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From Signal Space To STP-CS

Daizhan Cheng

TL;DR

A systematic analysis of the construction of sensing matrix based on balanced incomplete block design (BIBD) is presented and some characteristics of semi-tensor product based compressed sensing (STP-CS) are revealed.

Abstract

Under the assumption that a finite signal with different sampling lengths or different sampling frequencies is considered as equivalent, the signal space is considered as the quotient space of $\mathbb{R}^{\infty}$ over equivalence. The topological structure and the properties of signal space are investigated. Using them some characteristics of semi-tensor product based compressed sensing (STP-CS) are revealed. Finally, a systematic analysis of the construction of sensing matrix based on balanced incomplete block design (BIBD) is presented.

From Signal Space To STP-CS

TL;DR

A systematic analysis of the construction of sensing matrix based on balanced incomplete block design (BIBD) is presented and some characteristics of semi-tensor product based compressed sensing (STP-CS) are revealed.

Abstract

Under the assumption that a finite signal with different sampling lengths or different sampling frequencies is considered as equivalent, the signal space is considered as the quotient space of over equivalence. The topological structure and the properties of signal space are investigated. Using them some characteristics of semi-tensor product based compressed sensing (STP-CS) are revealed. Finally, a systematic analysis of the construction of sensing matrix based on balanced incomplete block design (BIBD) is presented.

Paper Structure

This paper contains 15 sections, 37 theorems, 108 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Left and Right STP - Preliminaries
  3. Signal Space
  4. Vector Space Structure on Signal Space
  5. Topological Structure on $\Omega$
  6. Linear Mapping on $\Omega$
  7. Structure of Signal Space
  8. Orthonormal Basis of Signal Space
  9. Norms of Signal Space
  10. Dimension-free STP-CS
  11. BIBD-Based Sensing Matrix
  12. BIBD-Based Construction
  13. A modified BIBD-Based Sensing Matrix
  14. Constructing $\mathbb{O}$ and $\mathbb{U}$
  15. Conclusion

Key Result

Proposition 1.1

don03 Consider equation (1.1). Assume $spark(A)>2k$, then the equation has at most one solution $x\in \Sigma^n_k$.

Figures (2)

  • Figure 1: Signal Addition
  • Figure 2: Projection

Theorems & Definitions (70)

  • Proposition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Proposition 1.5
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • ...and 60 more