and breaks a traditional antenna-array concept. LIS allows for an unprecedented focusing of energy in three-dimensional (3D) space, remote sensing with extreme precision, and unprecedented data-transmissions. The abundant signal dimensions with LIS can also facilitate potential applications using artificial intelligence (AI).
Earlier proposed LISs are in planar forms such as squared and disk shapes. In its fundamental form, LIS uses an entire surface for transmitting and receiving radiating signals, where each point-element is controllable. Under ideal assumptions, fundamental limits on the number of independent signal dimensions per unit surface-area are derived in , where we show that inter-user interference of two users is close to a sinc-function. In , Cramér-Rao lower-bounds (CRLB) for positioning with LIS are derived, which can decrease in the third-order of surface-area in near-field.
Implementing LIS in practical is however, challenging and requires new technologies in fields such as material, radio-frequency, hardware integration etc. One important observation form  is that after applying match-filtering, the received signal on LIS has a band-limit property, and thusly LIS can be implemented in a discrete-form based on sampling theory. Even so, since a LIS typically has a much larger scale of antenna-elements than Massive MIMO, it is still challenging. Followed by [1, 2], there are some simplifications by implementing LIS as reflecting (passive) surfaces [4, 5], which is similar to reflectarray antennas .
In this letter we extend the planar LIS into a spherical design depicted in Fig. 1, to cooperate with future networks that from the ground to space comprise of terrestrial cellular networks, unmanned aerial vehicle (UAV) networks, and satellites. A spherical LIS has many advantages compare to a planar one due to its geometric shape. It can also be seen as an extension of traditional spherical arrays .
Ii Spherical LIS
Ii-a Received Signal Model
The radiating signal from a terminal to the spherical LIS is depicted in Fig. 2. Expressed in Cartesian coordinates, the center of the spherical LIS is located at , while the terminal is located at . Due to the isotropic property of a spherical LIS, we can equivalently let the terminal located at position (by rotating the ball). For simplicity we still assume the terminal-position as . and for analytical tractability a perfect line-of-sight (LoS) propagation scenario is considered. Following similar analysis in  and under line-of-sight (LoS), the received signal at location on LIS (with a unit transmit-power and without receiving noise) can be modeled as
where the path-attenuation is , and is angle-of-arriving (AoA). We assume a narrow-band (NB) system at a carrier-frequency is whose wavelength is , and is the transmit-time from the terminal to the LIS with a distance
Seen from Fig.2, it holds from cosine theorem that
where is the radius of the spherical LIS. Transforming to spherical coordinates by setting
and combining (1)-(5) yields
with a normalized distance
The surface-region where the LIS can receive (transmit) signals from (to) the terminal is , and
Ii-B Advantages of Spherical LIS over a Planar LIS
There are several advantages of a spherical LIS compared to a planar LIS. The most fundamental advantage is the isotropic property of the spherical LIS. That is, rotating a terminal around the center of LIS will not change the information-theoretical properties. This brings several substantial gains in average received signal strength (RSS). It also has the advantage that a simple positioning method can be based on the boundary on the surface where LIS can receive signal, i.e., measuring as in (8), which utilizes the geometric shape of the LIS. In ideal implementation,
can be accurately estimated and the terminal can be positioned . Another advantage is a feasible design of reflecting surface where partial of the spherical LIS is used for collecting signal, while the other part is used for redirection the signal to a targeted terminal. Moreover, the phases can be compensated both at receiving and transiting based on the estimated position of the terminal. We will elaborate these nice properties of a spherical LIS in what follows.
