Algebraic Signal Processing Theory

@article{Pschel2006AlgebraicSP,
  title={Algebraic Signal Processing Theory},
  author={Markus P{\"u}schel and Jos{\'e} M. F. Moura},
  journal={ArXiv},
  year={2006},
  volume={abs/cs/0612077}
}
This paper presents an algebraic theory of linear signal processing. At the core of algebraic signal processing is the concept of a 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, and phi generalizes the concept of the z-transform to bijective linear mappings from a vector space of, e.g., signal samples, into the module M. A signal model provides the structure for a… 
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Novel O(nlogn) general-radix algorithms for the discrete Fourier transform and the discrete cosine transform of type 4 are derived by decomposing the regular modules of these algebras as a stepwise induction.
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A general-radix algorithm for the DTT of an n × n 2-D signal, focusing on the radix-2 × 2 case is derived, which shows that the obtained DTT algorithm is the precise equivalent of the well-known Cooley–Tukey fast Fourier transform, which motivates the title of this paper.
Research Progress of The Algebraic and Geometric Signal Processing
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References

SHOWING 1-10 OF 145 REFERENCES
The Algebraic Approach to the Discrete Cosine and Sine Transforms and Their Fast Algorithms
TLDR
This paper presents an algebraic characterization of the important class of discrete cosine and sine transforms as decomposition matrices of certain regular modules associated with four series of Chebyshev polynomials.
Automatic generation of fast discrete signal transforms
TLDR
An algorithm that derives fast versions for a broad class of discrete signal transforms symbolically by finding fast sparse matrix factorizations for the matrix representations of these transforms by using the defining matrix as its sole input.
Algebraic Theory of Finite Fourier Transforms
A wreath product group approach to signal and image processing .I. Multiresolution analysis
We propose the use of spectral analysis on certain noncommutative finite groups in digital signal processing and, in particular, image processing. We pay significant attention to groups constructed
Gauss and the history of the fast fourier transform
TLDR
The algorithm developed by Cooley and Tukey clearly had its roots in, though perhaps not a direct influence from, the early twentieth century, and remains the most Widely used method of computing Fourier transforms.
Diagonalizing properties of the discrete cosine transforms
TLDR
The decorrelating power of the DCTs is studied, obtaining expressions that show the decor Relating behavior of each DCT with respect to any stationary processes, and it is proved that the eight types of D CTs are asymptotically optimal for all finite-order Markov processes.
Decomposing Monomial Representations of Solvable Groups
TLDR
An efficient algorithm that decomposes a monomial representation of a solvable group G into its irreducible components is presented and well-known theorems in a constructively refined form are presented and derive new results on decomposition matrices of representations.
A More Symmetrical Fourier Analysis Applied to Transmission Problems
The Fourier identity is here expressed in a more symmetrical form which leads to certain analogies between the function of the original variable and its transform. Also it permits a function of time,
Fast Fourier analysis for abelian group extensions
The fast generalized discrete Fourier transforms: A unified approach to the discrete sinusoidal transforms computation
...
1
2
3
4
5
...