WO2023037243A1 - Reconfigurable intelligent surface enabled leakage suppression and signal power maximization system and method - Google Patents

Abstract

A base station (202) includes a transceiver (205) configured to exchange information with a desired wireless communication device (208) and with an undesired wireless communication device (212), and a processor (203) connected to the transceiver (205) and configured to calculate phase shifts of a reflecting beamforming vector to be applied at a reconfigurable intelligent surface, RIS, (110). The phase shifts are calculated using a Kronecker decomposition of the reflecting beamforming vector so that a first signal (402) emitted by the transceiver (205) and reflected from the RIS (110) toward the desired wireless communication device (208) is enhanced and a second signal (404) emitted by the transceiver (205) and reflected from the RIS (110) toward the undesired wireless communication device (212) is cancelled.

Description

RECONFIGURABLE INTELLIGENT SURFACE ENABLED LEAKAGE SUPPRESSION AND SIGNAL POWER MAXIMIZATION SYSTEM AND METHOD CROSS-REFERENCE TO RELATED APPLICATIONS [0001] This application claims priority to U.S. Provisional Patent Application No.63/241,172, filed on September 7, 2021, entitled “RECONFIGURABLE INTELLIGENT SURFACE ENABLED INTERFERENCE NULLING AND SIGNAL POWER MAXIMIZATION IN HIGH-FREQUENCY BANDS,” the disclosure of which is incorporated herein by reference in its entirety. BACKGROUND TECHNICAL FIELD [0002] Embodiments of the subject matter disclosed herein generally relate to a system and method for using a reconfigurable intelligent surface (RIS) to enhance wireless transmission, and more particularly, to a system and algorithm for calculating phase shifts of each element of the RIS for suppressing channel leakage to an undesired user and enhancing the transmitted power to a desired user. DISCUSSION OF THE BACKGROUND [0003] During the past few years, the fifth-generation (5G) wireless networks were being deployed in various countries worldwide. Three key application scenarios, including enhanced mobile broadband (eMBB), ultra-reliable and low- latency communications (URLLC), and massive machine-type communications (mMTC) are expected to support 5G services. More stringent requirements such as global connectivity, extremely high reliability, and ultra-data rate are required as users become more and more network dependent. However, these requirements cannot be fully satisfied by the existing technologies, which motivates experts to set their sights on the future beyond 5G networking and develop disruptively new and innovative technologies. Therefore, various promising technologies are proposed and thoroughly investigated, such as ultra-dense networking, Terahertz communications, and massive multiple-input multiple-output (MIMO). Among these techniques, the reconfigurable intelligent surface (RIS) has emerged as the most prominent technique to enable smart and programmable configuration of wireless propagation environments [1, 2]. [0004] Specifically, RIS is a planar artificial surface composed of a large number (in the order of thousands or more) of passive, small (mm range), and low- cost reflecting elements, which are supported by newly developed advanced micro- electrical-mechanical systems (MEMS) and metamaterials [3]. Each reflecting element is able to reflect an incident signal (e.g., a telecommunication signal from a base station or from a smart device of a user) with controllable amplitude and phase shift independently, thereby enhancing the communication quality of desired propagation and mitigating undesired information transmission [4, 5]. In addition to providing performance gains, RIS is an appealing technology for its high flexibility and superior compatibility for practical implementation. It presents the advantage of not requiring any signal processing hardware components for RF processing, encoding/decoding, and retransmission, thus leading to orders-of-magnitude lower hardware and energy consumption compared to traditional active antenna arrays. [0005] On the other hand, RIS can be integrated into the existing networks and be transparent to users without any change in the hardware and software of existing devices by deploying on various structures including but not limited to facades of buildings, ceilings and walls, aerial platforms, as well as pedestrian’s clothes. Due to these promising advantages, the performance of several enabling technologies can be improved when integrated together with RIS. [0006] One of the key enabling technologies that can be greatly enhanced by RIS is millimeter-wave (mmWave) communications. Due to the enormous amount of spectrum resource available in mmWave frequencies, gigabit-per-second data rates can be supported for rate-demanding wireless applications. However, as a result of severe propagation environment absorption and high directivity, the frequencies of mmWave are subjected to significant path attenuation and are much more susceptible to blockages, which leads to coverage problems and undermines the potential of mmWave. However, these critical challenges to the mmWave technology can be efficiently addressed by adopting RIS in mmWave systems to create additional connections between devices [6]. The additional paths reflected by RIS can bypass obstacles between devices, thereby allowing for higher spectral efficiency and coverage area along with more efficient use of the consumed energy. [0007] Several authors investigated the performance of RIS-assisted mmWave systems from the perspective of the received signal power [7, 8] and the achievable rate [9] to [12]. Focusing on the maximization of the received signal power, the authors in [7] and [8] proposed a joint design of the transmit precoding vector at the base station (BS) and the phase shift coefficients at RIS. An analytical near-optimal solution of it was provided in [8] assuming low-resolution phase shifters at RIS. A second line of research works focused on analyzing the achievable rate of RIS aided systems. In [9], the authors analyzed the impact of low-resolution analog- digital converters and the number of RIS reflecting elements on the uplink achievable rate of RIS-aided mmWave systems, while in [11], the authors studied the achievable rate of a RIS assisted MIMO system operating over mmWave bands when the RIS is composed of multiple subsurfaces. Targeting downlink transmissions, the authors in [10] proposed a joint design of the active beamforming, the passive beamforming as well as the power allocation for optimized downlink rate of RIS-assisted mmWave non-orthogonal multiple access (NOMA) systems. The performance of the mmWave networks consisting of multiple devices and RISs was analyzed in [12] by adopting stochastic geometry. The obtained results showed the important role that RIS can play to improve area spectral efficiency provided that its density is higher than that of the BSs. [0008] Integrating RIS in current systems does not only improve the communication quality between transmitters and desired users, but also it helps suppress signal leakage at undesired users. Undesired users can be commonly found in heterogeneous networks and underlay cognitive radio networks, which are typically composed of different kinds of BSs serving different users using the same frequency resources. RISs have the potential to mitigate the inter-cell leakage and intra-cell leakage received at users in heterogeneous networks, and the interference received at the primary user in underlay cognitive radio networks. However, this kind of problem has not been investigated in mmWave communication systems. In this regard, most of the literature investigating leakage/interference suppression targets wiretap channels in which the aim is to protect transmissions from eavesdroppers which represent undesired users. In such systems, serious information leakage has always been a fundamental, yet challenging concern, especially in mmWave communications, characterized by their broadcast nature and extremely narrow beams. Some recent works have thus far investigated the use of the RIS technology to handle physical layer security issues. For example, the authors in [13], [14] proposed a joint design of the beamforming vector at the transmitter and the phases shift at the RIS to maximize the secrecy rate under perfect knowledge of the eavesdropper’s channel. Targeting the optimization of the achievable secrecy of rate of RIS-assisted mmWave systems, the authors in [13] proposed several solutions that go from artificial-noise-added schemes aiming to jam the eavesdropper, when the eavesdropper’s channel is unknown, to the use of distributed aided RIS systems with a joint design of the transmit beamforming and RIS control. As far as wiretap channels are concerned, the imperfect channel state information (CSI) has been recently accounted for in [15], wherein the worst achievable secrecy rate was optimized under a specified uncertainty model for the eavesdropper’s channel. All aforementioned designs be they for maximizing the received signal power, the sum rate or the achievable accuracy, require perfect or partial knowledge of the CSI, which poses new challenges in the design and the implementation of RIS-assisted mmWave wireless systems. [0009] The channel estimation of the RIS-assisted system is a much more difficult task than in traditional systems because of the passive reflecting elements at RIS lacking transmit/receive RF chains. Benefiting from the unique mmWave channel characteristics, various and efficient channel estimation schemes appeared in recent works [16]–[19]. Most of the proposed methods capitalized on the sparsity of the mmWave cascaded channels of RIS aided systems, which allows for the use of existing statistical signal processing methods. In this context, the work in [16] transformed the sparse channel estimation problem into a sparse recovery problem, while the work in [18] formulated the channel estimation problem as a fixed-rank constrained non-convex optimization problem. Other solutions include leveraging the topology of the RIS to estimate the channel with low complexity or resorting to deep learning-based techniques to estimate the channels between RIS and the transmitter/receiver. A close look at the proposed channel estimation schemes reveals that they present the common denominator of focusing on the total channel response, while overlooking their underlying structure. As far as mmWave or higher frequency channels are concerned, it is well known that they result from the combination of a finite number of paths, each specified by its amplitude and their angles of departure and arrivals in the azimuth and elevation planes, as discussed in [20]. In some circumstances, estimating these parameters rather than the total channel response is not only more efficient but also it can simplify the optimization of RIS phase shifts. [0010] Thus, there is a need for a new method and system to implement the method that are capable of suppressing leakages to unwanted users and enhancing the transmitted signal to desired users by reconfiguring a RIS.

