# Transfer Functions and Linear Distortions in Ultra-Wideband Channels Faded by Rain in GeoSurf Satellite Constellations

^{*}

## Abstract

**:**

## 1. GeoSurf Satellite Constellations

## 2. Passband and Equivalent Baseband Transfer Functions

## 3. Attenuation, Phase Shift, and Time Delay Due to Rainfall in Zenith Paths

## 4. Passband and Baseband Transfer Functions

## 5. Long-Term Results

## 6. Probability of Bit Error

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Long-Term Equivalent Baseband Transfer Functions at Madrid, Prague, and Tampa

**Figure A1.**Madrid. Average value (continuous line) and $\pm 1$ standard deviation (dashed lines) baseband equivalent transfer functions in the range $2.5\le $ A < 3.5 dB.

**Left column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the direct channel;

**Right column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the orthogonal channel.

**Figure A2.**Madrid. Average value (continuous line) and $\pm 1$ standard deviation (dashed lines) baseband equivalent transfer functions in the range $9\le $ A < 11 dB dB.

**Left column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the direct channel;

**Right column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the orthogonal channel.

**Figure A3.**Madrid. Average value (continuous line) and $\pm 1$ standard deviation (dashed lines) baseband equivalent transfer functions in the range $18\le $ A < 22 dB.

**Left column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the direct channel;

**Right column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the orthogonal channel.

**Figure A4.**Average value (continuous line) and $\pm 1$ standard deviation (dashed lines) baseband equivalent transfer functions in the range $27\le $ A < 33 dB.

**Left column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the direct channel;

**Right column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the orthogonal channel.

**Figure A5.**Prague. Average value (continuous line) and $\pm 1$ standard deviation (dashed lines) baseband equivalent transfer functions in the range $2.5\le $ A < 3.5 dB.

**Left column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the direct channel;

**Right column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the orthogonal channel.

**Figure A6.**Prague. Average value (continuous line) and $\pm 1$ standard deviation (dashed lines) baseband equivalent transfer functions in the range $9\le $ A < 11 dB.

**Left column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the direct channel;

**Right column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the orthogonal channel.

**Figure A7.**Prague. Average value (continuous line) and $\pm 1$ standard deviation (dashed lines) baseband equivalent transfer functions in the range $18\le $ A < 22 dB.

**Left column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the direct channel;

**Right column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the orthogonal channel.

**Figure A8.**Prague. Average value (continuous line) and $\pm 1$ standard deviation (dashed lines) baseband equivalent transfer functions in the range $27\le $ A < 33 dB.

**Left column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the direct channel;

**Right column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the orthogonal channel.

**Figure A9.**Tampa. Average value (continuous line) and $\pm 1$ standard deviation (dashed lines) baseband equivalent transfer functions in the range $2.5\le $ A < 3.5 dB.

**Left column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the direct channel;

**Right column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the orthogonal channel.

**Figure A10.**Tampa. Average value (continuous line) and $\pm 1$ standard deviation (dashed lines) baseband equivalent transfer functions in the range $9\le $ A < 11 dB.

**Left column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the direct channel;

**Right column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the orthogonal channel.

**Figure A11.**Tampa. Average value (continuous line) and ±1 no standard deviation (dashed lines) baseband equivalent transfer functions in the range $18\le $ A < 22 dB.

**Left column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the direct channel;

**Right column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the orthogonal channel.

**Figure A12.**Tampa. Average value (continuous line) and $\pm 1$ standard deviation (dashed lines) baseband equivalent transfer functions in the range $27\le $ A < 33 dB. dB.

**Left column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the direct channel;

**Right column**: Real (

**upper panel**) and Imaginary (

**lower panel**) transfer functions of the orthogonal channel.

## Appendix B. Long-Term Probability Ratio at Madrid, Prague, and Tampa

**Figure A13.**Madrid. Probability ratio $\rho $ between the probability of bit error with rain and the probability of bit error with no rain (ideal case): $P\left(\epsilon \right)={10}^{-2}\left(blue\right),{10}^{-3}\left(green\right)$, ${10}^{-4}\left(magenta\right)$, ${10}^{-5}\left(cyan\right)$, ${10}^{-6}\left(black\right)$. Direct case: continuous lines; with interference: dashed lines.

