The multiplicative complexity of discrete cosine transforms (original) (raw)

On the multiplicative complexity of discrete cosine transforms

IEEE Transactions on Information Theory, 1992

The multiplicative complexity of discrete cosine transforms (DCT's) of arbitrary dimensions on inputs sizes, which are powers of two, are obtained. New upper bounds on the multiplicative complexity of scaled DCT's on input sizes, which are powers of two, are also obtained.

On relating discrete Fourier, sine, and symmetric cosine transforms

Acoustics, Speech and Signal Processing, IEEE …, 1985

The relationship among a real formalism of the discrete be needed: Fourier transform, discrete sine transform, and discrete symmetric cosine transform is discussed. it is shown that the real formalism of I i (N) = 2'11 (N) or 0 modulo N (1) the discrete Fourier transform is basically equivalent to the direct sum of the other two transforms, with modifications in the pre-and post-= 2Z12 (N) (2) computations with the data vector. where I is a natural number between 0 and log, N-1 and I. INTRODUCTION I , (N) = 1 + 4 k modulo N (3) k E integer.

Comments on "Canonical transformations of the discrete cosine transform

Signal Processing, 2008

In the paper ''Canonical transformations of the discrete cosine transform'' two relations between discrete cosine transforms (DCTs) of types 2 and 4 were presented. One relation is already known, the second was derived by the author of the paper. We would like to point out that the common theory published recently by M. Pu¨schel and J. Moura allows the discovery of existing and new relations between all 16 known types of DCTs and discrete sine transforms (DSTs). In these comments we show that above mentioned relations between DCT-II and DCT-IV can be easily derived from this universal theory. Note also that the second relation was implicitly presented in the fast algorithm of inverse DCT-II derived by B.G. Lee. r

Small-Size Algorithms for the Type-I Discrete Cosine Transform with Reduced Complexity

Electronics

Discrete cosine transforms (DCTs) are widely used in intelligent electronic systems for data storage, processing, and transmission. The popularity of using these transformations, on the one hand, is explained by their unique properties and, on the other hand, by the availability of fast algorithms that minimize the computational and hardware complexity of their implementation. The type-I DCT has so far been perhaps the least popular, and there have been practically no publications on fast algorithms for its implementation. However, at present the situation has changed; therefore, the development of effective methods for implementing this type of DCT becomes an urgent task. This article proposes several algorithmic solutions for implementing type-I DCTs. A set of type-I DCT algorithms for small lengths N=2,3,4,5,6,7,8 is presented. The effectiveness of the proposed solutions is due to the possibility of fortunate factorization of the small-size DCT-I matrices, which reduces the compl...

Small-Size Algorithms for Type-IV Discrete Cosine Transform with Reduced Multiplicative Complexity

Radioelectronics and Communications Systems, 2020

Discrete cosine transforms are widely used in smart radioelectronic systems for processing and analysis of incoming information. The popularity of using these transform is explained by the presence of fast algorithms that minimize the computational and hardware complexity of their implementation. Type-IV discrete cosine transform occupies a special place in the list of the specified transformations. This article proposes several algorithmic solutions for implementing the type-IV discrete cosine transform. The effectiveness of the proposed solutions is explained by the possibility of factorization of the DCT-IV matrix, which leads to a decrease in computational and implementation complexity. A set of completely parallel type-IV DCT algorithms for small lengths of signal sequences (N

The Discrete Cosine Transform over Finite Prime Fields

Lecture Notes in Computer Science, LNCS 3124, 2004

This paper examines finite field trigonometry as a tool to construct trigonometric digital transforms. In particular, by using properties of the k-cosine function over GF(p), the Finite Field Discrete Cosine Transform (FFDCT) is introduced. The FFDCT pair in GF(p) is defined, having blocklengths that are divisors of (p+1)/2. A special case is the Mersenne FFDCT, defined when p is a Mersenne prime. In this instance blocklengths that are powers of two are possible and radix-2 fast algorithms can be used to compute the transform.

Fast computing of discrete cosine and sine transforms of types VI and VII

Applications of Digital Image Processing XXXIV, 2011

We propose fast algorithms for computing Discrete Sine and Discrete Cosine Transforms (DCT and DST) of types VI and VII. Particular attention is paid to derivation of fast algorithms for computing DST-VII of lengths 4 and 8, which are currently under consideration for inclusion in ISO/IEC/ITU-T High Efficiency Video Coding (HEVC) standard.

On compatibility of order-8 integer cosine transforms and the discrete cosine transform

The author shows that integer cosine transforms (ICTs) are functionally compatible with discrete cosine transforms (DCTs) that are used in image coding. It is provided that the w-bit ICT can inversely transform exactly represented DCT coefficients with less mean-square-error than the w-bit DCT for w equal to 4, 3 and 2 and scaling factors of ICTs implemented using 8 bits. Conversely, an exactly represented DCT can inversely transform the coefficients from the w-bit ICT with less mean-square-error than the w-bit DCT. Therefore, the ICTs can be said to be compatible with the DCT. These ICTs, while being considered as new transforms, can also be regarded as alternative and better ways to implement the DCT when the number of bits for representing kernel components are restricted to 4, 3, and 2

Canonical transformations of the discrete cosine transform

Signal Processing, 2007

We provide different transformation formulae between the different discrete cosine transform (DCT) types of the same size. The transformations use only diagonal and special lower/upper triangular matrices that minimize the overhead of transformation. These transformations provide a tool for using any of the DCT types as a core module for computing all other types. r

The Discrete Cosine Transform over Prime Finite Fields

Lecture Notes in Computer Science, 2004

Split-Radix Algorithms for Discrete Trigonometric Transforms

Abstract: In this paper, we derive new split{radix DCT{algorithms of radix{2 length, which arebased on real factorization of the corresponding cosine matrices into products of sparse,orthogonal matrices. These algorithms use only permutations, scaling with2, butteryoperations, and plane rotations/rotation{reections. They can be seen by analogy withthe well{known split{radix FFT. Our new algorithms have a very low arithmetical complexitywhich compares with the best known fast DCT{algorithms....