BRIEF SUMMARY OF THE INVENTION [0011] According to an embodiment, there is a base station that includes a transceiver configured to exchange information with a desired wireless communication device and with an undesired wireless communication device, and a processor connected to the transceiver and configured to calculate phase shifts ^ of a reflecting beamforming vector ^ to be applied at a reconfigurable intelligent surface, RIS. The phase shifts ^ are calculated using a Kronecker decomposition of the reflecting beamforming vector ^ so that a first signal emitted by the transceiver and reflected from the RIS toward the desired wireless communication device is enhanced and a second signal emitted by the transceiver and reflected from the RIS toward the undesired wireless communication device is cancelled. [0012] According to another embodiment, there is a communication system that includes a base station having a processor and a transceiver configured to exchange information with a desired wireless communication device and with an undesired wireless communication device, and a reconfigurable intelligent surface, RIS, configured to reflect signals from the base station toward the desired wireless communication device. The processor is connected to the transceiver and is configured to calculate phase shifts ^ of a reflecting beamforming vector ^ at the RIS using a Kronecker decomposition so that a first signal emitted by the transceiver and reflected from the RIS toward the desired wireless communication device is enhanced and a second signal emitted by the transceiver and reflected from the RIS toward the undesired wireless communication device is cancelled. [0013] According to yet another embodiment, there is a method for reconfiguring a reconfigurable intelligent surface, RIS, that reflects communication signals from a base station toward plural communication devices. The method includes calculating frequencies associated with a communication channel between the RIS and an undesired wireless communication device, calculating frequencies and gains associated with a communication channel between the RIS and a desired wireless communication device, calculating, at the base station, phase shifts of a reflecting beamforming vector at the RIS using a Kronecker decomposition so that a first signal reflected from the RIS toward the desired wireless communication device is enhanced and a second signal reflected from the RIS toward the undesired wireless communication device is cancelled, and reconfiguring a surface of the RIS based on the phase shifts .

