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Multiuser communications for lineofsight large intelligent surface systems
EURASIP Journal on Advances in Signal Processing volume 2023, Article number: 129 (2023)
Abstract
In this paper, we propose a new interferencesuppression beamformer design and a DirectionofArrival (DoA) estimation method for LineofSight (LoS) multiuser communication systems with circular Large Intelligent Surfaces (LISs). This symmetrical circularshaped LIS allows signals defined on it to be formed by combining Fourier and Bessel series. The Fourier harmonics are orthogonal along the rotational direction and the Bessel functions constituting the Bessel series are orthogonal along the radial direction. Synthesizing basis functions from these two series results in the proposed interferencesuppression beamformer, with their coefficients designed to cancel interuser interference. Also by exploiting signal structure of a circular LIS communicating with Mobile Stations (MSs), we devise a DoA estimation method to obtain directional information of incoming signals. Under LoS channel, the DoA corresponds to elevation and Azimuth angles of MSs relative to the LIS. This method is numerically demonstrated to have promising performance with highfrequency carriers.
1 Introduction
Large Intelligent Surface (LIS) is a spatial continuous aperture packed with a large number of antennas [1]. Several advantages that help improving existing wireless communication systems come with this new configuration. First, the continuous aperture of a LIS enables fine tuning of electromagnetic fields on the surface. This provides flexibilities in designing beamformers to achieve spatial diversities. Second, a continuous aperture can focus transmitting energy on a small area with high spatial resolution. This property not only can boost transmission efficiency, but also is favorable in precise positioning and wireless charging. Third, LISbased systems can achieve high energy efficiency with its surface structure for transmitting and receiving signals, as shown in [1,2,3]. Fourth, utilizing advanced metamaterial technology, LISs can be installed integrating with surfaces such as windows, walls, and advertising boards on streets and highways [4]. These advantages coupled with recent developments of metamaterials and programmable metasurfaces that makes implementing a practical LIS system possible [5,6,7,8,9], has promoted LIS as a promising technology to serve future wireless communication systems with everincreasing demand of highspeed, highvolume data transmissions [2, 3, 10].
In existing literature, LISs can be used in either of the following two ways. The first is to use LIS as a constructive signal reflector so that the composite channel between a transmitter and a receiver becomes favorable. This is also known as Reconfigurable Intelligent Surface (RIS) or Intelligent Reflecting Surface (IRS) [7, 11,12,13]. The second is to use LIS as a transmitter and/or receiver. In this paper, we refer to LIS as in the second way.
One fundamental problem about LIS systems is what capacity they can achieve. This line of researches is conducted by applying electromagnetic theory to model propagation channels between two transmitting/receiving LIS volumes/surfaces. Then, the DegreeofFreedoms (DoFs), which can be interpreted as the number of orthogonal channels or communication modes, are obtained by solving for the number of eigenfunctions of the channel. In [14], the authors obtain a formal solution of orthogonal modes between one transmitting volume and one receiving volume communicating in free space. Specifically, the result is applied to calculate concrete communication modes between two volumes with rectangular shape. Asymptotic DoFs for two communicating LISs are derived in [4]. The result shows that largerthanone DoF can be achieved in LineofSight (LoS) channel condition, showcasing the possible benefit of deploying LIS systems. Authors in [15] take effects of multipath channel into account by incorporating an experimental scattering channel model in their analysis of DoF for systems with transmit and receive antenna arrays. The resulting spatial DoF is the product of effective array aperture and angular spread of the physical channel using spherical unipolarized antenna arrays. These theoretical results provides insightful observations for designing LIS systems. Under practical channel condition, solving for orthogonal communication modes is a challenge task because it involves complex differential equations, which is usually complicated to solve. As an effort to address this challenge, authors in [16] propose a practical method for deriving nearfield communication modes between linear aperture antennas in free space.
Another line of LISrelated researches includes performance analysis and transceiver design. In [3], the asymptotic capacity per unitvolume as well as DoF are analyzed for one and twodimensional (rectangular) LISs with matchedfilter receivers, showing the promising advantages of the system for data transmission. In [17], positioning using LIS is investigated and CramerRao bounds are computed under various scenarios. These two seminal works investigate the possibilities of LISs as a new technology for communications and positioning. In [2], authors consider circularshaped LISs communicating in the uplink, and calculate array gain, spatial resolution and capacity. This work first explore the effect of shape in implementing a LIS system. Performance of LIS system under practical channel environments with channel estimation error and hardware impairments is studied in [18, 19]. Asymptotic rates are analyzed and channel hardening effects discussed. This study shows benefits of LIS under practical channel, but does not propose new transceiver schemes. In [20], authors propose a channel model taking mutual coupling, superdirectivity and nearfield effects into account, and develop matched filter and weighted MMSE receivers for this model. In [21], authors ultilize electromagnetic information theory to formulate the continuousapertureMIMO pattern function design problem into an projection optimization problem by projecting continuous functions on finite orthogonal bases.
As can be seen from the above survey, most existing literatures study performance of Matched Filter (MF) or Minimum MeanSquare Error (MMSE) transceivers applied to LIS systems. These conventional techniques might as well be employed in LIS systems. However, they do not exploit all specific features of channels and transmitting signals of a LIS system. First, LIS is usually studied under LoS channels. Indeed, this is a reasonable channel model for LIS systems, as is argued in [3]. On one hand, LIS surfaces are deployed high up on top of buildings, resulting in dominant LoS path to Mobile Stations (MSs). On the other hand, employment of highfrequency spectrum such as millimeter wave, due to highcapacity demands and spectrum scarcity, also leads to LoS predominant propagations [22]. Under LoS model, channel coefficients are related to parameters such as DirectionofArrival (DoA) and the relative distance between transmitters and receivers. These parameters are implied in the signal structure, which might be exploited for transceiver design. Second, LIS can be manufactured in different shapes. Symmetric ones such as circular are preferable as spatial harmonics, e.g., Fourier series, can be defined on them. These harmonics provide a set of bases to synthesize signal waveforms. Third, LISs are continuous apertures, which is more mathematically tractable than discrete configurations (e.g., Massive MIMO). Inspired by these observations, we propose a new transceiver design method for multiuser LIS systems.
In this paper, we consider circular LISs communicating with multiple MSs in LoS channel. Exploiting the symmetric property of a circular LIS, we can synthesize Bessel and Fourier series defined on it to achieve interferencefree communications. Although [2] also studies circular LIS, we are different in terms of main content and methodology. The majority of [2] is on analysis of spectrum efficiencies and properties of LIS systems, while this paper focus on methods of designing multiuser transceiver and estimating DoA. In terms of methodology, we adopt new analysis techniques and propose new transceiver design method.
Our contributions in this paper can be summarized as follows.

