# A Method of Constructing Measurement Matrix for Compressed Sensing by Chebyshev Chaotic Sequence

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## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

- We analyze the high-order correlations among the elements sampled from the Chebyshev chaotic sequence.
- We use the sampled elements to construct a measurement matrix, termed the Chebyshev chaotic measurement matrix. Based on the assumption that the elements are statistically independent, we prove that the Chebyshev chaotic measurement matrix satisfies the RIP with high probability.

## 2. Chebyshev Chaotic Sequence and Sample Distance

#### 2.1. Chebyshev Chaotic Sequence

#### 2.2. The Internal Randomness of Chebyshev Chaos

#### 2.3. Statistical Property and Sample Distance

**Theorem**

**1.**

**Proof.**

## 3. Construction of Chebyshev Chaotic Measurement Matrix and RIP Analysis

#### 3.1. Construction of Chebyshev Chaotic Measurement Matrix

**Algorithm 1**.

Algorithm 1. The method of constructing the Chebyshev chaotic measurement matrix. | |

Input: the number of rows $M$, the number of columns $N$, order $q$, initial value ${x}_{0}$.Output: measurement matrix$\Phi $ | |

1. | Determine the sample distance $d$; |

2. | Generate the Chebyshev chaotic sequence $X$ of length $1+\left(MN-1\right)d$; |

3. | Sample $X$ with $d$ and get $Z=\left\{{z}_{1},{z}_{2}\cdots {z}_{MN}\right\}$; |

4. | Use $Z$ to construct a Chebyshev chaotic measurement matrix as $\Phi =\sqrt{\frac{2}{M}}\left(\begin{array}{cccc}{z}_{1}& {z}_{M+1}& \cdots & {z}_{\left(N-1\right)M+1}\\ {z}_{2}& {z}_{M+2}& \cdots & {z}_{\left(N-1\right)M+2}\\ \vdots & \vdots & \ddots & \vdots \\ {z}_{M}& {z}_{2M}& \cdots & {z}_{MN}\end{array}\right)$ |

5. | Return measurement matrix $\Phi $ |

#### 3.2. RIP Analysis

**Theorem**

**2.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

## 4. Results and Discussion

**Case**

**1.**

**Case**

**2.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Theorem**

**1.**

- ${m}_{1}$ is odd. According to Equation (5), $E\left({x}_{n}^{{m}_{0}}\right)E\left({x}_{n+d}^{{m}_{1}}\right)=0$ holds. We hypothesize that $E\left({x}_{n}^{{m}_{0}}{x}_{n+d}^{{m}_{1}}\right)$ has non-zero values, which means there exists a positive integer $j$ that satisfies $j={m}_{1}/2-\left(h-{m}_{0}/2\right){q}^{-d}$. When ${m}_{1}$ is odd, there is no integer $j$ satisfying $j\in \left(\left({m}_{1}-1\right)/2,\left({m}_{1}+1\right)/2\right)$. Therefore, the hypothesis fails and $E\left({x}_{n}^{{m}_{0}}{x}_{n+d}^{{m}_{1}}\right)=0$ holds. In this case, $E\left({x}_{n}^{{m}_{0}}{x}_{n+d}^{{m}_{1}}\right)$ is equal to $E\left({x}_{n}^{{m}_{0}}\right)E\left({x}_{n+d}^{{m}_{1}}\right)$.
- ${m}_{1}$ is even. According to Equation (5), we have $E\left({x}_{n}^{{m}_{0}}\right)E\left({x}_{n+d}^{{m}_{1}}\right)={2}^{-\left({m}_{0}+{m}_{1}\right)}{{\displaystyle C}}_{{m}_{0}}^{{m}_{0}/2}{{\displaystyle C}}_{{m}_{1}}^{{m}_{1}/2}$. When $h={m}_{0}/2$, there is an integer $j={m}_{1}/2$ that makes $E\left({x}_{n}^{{m}_{0}}{x}_{n+d}^{{m}_{1}}\right)$ be equal to ${2}^{-\left({m}_{0}+{m}_{1}\right)}{{\displaystyle C}}_{{m}_{0}}^{{m}_{0}/2}{{\displaystyle C}}_{{m}_{1}}^{{m}_{1}/2}$. In this case, $E\left({x}_{n}^{{m}_{0}}{x}_{n+d}^{{m}_{1}}\right)=E\left({x}_{n}^{{m}_{0}}\right)E\left({x}_{n+d}^{{m}_{1}}\right)$ holds.