**Left**: negative interfering impulse;

**Right**: positive interfering impulse.

**Figure A14.**Prague. Probability ratio $\rho $ between the probability of bit error with rain and the probability of bit error with no rain (ideal case): $P\left(\epsilon \right)={10}^{-2}\left(blue\right),{10}^{-3}\left(green\right)$, ${10}^{-4}\left(magenta\right)$, ${10}^{-5}\left(cyan\right)$, ${10}^{-6}\left(black\right)$. Direct case: continuous lines; with interference: dashed lines.

**Left**: negative interfering impulse;

**Right**: positive interfering impulse.

**Figure A15.**Tampa. Probability ratio $\rho $ between the probability of bit error with rain and the probability of bit error with no rain (ideal case): $P\left(\epsilon \right)={10}^{-2}\left(blue\right),{10}^{-3}\left(green\right)$, ${10}^{-4}\left(magenta\right)$, ${10}^{-5}\left(cyan\right)$, ${10}^{-6}\left(black\right)$. Direct case: continuous lines; with interference: dashed lines.

**Left**: negative interfering impulse;

**Right**: positive interfering impulse.

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**Figure 1.**Cartesian plane in which the phasors (vectors) at angular speed ${\omega}_{c}$ are fixed (non-rotating). The two side-bands of Equation (1) can be represented by two vectors rotating in phase counterclockwise (upper side-band, positive rotation) and clockwise (lower side-band, negative rotation) with angular speed $+\omega $ and $-\omega $, respectively.

**Figure 2.**Cartesian plane in which the phasors (vectors) at angular speed ${\omega}_{c}$ are fixed (non-rotating). The two side-bands of Equation (1) are shown at the output of the medium transfer function. ${H}_{r}\left(\psi \right)$ acts on the two side-bands in phase (real axis), while ${H}_{i}\left(\psi \right)$ acts on them by producing orthogonal side-bands (imaginary axis).

**Figure 3.**Annual probability distributions (%) $P\left(R\right)$ at Spino d’Adda, Madrid, Tampa, and Prague.

**Figure 4.**Baseband receiver in ideal conditions. $S\left(f\right)$ is the two-sided spectrum of the Nyquist impulse, $\sqrt{S\left(f\right)}$ is its matched filter, and $n\left(t\right)$ is the receiver total additive Gaussian white noise.

**Figure 5.**Quadrature baseband receiver in rain attenuation. $S\left(f\right)$ is the two-sided spectrum of the Nyquist reference impulse assumed to be positive, $\sqrt{S\left(f\right)}$ is the matched filter, and $n\left(t\right)$ is the receiver total additive Gaussian white noise.

**Figure 6.**SST-simulated rain attenuation event at Spino d’Adda at 80 GHz (8 May 2000, starts at 0:48 AM local time).

**Upper panel**: Rain attenuation at 80 GHz (circular polarization).

**Lower panel**: Relative attenuation at the extreme of a 10-GHz bandwidth.

**Figure 7.**SST-simulated rain attenuation event at Spino d’Adda at 80 GHz GHz (8 May 2000, starts at 0:48 AM local time).

**Upper panel**: Time delay at 80 GHz (circular polarization).

**Lower panel**: Relative time delay at the extreme of a 10-GHz bandwidth.

**Figure 9.**Spino d’Adda. Average value (continuous line) and $\pm 1$ standard deviation (dashed lines) baseband equivalent transfer functions in the range $2.5\le $ A < 3.5 dB.

**Left column**: Real (

**upper panel**) and imaginary (

**lower panel**) transfer functions of the direct channel;

**Right column**: Real (

**upper panel**) and imaginary (

**lower panel**) transfer functions of the cross-channel.

**Figure 10.**Spino d’Adda. Average value (continuous line) and $\pm 1$ standard deviation (dashed lines) baseband equivalent transfer functions in the range $2.5\le $ A < 3.5 dB.

**Left column**: Real (

**upper panel**) and imaginary (

**lower panel**) transfer functions of the direct channel;

**Right column**: Real (

**upper panel**) and imaginary (

**lower panel**) transfer functions of the cross-channel. $9\le $ A < 11 dB.

**Figure 11.**Spino d’Adda. Average value (continuous line) and $\pm 1$ standard deviation (dashed lines) baseband equivalent transfer functions in the range $2.5\le $ A < 3.5 dB.