We develop a new analysis of performance of LIS with MF transceiver. The analysis reveals a new observation of the interuser interference terms. The strength of this interference is not determined by physical interuser distance, as is conventionally believed, but by projected distance (see Sect. 3).

We propose a new interferencesuppression transceiver based on BesselandFourier–Series Synthesis (BFSS). The orthogonal Bessel series form a basis for signals along the radial direction and the Fourier series for signals along the rotational direction. By combining these two and tuning coefficients of the synthesized series, we obtain a beamformer that suppresses interference. The proposed beamformer is also numerically demonstrated to perform well in high SignaltoNoise Ratio (SNR) regime.

We propose a new DoA estimation method to obtain directional information of signals coming from a MS. Under LoS channels, the estimated DoAs are elevation and azimuth angles of a MS with respect to the LIS. We numerically demonstrate the promising performance of the proposed estimation method for carrier frequency in mmWave spectrum and higher.
The rest of the paper are organized as follows. System model and preliminary introduction of spatial harmonics are presented in Sect. 2. New interferencesuppressing beamformer is proposed in Sect. 3 along with an analysis of MF transceiver. DoA estimation method is presented in Sect. 4. Numerical results are shown and analyzed in Sect. 5. Conclusions are drawn in Sect. 6.
1.1 Notation and definition
In this paper, boldface upper and lowercase symbols represent matrices and vectors, respectively. The identity matrix is \({\textbf{I}}\) without specifying its size, as it is clear from context. The conjugate, transpose, and Hermitian transpose operators are denoted by \((\cdot )^*\), \((\cdot )^T\), and \((\cdot )^H\), respectively. Componentwise multiplication is denoted by \(\odot\). Expectation operation and absolute value are represented by \({\mathbb {E}} [\cdot ]\) and \(\cdot \), respectively. The norm of a vector \({\textbf{x}}\) is denoted by \(\Vert {\textbf{x}}\Vert\). \(\text {span}({\textbf{A}})\) means a linear space spanned by columns of matrix \({\textbf{A}}\).
2 System model
As shown in Fig. 1, we consider a base station equipped with a circular LIS of radius R, denoted by \({\mathcal {D}}\), situated on the \(xy\) plane, with its center at the origin o of a 3D coordinate system. The LIS communicates with K MSs situated in front of it. Denote the kth (\(k=1,2,\cdots ,K\)) MS’s position at \(M_k\) with spherical coordinate represented by \((\rho _k, \theta _{k},\gamma _{k})\), where \(\rho _k\) is the distance from MS k to the coordinate origin o, \(\theta _k\) is the elevation angle, measured with respect to the z axis and \(\gamma _k\) the azimuth angle, measured counterclockwise with respect to the xaxis. All MSs are equipped with single antenna.
2.1 Channel model
In practical deployment, LISs can be installed outside the wall of a top building, covering MSs roaming down on streets. The distance between a LIS and a MS is usually much larger than the size of a LIS, and the signal is dominated by Lineofsight (LoS) component. In this paper, a farfield LoS propagation channel is assumed with \(\rho _k\gg R\), \(k=1,2,\cdots ,K\), and \(\rho _{k}\) is also larger than the Rayleigh distance \(2(2R)^2/\lambda\) [2]. Consider an arbitrary infinitesimal element \(A\in {\mathcal {D}}\) (on \(xy\) plane) on the LIS with coordinate \((\rho , \pi /2, \varphi )\). The channel coefficient between A and \(M_k\) can be modeled as [2, 23, 24]
where \(\lambda\) is the carrier wavelength. Also note that pathloss factor in the above equation can be well approximated as \(\frac{1}{M_kA}\approx \frac{1}{\rho _k}\) for \(A\in {\mathcal {D}}\), since \(\rho _k\gg R\). The channel’s effect on phase is accounted for by \(h_k(\rho ,\varphi )\), which is given by
The approximation of \(M_kA\) by \(\rho _{k}\) in (1) is too crude to be applied to the phase term in the above equation. A secondorder approximation is needed here to take into account of phase variations on \({\mathcal {D}}\).
Equation (3) is obtained by omitting second and higher order terms due to \(\rho _k\gg \rho\) for \(\rho \leqslant R\). It can be seen that the first term in (3) is constant for all \(A\in {\mathcal {D}}\). The phase shift it induced can be compensated by each MS, thus this term can be omitted for succinct expressions. From here on, \(h_k(\rho ,\varphi )\) is expressed in polar coordinate as
2.2 Signal model
The LIS transmits signal \(s_k\) intended for MS k, precoded by filter \(f_k(\rho ,\varphi )\). The transmitted signals are zeromean and independent of each other, i.e.,
The received signal at the lth MS is a summation of signals from all of \({\mathcal {D}}\), which can be written as
for \(l=1,\cdots ,K\). The filters \(f_k(\rho ,\varphi )\) are normalized to have unit power. The noise term is a additive white Gaussian noise with mean zero and variance \(\sigma _l^2\). Note that this noise is an average of noises of elements on the LIS.
2.3 Orthogonal bessel series
This section briefly introduces orthogonal Bessell series and related expressions that will be used in later sections.
A circular LIS, like the one shown in Fig. 1, possesses rotational symmetry as well as radial symmetry. Signals defined on domains with circular symmetry can be decomposed into circular harmonics of integer frequencies, i.e., Fourier series. Along the radial direction, it is natural to decompose electromagnetic signals into orthogonal Bessel series as in [25, 26].