## Appendix B

**Proof**

**of**

**Lemma**

**1.**

## Appendix C

**Proof**

**of**

**Lemma**

**2.**

## Appendix D

**Proof**

**of**

**Lemma**

**3.**

## References

- Donoho, D.L. Compressed sensing. IEEE Trans. Inf. Theory
**2006**, 52, 1289–1306. [Google Scholar] [CrossRef] - Hashimoto, F.; Ote, K.; Oida, T.; Teramoto, A.; Ouchi, Y. Compressed-Sensing Magnetic Resonance Image Reconstruction Using an Iterative Convolutional Neural Network Approach. Appl. Sci.
**2020**, 10, 1902. [Google Scholar] [CrossRef] [Green Version] - Sabahi, M.F.; Masoumzadeh, M.; Forouzan, A.R. Frequency-domain wideband compressive spectrum sensing. IET Commun.
**2016**, 10, 1655–1664. [Google Scholar] [CrossRef] - Bai, Z.; Kaiser, E.; Proctor, J.L.; Kutz, J.N.; Brunton, S.L. Dynamic Mode Decomposition for Compressive System Identification. AIAA J.
**2020**, 58, 561–574. [Google Scholar] [CrossRef] - Candés, E.J. The restricted isometry property and its implications for compressed sensing. Comptes Rendus Math.
**2008**, 346, 589–592. [Google Scholar] [CrossRef] - Candes, E.; Tao, T. Decoding by Linear Programming. IEEE Trans. Inf. Theory
**2005**, 51, 4203–4215. [Google Scholar] [CrossRef] [Green Version] - Candès, E.J.; Tao, T. Near-Optimal Signal Recovery from Random Projections: Universal Encoding Strategies? IEEE Trans. Inf. Theory
**2006**, 52, 5406–5425. [Google Scholar] [CrossRef] [Green Version] - Candès, E.J.; Romberg, J.K.; Tao, T. Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math.
**2006**, 59, 1207–1223. [Google Scholar] [CrossRef] [Green Version] - Ravelomanantsoa, A.; Rabah, H.; Rouane, A. Fast and efficient signals recovery for deterministic compressive sensing: Applications to biosignals. In Proceedings of the 2015 Conference on Design and Architectures for Signal and Image Processing (DASIP), Krakow, Poland, 23–25 September 2015; pp. 1–6. [Google Scholar]
- Devore, R.A. Deterministic constructions of compressed sensing matrices. J. Complex.
**2007**, 23, 918–925. [Google Scholar] [CrossRef] [Green Version] - Xu, G.; Xu, Z. Compressed Sensing Matrices from Fourier Matrices. IEEE Trans. Inf. Theory
**2015**, 61, 469–478. [Google Scholar] [CrossRef] [Green Version] - Yu, L.; Barbot, J.P.; Zheng, G.; Sun, H. Compressive Sensing With Chaotic Sequence. IEEE Signal Process. Lett.
**2010**, 17, 731–734. [Google Scholar] [CrossRef] [Green Version] - Al-Azawi, M.K.M.; Gaze, A.M.; Al-Azawie, M.; Al Shaty, A. Combined speech compression and encryption using chaotic compressive sensing with large key size. IET Signal Process.
**2018**, 12, 214–218. [Google Scholar] [CrossRef] - Lu, W.; Liu, Y.; Wang, D. A Distributed Secure Data Collection Scheme via Chaotic Compressed Sensing in Wireless Sensor Networks. Circuits Syst. Signal Process.
**2013**, 32, 1363–1387. [Google Scholar] [CrossRef] - Frunzete, M.; Yu, L.; Barbot, J.-P.; Vlad, A. Compressive sensing matrix designed by tent map, for secure data transmission. In Proceedings of the Signal Processing Algorithms, Architectures, Arrangements, and Applications SPA 2011, Poznan, Poland, 29–30 September 2011; pp. 1–6. [Google Scholar]
- Zhou, W.; Jing, B.; Zhang, H.; Huang, Y.-F.; Li, J. Construction of Measurement Matrix in Compressive Sensing Based on Composite Chaotic Mapping. Acta Electron. Sin.
**2017**, 45, 2177–2183. [Google Scholar] - Vlad, A.; Luca, A.; Frunzete, M. Computational Measurements of the Transient Time and of the Sampling Distance That Enables Statistical Independence in the Logistic Map. In Proceedings of the International Conference on Computational Science and Its Applications, Seoul, Korea, 29 June–2 July 2009; pp. 703–718. [Google Scholar]
- Vaduva, A.; Vlad, A.; Badea, B. Evaluating the performance of a test-method for statistical independence decision in the context of chaotic signals. In Proceedings of the 2016 International Conference on Communications (COMM), Bucharest, Romania, 9–10 June 2016; pp. 417–422. [Google Scholar] [CrossRef]
- Zhu, S.; Zhu, C.; Wang, W. A Novel Image Compression-Encryption Scheme Based on Chaos and Compression Sensing. IEEE Access
**2018**, 6, 67095–67107. [Google Scholar] [CrossRef] - Martino, L.; Luengo, D.; Míguez, J. Independent Random Sampling Methods; Springer: Cham, Switzerland, 2018; pp. 9–12. [Google Scholar]
- Öztürk, I.; Kılıç, R. Digitally generating true orbits of binary shift chaotic maps and their conjugates. Commun. Nonlinear Sci. Numer. Simul.
**2018**, 62, 395–408. [Google Scholar] [CrossRef] - Geisel, T.; Fairén, V. Statistical properties of chaos in Chebyshev maps. Phys. Lett. A
**1984**, 105, 263–266. [Google Scholar] [CrossRef] - Pesin, Y.B. Characteristic Lyapunov Exponents and Smooth Ergodic Theory. Russ. Math. Surv.
**1977**, 32, 55–114. [Google Scholar] [CrossRef] - Tropp, J.A.; Gilbert, A.C. Signal Recovery from Random Measurements Via Orthogonal Matching Pursuit. IEEE Trans. Inf. Theory
**2007**, 53, 4655–4666. [Google Scholar] [CrossRef] [Green Version] - Wang, Y.; Zeng, J.; Peng, Z.; Chang, X.; Xu, Z. Linear Convergence of Adaptively Iterative Thresholding Algorithms for Compressed Sensing. IEEE Trans. Signal Process.
**2015**, 63, 2957–2971. [Google Scholar] [CrossRef] [Green Version] - Meena, V.; Abhilash, G. Robust recovery algorithm for compressed sensing in the presence of noise. IET Signal Process.
**2016**, 10, 227–236. [Google Scholar] [CrossRef] - Achlioptas, D. Database-friendly random projections: Johnson-Lindenstrauss with binary coins. J. Comput. Syst. Sci.
**2003**, 66, 671–687. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**The change tendency of $\lambda $ with system parameters: (

