Cyclic Convolution Research Articles

New transform using the Mersenne numbers

Number theoretic transforms (NTTs) find applications in the calculation of convolutions and correlations. They can perform these calculations without introducing additional noise in the processing due to rounding or truncation. Among all NTTs, Fermat and Mersenne number transforms have been given particular attention. However, the main drawback of these transforms is the inconvenient word length for the Fermat number transforms, and lack of fast algorithms for the Mersenne number transforms. The authors aim to introduce a new real transform defined modulo Mersenne numbers with long transform length equal to a power of two. This is achieved by dropping the condition that /spl alpha/_ should be /spl plusmn/2 and using a new definition for NTTs that departs from the usual Fourier-like definition. The new transform is suitable for fast algorithms. It has the cyclic convolution property and hence can be applied to the calculation of convolutions and correlations. The transform is extended to the two-dimensional case and then generalised to the multidimensional case. Examples are given for the one-dimensional and two-dimensional cases.

Rings, fields, the Chinese remainder theorem and an extension-Part II: applications to digital signal processing

For pt. I, see ibid., vol. 41, no. 10, p. 641-55 (1994). In Part I of the research work, we introduced an extension to the well known Chinese remainder theorem for processing polynomials with coefficients defined over a finite integer ring. We term this extension as the American-Indian-Chinese extension of the Chinese remainder theorem. A systematic procedure for factorizing a monic polynomial into pairwise relatively prime monic factor polynomials over integer rings was presented. This factorization is based on the corresponding factor polynomials, monic and relatively prime, over the associated finite field containing prime number of elements. In this paper, we study the application of the theory developed in Part I to deriving computationally efficient algorithms for performing tasks having multilinear form. Especially, we focus on the cyclic and acyclic convolution as they are two of the most frequently occurring computationally intensive tasks in digital signal processing. >

Real-valued decimation-in-time and decimation-in-frequency algorithms

The decimation-in-time (DIT) and the decimation-in-frequency (DIF) algorithms are the typical forms of the fast Fourier transform (FFT) algorithm. Many hardware and software implementations are based on these algorithms. One class of fast algorithms for computing the discrete Fourier transform (DFT) is based on a recursive factorization of the polynomial 1-z/sup N/. This paper introduces a simple recursive factorization of 1-z/sup N/ over the real numbers and a mathematical framework that generalizes the form of the DFT. Using the recursive factorization, efficient algorithms are derived to compute the DFT and the cyclic convolution of sequences of length with a power of two. Real-valued DIT and real-valued DIF algorithms are developed so that the accumulated FFT technologies can be fully utilized for real sequences. Introducing a real-valued butterfly, the algorithmic structures of the DIT and the DIF algorithms are shown to be equally applicable for the real-valued algorithms by systematic modifications. The computational complexity is fairly comparable with other available fast algorithms.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Two algorithms of the fundamental harmonic complex RMS value calculation

This paper discusses two algorithms for the complex rms valueX1 of the fundamental harmonic of the nonsinusoidal quantities calculation which are much more computationally effective than the traditional FFT algorithms. They mitigate, moreover, the requirement regarding the sampling rate of the A/D converters as compared with the method suggested by Sawicki in [4]. One of these algorithms is based on the DFT direct calculation, the second applies the cyclic convolution. The paper shows also how to evaluate the aliasing error and the error caused by the lack of the sampling frequency synchronization with the frequency of the sampled quantity.

New multipliers modulo 2/sup N/-1

Techniques for computing the product of two N-bit integers modulo 2/sup N/-1 from their k-bit byte decompositions are presented. A modulus 2/sup N/-1 is chosen, as multiplication performed in this modulus can be reconstructed from the cyclic convolution between the sequences of the k-bit bytes of the decomposed numbers. It is shown that cyclic convolutions can be computed using only additions and squaring operations but not two-operand multiplications. Since the squaring operation is a one-operand operation, significant savings in ROM bits can be obtained if look-up tables are used.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Decomposition of complex multipliers using polynomial encoding

