### Abstract

We derive a signal processing framework, called space signal processing, that parallels time signal processing. As such, it comes in four versions (continuous/discrete, infinite/finite), each with its own notion of convolution and Fourier transform. As in time, these versions are connected by sampling theorems that we derive. In contrast to time, however, space signal processing is based on a different notion of shift, called space shift, which operates symmetrically. Our work rigorously connects known and novel concepts into a coherent framework; most importantly, it shows that the sixteen discrete cosine and sine transforms are the space equivalent of the discrete Fourier transform, and hence can be derived by sampling. The platform for our work is the algebraic signal processing theory, an axiomatic approach and generalization of linear signal processing that we recently introduced.

Original language | English (US) |
---|---|

Article number | 5204282 |

Pages (from-to) | 242-257 |

Number of pages | 16 |

Journal | IEEE Transactions on Signal Processing |

Volume | 58 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2010 |

### Fingerprint

### Keywords

- Algebra
- Convolution
- Discrete cosine and sine transforms
- Fourier cosine transform
- Module
- Signal model
- Space shift

### ASJC Scopus subject areas

- Signal Processing
- Electrical and Electronic Engineering

### Cite this

*IEEE Transactions on Signal Processing*,

*58*(1), 242-257. [5204282]. https://doi.org/10.1109/TSP.2009.2029718

**Algebraic signal processing theory : Sampling for infinite and finite 1-D space.** / Kovacevic, Jelena; Püschel, Markus.

Research output: Contribution to journal › Article

*IEEE Transactions on Signal Processing*, vol. 58, no. 1, 5204282, pp. 242-257. https://doi.org/10.1109/TSP.2009.2029718

}

TY - JOUR

T1 - Algebraic signal processing theory

T2 - Sampling for infinite and finite 1-D space

AU - Kovacevic, Jelena

AU - Püschel, Markus

PY - 2010/1/1

Y1 - 2010/1/1

N2 - We derive a signal processing framework, called space signal processing, that parallels time signal processing. As such, it comes in four versions (continuous/discrete, infinite/finite), each with its own notion of convolution and Fourier transform. As in time, these versions are connected by sampling theorems that we derive. In contrast to time, however, space signal processing is based on a different notion of shift, called space shift, which operates symmetrically. Our work rigorously connects known and novel concepts into a coherent framework; most importantly, it shows that the sixteen discrete cosine and sine transforms are the space equivalent of the discrete Fourier transform, and hence can be derived by sampling. The platform for our work is the algebraic signal processing theory, an axiomatic approach and generalization of linear signal processing that we recently introduced.

AB - We derive a signal processing framework, called space signal processing, that parallels time signal processing. As such, it comes in four versions (continuous/discrete, infinite/finite), each with its own notion of convolution and Fourier transform. As in time, these versions are connected by sampling theorems that we derive. In contrast to time, however, space signal processing is based on a different notion of shift, called space shift, which operates symmetrically. Our work rigorously connects known and novel concepts into a coherent framework; most importantly, it shows that the sixteen discrete cosine and sine transforms are the space equivalent of the discrete Fourier transform, and hence can be derived by sampling. The platform for our work is the algebraic signal processing theory, an axiomatic approach and generalization of linear signal processing that we recently introduced.

KW - Algebra

KW - Convolution

KW - Discrete cosine and sine transforms

KW - Fourier cosine transform

KW - Module

KW - Signal model

KW - Space shift

UR - http://www.scopus.com/inward/record.url?scp=72949123297&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=72949123297&partnerID=8YFLogxK

U2 - 10.1109/TSP.2009.2029718

DO - 10.1109/TSP.2009.2029718

M3 - Article

VL - 58

SP - 242

EP - 257

JO - IEEE Transactions on Signal Processing

JF - IEEE Transactions on Signal Processing

SN - 1053-587X

IS - 1

M1 - 5204282

ER -