An orthogonal Bessel series is of the form [26]
where \(a_{vn }\) denotes the nth zero of vth order Bessel function \(J_v(\cdot )\) of the first kind, i.e., \(J_v(a_{vn })=0\), and \(c_n\) the series coefficients. In the decomposition, the Bessel functions with different zeros in their arguments are orthogonal, as the following orthogonality equation shows.
By this orthogonal property, the coefficients \(c_n\) can be obtained as
In the above equation, the factor \(\frac{2}{R^2[J_{v+1}(a_{vn})]^2}\) is to normalize the result, since
Utilizing Bessel series along radial direction and Fourier harmonics along rotational direction, we can synthesize beamforming patterns with desirable properties. Based on this observation, we will investigate into designing interferencesuppressing transceiver in next section.
3 BesselandFourier series synthesis (BFSS) transceiver
In this section, we first develop a new analysis of MF transceiver, and then propose the design of a interferencesuppressing transceiver for multiuser LIS systems.
3.1 MF transceiver
As is well known, MF maximizes individual receiver’s signal power, ignorant of interuser interference. For MS k, its MF transmit filter is given by
where \(\frac{1}{\sqrt{\pi }R}\) is a normalization factor to keep the filter power being 1 when integrating over the LIS. The received signal for MS l can then be obtained as
where \(\tau _l =\frac{\lambda R}{2\sqrt{\pi }\rho _l }\). The received signal consists of the desired signal for MS l, the interuser interference and the noise. The normalized effective channel coefficient \(\Phi _{lk}\) in the interuser interference term is given by
This coefficient signifies interuser interference level and its analytical expression is obtained in the following proposition.
Proposition 1
The normalized effective channel coefficients \(\Phi _{lk}\) are calculated as
where the projected distance \(\Delta (k, l)\) between MS k and l is calculated as
Proof
See Appendix A. \(\square\)
Remark 1
Note that the result obtain in Proposition 1 is the same as that in [2], only differing in a constant phase factor. This factor is precompensated as explained in Sect. 2.1 ( after (3)), resulting in succinct expressions without restricting the applicability of our result. Note that the derivation of the normalized effective channel coefficient \(\Phi _{lk}\), as shown in Appendix A, is different from [2]. This result leads to a physical interpretation of the variable \(\Delta (k, l)\) in its argument. We see from (17) that \(\Delta (k, l)\) is the distance of two points \((\sin \theta _{k}, \gamma _{k})\) and \((\sin \theta _{l}, \gamma _{l})\) on the \(xy\) plane. These two points are projections of MS k and l’s locations, i.e., \((\rho _k, \gamma _{k}, \theta _{k})\) and \((\rho _l, \gamma _{l}, \theta _{l})\), onto the \(xy\) plane and normalized by their, respectively, distance from origin o.
Note that when two MSs’ projected positions on \(xy\) plane are very close, i.e., \(\Delta (k, l)\rightarrow 0\), we have
where the derivative of Bessel function \(J_1(x)\) is utilized, i.e., \(J'_n(x) = \frac{1}{2}\left[ J_{n1}(x)  J_{n+1}(x)\right]\). This channel coefficient achieves maximum value when \(\Delta (k, l)=0\) (ref. to Fig. 2).
The normalized channel coefficient is a function of projected distance between different MSs. When two MSs’ projected positions overlapped, the received interuser interference reaches maximum power level, the same as the power of the desired signal component, as seen from (14). This is the case when two MSs locate on the same radial from the origin o (with the same elevation and azimuth angles). This means one MS is behind the other one as seen from o, which is very unlikely since we consider LoS path and MSs are usually roaming on a surface under the LIS.
Based on Eq. (14), we derive the sum rate for MF transmitter as shown in the following proposition.
Proposition 2
The sum rate for MF transceiver is given by
From the above analysis, we see that the sum rate is dependent on normalized effective channel coefficients \(\Phi _{lk}\) between interfering terminals. These coefficients are closely related to the projected distance \(\Delta (k, l)\) between two interfering MSs. The envelop of \(\Phi _{lk}\) decreases as \(\Delta (k, l)\) increases as seen from Fig. 2. Spatial proximity of MSs results in large interference, especially when two MSs situated on the same line going through the origin of the LIS.
If all MSs have the same power constraint, we have the following corollary.
Corollary 1
For equal power transmissions, i.e., \(P_k = P, \;\forall k\), the sum rate can be written as
where \(\varrho _l = \frac{PR^2\lambda ^2}{4\pi \rho _l^2\sigma _l^2}\). And
At high SNR, sum rate is dominantly determined by normalized effective channel coefficients \(\Phi _{lk}\), which are functions of the radius R of the LIS and carrier wavelength \(\lambda\), as seen from (16). An interesting observation is that, given other conditions the same, increasing R or decreasing \(\lambda\) decreases the normalized effective channel coefficients and thus interuser interferences. In consequence, the sum rate increases. This is the case because interuser distance affects interference through its normalization by carrier wavelength. Smaller wavelength results in large normalized interuser distance. Another observation is that the sum rate saturates at high SNR, as (21) shows. This is due to the MF transceiver not considering suppressing interuser interference.
