fields.

Exponential signals with time-varying amplitude: Parameter estimation via polar decomposition

This paper addresses the estimation of the center frequency of complex exponential signals with time-varying amplitude. A method which requires few assumptions regarding the signal’s envelope is proposed. It is based on the polar decomposition of a certain covariance matrix. The polar decomposition, a generalization to matrices of the complex number representation z= re i θ with r>0, is particularly suitable for the application considered. The notion of truncated polar decomposition is introduced. Simple schemes for estimating the signal’s frequency are presented, based on these decompositions. In contrast to most existing methods, the methods presented herein do not rely on any assumed structure for the time-varying amplitude, and they are shown to possess good performances in a large class of signals. The effectiveness and robustness of our approach are demonstrated on real radar data.

A property of linear fuzzy differential equations

In this paper, we investigate a property of linear differential equations where the initial state is described by a vector of fuzzy numbers. The property is related directly to the matrix defining the original nonfuzzy system by passing to a complex number representation of the α-level sets of the fuzzy system.

Testing quantum mechanics using third-order correlations.

Semiclassical theories similar to stochastic electrodynamics are widely used in optics. The distinguishing feature of such theories is that the quantum uncertainty is represented by random statistical fluctuations. They can successfully predict some quantum-mechanical phenomena; for example, the squeezing of the quantum uncertainty in the parametric oscillator. However, since such theories are not equivalent to quantum mechanics, they will not always be useful. Complex number representations can be used to exactly model the quantum uncertainty, but care has to be taken that approximations do not reduce the description to a hidden variable one. This paper helps show the limitations of ``semiclassical theories,'' and helps show where a true quantum-mechanical treatment needs to be used. Third-order correlations are a test that provides a clear distinction between quantum and hidden variable theories in a way analogous to that provided by the ``all or nothing'' Greenberger-Horne-Zeilinger test of local hidden variable theories. \textcopyright{} 1996 The American Physical Society.

Efficient FFT structures using a mixed complex number system

Abstract This paper describes a hardware structure for fast Fourier transform (FFT) computation thai is particularly suited to input data presented sequentially. The algorithm allows different representations of complex numbers to be used in the same processing system so that the FFT can be computed by using multiplication-free butterfly elements based on the radix numbers of 2, 3, 4 and 6. In comparison with previous designs in the literature, the new algorithm provides significant hardware savings on the requirements of data memory and multipliers. Furthermore, the FFT size N, which is usually a composite number of 2 or 4, can be more flexible. Efficient data format conversions between different number systems are also provided.

New redundant representations of complex numbers and vectors

The authors present a redundant representation for complex numbers called polygonal representation. This representation enables fast carry-free addition, in a way quite similar to the carry-free addition in signed-digit number systems and is convenient for multiplication. The technique is generalized in order to handle n-dimensional vectors.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

M-layer: NPS version

M-layer is a FORTRAN program for computing the over-the-horizon propagation factor of an electromagnetic (EM) wave in a stratified atmosphere when ducting is present. The M-layer program developed by the Naval Ocean Systems Center (NOSC) has been substantially revised into a Naval Postgraduate School (NPS) version. The mode locating algorithm was rewritten to take advantage of local analyticity of the mode function. The exponential complex number representation was adopted to simplify multiplication and to improve accuracy when adding two complex numbers which nearly cancel each other. A consistency condition was formulated to assure accurate evaluation of the height-gain function. As a result, this NPS version executes faster with greater stability.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Representation of complex numbers in number systems

Representation of complex numbers in number systems

Total and n=1 partial photoionisation cross sections of helium in the vicinity of the 3s3p 1P autoionising state

Abstract The total and n =1 partial photoionisation cross sections of helium in the vicinity of the 3s3p 1 P resonance are considered using the method of configuration interaction in the complex number representation and the R -matrix approach. A comparison is made with the experimental and theoretical results.

Optical processors using modular number codes for complex signals

AbstractTechniques are described for implementing an optical matrix‐vector processor for complex signals. A representation of complex numbers based on modular algebra is utilized to decouple real and imaginary channels, thereby simplifying the implementations. The pipelined architecture of an optical matrix‐vector processor is described.

