Digit Set Research Articles

Implementation of Fast Radix-10 BCD Multiplier in FPGA

Background: Multiplication is the basic operation in any signal processing systems and financial applications, all these applications requires multiplication to be performed in a faster and efficient manner on a silicon chip. Methods: This paper describes the algorithm and architecture of a BCD parallel multiplier. The design exploits two properties of redundant BCD codes to speed up its computation. Namely, the redundant BCD excess-3 code (XS-3), and the overloaded BCD representation (ODDS). In addition to this, number of new techniques are used in order to reduce significantly latency, area and for implementation on FPGA compared to existing implementations. Findings: Parallel architecture is used for generating partial products using radix-10 recoding technique for signed-digit of a BCD multiplier having set of digits between the range [–5, 5] and a positive set of multiplicand multiples coded in XS-3. Use of this encoding has various advantages like, as XS-3 is a self-complementing code, finding a negative of it is by just complementing the bits of respective number. Also the redundancy in XS-3 code is utilized for generating multiplicand multiples in a simple, faster and a carry-free way. Implemented design has three stages. Partial product generation, reduction and final conversion to BCD. For to implement the design in hardware the partial product reduction architecture is modified here to use a bank of ripple carry adder trees. ODDS representation uses 4-bit binary encoding technique which is similar to non-redundant BCD code, for this reason conventional VLSI circuit techniques such as carry-save adders and compression trees can be used effectively to perform operations on decimal numbers. Conclusion: To show the advantages of the resulted design, RTL model for 8 × 8-digit and 16 × 16-digits multiplication has been synthesized and implemented in Virtex-5 FPGA device. Results shows that the multiplier is about 10–15% delay efficient with existing work and about 14–18% area efficient.

Digit sets for self-similar tiles

A new class of digit sets, which we called very weak product-form digit sets, is introduced and it is shown that they are tile digit set for self-similar tiles. This extends previous results about product-form and weak product-form digit sets.

A necessary and sufficient condition for the finite µ M,D -orthogonality

The self-affine measure µM,D associated with an expanding matrix M ∈ Mn(ℤ) and a finite digit set D ⊂ ℤn is uniquely determined by the self-affine identity with equal weight. The set of orthogonal exponential functions E(Λ) := {e2πi〈λ, x〉 : λ ∈ Λ} in the Hilbert space L2(µM,D) is simply called µM,D-orthogonal exponentials. We consider in this paper the finiteness of µM,D-orthogonality. A necessary and sufficient condition is obtained for the set E(Λ) to be a finite µM,D-orthogonal exponentials. The research here is closely connected with the non-spectrality of self-affine measures.

PharmGKB summary: very important pharmacogene information for human leukocyte antigen B.

PharmGKB summary: very important pharmacogene information for human leukocyte antigen B.

Truncated ternary multipliers

© The Institution of Engineering and Technology 2015. Balanced ternary number representation and arithmetic, based on the symmetric radix-3 digit set {-1, 0, +1}, has been studied at various times in the history of computing. Among established advantages of balanced ternary arithmetic are representational symmetry, favourable error characteristics and rounding by truncation. In this study, we show an additional advantage: that of lower-error truncated multiplication with the same relative cost reduction as in truncated binary multipliers.

An Unsupervised Approach to Learning and Early Detection of Spatio-Temporal Patterns Using Spiking Neural Networks

This paper addresses the problem of learning and recognizing spatio-temporal patterns, which are typically encountered when representing gestures or other human actions. Existing approaches to learning such patterns are typically supervised, rely on extensive amounts of training data and require the observation of the entire pattern for recognition. We propose an approach that brings the following main contributions: i) it learns the patterns in an unsupervised manner, ii) it uses a very small number of training samples, and iii) it enables early classification of the pattern from observing only a small fraction of the pattern. The proposed method relies on spiking networks with axonal conductance delays, which learn encoding of individual patterns as sets of polychronous neural groups. Classification is performed using a similarity metric between sets, based on a modified version of the Jaccard index. The approach is evaluated on a data set of hand-drawn digits that encode the temporal information on how the digit has been drawn. In addition, the method is compared with three other standard pattern classification methods: support vector machines, logistic regression with regularization and ensemble neural networks, all trained with the same data set. The results show that the proposed approach can successfully learn these patterns from a significantly small number of training samples, can identify patterns before their completion, and it performs better than or comparable with the three other supervised methods.

Philosophical Considerations on Teacher Presence

IntroductIon Contemporary discourses about schooling and teachers are marked by a severe reductionism. Teaching has been operationalized and reduced to, in the best cases, a collection of indicators on checklists for principals, or even to merely a set of digits that should measure students’ learning and thus teachers’ effectiveness. Lorraine Code individuates an “instituted imaginary”1 according to which, in Clarence Joldersma’s words, teachers are viewed as “universal, interchangeable rational atoms, measurable against an impersonal standard, including guaranteed results.”2 A major consequence of this way of considering teachers is that individual differences are considered “deviations from the standard” and “blameworthy deficits.”3 The flattening brought about by this instituted imaginary is the necessary outcome of a reductionist mistake in conceiving what it is to teach and to be a teacher.