BRIEF DESCRIPTION OF THE DRAWINGS [0014] For a more complete understanding of the present invention, reference is now made to the following descriptions taken in conjunction with the accompanying drawings, in which: [0015] Figure 1 is a schematic diagram of a RIS-assisted secrecy communication system; [0016] Figure 2 is a schematic diagram of a RIS-assisted heterogeneous communication network; [0017] Figure 3 is a schematic diagram of a RIS-assisted underlay cognitive radio network; [0018] Figure 4 is a schematic diagram of a system model describing a wireless communication network; [0019] Figure 5 is a flow chart of a method for calculating phase shifts associated with element surfaces of the RIS; [0020] Figure 6 is a flow chart of a method for calculating frequencies and gains associated with communication channels between the base station, RIS, and the users; [0021] Figure 7 illustrates the received signal power versus the transmit power for various algorithms implemented in a wireless communication network; [0022] Figure 8 illustrates the secrecy capacity versus the transmit power for various numbers of elements of the NIS, implemented in the secrecy communication network; [0023] Figure 9 illustrates the secrecy capacity versus the transmit power for various number of paths, for the wireless communication network; [0024] Figure 10 illustrates the sum rate versus the transmit power for various numbers of elements of the NIS, for a wireless communication network; [0025] Figure 11 illustrates the sum rate versus the transmit power for various number of paths, for the wireless communication network; [0026] Figure 12 illustrates the normalized mean square error (NMSE) versus the transmit power for various number of elements of the NIS; [0027] Figure 13 illustrates the received signal power versus the transmit power for perfect CSI and estimated CSI; [0028] Figure 14 illustrates the secrecy capacity versus the transmit power with perfect CSI and estimated CSI; [0029] Figure 15 illustrates the sum rate versus the transmit power with perfect CSI and estimated CSI; [0030] Figure 16 is a flow chart of a method for reconfiguring the RIS so that communication to a desired user is enhanced and communication to an undesired user is suppressed; and [0031] Figure 17 is a schematic illustration of a computing system in which one or more of the above noted methods may be implemented. DETAILED DESCRIPTION OF THE INVENTION [0032] The following description of the embodiments refers to the accompanying drawings. The same reference numbers in different drawings identify the same or similar elements. The following detailed description does not limit the invention. Instead, the scope of the invention is defined by the appended claims. The following embodiments are discussed, for simplicity, with regard to a single undesired user and a single user that are served by a base station via a RIS in mmWave bands. However, the embodiments to be discussed next are not limited to a single undesired user, or a single desired user, or mmWave band, but may be applied to plural users and/or higher frequency bands. [0033] Reference throughout the specification to “one embodiment” or “an embodiment” means that a particular feature, structure or characteristic described in connection with an embodiment is included in at least one embodiment of the subject matter disclosed. Thus, the appearance of the phrases “in one embodiment” or “in an embodiment” in various places throughout the specification is not necessarily referring to the same embodiment. Further, the particular features, structures or characteristics may be combined in any suitable manner in one or more embodiments. [0034] According to an embodiment, a communication system is configured to adjust the phase shifts of the elements of a RIS so as to maximize the signal power at the desired user while cancelling out the leakage signal power received at the undesired user. This embodiment discussed an mmWave communication system where a BS serves a user via the assistance of a RIS in the presence of an undesired user in the transmission range of the BS. All devices in the system are assumed, for simplicity, to be equipped with a single antenna. The proposed system model are applicable to many practical networks such as secrecy communication systems, heterogeneous networks and underlay cognitive radio networks. [0035] A RIS-assisted point-to-point secrecy communication system 100 is shown in Figure 1, where an eavesdropper 102 tries to intercept the information sent by the transmitter 104 to a legitimate receiver 106 through an RIS 110. By reference to the system model, the legitimate receiver 106 represents the desired user, while the eavesdropper stands for the undesired user 102. The proposed RIS design in this case serves to maximize the signal power at the legitimate receiver 106 while cancelling out the received signal by the eavesdropper 102. This will offer full protection of the sent signal to the desired user from any form of interception by the eavesdropper. [0036] In another implementation, a heterogeneous network is shown in Figure 2, where a Macro BS 202 and the Micro BSs 204 share the same frequency, leading to intra-cell 206-1 and inter-cell (206-1 to 206-2) interference to desired communication devices 208 of desired users located in the coverage of Macro BS and the Micro BSs. The system model applies to the situation in which the Macro BS and the Micro BS serve only one desired user 208 at each time resource. A major concern in these systems is the management of the downlink interference that the Micro BS causes to the downlink of an undesired communication device 212 served by the Macro BS 202. In this context, the desired user is the Micro BS’s user 208, while the Macro BS’s user 212 represents the undesired user. Note that the term “user” is used interchangeable herein with the term “communication device.” A communication device may be any device that is capable to exchange in a wireless manner information with the BS, for example, a smart device, a camera, a sensor, etc. [0037] Cognitive radio networks is another ground where the invention to be discussed can be applied. Cognitive radio networks is a concept of modern wireless systems that has been introduced to ensure effective use of radio spectrum resources. In underlay cognitive radio networks, users are categorized into primary users that have the priority to access the whole spectrum and secondary users that could communicate with each other if the interference they cause to primary users is less than a certain threshold. This application is within the scope of the system model as shown in Figure 3, by considering the receiving secondary user 208 as the user of interest and the primary user 212 as the undesired user that should be protected by cancelling out the interfering signal caused by the communicating secondary users 208. All these networks illustrated in Figures 1 to 3 are good candidates for the new approach now discussed. [0038] In an embodiment, a physical, accurate, and widely applicable RIS channel model for mmWave frequencies is adopted [20], which is characterized by finite paths between the transmitter and the receiver, which are specified by different path amplitudes, azimuth angles of arrival (AoAs), and angles of departure (AoDs). The challenge for the existing systems for developing a suitable RIS reflecting scheme lies in designing the reflecting vector under the uni-modulus constraints on the vector elements. The term “uni-modulus” is defined, for a vector parameter, as a vector having the modulus of each component equal to one. As a result of the channel sparsity of the mmWave band, the propagation channels and reflecting vector at RIS can be equivalently represented by a Kronecker-product, which paves the way for the RIS design. Moreover, a novel channel estimation scheme to acquire the involved channel information within the RIS design of this embodiment is also proposed in a later embodiment. [0039] Thus, the embodiments discussed herein consider a RIS-assisted MIMO system 400 (see Figure 4) composed of a single-antenna BS 202, a single- antenna desired user 208, a single-antenna undesired user 212, and a RIS 110. An objective of the RIS parameters is to maximize the signal 402’s power at the desired user 208 under the constraint of signal 404’s leakage suppression at the undesired user 212. To achieve this objective, a systematic method for designing the reflecting beamforming at RIS 110 is proposed based on decomposing the reflecting vector as well as the desired propagation path 410-412 (please note that reference number 410 and 412 are intended to refer to multiple paths due to the reflections of various obstacles between the BS and RIS and/or the RIS and the user) between the BS 202 and the desired user 208, and the undesired propagation path 410-414 between the BS 202 and the undesired user 212, into corresponding Kronecker products of phase-shift factors under the assumption that the AoAs and AoDs of the channels between the BS and the desired user/undesired user are known (this information or corresponding frequencies associated with the angles are obtained from the CSI method discussed later). The novel idea of the RIS configuration in this embodiment is to allocate part of the factors of the reflecting vector at the RIS 110 for suppressing the undesired paths 414 and the remaining factors of the reflecting vector for enhancing the data transmission along the desired paths 412, between the BS 202 and the desired user 208. The mixed product property of the Kronecker products between the decomposed reflecting vector and the path-vectors lays the ground for the proposed novel framework. Since all the decomposed phase-shift vectors are composed of uni-modulus elements, the RIS design satisfies the uni-modulus constraints on each reflecting element assuming RIS is passive without any amplitude gains. Therefore, the desired reflecting beamforming at RIS is obtained to achieve the design objective discussed above. [0040] Before providing the mathematical foundations for selecting the phase shifts for beamforming at the RIS and an efficient channel estimation for providing the gains and frequencies used by the beamforming, a couple of mathematical conventions and notations are introduced. Thus, in the following embodiments, denotes a scalar, represents a vector, and stands for a matrix. denotes the th entry of the vector . Operation represents the expected operation, operation ⊙ stands for the Hadamard product, operation denotes the Kronecker product, and operation represents the left Kronecker product. Also symbol is the conjugate transpose, denotes the transpose, and conj (⋅) represents the conjugate. Moreover, matrix represents the all-one vector with length of , diag is a diagonal matrix whose diagonal entries are from the vector . ℂ stands for the complex domain and symbol “mod” denotes the modulo operator. Big O notation, represented as # = %(&), serves as a flexible abbreviation for |#| ≤ ^&, where is a generic constant. [0041] The system model used for the following embodiments is now discussed. Consider an mmWave RIS-assisted downlink system model as shown in Figure 4, where the RIS 110 is deployed to assist the data transmission from the single-antenna BS 202 to a single-antenna user 208 while the direct link is assumed to be blocked. The focus in this embodiment is on the downlink, but the proposed beamforming design can be straightforwardly extended to the uplink by exploiting the downlink-uplink duality. For the sake of simplicity and clarity of the calculations, it is assumed additionally that there is only one undesired user 212. Extensions of the methods discussed herein to the case of multiple users is possible. Suppose the RIS 110 is an uniform planar array (UPA) with N reflecting elements, which means . Due to the dense-device deployment and aggressive frequency reuse, the considered transmission causes signal leakage to other undesired users. Therefore, the received signal 402 at the desired user 208 and the received signal 404 at the undesired user 212 can be mathematically expressed as follows:

where denotes the channel from the BS 202 to the RIS 110, and denotes the channel from the RIS 110 to the desired user 208. Matrix , = denotes the phase-shift matrix of the RIS 110 with

phase shift =_> ∈ ?0,2AB. Moreover, the leakage channel 414 between the RIS 110 and the undesired user 212 is denoted as

The symbol transmitted by the BS is represented by ^ satisfying the expected power

where C is correspondingly the transmission power of the BS, while

are the received additive white Gaussian noise samples assumed to follow a standard normal distribution with mean zero and variance

[0042] The models for the channels 410, 412, and 441 are now discussed. The transmission between the terminals 208, 212 and RIS 110 is assumed to happen under a generic propagation environment with obstacles or reflecting/scattering elements (interacting objects, IOs). The multiple IOs for BS-RIS, RIS-desired user and RIS-undesired user are grouped under clusters

respectively, which can be defined as a group of sub-rays that share the same spatial and/or temporal characteristics. Each cluster between BS and RIS has

sub-rays for J = 1, … , F, while each cluster between RIS and desired user (undesired user) has sub-rays) for The channel vectors

are assumed to follow the model in [20], and are given by:

represents the complex Gaussian distributed path gain, is the rotationally

symmetric RIS element pattern in the direction of the (1,7) scatterer, and stands

for the attenuation. The term

} is the normalization factor for each channel, I_p characterizes

the existence of a line of sight (LoS) link, which takes values from the set [0,1], p_p = is the RIS element gain in the LoS direction,

denotes the attenuation of the LoS link, and

is the random phase term.

Here is the array response vectors of the RIS for the considered

azimuth arrival angles </>?• and elevation arrival angles which are given as follows

for uniformly distributed RIS elements with an inter-element spacing of d;

[0043] With these models for the various channels, it is possible now to formulate a phase-shift optimization problem at the elements of the RIS. A design objective in this embodiment is to maximize the signal power at the desired user under the constraint of leakage cancellation at the undesired user. The optimization problem can be expressed as:

where the first constraint arises from the leakage cancellation at the undesired user, and the second one aims to satisfy that no power is consumed at the RIS elements. Since , is a diagonal matrix can be rewritten as

is the reflect beamforming vector at RIS. The embodiment starts by showing that the expression diag(conj can be viewed as

a weighted sum of Kronecker products of two virtual array response vectors b. Indeed,

It appears from the expression of diag5conj that it is

formed by the sum of all possible paths 410-412 and 410-414. The total number of paths is

h. Note that for = 0 the total number of paths describes the paths between the BS and the desired user and for = 1 the total number of paths ³

describes the paths between the BS and the undesired user, i.e., there is a total number of paths for the channel diag(con and a total number of paths for the channel

diag(con

due to the obstacles being present between the main BS and the RIS and between the RIS and each user. For convenience, it is possible to change the index in the sum above to write diag5

where it is assumed that there is a one-to-one map that associates each

where ¡˜_x = diag5conj(¡_x)<¢, and U_> is the /-th element of the reflect beamforming vector ^. [0044] Mathematically, the optimization problem is non-convex and the conventional optimization approach based on linear algebra is inapplicable here due to the uni-modulus constraints. As a novel solution, according to this embodiment, the Kronecker decomposition technique is used to efficiently solve this optimization problem P2. The proposed strategy only requires partial CSI knowledge rather than the full knowledge of the channel vectors. A novel partial CSI approach is discussed later. Specifically, according to this embodiment, only the channel coefficients between the BS and the target user, and azimuth and elevation arrival

angle of all channels are necessary, but not the channel

coefficients between the BS and the undesired user. As later discussed, the channel coefficients corresponding to the gains and the azimuth and elevation arrival angles are substituted by corresponding frequencies calculated from the CSI approach. This proposed strategy is discussed later under the partial CSI knowledge, prior to handling the channel estimation problem. [0045] The reflecting beamforming coefficients are now selected. In this embodiment, a solution for the problem (¸2) is based on leveraging the Kronecker decomposition of (1) the reflecting beamforming vector and (2) the channel vectors. It is noted that a similar technique was adopted in [21] to design an analog spatial canceller for a wirelessly powered communications system and in [22] to design a hybrid beamforming for a mmWave MIMO system. In this embodiment, the proposed framework of Kronecker decomposition based reflecting beamforming design that solves (P2) comprises the following steps. First, the reflecting beamforming vector and the transmission path vectors between the BS and the desired/undesired user are decomposed into Kronecker products of uni-modulus phase shift vectors. Then, the phase shift vectors are designed to cancel the leakage signal at the undesired user and then to enhance the signal strength at the desired user. [0046] The Kronecker decomposition of the transmission path vector is now discussed. The proposed technique relies on the method of vector Kronecker decomposition, which is briefly reviewed here for the sake of completeness. Let ¹ be a ^ × 1 vector with uni-modulus elements and possessing the Vandermonde structure, i.e., a vector or a matrix with the terms of a geometric progression in each for some Θ fixed. Then the following Lemma holds.