**a**) Logistic; (

**b**) Tent; (

**c**) Chebyshev.

**Figure 2.**Logistic and Tent chaotic sequences. (

**a**) Logistic $d=5$; (

**b**) Tent $d=5$; (

**c**) Logistic $d=15$; (

**d**) Tent $d=15$.

**Figure 3.**$\rho \left({x}_{n},{x}_{n+d}\right)$ in the 8-order Chebyshev chaotic sequence with $d=5$.

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**MDPI and ACS Style**

Yi, R.; Cui, C.; Miao, Y.; Wu, B.
A Method of Constructing Measurement Matrix for Compressed Sensing by Chebyshev Chaotic Sequence. *Entropy* **2020**, *22*, 1085.
https://doi.org/10.3390/e22101085

**AMA Style**

Yi R, Cui C, Miao Y, Wu B.
A Method of Constructing Measurement Matrix for Compressed Sensing by Chebyshev Chaotic Sequence. *Entropy*. 2020; 22(10):1085.
https://doi.org/10.3390/e22101085

**Chicago/Turabian Style**

Yi, Renjie, Chen Cui, Yingjie Miao, and Biao Wu.
2020. "A Method of Constructing Measurement Matrix for Compressed Sensing by Chebyshev Chaotic Sequence" *Entropy* 22, no. 10: 1085.
https://doi.org/10.3390/e22101085