3.2 BFSS transceiver
Interuser interference is a major cause of performance degradation in future wireless communication systems with massive connections sharing a same spectrum. To mitigate interuser interference, new transceivers have to be designed for LIS systems. In the following, we develop a interferencesuppressing transceiver by synthesizing Bessel and Fourier series defined on the LIS.
Denote by \(f^{(\text {BFSS})}_k(\rho ,\varphi )\) the BFSS filter for MS k. We consider interferencemitigating beamformer of the following form
where \(a_{vm}\) is defined in (9).
Remark 2
Note that in Eq. (22), only Fourier harmonics of nonnegative frequencies are included, i.e., \(e^{j(m1)\varphi }\), for \(m=1, 2 \cdots\). As a result, the synthesized signals do not represent all possible signals that can be formed when employing all of available harmonics. This will not restrict our design of transceiver that follows. As it turns out, it is enough for the purpose of a interference suppression transceiver.
The filter coefficients \(c_{km}\) in (22) are to be designed such that the following normalization conditions are satisfied
where \(\beta _m = \sqrt{\pi }RJ_{v+1}(a_{vm})\). Using \(f^{(\text {BFSS})}_k(\rho ,\varphi )\), the received signal at MS l can be written as
We summarize the resulting formula for signal \(r_{l}\) in the following proposition.
Proposition 3
The received signal \(r_{l}\) can be rewritten as
where
and
and \(\alpha _{k l}\) is defined as
Proof
See Appendix B. \(\square\)
To suppress interuser interference, we set the undetermined coefficients \(c_{km}\) such that the following system of linear equations are satisfied
for \(k=1,2,\cdots , K\). The above equations can be written in matrix form as
where
The solution \({\textbf{c}}_k\) is in the null space of \(\mathbf {\Pi }_k\),
Proposition 4
The BFSS filter coefficients \({\textbf{c}}_k\) are chosen according to (33). The maximum SNR achieved by MS k is given by
where \({\textbf{e}}_1=[1\;0\;\cdots \;0]^T\) and \({\textbf{b}}= [\beta _0\;\; \beta _1\;\;\cdots \;\;\beta _M]^T\) (ref. to (23)). The sum rate achieved is given by
Proof
The coefficients can be obtained by picking one vector from the above null space and normalized. Of these solutions, the one with maximum \(c_{k1}\) is desirable, since it’s a gain of the desirable signal component in MS k’ received signal. This can be achieved by projecting vector \({\textbf{e}}_1=[1\;0\;\cdots \;0]^T\) onto the null space, resulting in \(\left( {\textbf{I}}  \mathbf {\Pi }_k^\dag (\mathbf {\Pi }_k\mathbf {\Pi }_k^\dag )^{1}\mathbf {\Pi }_k\right) {\textbf{e}}_1\). Taking the normalization condition (23) into account, we have
where \({\textbf{b}}= [\beta _0\;\; \beta _1\;\;\cdots \;\;\beta _M]^T\). \(\square\)
From (35), it can be seen that sum rate grows linearly with SNR in high SNR regime. It is also seen that the signal strength of BFSS transceiver is discounted compare to that of using MF transceiver. The discount factor is determined by the angle subtended by vector \({\textbf{e}}_1=[1\;0\;\cdots \;0]^T\) and the null subspace in (33). The size of this angle in turn depends on the dimension of the null space. But the null space’s dimension is limited by the number of interference signals to be suppressed, given fixed M. The more interferers there are, the “smaller” the null space, resulting in small signal strength. The BFSS transceiver trades signal strength for zero interference.
4 DoA estimation
From analyses in the above sections, we know that DoA information of MS k, i.e., the elevation angle \(\theta _{k}\) and the azimuth angle \(\gamma _{k}\) are “encoded” in the series expansions of its channel coefficient \(h_k(\rho , \varphi )\). These information can be extracted by taking correlations of received signals. This extracted DoA only reveals in which direction a MS’s signal comes from, and under LoS, this is also the direction a MS locates with respect to the LIS. To pinpoint a MS’s position, a third coordinate, the distance \(\rho _{k}\) from the origin have to be determined. This can be achieved by utilizing offtheshelf algorithm to estimate this distance [27,28,29]. This problem is well studied in literature, and thus we assumed the distances between the LIS and MSs are known in the following.
During training phase, each MS transmits a pilot symbol \({\bar{s}}_k\) with power that is proportional to square of its distance to the LIS, which makes received signal power levels from different MSs more or less the same. Then, the received aggregate signal at the LIS is given by
where \(h_0(\rho ,\varphi ) = \frac{4\pi }{\lambda }e^{j\frac{2\pi }{\lambda }\rho \sin (\theta _0)\cos (\varphi \gamma _0) }\) and the aggregate noise \(n_0\) follows a white Gaussian distribution of mean zero and variance \(\sigma _0\). The integration is normalized by area of the LIS to get succinct expressions in the following. Since MSs reside on one side of the LIS, the scanning variables \(\theta _0\) and \(\gamma _0\) and directional angles of MSs are physically constrained by
Following similar derivations in Sect. 3, we have
where