Ii-C Array Gain with Spherical LIS
We first compare RSS difference between spherical and planar LISs,. Note that the maximal effective surface-area of a spherical LIS that can receive signals from a terminal is half the surface-area, i.e., (). Therefore, we shall compare to a planar LIS (assume to have a disk-shape) with the same surface-area, i.e., a radius of . The RSS on a spherical LIS of radius , for a given and a maximal integral angle , is equal to (since the Jacobian of transform from Cartesian to spherical coordinates is )
When receiving at the maximum , it holds that
On the other hand, the RSS of a planar LIS in a disk-shape and of radius , for a given user located at (for a disk-shaped planar LIS, we can always rotate the disk such that the terminal located on ), which from  it can be shown that
When , i.e., the terminal is located at , the equality in (11) is exact. Further, we point out the fact that for a planar LIS, can be as small as possible. However, for a spherical LIS, the minimal is 1 when the terminal is infinitely close to the surface. Under the condition that , i.e., , when a terminal moves around the spherical LIS with a fixed radius , i.e., , the RSS measured on the spherical LIS are almost equal. But for a planar LIS, the RSS changes approximately by a factor seen from (11). Averaging over , the RSS ratio of a spherical LIS of radius over a planar LIS of radius is
Note that when is sufficiently large, it holds that , and half of the transmitting power propagating towards the LIS are received both at a spherical and a planar LIS. Further, to see gains with a spherical LIS, we let and it holds that
On the other hand, letting yields
Ii-D Positioning with Spherical LIS
With spherical LIS, the positioning becomes simpler and only -dimension needs to be estimated. Considering an idea LoS case, based on the measured angle on the LIS, the direction of the terminal aligns with the line that connects the center of the sphere and the center of the cross-section circle corresponding to the cone that can receive signals. Meanwhile, the distance can be estimated from (8) as
This yields a simple positioning technique.
As an auxiliary method, we can also set
and measure a series of RSS, which holds from (9) that
Based on which, we can obtain a series of estimates for , where the estimate of can be obtained.
Under Non-LoS (NLOS) scenario, there could be multiple paths reaching the LIS. Then, similar to positioning in traditional antenna-array systems, one needs to estimate the multiple-path components (MPC) and find the strongest LoS path. Afterwards, similar positioning can be carried out as under LoS case. Further, if there are multiple LIS deployed around, estimates from different LISs can be combined to refine the positioning.
Nevertheless, we consider the CLRB for positioning . A direct approach would be evaluating the Fisher-information on received signal model (5) following similar approaches as in . However, since we are more interest in the decreasing rate of CRLB in relation to the radius , we use (10) to evaluate the CRLB and assuming the noise is additional white Gaussian noise (AWGN) due to potential hardware impairments.
it shows that the RSS based positioning of , i.e., decreases in , or cubicly in the surface-area. This shall compared to the case with a planar LIS when the terminal is located at . From (11) it holds that
As expected, the CRLB with a spherical LIS is smaller than that with a planar LIS. But surprisingly, we see that with a spherical LIS the CRLB goes to zero when , however, with a planar LIS it has a fundamental limit equal to 4 as .
Ii-E Using Spherical LIS as Reflecting surfaces
As depicted in Fig.3, spherical LIS can also be used as reflecting surfaces aiming at a low-cost design. In which case, the received signal from a traditional base-station can be collected using partial of the surface towards it, and then redirected using another partial surface facing a terminal that is blocked out to the base-station. Due to the simple positioning as introduced earlier with spherical LIS, both the positions of base-station and terminal can be estimated based on the positions, and phases can be compensated when receiving from the base-station and reflecting the received signal. This yields a very flexible and effective designs of reflecting surfaces.
Ii-F Numerical Results
Below we show some simulation results with the spherical LIS. In Fig. 4, the averaged RSS are measured with (in meter), where the radius and changes. As can be seen, when , the approximation of RSS for planar LIS in (11) are quite accurate and aligned with numerical results. Further, spherical LIS provides substantial RSS gains. The ratio is plotted in Fig. 5, where we can see that as , converges to both for numerical and analytical results. In Fig. 6, we plotted the CRLBs obtained in (18) and (19), respectively, where when is small the CRLBs are close and decrease in 6th order of (i.e., ). As increases, spherical LIS outperforms planar LIS.
In this letter we have extended current planar LIS into a spherical design, which yields many advantages. We have shown analytically that a spherical LIS can yield higher RSS and lower positioning CRLB compared to a planar one. Asymptotic properties are also analyzed which are aligned with simulation results. In addition, spherical LIS can also be used as flexible and low-cost reflecting surfaces.
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