Two Lower Bounds for Shortest Double-Base Number System

In this letter, we derive two lower bounds for the number of terms in a double-base number system (DBNS), when the digit set is {1}. For a positive integer n, we show that the number of terms obtained from the greedy algorithm proposed by Dimitrov, Imbert, and Mishra [1] is $\Theta\left(\frac{\log n}{\log \log n}\right)$. Also, we show that the number of terms in the shortest double-base chain is Θ(log n).

Connectedness of planar self-affine sets associated with non-collinear digit sets

We study a class of planar self-affine sets \(T(A,{\mathcal {D}})\) generated by the integer expanding matrices \(A\) with \(|\det A|=3\) and the non-collinear digit sets \({\mathcal {D}}=\{0, v, kAv\}\) where \(k\in {\mathbb {Z}}\setminus \{0\}\) and \(v\in {\mathbb {R}}^{2}\) such that \(\{v, Av\}\) is linearly independent. By examining the characteristic polynomials of \(A\) carefully, we prove that \(T(A,{\mathcal {D}})\) is connected if and only if the parameter \(k=\pm 1\).

A class of spectral self-affine measures with four-element digit sets

The self-affine measure μM,D associated with an expanding matrix M∈Mn(Z) and a finite digit set D⊂Zn is uniquely determined by the self-affine identity with equal weight. In this paper we construct a class of self-affine measures μM,D with four-element digit sets in the higher dimensions (n≥3) such that the Hilbert space L2(μM,D) possesses an orthogonal exponential basis. That is, μM,D is spectral. Such a spectral measure cannot be obtained from the condition of compatible pair. This extends the corresponding result in the plane.

On the connectedness of planar self-affine sets

In this paper, we consider the connectedness of planar self-affine set T(A,D) arising from an integral expanding matrix A with characteristic polynomial f(x)=x2+bx+c and a consecutive collinear digit set D={0,1,…,m}v. The necessary and sufficient conditions only depending on b,c,m are given for the T(A,D) to be connected. Moreover, we also consider the case that D is non-consecutively collinear.

Two Challenges of Correct Validation in Pattern Recognition

Supervised pattern recognition is the process of mapping patterns to class labels that define their meaning. The core methods for pattern recognition have been developed by machine learning experts but due to their broad success an increasing number of non-experts are now employing and refining them. In this perspective I will discuss the challenge of correct validation of supervised pattern recognition systems, in particular when employed by non-experts. To illustrate the problem I will give three examples of common errors that I have encountered in the last year. Much of this challenge can be addressed by strict procedure in validation but there are remaining problems of correctly interpreting comparative work on exemplary data sets, which I will elucidate on the example of the well-used MNIST data set of handwritten digits.

Extensions of Laba-Wang's condition for spectral pairs

Let be a self-affine measure associated with an expanding matrix and a finite digit set . We consider in this paper the spectrality of . In the case when is a compatible pair for some , a necessary condition is obtained for the spectral pair . This condition is shown to be equivalent to the known necessary conditions for the same spectral pair. Moreover, we prove that all these necessary conditions are not sufficient in the higher dimensions, but they are sufficient in the dimension one. This extends Laba-Wang's condition for spectral pairs.

Common expansions in noninteger bases

In this paper we study the existence of simultaneous representations of real numbers in bases p > q > 1 with the digit set A = {−m, . . . , 0, . . . ,m}. Among other results, we prove if m = 1 and q < 2, then there is a continuum of sequences (ci) ∈ A∞ satisfying ∑∞ i=1 ciq −i = ∑∞ i=1 cip −i. On the other hand, if m = 1 and q ≥ 2 + √ 2, then only the trivial sequence (ci) = 0∞ satis es the former equality.

A Parallel and Uniform $k$-Partition Method for Montgomery Multiplication

A way to speed up the Montgomery Multiplication by distributing the multiplier operand bits into <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\mbi {k}}$</tex> </formula> partitions is proposed. All of them process in parallel and use an identical algorithm. Each partition executes its task in <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\mbi {n/k}}$</tex> </formula> steps. Even though the computationstep operates in radix <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\mbi {2^k}}$</tex> </formula> , the complexity is reduced by the use of a limited digit set. Experiments with a 90-nm cell library show that the hardware cost and its complexity have a linear growth according to the number of partitions. Besides the gain in speed, the proposal reduces power consumption for multiplication operands with 256, 512, 1024, and 2048 bits. The uniform treatment of partition hardware design enables the realization of a fault-tolerant hardware.

Digit systems over commutative rings

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be a polynomial. In the present paper we consider digit representations in the residue class ring [Formula: see text]. In particular, we are interested in the question whether each [Formula: see text] can be represented modulo P in the form e0+ e1x + ⋯ + ehxh, where the [Formula: see text] are taken from a fixed finite set of digits. This general concept generalizes both canonical number systems and digit systems over finite fields. Due to the fact that we do not assume that 0 is an element of the digit set and that P need not be monic, several new phenomena occur in this context.

