# Assessment of Various Window Functions in Spectral Identification of Passive Intermodulation

## Abstract

**:**

## 1. Introduction

## 2. Modeling of Passive Intermodulation

_{1}and f

_{2}is unique as it makes the analysis more tractable and; at the same time, this enables nonlinear distortion to be quantified and predicted at IMD frequencies in a wideband carrier aggregated wireless system. For a two-signal input, the passive nonlinearity results in IMD components at the IMD frequencies. In this scenario, the input to the passive nonlinearity is represented as

_{0}can be found as [17]

_{0}represents the spectral regrowth and cross modulation while the spectral components at the IMD frequencies which may interfere with other channels in a carrier aggregation system or with the receive bands.

## 3. Estimation of Power Spectral Density of the Output of a Passive Nonlinearity

_{0}, k

_{1}and k

_{2}are the model parameters which can be adjusted to fit a wide variety of nonlinear characteristics. Note that these characteristics can model the less than 3 dB/dB behavior of a passive nonlinearity by the proper selection of the model parameters. The passive nonlinearity in the simulations that will follow was generated using a hyper tangent model with parameters k

_{1}= 6 × 10

^{−5}, k

_{2}= 3 × 10

^{−3}and g

_{0}= 1. Figure 4a shows the I/V characteristics of the hyperbolic tangent model and Figure 4b shows the input/output power characteristics of the model.

## 4. Results

## 5. Discussion

## 6. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Kearney, F.; Chen, S. Passive Intermodulation (PIM) Effects in Base Stations: Understanding the Challenges and Solutions; Analog Devices: Norwood, MA, USA, 2017; p. 25. Available online: www.analogdialogue.com (accessed on 18 April 2021).
- Al-Kanan, H.; Li, F. A Simplified Accuracy Enhancement to the Saleh AM/AM Modeling and Linearization of Solid-State RF Power Amplifiers. Electronics
**2020**, 9, 1806. [Google Scholar] [CrossRef] - Zhang, L.; Wang, H.; Shen, J.; Wei, H.; Wang, X.; Li, Y.; Liu, C. A composite exponential model to characterize nonlinearity causing passive intermodulation interference. IEEE Trans. Electromagn. Compat.
**2018**, 61, 590–594. [Google Scholar] [CrossRef] - Kozlov, D.S.; Shitvov, A.P.; Schuchinsky, A.G. Characterization of passive intermodulation in passive RF devices with X-parameters. In Proceedings of the Antennas and Propagation Conference, Loughborough, Leicestershire, UK, 10 November 2014; pp. 64–67. [Google Scholar]
- Zhang, L.; Wang, H.; He, S.; Wei, H.; Li, Y.; Liu, C. A Segmented Polynomial Model to Evaluate Passive Intermodulation Products from Low-Order PIM Measurements. IEEE Microw. Wirel. Compon. Lett.
**2018**, 29, 14–16. [Google Scholar] [CrossRef] - Lui, P.L. Passive intermodulation interference in communication systems. Electron. Commun. Eng. J.
**1990**, 2, 109–118. [Google Scholar] [CrossRef] - Kozlov, D.S.; Shitvov, A.P.; Schuchinsky, A.G.; Steer, M.B. Passive Intermodulation of Analog and Digital Signals on Transmission Lines with Distributed Nonlinearities: Modelling and Characterization. IEEE Trans. Microw. Theory Tech.
**2016**, 64, 1383–1395. [Google Scholar] [CrossRef] - Henrie, J.J.; Christianson, A.J.; Chappell, W.J. Linear–nonlinear interaction and passive intermodulation distortion. IEEE Trans. Microw. Theory Tech.
**2010**, 58, 1230–1237. [Google Scholar] [CrossRef] - Kozlov, D.S.; Shitvov, A.P.; Schuchinsky, A.G. Polynomial model for high-order and multi-carrier passive intermodulation products. In Proceedings of the 46th European Microwave Conference (EuMC), London, UK, 4 October 2016; pp. 631–634. [Google Scholar]
- Jin, Q.; Gao, J.; Wu, Y.; Xie, G. Behavior modeling of passive intermodulation distortion with multiple nonlinear sources. Microw. Opt. Technol. Lett.
**2018**, 60, 2182–2185. [Google Scholar] [CrossRef] - Prabhu, K.M. Window Functions and Their Applications in Signal Processing; Taylor & Francis: Abingdon, UK, 2014; p. 404. [Google Scholar]
- Wen, H.; Teng, Z.; Guo, S. Triangular Self-Convolution Window With Desirable Sidelobe Behaviors for Harmonic Analysis of Power System. IEEE Trans. Instrum. Meas.
**2009**, 59, 543–552. [Google Scholar] [CrossRef] - Jwo, D.J.; Wu, I.H.; Chang, Y. Windowing Design and Performance Assessment for Mitigation of Spectrum Leakage. In E3S Web of Conferences; EDP Sciences: Les Ulis, France, 2019; Volume 94, p. 03001. [Google Scholar]
- Heinzel, G.; Rüdiger, A.; Schilling, R. Spectrum and Spectral Density Estimation by the Discrete Fourier transform (DFT), Including a Comprehensive List of Window Functions and Some New at-Top Windows; Technical Report; Albert-Einstein-Institut: Hannover, Germany, 2002. [Google Scholar]
- Steer, M.B.; Gharaibeh, K.M. Volterra Modeling in Analog, RF and Microwave Engineering. In Encyclopedia of RF and Microwave Engineering; Wiley: Hoboken, NJ, USA, 2005. [Google Scholar]
- Gharaibeh, K.M. The Combined Effect of Various Receiver Nonlinearities on Spectrum Sensing in Cognitive Radio Systems. J. Commun.
**2020**, 15, 350–358. [Google Scholar] [CrossRef] - Gharaibeh, K.; Steer, M. Modeling distortion in multichannel communication systems. IEEE Trans. Microw. Theory Tech.
**2005**, 53, 1682–1692. [Google Scholar] [CrossRef] - Schuck, A., Jr.; Bodmann, B.E.J. Audio Nonlinear Modeling through Hyperbolic Tangent Functionals. In Proceedings of the 19th International Conference on Digital Audio Effects (DAFx-16), Brno, Czech Republic, 5–9 September 2016; pp. 1–6. [Google Scholar]
- Henrie, J.; Christianson, A.; Chappell, W.J. Prediction of Passive Intermodulation from Coaxial Connectors in Microwave Networks. IEEE Trans. Microw. Theory Tech.
**2008**, 56, 209–216. [Google Scholar] [CrossRef]