[0047] Assume

being positive integers. Then vector ¹ can be decomposed as:

Kronecker product. [0048] This Lemma is applied to the azimuth and elevation arrival angle vectors both of which follow the Vandermonde structure. It is

assumed that and can be decomposed as

being positive integers. In case the

number of columns or rows of the RIS is a prime number, one can for instance select a subset of RIS elements, the number of which is the closest possible to the non- prime number of columns or rows. These elements may for instance be selected such that they satisfy a given selection criterion. [0049] Then, according to the Lemma, the vectors can

be decomposed as follow:

whereas are referred to as the Kronecker decomposed factors, and

are given by:

[0050] The reflecting beamforming vector ^ is selected so that it satisfies the Kronecker decomposition of phase shift vectors, i.e.,:

This selection of the reflecting beamforming decomposition is to

exploit the mixed product property of the Kronecker product and adjust the phase shift variables to achieve leakage cancellation at the undesired user and signal strength improvement at the desired user. [0051] It should be noted that different factorizations of ^_^ and ^_^ lead to different decompositions of the reflecting beamforming vectors. However, the prime factorization of ^_^ and ^_^ should be used in a preferred embodiment. The reason for this factorization selection is that the largest number of factors can be achieved when all and are prime numbers, which maximizes the degree of freedom (DoF) in the resultant reflecting beamforming vectors. [0052] Next, the process of leakage suppression for the undesired user is discussed. The leakage suppression constraints in can be rewritten using the mixed-product property of Kronecker product as

where can be re-expressed based on the Kronecker

decomposition expressions shown in (14) and (15) as,

[0053] An advantage that motivates the proposed decomposition is that a particular leakage path can be cancelled by setting any single factor in equation (20) to be zero, namely,

[0054] In other words, cancelling the leakage path from all paths of the undesired user requires factors out of the total factors (or DoF) of the reflecting beamforming vector, which is possible when is larger than . [0055] It is noted that the selection of any arbitrary hase shift Kronecker

decomposed factors that satisfy equations (21) and (22) allows for leakage cancellation regardless of their associated lengths. However, in order to leave the maximum number of degrees of freedom to enhance the signal energy for the desired user, it is desirable to choose these

factors among the ones that have the shortest length. [0056] In this regard, without loss of generality, assume the factors of ^_^ and are monotone increasing:

Then the factors having the shortest lengths are chosen for leakage suppression. Thus, it is possible that the first c_^factors of the É_^ sequence and the first c_^ factors of the sequence are chosen for leakage suppression, which means

[0057] To satisfy the leakage suppression, it is possible to show that for equations (21) or (22) to be satisfied, it suffices to select either ^^{(Ê^)} as

with being row vectors arbitrarily selected from the rows of the

Fourier matrix and the _^ Fourier matrix, respectively, except the first one

(an all-one vector). Indeed, by doing so, the constraints (21) and (22) can be re- expressed as:

where

n (26) represent the all-one vectors with length of and

respectively. It can be seen that the expressions (25) and (26) are both satisfied

since Ë Ë are orthogonal to the vector of all ones.

[0058] The leakage suppression is performed by using the first

factors in the phase shift vector that have the shortest lengths. The remaining

factors in the phase shift vector are allocated for signal enhancement at the desired user as now discussed. [0059] Next the signal enhancement for the desired user is discussed. In the following, the remaining factors in the phase shift vector are designed for maximizing the data-signal strength for the intended user. Instead of designing these factors one by one, it is more convenient to directly design their Kronecker product. The objective function can be expressed as:

where c_W^ and c_W^ denote the vector length o espectively. From here,

it is possible to obtain the final solution for

by plugging the obtained values for

[0064] Overall, the reflecting beamforming design procedure discussed above can be summarized as illustrated in Figure 5, to include a step 500 of decomposing the size of RIS ^_^ and ^_^ into a product of integers having É_^ and É_^ elements, respectively, a step 502 of performing a Kronecker decomposition of the +_^ path vector for the undesired user using equations (14) and (15) and performing another Kronecker decomposition of the reflecting beamforming vector ^ through equations (18), a step 504 of selecting the total

Kronecker decomposed factors of reflecting beamforming vector or leakage suppression for the undesired user using

equations (23) and (24), a step 506 of selecting the remaining

Kronecker decomposed factors of the reflecting beamforming vector for signal

enhancement for the desired user using equation (34), a step 508 of calculating the final reflecting beamforming vector by combining the ³ Kronecker decomposed

_^ factors for leakage suppression with the Kronecker decomposed

factors for signal enhancement using equation (35), and a step 510 in which the BS 202 transmits the final reflecting beamforming vector to the RIS 110 and the RIS

