Diagonalizing properties of the discrete cosine transforms

Sánchez, Victoria; Garcı́a, Pilar Garcı́a; Peinado, Antonio M.; Segura, José C.; Rubio, Antonio J.

doi:10.1109/78.482113

Cited by 95 publications

(63 citation statements)

References 9 publications

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“…This method, however, involves the estimation of the a posteriori distributions, which is one of common problems in statistical analysis. Graph theory provides a number of ways to solve problems in a variety of disciplines and has also been used for clustering [4]- [6]. In a recent paper [7], a method to the data reduction was proposed, which automatically selects the number of exemplars for each class.…”

Section: Introductionmentioning

confidence: 99%

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DCT/DST and Gauss-Markov fields: conditions for equivalence

Moura

Bruno

1998

IEEE Trans. Signal Process.

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Abstract-The correspondence addresses the intriguing question of which random models are equivalent to the discrete cosine transform (DCT) and discrete sine transform (DST). Common knowledge states that these transforms are asymptotically equivalent to first-order Gauss causal Markov random processes. We establish that the DCT and the DST are exactly equivalent to homogeneous one-dimensional (1-D) and two-dimensional (2-D) Gauss noncausal Markov random fields defined on finite lattices with appropriate boundary conditions. I. INTRODUCTIONIn this correspondence, we establish the second-order equivalence between the discrete sine transform (DST) and the discrete cosine transform (DCT) and arbitrary order noncausal Gauss-Markov random fields (GMrf's) defined on a finite lattice. We prove this by showing that the DST and the DCT diagonalize the covariance matrix associated with these fields. Following [1], we work with the inverse of the covariance matrix, which is called the potential matrix, that is highly structured; for homogeneous noncausal GMrf's of arbitrary order, it is given by a Toeplitz canonical matrix plus a boundary matrix.Section II expresses the Toeplitz component of the potential matrix as matrix polynomials that are diagonalizable by either the DST or the DCT plus a perturbation matrix. Section III shows that for a given arbitrary order one-dimensional (1-D) GMrf, particular choices of boundary conditions (bc's) lead to a boundary matrix that cancels the perturbation term in the expansion of the Toeplitz canonical matrix. The final result is then an overall potential matrix that is Manuscript received September 24, 1996; revised March 6, 1998. This work was supported in part by DARPA under Grant DABT 63-98-1-0004. The work of M. G. S. Bruno was also supported in part by CNPq-Brazil. The associate editor coordinating the review of this paper and approving it for publication was Dr. Jitendra K. Tugnait.The The coefficients j i are computed according to the recursion in Table I.The proof of Lemma II.1 is found in a technical report available from the authors. III. DCT/DST AND 1-D GMRF'SWe establish the second-order equivalence between the DCT (or DST) and 1-D noncausal GMrf's by determining the GMrf's for which the DCT (or the DST) is their Karuhnen-Loève transform (KLT). We do this for arbitrary-order, finite-length, spatially homogeneous, noncausal GMrf's with appropriately chosen bc's.The KLT of a GMrf diagonalizes its covariance matrix and is determined by the eigenvectors of the GMrf covariance matrix. Finding this eigenstructure is, in general, hard because there is no direct parametrization of the GMrf covariance. Reference [1] shows, however, that for arbitrary-order GMrf's on finite lattices, the inverse of the covariance (the potential matrix) has a well-defined parametrized structure. We work with the potential matrix and use this parametrization to show when the DCT or the DST diagonalize the potential matrix, hence, the covariance matrix of the GMrf.Consider a zero-mean spatially homo...

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Section: Introductionmentioning

confidence: 99%

“…43, pp. 2631-2641, Nov. 1995 V. Sánchez, A. Peinado, J. Segura, P. García, and A. Rubio, "Generating matrices for the discrete sine transforms, " IEEE Trans. Signal Processing, vol.…”

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confidence: 99%

DCT/DST and Gauss-Markov fields: conditions for equivalence

Moura

Bruno

1998

IEEE Trans. Signal Process.

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“…D ∈ R N ×N is power normalized DCT matrix in type II which is the most popular member of the DCT family [15]. C ∈ R (N +Lp+Ls)×N is the matrix implementation of adding prefix (length L p ) and suffix (length L s ) in symmetric form defined as…”

Section: Notations: [·] H and [·]mentioning

confidence: 99%

“…To the best of our knowledge, current work related to DCT- based multicarrier system mostly discusses the corresponding properties and system implementation methods, e.g., in [13], [15]. The more in-depth study, such as how output signal-tonoise ratio (SNR) variates among subcarriers by the filtering process in DCT-MCM, however, has not been investigated.…”

Section: Introductionmentioning

confidence: 99%

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Output SNR analysis and detection criteria for optimum DCT-based multicarrier system

Zhang

Mao

et al. 2016

2016 International Symposium on Wireless Communication Systems (ISWCS)

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Abstract-The discrete cosine transform (DCT) based multicarrier system is regarded as one of the complementary multicarrier transmission techniques for 5th Generation (5G) applications in near future. By employing cosine basis as orthogonal functions for multiplexing each real-valued symbol with symbol period of T , it is able to reduce the minimum orthogonal frequency spacing to 1/(2T ) Hz, which is only half of that compared to discrete Fourier transform (DFT) based multicarrier systems. Critical to the optimal DCT-based system design that achieves interference-free single-tap equalization, not only both prefix and suffix are needed as symmetric extension of information block, but also a so-called front-end pre-filter is necessarily introduced at the receiver side. Since the pre-filtering process is essentially a time reversed convolution for continuous inputs, the output signal-to-noise ratio (SNR) for each subcarrier after filtering is enhanced. In this paper, the impact of pre-filtering on the system performance is analyzed in terms of ergodic output SNR per subcarrier. This is followed by reformulated detection criterion where such filtering process is taken into consideration. Numerical results show that under modified detection criteria, the proposed detection algorithms improve the overall bit error rate (BER) performance effectively.

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Lateral Movement in Undergraduate Research: Case Studies in Number Theory

Garcia

2020

Foundations for Undergraduate Research in Mathematics

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Diagonalizing properties of the discrete cosine transforms

Cited by 95 publications

References 9 publications

DCT/DST and Gauss-Markov fields: conditions for equivalence

DCT/DST and Gauss-Markov fields: conditions for equivalence

Output SNR analysis and detection criteria for optimum DCT-based multicarrier system

Lateral Movement in Undergraduate Research: Case Studies in Number Theory

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