A method for complex multiplication that relies on encoding 2n-bit complex numbers as polynomials of degree 7 in the ring of polynomials modulo x/sup 8/-1 with n/4-bit coefficients is introduced. Complex multiplication can then be performed with an 8-point cyclic convolution plus some conversion overhead and, with care, this can be done without introducing any errors. The technique is suitable for designs using systolic arrays.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The efficient memory-based VLSI array designs for DFT and DCT

Efficient memory-based VLSI arrays and a new design approach for the discrete Fourier transform (DFT) and discrete cosine transform (DCT) are presented. The DFT and DCT are formulated as cyclic convolution forms and mapped into linear arrays which characterize small numbers of I/O channels and low I/O bandwidth. Since the multipliers consume much hardware area, the designs utilize small ROMs and adders to implement the multiplications. Moreover, the ROM size can be reduced effectively by arranging the data in the designs appropriately. The arrays outperform others in the architectural topology (local and regular connection), computing speeds, hardware complexity, the number of I/O channels, and I/O bandwidth. They benefit from the advantages of both systolic array and the memory-based architectures. >

Adaptation of group algebras to signal and image processing

Filtering techniques in digital signal and image processing typically use linear and/or cyclic convolutions, both of which are based on polynomial arithmetic. When the input data and the filter weights are treated as coefficients of polynomials, linear convolution is computed directly as a product of these polynomials. Cyclic convolution is used as an efficient means of computing the linear convolution. In this case, the input is organized into sections, each section treated as the coefficients of a polynomial. The output data are generated by selecting the last coefficients of the products of the section polynomials with the filter polynomial. We develop a more general methodology for filtering signals and images on the basis of the ideas of: (1) group algebras, a generalization of polynomials and polynomial arithmetic, and (2) value functions, a generalization of the coefficient selection used in generating the output data. Applications of these ideas to signal and image processing are given. Finally, the concepts are incorporated into the framework of image algebras.

Two-sided filters for frame-based prediction

A linear prediction model, based on a two-sided predictor which predicts on the basis of past and future samples within a frame, is presented. The linear prediction model may be applied wherever frame-based prediction is used. A stable synthesis procedure is derived by casting the prediction equation as a cyclic convolution in the time domain. When the filter order is the maximum possible, the synthesis filter is shown to have a frequency response proportional to the squared magnitude of the DFT of the frame. A symmetric two-sided predictor which has only half the number of coefficients to be coded as compared to a one-sided predictor of the same order is described. Two-sided prediction showed at least 5-dB improvement in prediction gain over one-sided prediction in simulations on speech data.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Multiplicative complexity of bilinear algorithms for cyclic convolution over finite fields

The multiplicative complexity of bilinear algorithms for cyclic convolution over finite fields is investigated. It is shown that mutually prime factor algorithms are inferior to directly designed algorithms for all lengths except those whose factors have relatively prime exponents. A previously described approach is proposed for directly designing algorithms which are highly structured and computationally efficient. Several complexity results are provided for factor lengths of specific form, and the manner in which cyclic convolution algorithms lead to linear algebraic error-correcting codes is discussed.

THE DISCRETE FOURIER TRANSFORM AND CYCLIC CONVOLUTION ON INTEGRAL LATTICES

A discrete Fourier transform and a cyclic convolution are constructed on an arbitrary integral lattice. The construction includes as a special case the usual discrete Fourier transform and the usual cyclic convolution. Applications to questions of interpolation of functions and digital signal processing are considered. Methods in the spectral theory of automorphic functions are used to investigate questions in approximation of arbitrary lattices by integral lattices.Bibliography: 14 titles.