110 implements the corresponding phase shits of the final reflecting beamforming vector to each element. [0065] The input to the above noted calculations include (1) the gains of the channel to the desired user and the arrival and departure angles (or corresponding frequencies) and (2) the arrival and departure angles (or corresponding frequencies) for the undesired user. Although some of the above equations used the arrival and departure angles, the following embodiments describe how to obtain the corresponding frequencies and also what is the relationship between the angles and the frequencies. This means that in a practical implementation, either the angles or the frequencies are necessary for being able to calculate the final reflecting beamforming vector ^ discussed above. All this information is obtained based on a CSI procedure. [0066] A problem with the CSI procedure is the computation complexity and system extensions, which are discussed in the following. As reflected in the proposed reflecting beamforming selection process discussed above, different levels of CSI for undesired user leakage suppression and desired user signal enhancement are required. Specifically, only the azimuth and elevation AoAs and AODs of the undesired user interference path ¡˜_^ are needed for leakage suppression, but both elevation arrival and departure angles and gains of the desired user data path are required for signal enhancement. The channel estimation for acquiring the necessary CSI is now addressed. [0067] The computation complexity of the proposed Kronecker reflecting beamforming design is %(^), which is composed by vector-wise computation, Fourier based construction in (23) and (24), signal enhancement in (34), and the final combination of Kronecker decomposed factors in (35). It performs very efficiently, especially in a RIS-assisted system with several hundred or even thousands of reflecting elements. Although the method mitigates the leakage caused by the BS at the undesired user, the Kronecker decomposition based reflecting beamforming selection can also be used to suppress the leakage received by the desired user. Moreover, the Kronecker phase shift selection can be adaptively allocated to satisfy different communication purposes. It can be extended to the case of multiple undesired users by allocating as many factors as needed to cancel the leakage at each user. However, this will be at the cost of a reduction in the number of DoF to maximize the signal strength, which may require adding more reflecting elements at the RIS. It can also efficiently handle the situation in which the aim is to either only enhance the signal of interest over the paths that cause the least of leakage or to suppress the leakage over the paths that correspond to the strongest leakage path. Another scenario where this method can be of interest is in the physical layer security problems to assist in suppressing the received signal at the eavesdropper. [0068] In this embodiment, a channel estimation method is proposed that is similar to the beamforming phase shift selection, in the sense that it leverages the particular structure of the channel. In this embodiment, it is assumed that the channel is estimated at the BS, since the RIS, being passive, is not able to process signals. In other words, the BS 202 in Figure 4 includes processing capabilities 203, e.g., a processor and a memory, and RF transmission capabilities 205 (e.g., transceiver) for processing all the equations discussed herein and communicating to the RIS, via RF channels, the beamforming phase shift selection for each element of the RIS. [0069] Based on the time division duplex (TDD) protocol and assuming channel reciprocity, the downlink channel can be estimated using the received pilot transmissions from the users 208 and/or 212. The transmitted pilots from the users are assumed to be mutually orthogonal. The total channel estimation period consists of à sub-phases, during which each the users transmit the same copy of á_O pilot symbols, where

The pilot sequences of the desired user and the

C_S is the transmit power of users. It is assumed that the RIS applies the reflecting

wher Therefore, by multiplying the pseudo-inverse of

and applying conjugation at both sides of equation (38), the processed signal at the BS becomes:

obtained, the remaining task is to acquire estimates of the channel gains as well as the AoAs and AODs or equivalent quantities (or corresponding frequencies as previously discussed) associated with each user. For that, the structure of the channel model characterizing mmWave propagation signals, which models the channel as a positive combination of complex sinusoids with arbitrary phases is used. Such a model allows for the use of gridless compressive sensing approaches based on atomic norm minimization (ANM) to estimate the amplitudes of paths and their associated angles or frequencies:

[0070] The ANM framework was principally applied to perform spectrum estimation of a signal that can be written as a mixed linear combination of sinusoid components. For instance, in 2D-line spectrum estimation, the observed signal has typically the following decomposition:

where is the complex amplitude (which corresponds to the gain of that path) for path

denotes a vectorized manifold vector of length

represent the frequencies noted above. In equation (42), the parameters and the frequencies are

assumed to be unknown. To perform a gridless estimation of these parameters, the basic idea behind the 2D-ANM framework is to consider the observed vector as a linear combination of elements in the following vector-form atomic set defined as the continuous collection of all normalized 2-D complex sinusoids:

where fo Under certain frequency

separation, that will be discussed later in more detail, the true decomposition in equation (42) of the observed channel ¡̅ is the unique one that achieves the atomic norm. The atomic norm for a signal ^ is defined as:

and returns the sparsest decomposition in the sense of norm measure of ^ over

[0071] Prior to applying the 2D-ANM framework, it is necessary to show how the observed vector ¡˜_x can be written in the form shown in equation (42). Recall that the channel model ¡˜_x is given by

In view of equation (

can also be

given by:

_^R^ where the frequencies are related to the associated angles of

departure and arrival as follows:

and mod denotes the modulo operator. The operations in equations (46) and (47) ensures that the frequencies associated with angles of departure and

arrival are in the interval ?0,1B as required by the 2D-ANM framework used for 2D spectrum estimation. Also, the operations in equations (46) and (47) ensures that the method discussed in Figure 5 may use as input, instead of the angles of departure and arrival, the corresponding frequencies which are calculated

next. [0072] The 2D-ANM framework involves computing the atomic norm decomposition, which can be formally expressed as:

[0073] It is worth mentioning that the 2D-ANM does not require a priori knowledge of ³_x, as an estimate of it will be implicitly provided when solving equation (48). For a general atomic set, computing the atomic norm is not possible. However, as known in the art, for some atomic sets like the one considered in equation (42), computing the atomic norm can be reduced to solving a convex optimization problem. [0074] Handling the noisy observations are also discussed. In practice, because of noise, the observation vector does not satisfy exactly the decomposition shown in equation (42). One solution to handle noisy observations is to modify the optimization algorithm by constraining to be within an -ball centered at the observed vector

, which is equivalent to solving:

[0075] What makes the ANM framework useful in practical scenarios is its ability to convert the computation of the atomic norm to a semi-definite programming (SDP) problem. Indeed, as far as 2D spectrum estimation is concerned, it has been shown that under certain separability conditions, the atomic norm can be calculated using SDP as follows:

where is a two-level Toeplitz matrix with being its first

row. The formulation in equation (50) stands as a direct generalization of the SDP formulation for the 1D spectrum estimation, with possible extension for higher dimensional line spectrum estimation. However, it presents a drawback of leading to an optimization problem with variables, the complexity of which can be prohibitively high in practical scenarios. As a solution, an alternative formulation has been proposed in the art, which reformulates the atomic norm optimization problem into a problem that involves optimization variables, without entailing any loss

in optimality. The idea behind this proposed technique is to reshape the vector in a matrix

as follows:

[0076] In the noiseless case, the observed matrix can be viewed as a linear decomposition over elements in the following matrix-form atom set :

with which the following matrix-form atomic norm is associated:

[0077]

Then if the following sufficient frequency condition holds:

then the atomic norm decomposition (that is the one that achieved the atomic norm) is unique and coincides with (51). This suggests that the amplitude and

the frequencies can be retrieved by solving the following optimization problem:

