The Tangent FFT

Signal processing had a significant impact on the development of many elements of modern life, including telecommunications, education, healthcare, industry, and security. The semiconductor industry is the primary driver of signal processing innovation, producing ever-more sophisticated electronic devices and circuits in response to global demand. In addition, the central processing unit (CPU) is described as the “brain” of a computer or all electronic devices and signal processing. CPU is a critical electronic device that includes vital components such as memory, multiplier, adder, etc. Also, one of the essential components of the CPU is the arithmetic and logic unit (ALU), which executes the arithmetic and logical operations within all types of CPU operations, such as addition, multiplication, and subtraction. However, delay, occupied areas, and energy consumption are essential parameters in ALU circuits. Since the recent ALU designs experienced problems like high delay, high occupied area, and high energy consumption, implementing electronic circuits based on new technology can significantly boost the performance of entire signal processing devices, including microcontrollers, microprocessors, and printed devices, with high-speed and low occupied space. Quantum dot cellular automata (QCA) is an effective technology for implementing all electronic circuits and signal processing applications to solve these shortcomings. It is a transistor-less nanotechnology being explored as a successor to established technologies like CMOS and VLSI due to its ultra-low power dissipation, high device density, fast operating speed in THz, and reduced circuit complexity. This research proposes a ground-breaking ALU that upgrades electrical devices such as microcontrollers by applying cutting-edge QCA nanotechnology. The primary goal is to offer a novel ALU architecture that fully utilizes the potential of QCA nanotechnology. Using a new and efficient approach, the fundamental gates are skillfully utilized with a coplanar layout based on a single cell not rotated. Furthermore, this work presents an enhanced 1-bit and 2-bit arithmetic logic unit in quantum dot cellular automata. The recommended design includes logic, arithmetic operations, full adder (FA) design, and multiplexers. Using the powerful simulation tools QCADesigner, all proposed designs are evaluated and verified. The simulation outcomes indicates that the suggested ALU has 42.48 and 64.28% improvements concerning cell count and total occupied area in comparison to the best earlier single-layer and multi-layer designs.

In computable analysis, sequences of rational numbers which effectively converge to a real number x are used as the (rho-) names of x. A real number x is computable if it has a computable name, and a real function f is computable if there is a Turing machine M which computes f in the sense that, M accepts any rho-name of x as input and outputs a rho-name of f(x) for any x in the domain of f. By weakening the effectiveness requirement of the convergence and classifying the converging speeds of rational sequences, several interesting classes of real numbers of weak computability have been introduced in literature, e.g., in addition to the class of computable real numbers (EC), we have the classes of semi-computable (SC), weakly computable (WC), divergence bounded computable (DBC) and computably approximable real numbers (CA). In this paper, we are interested in the weak computability of continuous real functions and try to introduce an analogous classification of weakly computable real functions. We present definitions of these functions by Turing machines as well as by sequences of rational polygons and prove these two definitions are not equivalent. Furthermore, we explore the properties of these functions, and among others, show their closure properties under arithmetic operations and composition.

T-sum of sigmoid-shaped fuzzy intervals

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.

Multiplication is an important fundamental function in arithmetic operations. Multiplication-based operations such as Multiply and Accumulate(MAC) and inner product are among some of the frequently used computation Intensive Arithmetic Functions(CIAF) currently implemented in many Digital Signal Processing (DSP) applications such as convolution, Fast Fourier Transform(FFT), filtering and in microprocessors m its arithmetic and logic unit. Since multiplication dominates the execution tune of most DSP algorithms, so there is a need of high speed multiplier. Currently, multiplication time is still the dominant factor in determining the instruction cycle time of a DSP chip. The demand for high speed processing has been increasing as a result of expanding computer and signal processing applications. Higher throughput arithmetic operations are important to achieve the desired performance in many real-time signal and image processing applications . One of the key arithmetic operations in such applications is multiplication and the development of fast multiplier circuit has been a subject of interest over decades. Multiplier based on Vedic Mathematics is one of the fast and low power multiplier. Employing this technique in the computation algorithms will reduce the complexity, execution time, power etc. This vedic based multiplier is compared with binary multiplier(partial products method).

In the recent years, numerous proof systems have improved enough to be used for formally verifying non-trivial mathematical results. They, however, have different purposes and it is not always easy to choose which one is adapted to undertake a formalization effort. In this survey, we focus on properties related to real analysis: real numbers, arithmetic operators, limits, differentiability, integrability and so on. We have chosen to look into the formalizations provided in standard by the following systems: Coq, HOL4, HOL Light, Isabelle/HOL, Mizar, ProofPower-HOL, and PVS. We have also accounted for large developments that play a similar role or extend standard libraries: ACL2(r) for ACL2, C-CoRN/MathClasses for Coq, and the NASA PVS library. This survey presents how real numbers have been defined in these various provers and how the notions of real analysis described above have been formalized. We also look at the methods of automation these systems provide for real analysis.

