Each discrete cosine transform (DCT) uses $N$ real basis vectors whose components are cosines In the DCT-4, for example, the $j$th component of $\boldv_k$ is $\cos (j + \frac{1}{2}) (k + \frac{1}{2}) \frac{\pi}{N}$ These basis vectors are orthogonal and the transform is extremely useful in image processing If the vector $\boldx$ gives the intensities along a row of pixels, its cosine series $\sum c_k \boldv_k$ has the coefficients $c_k=(\boldx,\boldv_k)/N$ They are quickly computed from a Fast Fourier Transform But a direct proof of orthogonality, by calculating inner products, does not reveal how natural these cosine vectors are 
We prove orthogonality in a different way Each DCT basis contains the eigenvectors of a symmetric "second difference" matrix By varying the boundary conditions we get the established transforms DCT-1 through DCT-4 Other combinations lead to four additional cosine transforms The type of boundary condition (Dirichlet or Neumann, centered at a meshpoint or a midpoint) determines the applications that are appropriate for each transform The centering also determines the period: $N-1$ or $N$ in the established transforms, $N-\frac{1}{2}$ or $N+ \frac{1}{2}$ in the other four The key point is that all these "eigenvectors of cosines" come from simple and familiar matrices

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The Discrete Cosine Transform

This paper discusses the use of symmetric convolution and the discrete sine and cosine transforms (DSTs and DCTs) for general digital signal processing. The operation of symmetric convolution is a formalized approach to convolving symmetrically extended sequences. The result is the same as that obtained by taking an inverse discrete trigonometric transform (DTT) of the product of the forward DTTs of those two sequences. There are 16 members in the family of DTTs. Each provides a representation for a corresponding distinct type of symmetric-periodic sequence. The author defines symmetric convolution, relates the DSTs and DCTs to symmetric-periodic sequences, and then use these principles to develop simple but powerful convolution-multiplication properties for the entire family of DSTs and DCTs. Symmetric convolution can be used for discrete linear filtering when the filter is symmetric or antisymmetric. The filtering will be efficient because fast algorithms exist for all versions of the DTTs. Conventional linear convolution is possible if one first zero-pad the input data. Symmetric convolution and its fast implementation using DTTs are now an alternative to circular convolution and the DFT. >

Symmetric convolution and the discrete sine and cosine transforms

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

SPIRAL is a generator for libraries of fast software implementations of linear signal processing transforms. These libraries are adapted to the computing platform and can be re-optimized as the hardware is upgraded or replaced. This paper describes the main components of SPIRAL: the mathematical framework that concisely describes signal transforms and their fast algorithms; the formula generator that captures at the algorithmic level the degrees of freedom in expressing a particular signal processing transform; the formula translator that encapsulates the compilation degrees of freedom when translating a specific algorithm into an actual code implementation; and, finally, an intelligent search engine that finds within the large space of alternative formulas and implementations the "best" match to the given computing platform. We present empirical data that demonstrate the high performance of SPIRAL generated code.

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Spiral: A Generator for Platform-Adapted Libraries of Signal Processing Algorithms

This paper introduces a general and axiomatic approach to linear signal processing (SP) that we refer to as the algebraic signal processing theory (ASP). Basic to ASP is the linear signal model defined as a triple (A,M,Phi) where familiar concepts like the filter space and the signal space are cast as an algebra A and a module M, respectively. The mapping Phi generalizes the concept of a z-transform to bijective linear mappings from a vector space of signal samples into the module M. Common concepts like filtering, spectrum, or Fourier transform have their equivalent counterparts in ASP. Once these concepts and their properties are defined and understood in the context of ASP, they remain true and apply to specific instantiations of the ASP signal model. For example, to develop signal processing theories for infinite and finite discrete time signals, for infinite or finite discrete space signals, or for multidimensional signals, we need only to instantiate the signal model to one that makes sense for that specific class of signals. Filtering, spectrum, Fourier transform, and other notions follow then from the corresponding ASP concepts. Similarly, common assumptions in SP translate into requirements on the ASP signal model. For example, shift-invariance is equivalent to A being commutative. For finite (duration) signals shift invariance then restricts A to polynomial algebras. We explain how to design signal models from the specification of a special filter, the shift. The paper illustrates the general ASP theory with the standard time shift, presenting a unique signal model for infinite time and several signal models for finite time. The latter models illustrate the role played by boundary conditions and recover the discrete Fourier transform (DFT) and its variants as associated Fourier transforms. Finally, ASP provides a systematic methodology to derive fast algorithms for linear transforms. This topic and the application of ASP to space dependent signals and to multidimensional signals are pursued in companion papers.

/pdf/algebraic-signal-processing-theory-foundation-and-1-d-time-3fztll81ha.pdf

B. R. Hunt

Papers

The discreteW transform