A Polynomial Approach to Fast Algorithms for Discrete Fourier-cosine and Fourier-sine Transforms
نویسنده
چکیده
The discrete Fourier-cosine transform (cos-DFT), the discrete Fourier-sine transform (sin-DFT) and the discrete cosine transform (DCT) are closely related to the discrete Fourier transform (DFT) of real-valued sequences. This paper describes a general method for constructing fast algorithms for the cos-DFT, the sin-DFT and the DCT, which is based on polynomial arithmetic with Chebyshev polynomials and on the Chinese Remainder Theorem.
منابع مشابه
The Discrete Trigonometric Transforms and Their Fast Algorithms: an Algebraic Symmetry Perspective
It is well-known that the discrete Fourier transform (DFT) can be characterized as decomposition matrix for the polynomial algebra C [x℄=(xn 1). This property gives deep insight into the DFT and can be used to explain and derive its fast algorithms. In this paper we present the polynomial algebras associated to the 16 discrete cosine and sine transforms. Then we derive important algorithms by m...
متن کاملAlgebraic Signal Processing Theory: Cooley-Tukey Type Algorithms for Real DFTs
In this paper we systematically derive a large class of fast general-radix algorithms for various types of real discrete Fourier transforms (real DFTs) including the discrete Hartley transform (DHT) based on the algebraic signal processing theory. This means that instead of manipulating the transform definition, we derive algorithms by manipulating the polynomial algebras underlying the transfo...
متن کاملThe Algebraic Approach to the Discrete Cosine and Sine Transforms and Their Fast Algorithms
It is known that the discrete Fourier transform (DFT) used in digital signal processing can be characterized in the framework of representation theory of algebras, namely as the decomposition matrix for the regular module C[Zn] = C[x]/(x − 1). This characterization provides deep insight on the DFT and can be used to derive and understand the structure of its fast algorithms. In this paper we pr...
متن کاملAlgebraic Signal Processing Theory: Cooley-Tukey Type Algorithms for DCTs and DSTs
This paper presents a systematic methodology based on the algebraic theory of signal processing to classify and derive fast algorithms for linear transforms. Instead of manipulating the entries of transform matrices, our approach derives the algorithms by stepwise decomposition of the associated signal models, or polynomial algebras. This decomposition is based on two generic methods or algebra...
متن کاملCooley-Tukey FFT like algorithms for the DCT
The Cooley-Tukey FFT algorithm decomposes a discrete Fourier transform (DFT) of size n = km into smaller DFTs of size k and m. In this paper we present a theorem that decomposes a polynomial transform into smaller polynomial transforms, and show that the FFT is obtained as a special case. Then we use this theorem to derive a new class of recursive algorithms for the discrete cosine transforms (...
متن کامل