Fast Radix-10 Multiplication Using Redundant BCD Codes

We present the algorithm and architecture of a BCD parallel multiplier that exploits some properties of two different redundant BCD codes to speedup its computation: the redundant BCD excess-3 code (XS-3), and the overloaded BCD representation (ODDS). In addition, new techniques are developed to reduce significantly the latency and area of previous representative high-performance implementations. Partial products are generated in parallel using a signed-digit radix-10 recoding of the BCD multiplier with the digit set [-5, 5], and a set of positive multiplicand multiples (0X, 1X, 2X, 3X, 4X, 5X) coded in XS-3. This encoding has several advantages. First, it is a self-complementing code, so that a negative multiplicand multiple can be obtained by just inverting the bits of the corresponding positive one. Also, the available redundancy allows a fast and simple generation of multiplicand multiples in a carry-free way. Finally, the partial products can be recoded to the ODDS representation by just adding a constant factor into the partial product reduction tree. Since the ODDS uses a similar 4-bit binary encoding as non-redundant BCD, conventional binary VLSI circuit techniques, such as binary carry-save adders and compressor trees, can be adapted efficiently to perform decimal operations. To show the advantages of our architecture, we have synthesized a RTL model for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math>$16\times 16$</tex-math></inline-formula> -digit and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math>$34\times 34$</tex-math></inline-formula> -digit multiplications and performed a comparative survey of the previous most representative designs. We show that the proposed decimal multiplier has an area improvement roughly in the range 20-35 percent for similar target delays with respect to the fastest implementation.

Analysis of the Binary Asymmetric Joint Sparse Form

We consider redundant binary joint digital expansions of integer vectors. The redundancy is used to minimize the Hamming weight,i.e., the number of non-zero digit vectors. This leads to efficient linear combination algorithms in abelian groups, which are used in elliptic curve cryptography, for instance.If the digit set is a set of contiguous integers containing zero, a special syntactical condition is known to minimize the weight. We analyse the optimal weight of all non-negative integer vectors with maximum entry less thanN. The expectation and the variance are given with a main term and a periodic fluctuation in the second-order term. Finally, we prove asymptotic normality.

Classifying chemical sensor data using GPU-accelerated bio-mimetic neuronal networks based on the insect olfactory system

Chemosensing “e-nose” technology has great potential applications in everyday life, ranging from drug detection to food quality assessment and even the diagnosis of illness. However, odour detection and classification remains a highly challenging domain, characterized by high dimensionality, unknown organization of the vast “odourant space” of volatile chemicals and complex, turbulent odour plumes. To compound these difficulties, current sensor technology continues to exhibit distinct shortcomings in speed, sensitivity, selectivity, recovery, and drift avoidance. In the research reported here we turn to a range of recent neuronal models [1-6] that were developed to describe the insect olfactory system, and have been shown to perform well across a range of classic, static classification tasks such as the MNIST handwritten digit set and the Sigma-Aldrich scent database. The insect olfactory system has been extensively studied and has been shown to be both fast and highly effective in complex natural conditions despite its limited size and complexity (when compared to the mammalian system) [2]. Insects such as moths, honey bees, locusts and fruit flies are capable of odour detection and classification tasks well beyond the abilities of current e-nose technology and machine learning algorithms [6]. We present results of applying an insect-inspired approach to the design of a learning spiking neural network that receives synchronized time series data from up to 12 metal-oxide based gas sensors, comprising an optimised [7] combination of classical doped tin oxide and novel zeolite-coated chromium titanium oxide sensors. We have collected sample data sets for classification tasks ranging from “easy” (single chemical identification presented under laboratory conditions) through to “hard” (identification of indicators of infectious diseases in breath samples taken from patients). To address slow sensor response we consider a range of transient-based processing before applying self-organisation techniques to most effectively locate “virtual receptors” (VR) in sensor space. We look to address decorrelation (separation in feature space) and the supervised association of responses with rewards by using correlates of the insect antennal lobe (AL) and mushroom body (MB) structures whilst applying reward-based spike-timing dependent plasticity mechanisms. Classification accuracy is compared with support vector machine (SVM) learning which we also look to match for speed through the use of GPU accelerated neural simulation [5] via the NVidia CUDA TM -based GeNN platform (http://sourceforge.net/projects/genn/).

The connectedness of some two-dimensional self-affine sets

In the paper, we mainly discuss the connectedness of two kinds of self-affine sets. One is generated by matrix A=(p0−aq) and digit set D={(is,jt)T:i=0,1,…,|q|−1,j=0,1,…,|p|−1}, where s,t≠0 and p,q∈Z with 3≤|p|+1<|q|<2|p|−1. The other is generated by matrix A=(p0−aq) and digit set D={(is,(di+j)t)T:i=0,1,…,|p|−1,j=0,1,…,|q|−1}, where s,t≠0, and p,q,d∈Z with |p|,|q|≥2. The sufficient or necessary conditions for their connectedness are revealed.