# Spectral Properties of Clipping Noise

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## Abstract

**:**

## 1. Introduction

## 2. System Model

## 3. Review of the Bussgang Theorem

#### 3.1. Mathematical Derivation of the Bussgang Theorem

#### 3.1.1. Calculation of the Linear Damping Factor K

#### 3.1.2. Calculation of the Noise Variance ${\sigma}_{u}^{2}$

#### 3.2. Symbol Error Probability Based on the Bussgang Theorem

## 4. Power Spectral Density of the Clipping Distortion

#### 4.1. Analytical Calculation of the Power Spectral Density of Clipping Noise

#### 4.1.1. Clipping Level Crossing

#### 4.1.2. Duration of an Overshooting

#### 4.1.3. Mathematical Description of an Overshooting

#### 4.1.4. Closed-Form Analytical Expression of the Power Spectral Density

#### 4.2. Symbol Error Probability Based on the Analytical Power Spectral Density of Clipping Noise

#### 4.3. Approximated Power Spectral Density of Clipping Noise

- The simulated and analytical curves intersect at the corner frequency ${f}_{\mathrm{c}}=B/2$;
- The gain from ${S}_{{n}_{\mathrm{c}}{n}_{\mathrm{c}},\mathrm{dB}}(B/2)$ to ${S}_{{n}_{\mathrm{c}}{n}_{\mathrm{c}},\mathrm{dB}}\left(0\right)$ in dB appears to similar for all A; and
- The shape of the analytical curves in dB inside the transmission bandwidth can be approximated by a quadratic function and does not depend on A as well.

#### 4.4. Symbol Error Probability Based on the Approximated Power Spectral Density of Uncorrelated Clipping Noise

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Detailed Calculations

#### Appendix A.1

#### Appendix A.2

## References

- Afgani, M.; Haas, H.; Elgala, H.; Knipp, D. Visible light communication using OFDM. In Proceedings of the 2nd International Conference on Testbeds and Research Infrastructures for the Development of Networks and Communities, Barcelona, Spain, 1–3 March 2006. [Google Scholar]
- Czylwik, A. Kanalkapazität intensitätsmodulierter optischer Übertragungssysteme mit Direktempfängern. Frequenz
**1996**, 50, 60–68. [Google Scholar] [CrossRef] - Kahn, J.M.; Barry, J.R. Wireless infrared communications. Proc. IEEE
**1997**, 82, 265–298. [Google Scholar] [CrossRef][Green Version] - Armstrong, J. OFDM for Optical Communications. J. Light. Technol.
**2009**, 27, 189–204. [Google Scholar] [CrossRef] - Huang, Y.; Liu, Y.; Shi, L.; Shi, D.; Zhang, X.; Aglzim, E.-H.; Zheng, J. On improving the accuracy of Visible Light Positioning system with SLM-based PAPR reduction schemes. In Proceedings of the 2020 IEEE International Symposium on Broadband Multimedia Systems and Broadcasting (BMSB), Paris, France, 27–29 October 2020. [Google Scholar]
- Offiong, F.B.; Sinanovic, S.; Popoola, W.O. On PAPR Reduction in Pilot-Assisted Optical OFDM Communication Systems. IEEE Access
**2017**, 5, 8916–8929. [Google Scholar] [CrossRef] - Dimitrov, S.; Sinanovic, S.; Haas, H. Clipping Noise in OFDM-Based Optical Wireless Communication Systems. IEEE Trans. Commun.
**2012**, 60, 1072–1081. [Google Scholar] [CrossRef] - Randel, S.; Breyer, F.; Lee, S.C.J.; Walewski, J.W. Advanced Modulation Schemes for Short-Range Optical Communications. IEEE J. Sel. Top. Quantum Electron.
**2010**, 16, 1280–1289. [Google Scholar] [CrossRef] - Tsonev, D.; Sinanovic, S.; Haas, H. Complete Modeling of Nonlinear Distortion in OFDM-Based Optical Wireless Communication. J. Light. Technol.
**2013**, 31, 3064–3076. [Google Scholar] [CrossRef] - Mazahir, S.; Chaaban, A.; Elgala, H.; Alouini, M.-S. Achievable Rates of Multi-Carrier Modulation Schemes for Bandlimited IM/DD Systems. IEEE Trans. Wirel. Commun.
**2019**, 18, 1957–1973. [Google Scholar] [CrossRef] - Jiang, Z.; Gong, C.; Xu, Z. Clipping Noise and Power Allocation for OFDM-Based Optical Wireless Communication Using Photon Detection. IEEE Wirel. Commun. Lett.
**2018**, 8, 237–240. [Google Scholar] [CrossRef] - Wang, T.Q.; Li, H.; Huang, X. Analysis and Mitigation of Clipping Noise in Layered ACO-OFDM Based Visible Light Communication Systems. IEEE Trans. Commun.
**2019**, 67, 564–577. [Google Scholar] [CrossRef] - Bussgang, J.J. Crosscorrelation Functions of Amplitude-Distorted Gaussian Signals; Tech. Rep. 216; Research Laboratory of Electronics, Massachusetts Institute of Technology: Cambridge, MA, USA, 1952. [Google Scholar]
- Mazo, J.E. Asymptotic distortion spectrum of clipped, DC-biased, Gaussian noise. IEEE Trans. Commun.
**1992**, 40, 1339–1344. [Google Scholar] [CrossRef] - Rice, S.O. Distribution of the Duration of Fades in Radio Transmission: Gaussian Noise Model. Bell Syst. Tech. J.
**1958**, 37, 581–635. [Google Scholar] [CrossRef] - Czylwik, A. Error Probability due to Clipping in Subcarrier Multiplexed Fiber-Optic Transmission Systems. Frequenz
**2000**, 54, 52–57. [Google Scholar] [CrossRef] - Elgala, H.; Mesleh, R.; Haas, H. Practical considerations for indoor wireless optical system implementation using OFDM. In Proceedings of the 10th International Conference on Telecommunications, Zagreb, Croatia, 8–10 June 2009. [Google Scholar]
- Ohm, J.-R.; Lüke, H.D. Mehrpegelübertragung. In Signalübertragung, 12. Auflage; Springer: Berlin/Heidelberg, Germany, 2014; pp. 309–313. [Google Scholar]
- Papoulis, A. The concept of a random variable. In Probability, Random Variables and Stochastic Processes, 4th ed.; McGraw-Hill Higher Education: New York, NY, USA, 2002; pp. 94–95. [Google Scholar]
- Proakis, J.G. Probability of Error for QAM. In Digital Communications, 4th ed.; McGraw-Hill Higher Education: New York, NY, USA, 2001; pp. 276–279. [Google Scholar]