From Eq. (39), we see that the received signal is a sum of functions of the same form \(\frac{J_1(x)}{x}\) with different arguments and noise. As Fig. 2 shows, this function oscillates with decaying envelop amplitude. At \(x=0\), it achieves maximum value of \(\lim _{x\rightarrow 0} \frac{J_1(x)}{x} =\frac{1}{2}\). Each transmitting MS “excites” a waveform \(\frac{J_1(x)}{x}\) in the received signal, and a “peak” appears at \((\theta _k,\gamma _k)\) if the argument \(\Delta (0, k)=0\) corresponding to \(\theta _0=\theta _k,\;\gamma _0=\gamma _k\). This is shown by the following lemma.
Lemma 1
Under the condition (38), the projected distance function \(\Delta (0, k) = 0\) if and only if \(\theta _{0} =\theta _{k}\) and \(\gamma _{0}=\gamma _{k}\), for some k.
Proof
See Appendix C. \(\square\)
This lemma establishes the correspondence between \(\Delta (0, k) = 0\) and the scanning variables \(\theta _0\) and \(\gamma _0\) equaling to the directional angles of some MS, i.e., \(\theta _0= \theta _l\) and \(\gamma _0=\gamma _l\) for some l. Hence, we can locate the “peak” signals to estimate DoAs of respective MSs. The accuracy of this DoA estimation process is affected by side lobes induced by interfering MSs.
Proposition 5
DoA of MSs, i.e., \(\theta _k\) and \(\gamma _k\), for \(k=1,2,\cdots , K\), can be estimated by locating the peaks of function \(r_0(\theta _0,\gamma _0)\), \(\theta _0\in \left( 0,\frac{\pi }{2}\right) ,\gamma _0\in [0,2\pi )\). The estimation resolution is given by
where \(a_{1m}\) is the mth zero of \(J_1(x)\), \(m=1,2,3,\cdots\).
The major source of interference of the proposed DoA estimation method comes from side lobes of \(\frac{J_1(x)}{x}\) induced by MSs. Fartheraway side lobes are smaller in amplitude and produces smaller disturbance. Selecting m in Proposition 5 is to determine how much disturbance will be considered acceptable. Larger m results in smaller disturbance and thus higher estimation accuracy. This leads to a tradeoff between accuracy and resolution in the proposed estimation method, given other parameters fixed. As shown in Proposition 5, the resolution is inverse proportional to m. Increasing m increases \(a_{1m}\) and decreases the resolution. At the same time, interuser projected distances are increased. The farther away from each other the MSs are, the smaller the mutual interference. As a result, estimation accuracy increases. Two other important parameters that also affect the resolution are R and \(\lambda\). Larger R and/or smaller \(\lambda\) results in higher resolution. However, size of the LIS R cannot get too large due to physical constraint. It is more promising to employ highfrequency carriers to obtain high estimation resolution, since mmWave and higher spectra are to be employed in future wireless communication systems [30, 31].
Algorithm 1 is proposed based on Proposition 5. The algorithm evaluates aggregate signal strength at different locations and obtains a location with maximum signal value as a DoA estimation for some MS. Then, the MS’s signal is regenerated using the estimated DoA and subtracted from \(r_0(\theta _0,\gamma _0)\). The major part of complexity comes from these evaluations. Thus, the complexity is proportional to \(\frac{2\pi K}{Res}\). In the above discussion, a onesymbol training scheme is assumed. This can be easily generalized to multiple training symbols. Estimation accuracy can be improved by exploiting diversity among different received symbols.
5 Numerical results
In this section, simulation results are presented to show the performance of using circular LIS for communicating and DoA estimation.
5.1 Capacity
In the following numerical evaluations of sum rates, the scenario is that a LIS is installed on a building surface, serving MSs down on streets with LoS channels. The MSs’ distance to the LIS is set to be larger than the Rayleigh distance \(2(2R)^2/\lambda\). Since the LIS covers terminals in front of the antenna surface, the elevation angle is within the range \(\theta _{k}\in (0,\pi /2)\) and azimuth angle is \(\gamma _{k}\in [0,2\pi )\), \(k=1,2,\cdots ,K\). These location coordinates are generated uniformly randomly in following simulations. Since each MS has a different received SNR, we define a reference SNR as \(\text {rSNR}=\frac{P \lambda ^2}{4{\bar{\rho }}\sigma ^2}\), where \({\bar{\rho }}\) is the distance from a reference point to the LIS. This reference distance is set to be in the middle of the range.
MF beamformer performances are shown in Figs. 3 and 4. Figure 3 shows sum rates achieved by MF beamformer under different sets of configuration parameters. The performance plotted in the figure is averaged over the random PointProcess (PP), marked as “PPaveraged”. From the figure, we see that the sum rate is increasing with SNR. Another observation is that larger LIS radius, i.e., larger aperture, result in better performance, corroborating analysis in Sect. 3.1. Simulation results of sum rate versus the number of MSs K are presented in Fig. 4. The sum rate is in general increasing with the number of MSs K. But the increasing rate becomes slower as K increases, because more simultaneous transmitting MSs result in an interferencelimited system, showing saturated performance.
Figure 5 illustrates the performance of BFSS beamformer under various configurations. The sum rate is larger with larger LIS radius R and/or more number of MSs. These are the same as observed in MF receiver. An interesting observation is that with BFSS, increasing the carrier wavelength \(\lambda\) slightly increases sum rate, in contrast to what happens in MF receiver. This is indeed the case because BFSS suppresses interuser interferences and its SNR is proportional to \(\lambda\). In MF, increasing \(\lambda\) reduces the projected distance between MSs, which results in stronger interference.