**Figure 1.**Spectra in a wideband receiver: (

**a**) Input spectrum of two RF signals to a passive nonlinearity and (

**b**) the Output spectrum; dashed line indicates PSD computed using FFT with rectangular window indicating spectral leakage.

**Figure 2.**PSD of the output of nonlinearity for an input that consists of the sum of two OFDM signals with frequency separation of 100 MHz.

**Figure 3.**Spectrum of various window functions: (

**a**) Rectangular window; (

**b**) Hanning window; (

**c**) Blackman window; (

**d**) Blackman-Harris window; (

**e**) Hamming window and (

**f**) Kaiser window.

**Figure 5.**PSD of the output of the passive nonlinearity using a Hanning window and a rectangular window.

**Figure 6.**Power in the intermodulation components vs. Input power computed from the periodogram with various window functions: Solid o: Hanning window Solid *: Blackman window, Solid ∆: Blackman Harris window, Dashed o: rectangular window, Dashed *: Hamming window and Dashed ∆: Kaiser window; (

**a**) IMD3, (

**b**) IMD5, (

**c**) IMD7 and (

**d**) IMD9.

**Figure 7.**Power in the intermodulation components vs. frequency separation computed from the periodogram with various window functions: Solid o: Hanning window Solid *: Blackman window, Solid ∆: Blackman Harris window, Dashed o: rectangular window, Dashed *: Hamming window and Dashed ∆: Kaiser window.

N | Upper Intermod (Band Pass) | Upper Intermod (Envelop) | Lower Intermod (Band Pass) | Lower Intermod (Envelop) |
---|---|---|---|---|

