## 1 GeoSurf Satellite Constellations  
The GeoSurf satellite constellations belong to the family of Walker Star Constellations, and they emulate the geostationary orbit with zenith paths for ground stations located at any latitude [1]. The GeoSurf constellations could be a good choice for future worldwide internet radio links because they have most of the advantages of current GEO (Geostationary), MEO (Medium Earth Orbit), and LEO (Low Earth Orbit) satellite constellations without having most of their drawbacks, as seen in Table 1 of [1]. In [2], we compared the tropospheric attenuation of GeoSurf zenith paths with those of GEO, MEO, and LEO satellites. In [3], we compared the relationship between the annual average probability distribution \(P(A)\) of exceeding a given rain attenuation \(A\) (dB) and the carrier frequency \(f_{\mathrm{c}}\) (GHz) in the GeoSurf paths at sites in different climatic regions with different rainfall statistics. All these studies refer to narrow-band channels, i.e., channels with negligible linear (amplitude and phase) distortions. In this paper, we estimate the transfer function and linear distortions (amplitude and phase) that are likely to be found in ultra-wideband radio-links to/from satellites orbiting in GeoSurf constellations working at very high frequencies. As a practical example, we consider a 10-GHz bandwidth centered at 80 GHz (W-Band). We study this very ultra-wideband--we refer to it as \"ultra-wideband\" to distinguish it from the actual \"wideband\" which, for the present study, should be considered as \"narrowband\"--because we think it might be useful in future worldwide internet radio links using spread spectrum modulation and Code Division Multiple Access (CDMA) [4, 5, 6, 7, 8, 9], once high-frequency ultra-wideband technology, now developed at lower frequencies [10], will also be available at W-Band. CDMA can provide a large processing gain and can be designed without considering issues related to the frequency and access coordination which, together with the advantagesof the GeoSurf constellations [1], can be an effective choice to provide high bit rates to users. For example, with a 5-GHz baseband, impulses with raised cosine spectrum and roll-off factor \(r=1\) (in the following, we refer to these impulses as the Nyquist impulses), and a coded information bit rate of 50 Mbits per second, being the chip-rate 5 Gbits per second with \(r=1\), the CDMA processing gain would be \(10\times\log_{10}(5/0.05)=20\) dB, which is enough to protect a carrier from the interference due to the other CDMA carriers. The vast literature on what today is termed \"wideband\" communication systems never refers to rain attenuation, or to the ultra-wideband radio links here supposed, but only to radio links in clear-sky conditions (e.g., faded by multipath, obstructions, etc. ), both for terrestrial and satellite systems [4; 11; 12; 13; 14; 15]; therefore, we are discussing these topics here for the first time, as our previous study involving rain attenuation and phase delay referred to narrow-band satellite systems and deep-space communications [16]. The purpose of this paper is to determine the amplitude and phase distortions produced by rain attenuation in very large bandwidths, in systems specially designed for using double-sideband suppressed-carrier (DSB-SC) modulation, i.e., bipolar phase-shift keying (BPSK), in both quadrature channels (QPSK). We assume that the total signal-to-noise ratio of the Gaussian channel is greater than the minimum required to provide a probability of bit error \(P(\varepsilon)\) smaller than the maximum value tolerated by users. In these conditions, we estimate how the in-band attenuation and phase delay due to rainfall affect the baseband digital signal. We evaluate both the output of a direct channel (e.g., the cosine channel) and the interference coming from the quadrature channel (the sine channel, referred to as the orthogonal). Because this the first approach to the topic, we use the average transfer function of direct and orthogonal channels.