Complexity of the Fourier transform on the Johnson graph

arXiv: Combinatorics, 2017

The set XXX of kkk-subsets of an nnn-set has a natural graph structure where two kkk-subsets are connected if and only if the size of their intersection is k−1k-1k−1. This is known as the Johnson graph. The symmetric group SnS_nSn acts on the space of complex functions on XXX and this space has a multiplicity-free decomposition as sum of irreducible representations of SnS_nSn, so it has a well-defined Gelfand-Tsetlin basis up to scalars. The Fourier transform on the Johnson graph is defined as the change of basis matrix from the delta function basis to the Gelfand-Tsetlin basis. The direct application of this matrix to a generic vector requires binomnk2\binom{n}{k}^2binomnk2 arithmetic operations. We show that --in analogy with the standard Fast Fourier Transform on the discrete circle-- this matrix can be factorized as a product of n−1n-1n−1 orthogonal matrices, each one with at most two nonzero elements in each column. This factorization shows that the number of arithmetic operations required to apply this...

Fast computing of discrete cosine and sine transforms of types VI and VII

2011

A New Algorithm for Discrete Cosine Transform of Arbitrary Number of Points

IEEE Transactions on Computers, 1980

An alternate algorithm to compute the discrete cosine transform (DCT) of sequences of arbitrary number of points is proposed. The algorithm consists of partitioning the DCT kernel into submatrices which by proper row and column shuffling and negations can be made equivalent to the group tables (or parts of them) of appropriate Abelian groups. The computations pertaining to the submatrices can be carried out using multidimensional cyclic convolutions. Algorithms are also developed to perform the computations associated with the submatrices that are parts of larger group tables. The new algorithms are more versatile and generally better in terms of the computational complexity in comparison with the existing algorithms.

Faible complexité et haute performance de la transformée de Fourier

2013

Le trav~il présenté par cette thèse porte sur l'amélioration de la transformation rapide de Fourier (TRF) et représente une contribution aux progrès dans le traitement numérique du signal et des algorithmes de calcul rapide. La réduction des temps de calcul offerte par la TRF proposée trouve des applications en traitement numérique du signal à temps réel et en analyse spectrale. C' est une contribution bien accueillie dans les domaines du traitement de la parole, les communications par satellite et terrestre, communications numériques avec ou sans fil , traitement du signal multidiffusion , détections et identifications des cibles, radar et systèmes de sonar, machine aux signaux surveillés, sismologie et biomédecine. En outre, les propositions peut être d'intérêt particulier dans les applications de communication sans fil , les cartes DSP (Digital Signal Processor) et FPGA (Field Programmable Gate Array ). Cette thèse développe et présente un algorithme de la TRF à radice-r qui réduit l'effort de calcul (telle que mesurée par le nombre d'opérations arithmétiques) par un facteur de r en comparaison avec la plupart des algorithmes de la TRF à radice-r. Le problème réside dans la définition du modèle mathématique de la phase de combinaison, dans laquelle la représentation de la TDF en termes de ses TDF partielles devrait être bien structuré pour obtenir le vrai modèle mathématique. L'algorithme qui en résulte, dans lequel les r processeurs en parallèles pourraient fonctionner simultanément avec une seule instruction. La clé conceptuelle du papillon modifié de la TRF à base r est la formulation de la TRF à radice-r comme r éléments de traitement élémentaires (BPE -Butterfly Processing Element) avec des structures identiques et un moyen systématique d'accéder les coefficients v Quantization Noise Ratio) en comparaison avec l'algorithme de Goertzel. Enfin, pour ce domaine nous avons développé le Low Complexity Input/output Pruning FFTs qui est une méthode utilisée pour calculer une DFT où un sous-ensemble des sorties sont nécessaires. VII

The fractional discrete cosine transform

IEEE Transactions on Signal Processing, 2002

Abstract The extension of the Fourier transform operator to a fractional power has received much attention in signal theory and is finding attractive applications. The paper introduces and develops the fractional discrete cosine transform (DCT) on the same lines, discussing multiplicity and computational aspects. Similarities and differences with respect to the fractional Fourier transform are pointed out

An order-16 integer cosine transform

IEEE Transactions on Signal Processing, 1991

It is possible to replace the real-numbered elements of a discrete cosine transform (DCT) matrix by integers and still maintain the structure, i.e., relative magnitudes and orthogonality, among the matrix elements. The result is an integer cosine transform (ICT). Thirteen ICT's have been found and some of them have performance comparable to the DCT.

Discrete cosine transforms on quantum computers

Image and Signal Processing …, 2001

A classical computer does not allow to calculate a dis-crete cosine transform on N points in less than linear time. This trivial lower bound is no longer valid for a computer that takes advantage of quantum mechanical superposition, entanglement, and inter$erence principles. In fact, ...