[0078] In practice, only a noisy version available and

this is obtained by applying the same reshape transformation to

and is given by:

where denotes the Toeplitz matrix whose first column is equal to and denotes the Toeplitz matrix whose first column is equal to [0079] The frequency and complex amplitude estimation can now be performed. It remains to estimate the frequencies and as well as their associated complex amplitudes from equation (57) as they will server as the input for the method illustrated in Figure 5. For this task, the present embodiment follows the approach used in [23], which is detailed hereafter for the sake of completeness. The estimated frequencies are related to matrices via their associated Vandermonde decompositions:

the frequencies arising in the Vandermonde decomposition from the Toeplitz matrix. They principally include the matrix pencil method, and the Prony method [24]. In this embodiment, the Prony method is selected, the steps of which are now summarized. Details are provided for only the set of frequencies

_, while the frequencies

can be similarly estimated. Define

the coefficients

Vandermonde decomposition of that the observable quantities

satisfy the following system of equations [40]:

[0080] Consider the annihilating filter 9(æ) given by:

where to obtain the last equality, it was used the fact t

are the roots of 9(æ). Using the fact that

by varying 8 in (62), a

linear system of equations with solutio can be obtained. Solving this

linear system provides estimates o These estimates can be used to

compute the filter and derive the frequencies from roots of After

extracting the frequencies from matrices and the following step is to identify the and frequency pairs, that is,

For simplicity, it is assumed that the frequencies do not overlap over any dimensions, i.e., there exists no

. In this case, the reshaped channel matrix satisfy:

a diagonal matrix. It is noted that any column permutation n be associated with a different pairing of the frequencies

from both dimensions and vice versa. For any other pairing obtained by applying a different permutation on and matrix satisfies:

where Λ˜ is not diagonal but it has only one non-zero element at each row and column, the position of which is determined by the correct frequency pairing. In view of this, to pair the frequencies, one may start with any possible re-ordering of the frequencies and compute the matrix:

where operation denotes the pseudo-inverse. By deefining ¸_x and neglecting the effect of noise, the following is obtained:

with matrix having a unique non-zero element close to 1 at the positions determined by the correct pairing. [0083] The details of the channel estimation procedure discussed above are summarized in the flow chart in Figure 6. In step 600, the received pilot signal at BS from the user is processed and reshaped to obtain from equation (55). In step 602, the problem of determining the gains and frequencies of the channels is formulated as the SDP problem shown in equation (57), to obtain the estimated channel and two Toeplitz matrices. In step 604, extract all frequencies by first computing the annihilating filter from equation (62) and then calculating its roots. In step 606, the obtained frequencies are paired by identifying the elements of that are closest to 1 and calculate their associated amplitudes which correspond to the gains. In step 608, the estimated angles or frequencies and the amplitude are output. These output values are used as the input values for the method illustrated in Figure 5. [0084] The performance of the above proposed Kronecker based reflecting beamforming selection (see Figure 5) and the channel estimation approach (see Figure 6) are evaluated. The simulation parameters are set as follows unless

undesired user are set as (0,25,20), (40,60,10), (50,50,1), and (45,65,1) respectively. In the following, the novel Kronecker based reflecting beamforming selection is referred to as the ‘Kronecker algorithm’. First, the proposed scheme is studied under perfect CSI. The effect of channel estimation error is discussed later. [0085] In this embodiment, the received signal power at the desired user and the undesired user using the ‘Kronecker algorithm’ are compared with that using the random reflecting beamforming based on random phase shifts, when and As can be seen from Figure 7, the use of random reflecting beamforming leads to higher leakage 700 at the undesired user, which is at the same level as the signal power 702 at the desired user. On the other hand, the ‘Kronecker algorithm’ allows for better signal power 704 at the desired user and at the same time achieves almost total suppression of the leakage 706 at the undesired user. Note that the leakage 706 it is not exactly zero as it should be in theory due to numerical errors. [0086] As discussed above with regard to Figures 1-3, possible applications of the novel reflecting beamforming method illustrated in Figure 5 are to enhance the physical layer security of wiretap channels and to mitigate the interference in heterogeneous networks and underlay cognitive radio networks. In this section, numerical simulations are performed to assess the performance of the proposed design for both applications by measuring the secrecy capacity achieved in wiretap channels (Figure 1) and evaluating the total sum rate in heterogeneous networks/cognitive radio networks (Figures 2 and 3). [0087] With regard to the secrecy communication network, the inventors have compared the secrecy capacity of the proposed design with the following two schemes: The first one, referred to as ‘Precoder design’, describes the situation in which there is no RIS but the BS is equipped with the same number of transmit antennas as the elements of the RIS. Letting ¡_y and ¡_` be the channels between the BS and the desired user and the BS and the eavesdropper, respectively, it is assumed that in the ‘Precoder design’ scheme, the transmitted signal is precoded using the beamforming vector The second scheme, denoted as

the ‘SNR maximization’ relies on the same considered RIS assisted model, but the phases of the RIS are designed so as to maximize the received signal power at the legitimate user. [0088] The secrecy capacity results are shown Figure 8. As can be seen, the proposed ‘Kronecker algorithm’ achieves the best performance in terms of secrecy capacity when However, the ‘SNR maximization’ outperforms both the

‘Kronecker algorithm’ and the ‘Precoder design’ in the low SNR regime and when the RIS is equipped with antennas. The reason is that in the low SNR

regime, the received signal power at the eavesdropper is too low to significantly impact the secrecy capacity rate. Moreover, with a high number of antennas, the RIS can exploit the excess in spatial degrees of freedom to better cope with low SNR channels. All these explain the efficiency of the ‘SNR maximization’ compared to the other schemes in low SNR environments and when the RIS has a large antenna array. Moreover, it is noted that in all cases, the proposed ‘Kronecker algorithm’ always outperforms the ‘Precoder design’ in terms of secrecy capacity. As a benchmark, Figure 8 displays the secrecy rate of the random reflecting beamforming for ^_^ = 8. Note that it achieves the worst secrecy performance, which again confirms the importance of appropriate tuning of the RIS phase shifts in practical scenarios. [0089] Finally, Figure 9 illustrates the impact of the number of clusters in the channel with the eavesdropper on the secrecy capacity performance of the ‘Kronecker algorithm’ and the ‘SNR maximization’ schemes when F