Through the mincing machine with a Boolean layer cake

The purpose of the present paper is threefold. Firstly, it introduces an abstract theory of straight-line (circuit) Boolean computations and complexity based on a strict separation of the syntax and semantics of Boolean circuits; the usual (standard) way to compute a circuit becomes thus only one of many other (nonstandard) operation modes over the circuit. Secondly, we construct and investigate three particular nonstandard operation modes, of which one, the mincing (M-)mode, is the superposition of two others, the injection and ejection modes respectively. The M-mode uses the circuit as a pipe-lining device, channeling simultaneously through each layer of the circuit (a layer comprises all nodes of a given individual depth) information stored in previous layers and renewing the input and storing the output information after every such parallel step; the number of steps is equal to the double depth of the circuit. The function computed by the M-mode is richer as a functional device than the standard functional (S-)representation of a circuit; in particular, it not only detects paths connecting individual inputs and outputs of a circuit, as the S-representation does, but their lengths as well. Thirdly, based on the M-mode, a new (in fact, the first known nontrivial) general lower bound to the relative circuit-size complexity of a Boolean function is presented, in the form of the inequality $$s(f, C') \geqq m(f, C')$$ , which means that the circuit-size (CS-)complexity of a Boolean function f with respect to a fixed subset C′ ⫅ C of the set C of all Boolean circuits is at least as big as the mincing complexity of f with respect to the same subset; choosing appropriate subsets, one arrives at the complete, monotone, synchronous, locally synchronous (and in algebraic computations which can be naturally treated in the same vein, linear, bilinear) or other CS-complexities. The corresponding mincing complexity is defined as the minimum, taken over all circuits from C′ standardly computing f, of the Boolean dimension of the function computing by the M-mode; the Boolean dimension of a function is equal by definition to the 2-logarithm of the cardinality of the function's image. The advantage to work with the new lower bound is that it is defined as a minimum taken over a relatively well defined functional subspace associated with f and C′, and not, as in the CS-complexity case, over a subset of circuits, i.e., a mixed discrete-continuum structure of an obscure nature (the real reason why the lower bound problem is so difficult). To demonstrate how the above inequality works, we prove an Ω(m·log m) lower bound to the locally synchronous CS-complexity of the m-dimensional cyclic convolution. This particular result is important for several reasons: (i) it improves the more than ten-years old non-linear lower bound to the synchronous CS-complexity of the same function, (ii) it can not be, apparently, proved by previously known methods, (iii) there exist functions (an example will be given in this paper) with the linear locally synchronous and a non-linear synchronous CS-complexities.

A new DSP‐oriented FFT algorithm

AbstractIn the traditional fast‐Fourier trans form (FFT) algorithm, the design for high‐ speed operation is made based on the number of operations, especially the number of multiplications, which is time‐consuming in the general‐purpose processor. This is not necessarily reasonable in the digital signal processor (DSP), where the multiplication can be performed in the time equal to that of other operations. From such a viewpoint, this paper proposes an algorithm which matches the feature of DSP, where the pro cessing is fast when the multiply‐accumulate operations are continued in execution. the algorithm is applicable when the number of data IV can be factorized into mutually prime factors. DFT for smaller number (factor) of data is modified into the cyclic convolution, and is executed at a high speed (small‐ N algorithm). the proposed algorithm is composed of the small‐N algorithms and DFT for a large number of data (multidimensional algorithm), which combines small‐N algorithms into a multidimensional DFT. the number of operations is not necessarily reduced in the proposed algorithm, compared with the traditional FFT, but it was verified by an actual run of the program on DSP that the execution time is reduced.

An FPT algorithm with a modularized structure for computing 2-D cyclic convolutions

The fast polynomial transform (FPT) for computing 2-D cyclic convolutions is modularized into identical modules. In the new method, the 1-D cyclic polynomial convolution is decomposed into cycle convolutions of polynomials, all of the same length. Thus, only FPTs and FFT's of the same length are required. As a consequence, the architecture is more regular and naturally suitable for a VLSI implementation. >

The Arithmetic Cube

We present the design of a VLSI processor which can be programmed to compute the discrete Fourier transform of a sequence of n points and which achieves the theoretical AT2 lower bound of ?(n2) for n ? n where n is an infinite set. Furthermore, since the set n is also sufficiently dense, the processor achieves for any n the theoretical AT2 lower bound of ?(n2) for computing the cyclic convolution of two sequences of n points. Uniquely, our design achieves this bound without the use of data shuffling or long wires. Also, the processor uses only approximately ?n multipliers, while many other designs need ?(n) multipliers to achieve the same time bounds. Since multipliers are usually much larger than adders, the processor presented in this paper should be smaller. The design also features layout regularity, minimal control, and nearest neighbor interconnect of arithmetic cells of a few different types. These characteristics make it an ideal candidate for VLSI implementation.