Another important question is to compare performance between MF and BFSS beamformers, which is shown in Fig. 6. Under the same setting, MF performs better than BFSS in low SNR regime, but is inferior in high SNR. There is a point where BFSS surpasses MF in performance. After this cross point, MF sum rate saturates, as it enters an interferencelimited regime. The performance gap between BFSS and MF becomes larger and larger. This shows the promising performance gain of BFSS transceiver for interferencelimited wireless networks. And as is envisioned, future cellular networks will be deeply interferencelimited with massive connections and aggressive spectrum reuse schemes [32, 33].
In the development of BFSS beamformer in Proposition 3, we truncate the Bessel and Fourier series and keep M terms. It is important to investigate the effects of such truncation on overall performance. To this aim, Fig. 7 presents sum rate versus the number of truncated terms M. We see that, as M increases, the sum rate first increases and then decreases, resulting in a maximum value for each parameter setting. This is due to two conditions imposing on M has competing effects on the resultant beamformer, specifically, on the first coefficient of the BFSS beamformer, as shown in Proposition 4. The two conditions on M are Eqs. (23) and (29), corresponding to the normalization and interferencesuppression requirements. On one hand, the normalization condition dictates that M coefficients of the BFSS beamformer have a fixed power. Increasing the number of coefficients reduces the power each one might get. On the other hand, the interferencesuppression condition demands that M is large enough to cancel all interferers’ signals. In the setting of Fig. 7, M must be larger than 8. Increasing M enlarges the null space of the interferers’ signals, as shown in Eq. (33), which might result in large coefficient values. But, the coefficient values are also constrained by the normalization condition. The overall result of these two effects is shown in Fig. 7.
5.2 DoA estimation
In LoS environment, DoA reveals a MS’s directional information of its position and also determines the phase of its channel coefficient to the LIS. In addition, estimating DoA under this scenario is part of the positioning procedure. The acquired DoA information can also be used in beamforming designs.
Figure 8 illustrates the function \(r_0(\theta _0,\gamma _0)\) with three MSs and LIS radius \(R=1.5\) and carrier wavelength \(\lambda =0.1\)m. The three “peaks” represents three signals present in the system. The coordinates of those “peaks” approximate the elevation and azimuth angles of corresponding signal’s DoA. The approximation error mainly comes from tail fluctuation of the function \(\frac{J_1(x)}{x}\) (refer to Eq. (39)), induced by presence of other MSs’ transmitting signals. When two MSs’ projected distance \(\Delta (k_1, k_2)\) is larger than, e.g., \(a_{12}\), the second zero of \(\frac{J_1(x)}{x}\), the estimation error can be greatly reduced if parameters R and \(\lambda\) are large enough.
To evaluate the proposed algorithm’s performance, we run simulations for different parameters and present the Mean Square Error (MSE) vs the number of MSs K in Fig. 9. We see that MSE increases with K, and increases only slightly when K is larger than ten. More MSs transmitting signals result in more interference in the estimation process, increasing the MSE. But the increasing rate becomes small. Comparing different curves in the figure shows that large value of radius R and small wavelength \(\lambda\) improve the proposed estimation algorithm’s performance. This is in accordance with that large aperture and highfrequency carriers improve resolution. Importantly, for \(R=2\) and \(\lambda =0.01\)m the MSE can be as low as about \(10^{9}\). This wavelength corresponds to frequency of 30GHz. This is about in the lower part of the mmWave spectrum, which will be employed in future wireless communication systems [30, 31]. Figure 10 presents how the MSE performance varies with the rSNR. It is seen that as signal strength increases and/or noise power reduces, MSE decreases slightly.
6 Conclusion
This paper proposes a new interferencesuppression transceiver for LoS multiuser LIS systems based on exploiting the signal structures resulting from geometric symmetries of the antenna surface. The signal can be decomposed into orthogonal communication modes using Bessel and Fourier series. The proposed beamformer tunes the coefficients of these communication modes to cancel interuser interferences. The proposed transceiver method BFSS’s performance is studied under varying parameters and compared with conventional MF receiver. The BFSS performs better than the MF in the high SNR regime. Another interesting result is that choosing more communication modes will not always increase the sum rate because power is thinly spread on these modes. Thus, we need to choose just enough modes but not too many. A DoA estimation method is also proposed for identifying directional information of MSs in LoS condition, following the same line of ideas. This method is demonstrated to have promising performance with highfrequency carriers, e.g., mmWave. The principle can be generalized to other symmetric geometries.
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Abbreviations
 LIS:

Large intelligent surface
 LoS:

Lineofsight
 DoA:

Directionofarrival
 MS:

Mobile station
 RIS:

Reconfigurable intelligent surface
 IRS:

Intelligent reflecting surface
 DoF:

Degreeoffreedom
 MMSE:

Minimum meansquare error
 BFSS:

BesselandFourierseries synthesis
 MF:

Matched filter
 MSE:

Meansquare error
 SNR:

Signaltonoise ratio
References
E. Björnson, L. Sanguinetti, H. Wymeersch, J. Hoydis, T.L. Marzetta, Massive mimo is a realitywhat is next?: five promising research directions for antenna arrays. Dig. Sig. Process. 94, 3–20 (2019)
J. Yuan, H.Q. Ngo, M. Matthaiou, Towards large intelligent surface (lis)based communications. IEEE Trans. Commun. 68(10), 6568–6582 (2020)
S. Hu, F. Rusek, O. Edfors, Beyond massive mimo: the potential of data transmission with large intelligent surfaces. IEEE Trans. Signal Process. 66(10), 274–2758 (2018)
D. Dardari, Communicating with large intelligent surfaces: fundamental limits and models. IEEE J. Sel. Areas Commun. 38(11), 2526–2537 (2020)
J. Hunt, T. Driscoll, A. Mrozack, G. Lipworth, M. Reynolds, D. Brady, D.R. Smith, Metamaterial apertures for computational imaging. Science 339(6117), 310–313 (2013)
L. Li, T. Jun Cui, W. Ji, S. Liu, J. Ding, X. Wan, Y. Bo Li, M. Jiang, C.W. Qiu, S. Zhang, Electromagnetic reprogrammable codingmetasurface holograms. Nat. Commun. 8(11), 197 (2017)
C. Liaskos, S. Nie, A. Tsioliaridou, A. Pitsillides, S. Ioannidis, I. Akyildiz, A new wireless communication paradigm through softwarecontrolled metasurfaces. IEEE Commun. Mag. 56(9), 162–169 (2018)
N. Shlezinger, O. Dicker, Y.C. Eldar, I. Yoo, M.F. Imani, D.R. Smith, Dynamic metasurface antennas for uplink massive mimo systems. IEEE Trans. Commun. 67(10), 6829–6843 (2019)
C.L. Holloway, E.F. Kuester, J.A. Gordon, J. O’Hara, J. Booth, D.R. Smith, An overview of the theory and applications of metasurfaces: the twodimensional equivalents of metamaterials. IEEE Antennas Propag. Mag. 54(2), 10–35 (2012)
A. Puglielli, N. Narevsky, P. Lu, T. Courtade, G. Wright, B. Nikolic, E. Alon, A scalable massive mimo array architecture based on common modules, in Proceeding of IEEE International Conference on Communication Workshop (ICCW), June 2015, pp. 1310–1315
Q. Wu, R. Zhang, Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming. IEEE Trans. Wireless Commun. 18(11), 5394–5409 (2019)
Z. Zhang, L. Dai, A joint precoding framework for wideband reconfigurable intelligent surfaceaided cellfree network. IEEE Trans. Signal Process. 69, 4085–4101 (2021)
H.U. Rehman, F. Bellili, A. Mezghani, E. Hossain, Modulating intelligent surfaces for multiuser mimo systems: Beamforming and modulation design, arXiv:2108.10505, Aug 2021. [Online]. Available: http://arxiv.org/abs/2108.10505
D.A.B. Miller, Communicating with waves between volumes: evaluating orthogonal spatial channels and limits on coupling strengths. Appl. Opt. 39(11), 1681–1699 (2000)
A. Poon, R. Brodersen, D. Tse, Degrees of freedom in multipleantenna channels: a signal space approach. IEEE Trans. Inf. Theory 51(2), 523–536 (2005)
N. Decarli, D. Dardari, Communication modes with large intelligent surfaces in the near field, arXiv:2108.10569, August 2021. [Online]. Available: http://arxiv.org/abs/2108.10569
S. Hu, F. Rusek, O. Edfors, Beyond massive mimo: the potential of positioning with large intelligent surfaces. IEEE Trans. Signal Process. 66(7), 1761–1774 (2018)
M. Jung, W. Saad, Y. Jang, G. Kong, S. Choi, Performance analysis of large intelligent surfaces (liss): asymptotic data rate and channel hardening effects. IEEE Trans. Wireless Commun. 19(3), 2052–2065 (2020)
M. Jung, W. Saad, G. Kong, Performance analysis of active large intelligent surfaces (liss): uplink spectral efficiency and pilot training. IEEE Trans. Commun. 69(5), 3379–3394 (2021)
R. Jess Williams, P. RamirezEspinosa, E. de Carvalho, T.L. Marzetta, Multiuser mimo with large intelligent surfaces: communication model and transmit design, in Proceeding on IEEE International conference on communications (ICC), June 2021, pp. 1–6
Z. Zhang, L. Dai, Continuousaperture mimo for electromagnetic information theory, arXiv:2111.08630, November 2021.
