An efficient method for computing the discrete cosine transform (DCT) is proposed. Based on direct decomposition of the DCT, the recursive properties of the DCT for an even length input sequence is derived, which is a generalization of the radix 2 DCT algorithm. Based on the recursive property, a new DCT algorithm for an even length sequence is obtained. The proposed algorithm is very structural and requires fewer computations when compared with others. The regular structure of the proposed algorithm is suitable for fast parallel algorithm and VLSI implementation.

Fast algorithm for computing discrete cosine transform

This chapter provides an overview of this book. The book presents the complete set of discrete cosine transforms (DCTs) and discrete sine transforms (DSTs) constituting the entire class of discrete sinusoidal unitary transforms, including their definitions, general mathematical properties, relations to the Karhunen-Loeve transform (KLT), with the emphasis on fast algorithms and integer approximations for their efficient implementations in the integer domain. The book covers various latest developments in DCTs and DSTs in a unified way, and it is essentially a detailed excursion on orthogonal/orthonormal DCT and DST matrices, their matrix factorizations, and integer approximations. For the DCT and DST to be viable, feasible, and practical, the fast algorithms are essential for their efficient implementation in terms of reduced memory, implementation complexity, and recursivity. Extensive definitions, principles, properties, signal flow graphs, derivations, proofs, and examples are provided throughout the book for proper understanding of the strengths and shortcomings of the spectrum of cosine/sine transforms and their application in diverse disciplines.

CHAPTER 1 – Discrete Cosine and Sine Transforms

In previous papers, the Fourier transform (FT) has been generalized into the fractional Fourier transform (FRFT), the linear canonical transform (LCT), and the simplified fractional Fourier transform (SFRFT). Because the cosine, sine, and Hartley transforms are very similar to the FT, it is reasonable to think they can also be generalized by the similar way. We introduce several new transforms. They are all the generalization of the cosine, sine, or Hartley transform. We first derive the fractional cosine, sine, and Hartley transforms (FRCT/FRST/FRHT). They are analogous to the FRFT. Then, we derive the canonical cosine and sine transforms (CCT/CST). They are analogous to the LCT. We also derive the simplified fractional cosine, sine, and Hartley transforms (SFRCT/SFRST/SFRHT). They are analogous to the SFRFT and have the advantage of real-input-real-output. We also discuss the properties, digital implementation, and applications (e.g., the applications for filter design and space-variant pattern recognition) of these transforms. The transforms introduced in this paper are very efficient for digital implementation. We can just use one half or one fourth of the real multiplications required for the FRFT and LCT to implement them. When we want to process even, odd, or pure real/imaginary functions, we can use these transforms instead of the FRFT and LCT. Besides, we also show that the FRCT/FRST, CCT/CST, and SFRCT/SFRST are also useful for the one-sided (t /spl isin/ [0, /spl infin/]) signal processing.

http://ntur.lib.ntu.edu.tw/bitstream/246246/142384/1/09.pdf

Fractional cosine, sine, and Hartley transforms

http://www-math.mit.edu/~stevenj/preprints/ShaoJo07-preprint.pdf

Type-II/III DCT/DST algorithms with reduced number of arithmetic operations

In this paper, we propose a fully pipelined two-dimensional (2-D) bit level systolic architecture for efficient implementation of discrete orthogonal transforms using a serial-parallel vector-matrix multiplication scheme based on the Baugh-Wooley algorithm. Apart from its regularity and simplicity, the proposed structure yields high throughput due to massive parallelism across the 2-D mesh. The area- and time-complexities of the proposed structure are (ON/sup 2/) and O(2nN/sup 2/), respectively, for implementation of N-point transform, where n is the wordlength.

High throughput VLSI implementation of discrete orthogonal transforms using bit-level vector-matrix multiplier

The discrete cosine transform (DCT) and the discrete sine transform (DST) have found wide applications in speech and image processing, as well as telecommunication signal processing for the purpose of data compression, feature extraction, image reconstruction, and filtering. In this paper, we present new recursive algorithms for the DCT and the DST. The proposed method is based on certain recursive properties of the DCT coefficient matrix, and can be generalized to design recursive algorithms for the 2-D DCT and the 2-D DST. These new structured recursive algorithms are able to decompose the DCT and the DST into two balanced lower-order subproblems in comparison to previous research works. Therefore, when converting our algorithms into hardware implementations, we require fewer hardware components than other recursive algorithms. Finally, we propose two parallel algorithms for accelerating the computation. >

Restructured recursive DCT and DST algorithms

The prime-factor decomposition is a fast computational technique for many important digital signal processing operations, such as the convolution, the discrete Fourier transform, the discrete Hartley transform, and the discrete cosine transform (DCT). The authors present a new prime-factor algorithm for the DCT. They also design a prime-factor algorithm for the discrete sine transform based on the prime-factor DCT algorithm. Hardware implementations for the prime-factor DCT are also studied. They are especially interested in the hardware designs which are suitable for the VLSI implementations. They show three hardware designs for the prime-factor DCT, including a VLSI circuit fabricated directly according to the signal-flow graph, a linear systolic array, and a mesh-connected systolic array. These three designs show the trade-off between cost and performance. The methodology, which deals with general (N/sub 1//spl times/ N/sub 2/)-point DCTs, where N/sub 1/ and N/sub 2/ are mutually prime, is illustrated by converting a 15-point DCT problem into a (3/spl times/5)-point 2D DCT problem. >

An efficient prime-factor algorithm for the discrete cosine transform and its hardware implementations

The input index mapping adopted is the Ruritanian mapping; the output index mapping is the same as B.G. Lee's (IEEE Trans. vol. ASSP.37, no.2, p.237-44, Feb. 1989). Hardware implementations for the prime-factor DCT are also studied. The methodology, which deals with general (N/sub 1/*N/sub 2/)-point DCTs, where N/sub 1/ and N/sub 2/ are mutually prime, is illustrated by converting a 15-point DCT problem into a (3*5)-point 2-D DCT problem. >

A New Method to Design Recursive Algorithms for Computing the 1-D and 2-D Dcts

This paper is concerned with designing efficient algorithms to process the appraisal books, which are used to estimate the amount of money lost due to a fire, in the Bank of Taiwan. First, we propose heuristic algorithms to pack appraisal books daily into bundles in the units of 1,000,000 dollars, 500,000 dollars, and 100,000 dollars. Second, we present efficient algorithms to take out certain appraisal books so that the amount of the remaining accumulated appraisal books is a multiple of one thousand dollars. We also introduce a software tool which integrates the functions of processing appraisal books. Experimental studies show that the software tool is very helpful for the clerks.

Fang-Yu Huang

Papers

Restructured recursive DCT and DST algorithms

An efficient prime-factor algorithm for the discrete cosine transform and its hardware implementations

An efficient prime-factor algorithm for the discrete cosine transform and its hardware implementations

A New Method to Design Recursive Algorithms for Computing the 1-D and 2-D Dcts

Efficient implementations of two variant subset sum problems: a case study of how to process appraisal books resulting from fire-destroyed money