Improved Fourier and Hartley transform algorithms: Application to cyclic convolution of real data

This paper highlights the possible tradeoffs between arithmetic and structural complexity when computing cyclic convolution of real data in the transform domain. Both Fourier and Hartley-based schemes are first explained in their usual form and then improved, either from the structural point of view or in the number of operations involved. Namely, we first present an algorithm for the in-place computation of the discrete Fourier transform on real data: a decimation-in-time split-radix algorithm, more compact than the previously published one. Second, we present a new fast Hartley transform algorithm with a reduced number of operations. A more regular convolution scheme based on FFT's is also proposed. Finally, we show that Hartley transforms belong to a larger class of algorithms characterized by their "generalized" convolution property.

Multidimensional cyclic convolution algorithms with minimal multiplicative complexity

This paper presents novel multidimensional cyclic convolution algorithms which achieve the theoretically minimum number of multiplications. The proposed algorithms are compared to the well-known and used polynomial transform and split nesting convolution algorithms.

Index transforms of multidimensional cyclic convolutions and discrete Fourier transforms

In this correspondence, general equations are described for computing the so-called index transforms which map the m-dimensional discrete Fourier transform (DFT) and cyclic convolution (CC) on n-dimensional DFT or CC, respectively. These transforms are based on mapping indexes of input and output sequences of DFT or CC operators. The general form of these transforms is found, and the necessary and sufficient conditions of their existence are described. Universality of the results is illustrated from the standpoint of practical applications.

Implementation of complex number theoretic transforms using quadratic residue number systems

Very recently, the Quadratic Residue Number System (QRNS) has been introduced [6], [7]. The QRNS is obtained from a mapping of Gaussian integers over a finite ring to a ring of conjugate elements. The conjugate ring has the remarkable property that both addition and multiplication are performed component-wise. Complex multiplication only requires two base field multiplications and zero additions; this contrasts to the requirement of four multiplications and two additions, or three multiplications and five additions for the Gaussian integer ring. The operations are performed over sub-rings, isomorphic to the conjugate ring, and the results mapped to the conjugate ring via the Chinese Remainder Theorem isomorphism. The primary restriction is the limited form of the moduli set for RNS computations. The QRNS has since been generalized for any type of moduli set with an increase in multiplications from 2 to 3 and the resulting number system has been termed the Modified Quadratic Residue Number System (MQRNS) [3], [4]. In [4] the direct FIR filter architecture and bit-slice architecture for FIR and recursive digital filters have been presented using the QRNS and MQRNS. In this paper, the computation of the Complex Number Theoretic Transform (CNTT) and the hardware implementation of a radix-2 butterfly structure, using high-density ROM arrays, are presented. This paper shows that both the QRNS and MQRNS require almost the same amount of hardware for the implementation of the butterfly structure. The computation of Cyclic Convolution in both the QRNS and MQRNS is discussed.

Voltammetric studies of the electrochemical behaviour of salicylidene-2-aminopyridine at a hanging-mercury-drop-electrode

Electrochemical behaviour of salicylidene-2-aminopyridine has been investigated in 0.1 M tetraethylammonium perchlorate–dimethylformamide solutions by cyclic voltammetric and convolution potential sweep voltammetric methods. It was found that the depolarizer exhibits two well-defined diffusion-controlled irreversible one-electron waves. The cyclic voltammetric characteristics and the convolution, deconvolution, and logarithmic convolution analyses reveal that salicylidene-2-aminopyridine in such conditions follows a set of two one-electron transfer reactions each followed by an irreversible chemical reaction. The values of the first-order rate constant of the irreversible chemical reaction and E1/2 were computed.