A 200-MHz complex number multiplier using redundant binary arithmetic

In the paper we make a comparative study of two techniques to design error-detectable array architectures. These techniques are the redundant binary representation (RBR) where the data is encoded in the 1-out-of-3 code; and the two-rail logic where the data is encoded in the 1-out-of-2 code. In recent work, the RBR has been used to achieve online error detection and localisation by checking the data on the array borders. Here we show that another approach is also possible, with less hardware cost, where the checking takes place on the local (processor) level. This provides immediate error detection without delay. The performance of the RBR approaches has been compared with the two-rail approach. The results show that the RBR approaches require more hardware overheads, for small word lengths (n). However, the hardware cost of the two techniques are approximately the same for large n, while the RBR approaches offer much faster arithmetics for all n.

A fast multiplier with a new Redundant Binary Booth's Encoding (RBBE) for Radix-8 has been developed. The new RBBE enables one to directly generate all Redundant Binary Partial Products (PP) including the hard multiple of 3/spl times/Multiplicand with no additional circuitry or time delay. Also, the use of TGs in Partial Product Selection Circuit permits the rapid generation of PPs. In this work, a word size of 54 bits is chosen to comply with the IEEE double-precision floating-point standard, which consists of one sign bit and 53 bits of fraction. Thus, the RBBE unit for Radix-8 generates 18 Redundant Binary PPs to be accumulated by a Redundant Binary Adder (RBA) tree. Due to the simplicity of the necessary interconnections between the adders, the RBA tree has a regular layout. The output of the RBA tree is in the Redundant Binary (RB) form, which needs to be converted to Standard Binary (SB) form. Although this conversion can be performed by a Carry Look Ahead (CLA) adder, a better conversion circuit has been designed to take advantage of the RB number system. The proposed design contains a converter of a higher speed and a lower transistor count. As a result, the proposed fast multiplier accepts two 54 bit SB numbers and produces a final product of 108 bit SB number.

A high-performance iterative quadratic convergence fixed-point, real and complex number, divider circuit using a previously developed redundant binary inner-product processor core is presented. The intermediate quotient coefficients are kept in redundant binary (RB) form without converting back to normal binary numbers. A unified multiplier using redundant binary representation for both signed and unsigned numbers is investigated. Goldschmidt and Newton-Raphson division methods are compared and the mathematical equivalence of these two methods is provided.

This paper presents the measurement and analysis of local oscillator isolation in an analog frontend based on monolithic microwave integrated circuits (MMICs) at a carrier frequency of 300 GHz. The transmit-receive wireless system uses a single-balanced resistive mixer with in-phase and quadrature channels that provides a very good isolation of the local oscillator. The dependency of the isolation on the baseband input is presented. The isolation obtained when a signal is transmitted only on the in-phase channel is higher than in the case of the signal transmitted on the quadrature channel. The paper presents the measurement method and a theoretical analysis which confirms the experimental observations.

Computer operations involving complex numbers, essential in such applications as digital signal processing and image processing, are usually performed in a divide-and-conquer approach dealing separately with the real and imaginary parts and then accumulating the results. There have been several proposals to treat complex numbers as a single unit but all seem to have floundered on the basic problem of the division process without which, of course, it is impossible to carry out all but the most basic arithmetic. This paper resurrects an early proposal to express complex numbers in a single binary representation and provides a fail-safe procedure for obtaining the quotient of two complex numbers expressed in this representation.

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.

There is no better way to learn than playing. After all, that is how children learn. In this appendix, we are going to provide the basic guidelines for “playing the quantum computing game” with the help of the MATLAB environment.

The method of linear theory for small perturbations is used to study the stability of heavy inclusions in the upper crust. The existence of such inclusions of increased density is associated with chemical heterogeneity or phase transitions. At small deformations considered in the linear stability theory, the material undergoes transient creep which is described by the Andrade rheological model. Under this rheology, the convective instability of thickened segments of the lithosphere is oscillatory: the increment is a complex number whose real part is close to zero (threshold instability). Heavy inclusions in the upper crust introduce a small complex correction to the increment that determines the convective instability of the lithosphere, slightly increasing the real and imaginary parts of the increment, enhancing the instability and increasing the oscillation frequency.

This paper proposes a unified depth model based on the light field gradient, in which estimated disparity is represented by the complex number. The complex-valued disparity by the proposed depth model can be represented in both the Cartesian and polar coordinates. In the Cartesian representation, the proposed depth model is represented by real and imaginary parts of the disparity. The real part can be used for disparity estimation with respect to the in-focus plane, whereas the imaginary part represents the non-Lambertian-ness. In the polar representation, the proposed depth model is expressed by the disparity magnitude and disparity angle. The disparity magnitude shows the relationship among depth from stereo, depth from focus, and depth from defocus, whereas the disparity angle shows whether or not the bundles of rays are flipped with respect to the in-focus plane. For disparity analysis, we present the real response, imaginary response, magnitude response, and angle response, which are represented by the three-dimensional volume. Experimental results on synthetic and real light field images show that the real and magnitude responses of the proposed depth model are valid for local disparity estimation.

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 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.

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.

10 - COMPLEX NUMBERS

On the interpretation of shear viscosity ultrasonic measurements

A new method for developing in-phase and quandrature samples of a band limited real signal is presented and compared with other methods. The proposed method improves the computational efficiency of a frequency-domain implementation of the required filtering by taking advantage of mirror-image symmetry between the filters on the in-phase and quandrature channels. This technique makes it possible to filteer both the in-phase and quadrature channels at once using one standard FFT for complex-valued input and one standard IFFT for complex-valued output without adding computational overhead. The proposed method is more computationally efficient than techniques using commonly available FFT routines for real-valued input. Specialized FFT routines derived specifically for real- valued input can achieve greater computational efficiency than the proposed method, but these routines are not as commonly available as the standard FFT routines used in the proposed method. For systems where the effective channel filter is short in length or has special structure, there are other implementations of the channel filters which are more efficient than the proposed method.