We studied an efficient software implementation of H.264/AVC sub-pixel motion estimation (ME) algorithm on a VLIW-SIMD digital signal processor, TMS320C6416. The sub-pixel ME algorithm demands large memory accesses while the required arithmetic operations are fairly simple. Although the CPU clock cycles for arithmetic operations can be reduced much by employing sub-word operations and applying software pipelining techniques, the limited memory bandwidth of the architecture restricts the overall performance. Moreover, aggressive VLIW-SIMD optimization results in the degradation of the performance by causing excessive CPU stalls during memory accesses. In this paper, we relieved the memory bandwidth requirements for creating quarter-pixel images by reducing the precision of image data, from 8 bits to 4 bits. As a result, the amount of memory accesses is much reduced at the cost of some increase of the arithmetic operations, which contributes to the balance of arithmetic and memory access operations. The experimental result shows that the memory stall cycles are decreased by 80% and the speed-up of 260% is obtained. The bit rate of the encoded video stream is increased slightly, about 2% on the average, due to the effects of quantization.

Research on learning through serious games has been gaining importance in recent years. We develop the serious game Tempoly, concerning the learning of the four arithmetic operations on polynomials. A quasi-experimental study was carried out with 95 students of 8th grade, to evaluate the reactions of the students to the game, the effects of the use of the game in the students’ learning, and the use of the game through time. According to data collected through a questionnaire, we may conclude that students liked to play the game, considered that the game helped them learning the polynomial operations and that what they learned in class was useful to play the game. It was concluded that students who played the game simultaneously with the classes improved their results in a pre/post-test more than the students who did not play the game and through a Mann-Whitney test, the results showed statistical differences. The study revealed that the students that had more time to play the game did play more out of the class.

A number system uses a set of digits and sign to represent all the numbers. Different number systems use different amount of digits and sign. To compare the leanness of a number system, the cardinality of the complete set of digits and sign are employed in this work. The 01 binary number system is used by almost all the modern computers. It can represent all the real numbers by two digits 0, 1 and a sign (-), which means its cardinality is 3. Thus, the 01 binary system is not a true binary system. After reviewing all the existing number systems it is found that no true binary system exists for real numbers. All the current binary number systems either don’t have radix 2, or have cardinality 3. In this paper, a true binary system for real numbers, with both radix and cardinality 2, is invented based on the principle of Yin-Yang. Surprisingly, this binary system can not only represent all real numbers without using a sign, but also perform arithmetic and Boolean operations without any problem. Furthermore, due to its radix being 2, this true binary system is compatible with the 01 binary system and many operations are interchangeable between the two systems. It is expected that this true binary system will be applied in the fields of number representation, computer science and cryptography.

The Real Number Model in Numerical Analysis

We consider probabilistic circuits working over the real numbers and using arbitrary semialgebraic functions of bounded description complexity as gates. In particular, such circuits can use all arithmetic operations (+, −, ×, ÷), optimization operations (min and max), conditional branching (if-then-else), and many more. We show that probabilistic circuits using any of these operations as gates can be simulated by deterministic circuits with only about a quadratical blowup in size. A slightly larger blowup in circuit size is also shown when derandomizing approximating circuits. The algorithmic consequence, motivating the title, is that randomness cannot substantially speed up dynamic programming algorithms.

Airborne critical systems that use real numbers in their algorithms can be considerably affected by the floating-point discretized representation. This limitation in the representation is the cause of many problems already known while running software with floating-point variables such as loss of precision in arithmetic operations. This work presents a method for detecting floating-point absorption and cancellation phenomena using static analysis in software design models. Our method uses a range analysis tool that operates over SCADE model to propagate the range of the internal variables and, then, each addition and subtraction operation in the model is analyzed to check the occurrence of the floating-point absorption and cancellation phenomena. The results are presented in an HTML report containing absorption alerts and cancellations charts providing the number of canceled bits for each sub-range in order to permit better assessment of the impact of each detected cancellation. In order to validate the proposed method, we compare the results of our proposed method with the results obtained by an alternative validation method developed using the source code as input and the Frama-C EVA to propagate internal ranges. The method can be used as an early design analysis enabling the cheaper detection and treatment of floating-point anomalies.

Geometric algorithms are usually described assuming that arithmetic operations are performed exactly on real numbers. A program implemented using a naive substitution of floating-point arithmetic for real arithmetic can fail, since geometric primitives depend upon sign-evaluation and may not be reliable if evaluated approximately. Geometric primitives are reliable if evaluated exactly with integer arithmetic, but this degrades performance since software extended-precision arithmetic is required.

Some Extensions of Markov's Constructive Continuum and Their Applications to the Theory of Constructive Functions

The usual arithmetic operations on real numbers can be extended to arithmetical operations on fuzzy intervals by means of Zadeh's extension principle based on a t-norm T. Dombi and Gyorbiro proved that addition is closed if the Dombi t-norm is used with two bell-shaped fuzzy intervals. Recently, Hong [Fuzzy Sets and Systems 158(2007) 739-746] defined a broader class of bell-shaped fuzzy intervals. Then he study t-norms which are consistent with these particular types of fuzzy intervals as applications of a result proved by Mesiar on a strict f-norm based shape preserving additions of LR-fuzzy intervals with unbounded support. In this note, we give a direct proof of the main results of Hong.