Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Signal Processing via Cross-Dimensional Projection

Daizhan Cheng

TL;DR

Using projection between Euclidian spaces of different dimensions, the signal compression and decompression become straightforward and it is shown that under the equivalence assumption the technique provides the best approximation with least square error.

Abstract

Using projection between Euclidian spaces of different dimensions, the signal compression and decompression become straightforward. This encoding/decoding technique requires no preassigned measuring matrix as in compressed sensing. Moreover, in application there is no dimension or size restrictions. General formulas for encoding/decoding of any finite dimensional signals are provided. Their main properties are revealed. Particularly, it is shown that under the equivalence assumption the technique provides the best approximation with least square error.

Signal Processing via Cross-Dimensional Projection

TL;DR

Using projection between Euclidian spaces of different dimensions, the signal compression and decompression become straightforward and it is shown that under the equivalence assumption the technique provides the best approximation with least square error.

Abstract

Using projection between Euclidian spaces of different dimensions, the signal compression and decompression become straightforward. This encoding/decoding technique requires no preassigned measuring matrix as in compressed sensing. Moreover, in application there is no dimension or size restrictions. General formulas for encoding/decoding of any finite dimensional signals are provided. Their main properties are revealed. Particularly, it is shown that under the equivalence assumption the technique provides the best approximation with least square error.

Paper Structure

This paper contains 14 sections, 15 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Semi-Tensor Product of Matrices
  4. Vector Space Structure on ${\mathbb R}^{\infty}$
  5. Topological Structure on ${\mathbb R}^{\infty}$
  6. Projection between Two Euclidian Spaces with Different Dimensions
  7. Projection-based Signal Compression/Decompression
  8. PS and PD for $x\in {\mathbb R}^{\infty}$
  9. PS and PD for Higher Dimensional Signals
  10. Right PC and PD
  11. Compressed Sensing vs PC/PD
  12. Compressed Sensing
  13. Various Topologies on ${\mathbb R}^{\infty}$
  14. Conclusion

Key Result

Proposition 2.2

che12 Let $A,B,C\in {\mathcal{M}}$, $x,y\in {\mathbb R}^{\infty}$.

Figures (1)

  • Figure 1: Projection

Theorems & Definitions (27)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Remark 2.6
  • Proposition 2.7
  • Definition 2.8
  • Remark 2.9
  • Proposition 2.10
  • ...and 17 more