Sensitivities of frequency response functions of linear dynamic systems to variations in design parameter values

Expressions for the sensitivities of the frequency response functions of linear constant coefficient systems with distinct eigenvalues to perturbations in the values of their design parameters are derived. The expressions are of each of two forms, the one involving complex number (phasor) representation of the responses and the sensitivities, the other involving separate amplitude and phase representations. The relations derived contain eigenvalue, eigenvector and adjoint eigenvector sensitivities. Engineering applications of the theory are suggested.

Infinite series representations for complex numbers

We introduce two new algorithms that lead to finite or infinite series expansions for complex number in terms of ‘integral digits’ within the complex quadratic fields ℚ\(\left( {\sqrt { - m} } \right)\), form=1, 2,…, 11. In particular, we derive complex number representations as sums of reciprocal of Gaussion integers and as sums of reciprocals of algebraic integers in ℚ\(\left( {\sqrt { - m} } \right)\), form=2, 3, 7 and 11. In addition to convergence of the various algorithms we investigate the representation of ‘rationals’ relative to the fields ℚ\(\left( {\sqrt { - m} } \right)\), form=1, 2, 3, 7 and 11.

The design of dual-mode complex signal processors based on quadratic modular number codes

It has been known for a long time that quadratic modular number codes admit an unusual representation of complex numbers which leads to complete decoupling of the real and imaginary channels, thereby simplifying complex multiplication and providing error isolation between the real and imaginary channels. This paper first presents a tutorial review of the theory behind the different types of complex modular rings (fields) that result from particular parameter selections, and then presents a theory for a complex signal processor based on the choice of augmented power-of-2 moduli. It is shown how a diminished-1 binary code, used by previous designers for the realization of Fermat number transforms, also leads to efficient realizations for dual-mode complex arithmetic for certain augmented power-of-2 moduli. Then a design is presented for a recursive complex filter based on a ROM/ACCUMULATOR architecture and realized in an augmented power-of-2 quadratic code, and a computergenerated example of a complex recursive filter is shown to illustrate the principles of the theory.

Incoherent optical processing: a tristimulus-based method

Optical systems which make use of incoherent illumination provide potential benefits in noise immunity over systems using coherent illumination. Because incoherent systems are linear with respect to intensity, however, complex number representations and operations are not readily obtainable. This paper proposes using the tristimulus (three-color) system as found in color television to establish a vector space, which then permits the complex number operations of polar-form multiplication and rectangular-form addition to be achieved with incoherent illumination. When these operations are combined in a hybrid (optical-electronic) processor using color television equipment, discrete approximations of the general 1-D and 2-D linear complex superposition integrals may be obtained.

Arithmetic in Complex Bases

In this note, we introduce a novel way of doing complex arithmetic that does not involve separating the complex numbers into their real and imaginary parts. This method uses the representation of complex numbers in positional notation using a complex base - n + i, for a positive integer n, with natural numbers as digits. Addition, subtraction and multiplication can be performed directly in this positional notation and is similar to real decimal arithmetic; the main difference is in the carry digits. However, division is more complicated and the construction of a good algorithm for long division is a challenging unsolved problem. We say that an integer z (real or complex) is represented in the base b with digits from the set 6D

On the Relationship between Rotations and Lorentz Transformations in Two, Three, and Four Dimensions

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 sinθ and cosθ, and the hyperbolic functions sinhφ and coshφ.

On the master equation and pure state representations for the quantized damped oscillator

A general master equation for the linearly damped harmonic oscillator is investigated. First, and extended version is presented of Hasse's existence condition for a pure state representation by means of a nonlinear frictional Schrödinger equation with a nonhermitian, but norm conserving hamiltonian. Secondly, reconsidering Dekker's phase space quantization, invoking complex hamiltonians, it is shown (i) how it involves the mirror image system, (ii) that its formulation is terms of a diagonal complex number representation of the hamiltonian allows for a free but nonobservable phase parameter, and (iii) that it leads to specific diffusion coefficients in the master equation which in the long time limit exactly satisfy the pure state condition.

Comments on "On the Floating Point Representation of Complex Numbers

A recent paper <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> discussed an integrated notation for complex numbers. A second choice of the radix point position will increase the number of values near the origin. An argument is also made for unnormalized notation of both parts of a complex number.

On the Floating Point Representation of Complex Numbers

It is shown that arithmetic operations are simplified if complex numbers are represented in an integrated form with real and imaginary parts sharing the same exponent. In the new representation, numbers with large magnitudes suffer increased rounding errors but errors for some other numbers decrease.