T.S. Rappaport, E. BenDor, J. N. Murdock, Y. Qiao, 38 ghz and 60 ghz angledependent propagation for cellular amp; peertopeer wireless communications, in Proceeding of IEEE International Conference on Communications (ICC), 2012, pp. 4568–4573
A. Sayeed, Deconstructing multiantenna fading channels. IEEE Trans. Signal Process. 50(10), 2563–2579 (2002)
R. Ertel, P. Cardieri, K. Sowerby, T. Rappaport, J. Reed, Overview of spatial channel models for antenna array communication systems. IEEE Pers. Commun. 5(1), 10–22 (1998)
L.C. Andrews, Special functions of mathematics for engineers, 2nd edn. (McGrawHill Inc, New York city, 1992)
G.B. Arfken, H.J. Weber, F. Harris, Mathematical methods for physicists: a comprehensive guide (Academic Press, Cambridge, 2005)
A. Zanella, Best practice in RSS measurements and ranging. IEEE Commun. Surv. Tutor. 18(4), 2662–2686 (2016)
K.I. Itoh, S. Watanabe, J.S. Shih, T. Sato, Performance of handoff algorithm based on distance and RSSI measurements. IEEE Trans. Veh. Technol. 51(6), 1460–1468 (2002)
G. Ghatak, On the placement of intelligent surfaces for RSSIbased ranging in mmwave networks. IEEE Commun. Lett. 25(6), 2043–2047 (2021)
I.A. Hemadeh, K. Satyanarayana, M. ElHajjar, L. Hanzo, Millimeterwave communications: physical channel models, design considerations, antenna constructions, and linkbudget. IEEE Commun. Surv. Tutor. 20(2), 870–913 (2018)
B. Ning, Z. Tian, Z. Chen, C. Han, J. Yuan, S. Li, Prospective beamforming technologies for ultramassive mimo in terahertz communications: a tutorial, [Online]. arXiv:2107.03032, (2021)
W. Nam, D. Bai, J. Lee, I. Kang, Advanced interference management for 5g cellular networks. IEEE Commun. Mag. 52(5), 52–60 (2014)
B. Soret, K.I. Pedersen, N.T.K. Jørgensen, V. FernándezLópez, Interference coordination for dense wireless networks. IEEE Commun. Mag. 53(1), 102–109 (2015)
G. Arfken, H. Weber, Mathematical methods for physicists, 7th edn. (Academic press, Cambridge, 2013)
L.J. Landau, Bessel functions: monotonicity and bounds. J. Lond. Math. Soc. 61(1), 197–215 (2000)
Acknowledgements
This work was supported by National Natural Science Foundation of China (61602317) and Natural Science Foundation of Guangdong Province (2016A030310066).
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Appendices
Appendix A: proof of proposition 1
The received signal for MS l can then be calculated as
The normalized effective channel coefficients \(\Phi _{lk}\), \(k\ne l\), can be calculated as
where step (41) follows from equation \(J_n(x) = \frac{j^{n}}{2\pi }\int _{0}^{2\pi } e^{j(x\cos \varphi +n\varphi )} d\varphi\) [34]. In above formulas, the normalized projected distance \(\Delta (k, l)\) between MS k and l is defined by
Appendix B: proof of proposition 3
Substituting (22) into (24), the received signal at MS l is given by
In (44), the summation term can be computed as follows. When \(k=l\), we have
Without loss of generality, let \(v=0\). Then,
When \(k\ne l\), we have
where \(\alpha _{k l}\) is defined as in (28).
Evoking upper bounds given in [35], the integration in the above equation can be bounded as
where \(\beta _0=0.674885...\) and \(\beta _1 = 0.7857\) are constants [35]. As order m increases, the integration approaches zero. When \(m> M \triangleq \max \{K, R^3\}\), the above integration can be omitted.
Collecting the above results together, equation (44) can be rewritten as
where
and
and
Appendix C: proof of lemma 1
If \(\theta _{0} =\theta _{k}\) and \(\gamma _{0}=\gamma _{k}\), then it is obvious \(\Delta (0, k) = 0\).
If \(\Delta (0, k) = 0\), then
But, we have inequality \(\frac{\sin \theta _{0} }{\sin \theta _{k}} + \frac{\sin \theta _{k}}{\sin \theta _{0}}\geqslant 2\), with equality being achieved when \(\sin \theta _{k} = \sin \theta _{0}\). This equality condition in turn leads to \(\gamma _{0}=\gamma _{k}\).
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Chen, J. Multiuser communications for lineofsight large intelligent surface systems. EURASIP J. Adv. Signal Process. 2023, 129 (2023). https://doi.org/10.1186/s13634023010841
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DOI: https://doi.org/10.1186/s13634023010841