Abstract
This paper presents factorizations of each discrete sine transform (DST) matrices of types I, II, III, and IV into a product of sparse, diagonal, bidiagonal, and scaled orthogonal matrices. Based on the proposed matrix factorization formulas, reduced multiplication complexity, recursive, and radix-2 DST-I/II/III/IV algorithms are presented. We will present the lowest multiplication complexity DST-IV algorithm in the literature. The paper fills a gap in the self-recursive, exact, and radix-2 DST-I/II/III algorithms executed via diagonal, bidiagonal, scaled orthogonal, and simple matrix factors for any input n = 2t (t ≥ 1). The paper establishes a novel relationship between DST-II and DST-IV matrices using diagonal and bidiagonal matrices. Similarly, a novel relationship between DST-I and DST-III matrices is proposed using sparse and diagonal matrices. These interweaving relationships among DST matrices enable us to bridge the existing factorizations of the DST matrices with the proposed factorization formulas. We present signal flow graphs to provide a layout for realizing the proposed algorithms in DST-based integrated circuit designs. Additionally, we describe an implementation of algorithms based on the proposed DST-II and DST-III factorizations within a double random phase encoding (DRPE) image encryption scheme.
Highlights
Discrete sine transforms are real-valued transform matrices and a subclass of the discrete Fourier transform (DFT) matrices, extracting the imaginary part of the complex trigonometric form
We note here that the factorization formulas to compute discrete sine transform (DST)-III/IV matrices in [14] are not included because the proposed DST-III/IV matrix factorization formulas are completely different from the factorization formulas in
We have proposed novel factorization formulas to compute DST I-IV matrices using the product of simple, sparse, diagonal, bidiagonal, and scaled orthogonal matrices
Summary
Discrete sine transforms are real-valued transform matrices and a subclass of the discrete Fourier transform (DFT) matrices, extracting the imaginary part of the complex trigonometric form. There are no simplest sparse factors, no explicit algorithms, and no complexity analysis to compute DCT and DST matrices by a vector in [14]. We obtain simple factors, i.e, diagonal, bidiagonal, sparse, and scaled orthogonal factors, to compute DST matrices by a vector. We have observed that the lowest multiplicative complexity, radix-2, and recursive DST algorithms having diagonal, bidiagonal, sparse, and scaled orthogonal matrices are missing elements in the literature. In this paper, we follow a similar approach in deriving DCT II/III algorithms in [41] and the DCT I/IV algorithms in [42] to obtain novel lowest multiplication complexity, recursive, and radix-2 DST algorithms execute via diagonal, bidiagonal, sparse, and scaled orthogonal matrices. We note here that the factorization formulas to compute DST-III/IV matrices in [14] are not included because the proposed DST-III/IV matrix factorization formulas are completely different from the factorization formulas in [14]
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