**Figure 2.**Transformation of the Gaussian probability density function of x over a non-linear characteristic to represent the probability density function of ${n}_{\mathrm{c}}$.

**Figure 3.**Theoretical versus simulated symbol error probability in a clipped OFDM transmission system, based on the Bussgang theorem. No low-pass filter on the receiving side is used here.

**Figure 4.**Theoretical versus simulated symbol error probability in a clipped OFDM transmission system, based on the Bussgang theorem. For this graph, the out-of-band distortion is filtered out.

**Figure 5.**Exemplary representation of the i-th overshooting of the Gaussian distributed signal x on the clipping level A.

**Figure 7.**The clipping distortion ${n}_{\mathrm{c}}\left(t\right)$ can be modeled as a sum of shifted parabolic arcs.

**Figure 8.**Simulated (solid line) and analytical (dashed line) power spectral density of the clipping distortion for ${\sigma}_{x}^{2}=1$, $B=200\phantom{\rule{1.33333pt}{0ex}}\mathrm{MHz}$ and different clipping levels A.

**Figure 9.**Power of the uncorrelated clipping noise that is located inside the transmission band, relative to the entire power of the uncorrelated clipping noise.

**Figure 10.**Simulated and analytical calculated symbol error probability for a ${2}^{M}$-QAM OFDM-transmission that suffers from clipping at level A.

**Figure 11.**Analytically calculated versus simulated power spectral density of the clipping distortion evaluated at $f=B/2$ for ${\sigma}_{x}^{2}=1$, $B=200\phantom{\rule{1.33333pt}{0ex}}\mathrm{MHz}$ and different clipping levels A.

**Figure 12.**Approximated analytical and simulated power spectral density of the uncorrelated part of the clipping distortion for ${\sigma}_{x}^{2}=1$, $B=200\phantom{\rule{1.33333pt}{0ex}}\mathrm{MHz}$ and different clipping levels A.

**Figure 13.**Simulated and calculated symbol error probability for a ${2}^{M}$-QAM OFDM-transmission that suffers from clipping at level the A based on the approximated power spectral density.

**Table 1.**Parameters used in this work for the evaluation of the analytical and theoretical expressions, as well as for the Monte Carlo simulations.

Parameter | Shortcut | Value |
---|---|---|

Subcarriers | N | 8192 |

Bandwidth | B | $200\phantom{\rule{1.33333pt}{0ex}}\mathrm{MHz}$ |

Oversampling factor | – | 50 |

Modulation order | M | 2:2:10 |

Signal power | ${\sigma}_{x}^{2}$ | 1 |

Clipping level | A | 0.1:0.1:4 |

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**MDPI and ACS Style**

Frömming, A.; Häring, L.; Czylwik, A. Spectral Properties of Clipping Noise. *Mathematics* **2021**, *9*, 2592.
https://doi.org/10.3390/math9202592

**AMA Style**

Frömming A, Häring L, Czylwik A. Spectral Properties of Clipping Noise. *Mathematics*. 2021; 9(20):2592.
https://doi.org/10.3390/math9202592

**Chicago/Turabian Style**

Frömming, Alexander, Lars Häring, and Andreas Czylwik. 2021. "Spectral Properties of Clipping Noise" *Mathematics* 9, no. 20: 2592.
https://doi.org/10.3390/math9202592