The Karhunen-Loeve transform for a class of signals is proven to be a set of periodic sine functions and this Karhunen-Loeve series expansion can be obtained via an FFT algorithm. This fast algorithm obtained could be useful in data compression and other mean-square signal processing applications.

A fast Karhunen-Loeve transform for a class of random processes

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

Digital Holography and Digital Image Processing

A fixed-point-error analysis for several 1D fast DCT algorithms is presented. For comparison, a direct-form approach is also included in the investigation. A statistical model is used as the basis for predicting the fixed-point error in implementing the algorithms, and a suitable scaling scheme is selected to avoid overflow. Closed-form expressions for both the mean and variance for fixed-point error are derived and compared with experimental results. Simulation results show close agreement between theory and experiment, validating the analysis. The results show that one of the algorithms is better than others in terms of average SNR performance. Based on the 1D analysis, attempts are made to investigate the fixed-point-error analysis of the two-column approach for 2D DCT. It is found that the fixed-point-error characteristics of the row-column approach for 2D DCTs is very similar to that of their 1D counterparts. >

On the fixed-point-error analysis of several fast DCT algorithms

The problem of digital signal and image resampling with discrete sinc interpolation is addressed. Discrete sinc interpolation is theoretically the best one among the digital convolution-based signal resampling methods because it does not distort the signal as defined by its samples and is completely reversible. However, sinc interpolation is frequently not considered in applications because it suffers from boundary effects, tends to produce signal oscillations at the image edges, and has relatively high computational complexity when irregular signal resampling is required. A solution that enables the elimination of these limitations of the discrete sinc interpolation is suggested. Two flexible and computationally efficient algorithms for boundary effects free and adaptive discrete sinc interpolation are presented: frame-wise (global) sinc interpolation in the discrete cosine transform (DCT) domain and local adaptive sinc interpolation in the DCT domain of a sliding window. The latter offers options not available with other interpolation methods: interpolation with simultaneous signal restoration/enhancement and adaptive interpolation with super resolution.

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Boundary effect free and adaptive discrete signal sinc-interpolation algorithms for signal and image resampling.

In this paper, a fixed-point round-off error analysis of the discrete cosine transform (DCT) has been carried out. A relationship between the range of the twiddle factor and the dimension of the DCT is first derived, whence a suitable scaling model is chosen for the DCT algorithm and the average output signal-to-noise ratio is calculated.

Fixed-point round-off error analysis for the discrete cosine transform

It is interesting to note that the results given in the paper of Cvetkovic and Popovic (see ibid., vol.40, no.8, p.2083, 1992) may be obtained from different derivations of previous algorithms using either forward or inverse DST/DCT. The authors discuss the discrete sine transform (DST) and discrete cosine transform (DCT) algorithms separately. >

Further results on "New fast recursive algorithms for the discrete cosine and sine transforms"

Suitable scaling schemes are chosen for the Gupta-Rao and Yao-Hsu fast sine transform algorithms and the relative fixed-point round-off error analyses have been carried out respectively. The average output signal-to-noise ratios are then calculated. It appears that in forward DST for N > 16, scaling stage-by-stage in the Gupta-Rao algorithm has the best performance while in inverse DST the overall scaling (in either the Gupta-Rao or Yao-Hsu algorithms) has the best performance.

Jui Chi Yao

Papers

Fixed-point round-off error analysis for the discrete cosine transform

Further results on "New fast recursive algorithms for the discrete cosine and sine transforms"

Comparative performance of discrete sine transforms with fixed-point round-off error analysis