**Left column**: Real (

**upper panel**) and imaginary (

**lower panel**) transfer functions of the direct channel;

**Right column**: Real (

**upper panel**) and imaginary (

**lower panel**) transfer functions of the cross-channel. $18\le $ A < 22 dB.

**Figure 12.**Spino d’Adda. Average value (continuous line) and $\pm 1$ standard deviation (dashed lines) baseband equivalent transfer functions in the range $2.5\le $ A < 3.5 dB.

**Left column**: Real (

**upper panel**) and imaginary (

**lower panel**) transfer functions of the direct channel;

**Right column**: Real (

**upper panel**) and imaginary (

**lower panel**) transfer functions of the cross-channel. $27\le $ A < 33 dB.

**Figure 13.**Spino d’Adda. Probability ratio $\rho $ between the probability of bit error with rain and the probability of bit error with no rain (ideal case): $P\left(\epsilon \right)={10}^{-2}\left(blue\right),{10}^{-3}\left(green\right)$, ${10}^{-4}\left(magenta\right)$, ${10}^{-5}\left(cyan\right)$, ${10}^{-6}\left(black\right)$. Direct case: continuous lines; with interference: dashed lines.

**Left**: negative interfering impulse;

**Right**: positive interfering impulse.

**Figure 14.**Probability ratio $\rho $ between the probability of bit error with rain and the probability of bit error with no rain (ideal case) vs. probability of bit error in the direct channel with no interference, according to the average SNR (continuous lines) and $-1$ standard deviation of SNR (dashed lines), at Spino d’Adda, Prague, Madrid, Tampa. Blue lines, $A=10$ dB; red lines, $A=20$ dB.

**Table 1.**Geographical coordinates, altitude (km), rain height ${H}_{R}$ (km), and number of years of continuous rain rate measurements at the indicated sites.

Site | Latitude N (°) | Longitude E (°) | $\mathbf{Altitude}{\mathit{H}}_{\mathit{S}}$ | $\mathbf{Precipitation}\mathbf{Height}{\mathit{H}}_{\mathit{R}}$ | Rain Rate Data Bank (Years) |
---|---|---|---|---|---|

Spino d’Adda (Italy) | 45.4 | 9.5 | 0.084 | 3.341 | 8 |

Madrid (Spain) | 40.4 | 356.3 | 0.630 | 3.001 | 8 |

Prague (Czech Republic) | 50.0 | 14.5 | 0.250 | 3.051 | 5 |

Tampa (Florida) | 28.1 | 277.6 | 0.050 | 4.528 | 4 |

**Table 2.**Local precipitation height (km) at the indicated sites to be used in the Synthetic Storm Technique. Notice that the depth of Layer B is 0.400 km, which, if added to the local precipitation height of layer A, gives the local precipitation height. For example, at the latitude and longitude of Spino d’Adda, the ITU-R gives 3.341 km, which becomes 3.257 km after subtracting its altitude of 0.084 km.

Site | $\mathbf{Altitude}{\mathit{H}}_{\mathit{S}}$ | $\mathbf{Precipitation}\mathbf{Height}{\mathit{H}}_{\mathit{R}},\mathbf{ITU}-\mathbf{R}$ | Local Precipitation Height | Local Height of Layer A |
---|---|---|---|---|

Spino d’Adda (Italy) | 0.084 | 3.341 | 3.257 | 2.857 |

Madrid (Spain) | 0.630 | 3.001 | 2.371 | 1.971 |

Prague (Czech Republic) | 0.250 | 3.051 | 2.801 | 2.401 |

Tampa (Florida) | 0.050 | 4.528 | 4.478 | 4.078 |

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**MDPI and ACS Style**

Matricciani, E.; Riva, C.
Transfer Functions and Linear Distortions in Ultra-Wideband Channels Faded by Rain in GeoSurf Satellite Constellations. *Future Internet* **2023**, *15*, 27.
https://doi.org/10.3390/fi15010027

**AMA Style**

Matricciani E, Riva C.
Transfer Functions and Linear Distortions in Ultra-Wideband Channels Faded by Rain in GeoSurf Satellite Constellations. *Future Internet*. 2023; 15(1):27.
https://doi.org/10.3390/fi15010027

**Chicago/Turabian Style**

Matricciani, Emilio, and Carlo Riva.
2023. "Transfer Functions and Linear Distortions in Ultra-Wideband Channels Faded by Rain in GeoSurf Satellite Constellations" *Future Internet* 15, no. 1: 27.
https://doi.org/10.3390/fi15010027