3 | $2{f}_{2}-{f}_{1}$ | $3{f}_{0}$ | $2{f}_{1}-{f}_{2}$ | $-3{f}_{0}$ |

5 | $3{f}_{2}-2{f}_{1}$ | $5{f}_{0}$ | $3{f}_{1}-2{f}_{2}$ | $-5{f}_{0}$ |

7 | $4{f}_{2}-3{\omega}_{1}$ | $7{f}_{0}$ | $4{f}_{1}-3{f}_{2}$ | $-7{f}_{0}$ |

9 | $5{f}_{2}-4{f}_{1}$ | $9{f}_{0}$ | $5{f}_{1}-4{f}_{2}$ | $-9{f}_{0}$ |

11 | $6{f}_{2}-5{f}_{1}$ | $11{f}_{0}$ | $6{f}_{2}-5{f}_{1}$ | $-11{f}_{0}$ |

Window | $\mathit{w}\mathbf{\left(}\mathit{n}\mathbf{\right)}$ | Width of the Main Lobe (3-dB BW) $\mathbf{\left(}\mathit{\pi}\frac{\mathbf{s}\mathbf{a}\mathbf{m}\mathbf{p}\mathbf{l}\mathbf{e}\mathbf{s}}{\mathbf{c}\mathbf{y}\mathbf{c}\mathbf{l}\mathbf{e}}\mathbf{\right)}$ | Max Side Lobe Level (dB) | Side Lobe Roll-Off Rate (dB/Octave) |
---|---|---|---|---|

Rectangular | $\{\begin{array}{cc}1& 0\le n\le N\\ 0& \mathrm{Otherwise}\end{array}$ | 0.011719 | −13.3 | −6 |

Hanning | $\frac{1}{2}\left(1-\mathrm{cos}\left(\frac{2\pi n}{N}\right)\right),0\le n\le N$ | 0.019531 | −31.6 | −18 |

Hamming | $0.54+0.46\mathrm{cos}\left(\frac{2\pi n}{N}\right),0\le n\le N$ | 0.019531 | −43.7 | −6 |

Blackman | $0.42-0.5\mathrm{cos}\left(\frac{2\pi n}{N}\right)+0.08\left(\frac{4\pi n}{N}\right),0\le n\le N$ | 0.023438 | −58.2 | −18 |

Blackman–Harris | $0.35875-0.48829\mathrm{cos}\left(\frac{2\pi n}{N}\right)$ $+0.14128\mathrm{cos}\left(\frac{4\pi n}{N}\right)-0.01168\mathrm{cos}\left(\frac{6\pi n}{N}\right)$ | 0.027344 | −92.2 | −6 |

Kaiser | $\frac{{I}_{0}\left[\beta \sqrt{1-{\left(1-\frac{2n}{N}\right)}^{2}}\right]}{{I}_{0}\left(\beta \right)},0\le n\le N$ ($\beta $ is a parameter that determines the tradeoff between main lobe width and side lobe levels) | 0.011719 | −14.7 | −6 |

No. of Sub-Carriers | 1705 |
---|---|

Modulation | 16 QAM Constellation type: Gray |

Pulse Shaping | Rectangular pulse |

PAPR | 10.8 dB |

Input Power range | 0 dBm to 35 dBm |

Pin = 0 dBm | Pin = 30 dBm | |||||||
---|---|---|---|---|---|---|---|---|

Window | IMD3 | IMD5 | IMD7 | IMD9 | IMD3 | IMD5 | IMD7 | IMD9 |

Rectangular | −53 | −58 | −60.9 | −62.8 | −6.8 | −25.7 | −31.4 | −33.7 |

Kaiser | −53.4 | −58.3 | −61.2 | −63.1 | −6.8 | −25.8 | −31.7 | −34 |

Hamming | −70.9 | −75.9 | −78.8 | −80.7 | −7 | −27.3 | −43.7 | −50.9 |

Hanning | −96.7 | −174.4 | −211 | −218 | −7 | −27.3 | −45.8 | −64.7 |

Blackman | −97.5 | −175.3 | −219 | −226 | −7.4 | −28 | −46.8 | −65.6 |

Blackman Harris | −98.1 | −136.5 | −139.4 | −141 | −7.8 | −28.8 | −47.9 | −66.8 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gharaibeh, K.
Assessment of Various Window Functions in Spectral Identification of Passive Intermodulation. *Electronics* **2021**, *10*, 1034.
https://doi.org/10.3390/electronics10091034

**AMA Style**

Gharaibeh K.
Assessment of Various Window Functions in Spectral Identification of Passive Intermodulation. *Electronics*. 2021; 10(9):1034.
https://doi.org/10.3390/electronics10091034

**Chicago/Turabian Style**

Gharaibeh, Khaled.
2021. "Assessment of Various Window Functions in Spectral Identification of Passive Intermodulation" *Electronics* 10, no. 9: 1034.
https://doi.org/10.3390/electronics10091034