16. It is expected that the eavesdropper’s channel quality gets improved with the increase in the number of clusters. As a result, the performances of both schemes deteriorate as the number of clusters with the eavesdropper increases. For the Kronecker algorithm, the degradation in secrecy is important and is caused by the fact that a higher number of Kronecker decomposed factors are required to mitigate the transmission leak over the eavesdropper cluster paths, leaving fewer degrees of freedom to enhance the signal power at the user of interest. On the other hand, for the ‘SNR maximization’ scheme, the degradation is less significant and is mainly caused by the eavesdropper possessing a better channel. [0090] For the Heterogeneous Network/Underlay Cognitive Radio Network, the proposed ‘Kronecker algorithm’ is used to mitigate the interference in a two-tier heterogeneous network composed of a macro cell overlaid with a micro cell. For this network the micro BS in the corresponding micro cell communicates with its user through an RIS, while the macro BS in the macro cell communicates with its user directly, without the assistance of the RIS. The considered system model can equally apply to underlay cognitive radio networks by treating the micro BS, micro user, macro BS, and macro user as secondary BS, secondary user, primary BS, and primary user, respectively. It is assumed that the RIS in the micro cell is used to enhance the received signal at the micro user, while mitigate the leakage received at the macro user at the same time. To achieve this, the cases in which the RIS uses the 'Kronecker algorithm' or employs the 'SNR maximization' that maximizes the received signal power at the target micro users, without considering the leakage caused at the macro user are considered. Additionally, a comparison is performed of both of these schemes with that of a standard MISO heterogeneous network in which the micro BS is equipped with ^ transmit antennas but are not assisted by RIS, which is also called ‘Precoder design’. For all the considered schemes, the macro BS is assumed to have the same number of antenna as the RIS. To mitigate the inter-tier leakage, the macro BS and the micro BS under ‘Precoder design’ use transmit precoding that projects the matched filter onto the subspace orthogonal to the channel of the user in the other tier. More specifically, consider that ¡_´F,´^, and ¡_´F,´^ are the channel vectors from the macro BS to macro user and micro user respectively, and ¡_´^,´F and ¡_´^,´^ are the channel vectors from the micro BS to macro user and micro user. The transmit precoding at the macro BS is thus C_´F =

schemes with that of a standard MISO heterogeneous network based on the spatial transmit precoding C_´F and C_´^. For all these simulations, the micro BS, RIS, micro user, macro BS, are macro user are set at (0,25,20), (40,60,10), (50,50,1), (60,90,20), (45,65,1), respectively. [0091] As shown in Figure 10, the ‘SNR maximization’ is the most efficient scheme in the low regime. However, as the transmit power increases, the power of the leakage signal increases, leading to a saturation of the performance at high transmit powers. The ‘Kronecker algorithm’ presents higher sum rate performances as the transmit power gets higher. Moreover, it is also interesting to note that in all cases, and while using only a single antenna, it outperforms the traditional MISO system that is based on large antenna arrays at both tiers. This finding is also in agreement with similar observations found in the physical layer security studied in the previous section. [0092] In Figure 11, the performances of all schemes are represented with respect to the transmit power for different number of clusters in the channel between the BS and the RIS. One can observe that the performance of the ‘SNR maximization’ is not significantly affected by the increase in the number of clusters in the first hop, when the SNR is not very high or not very low. In particular, the sum rate achieved by ‘SNR maximization’ increases as the number of clusters in the first hop increase in the low SNR regime, but performs inversely in the high SNR regime. The reason is the number of clusters in the first hop influences both the desired channel and leakage channel. The undesired transmission does not impair the sum rate apparently in the low SNR regime, but it becomes more important as the SNR increases. On the other hand, the performance of the ‘Kronecker algorithm’ decreases when the number of channel clusters of the first hop increases, presenting around 2 dB loss in the transmit power when the number of clusters increases by 1. The reason lies in the fact that as the number of clusters increases, more Kronecker decomposed factors need to be allocated to cancel the leakage over the undesired paths, leaving less degrees of freedom to enhance the power signal. [0093] An investigation of the results in Figures 8 to 11 reveals that a solution to enhance the robustness of the proposed scheme towards the number of paths of undesired users is to cancel out the leakage over only the strongest undesired paths, while ignoring the weak leakage paths. By doing so, it is possible to save more Kronecker decomposed factors for the desired signal power maximization. [0094] The performance of the proposed atomic norm minimization channel estimation scheme was also investigated. Figure 10 shows the normalized mean square error (NMSE) with respect to the transmit power for various number of reflecting elements of the RIS. It can be observed that the NMSE decreases as the number of reflecting elements increases. By doubling the number of reflecting elements, for instance, from = 16, the proposed channel estimation method of Figure 6 achieves about 10 dB power gain. This result shows that the channel estimation performance can be greatly improved by increasing the number of reflecting elements. However, increasing the number of elements requires a longer channel estimation period, which may result in lower spectral efficiency. A trade-off between channel estimation quality and time allocation resources should be thus solved in practice. [0095] The received signal powers at the desired user and undesired user, achieved with the ‘Kronecker algorithm’ under perfect and imperfect CSI, are compared in Figure 13. It is observed that the desired signal power with estimated CSI matches the one under perfect CSI very well. The cancellation of the leakage signal is not perfect, as a result of the imperfect CSI. However, it remains at a much smaller level than the desired signal power. [0096] As a further experiment, the inventors investigated the use of the proposed channel estimation method in both applied scenarios studied previously, namely, the secrecy capacity in wiretap channels and the leakage mitigation in heterogeneous networks/underlay cognitive radio networks. Figure 14 represents the secrecy capacity for the ‘Kronecker algorithm’ for perfect and estimated CSI, in combination with the capacity of the channel between the BS and the desired user. As can be seen, the channel capacity represented by the line ‘capacity’ coincides with the ‘secrecy capacity’ referred to as ‘secrecy capacity’ with perfect CSI. This is so because the channel between the BS and the eavesdropper is almost zero. Moving forward to the heterogeneous network/underlay cognitive radio network, the sum rate calculated based on the received SINR at each user or at the received SNR (without interference) is represented. As can be seen from the figure, quantities with perfect CSI coincide with each other, revealing efficient cancellation of the leakage at the undesired user. Also, as can be seen from Figures 14 and 15, there is only a very small performance loss with the estimated CSI, which proves the high efficiency of the proposed channel estimation scheme. The performance loss is caused by the imperfect leakage cancellation, which is also reflected by the performance gap between ‘capacity’/’SNR’ and ‘secrecy capacity’/’SINR’ with estimated CSI. This finding is also in agreement with the results shown in Figure 13. [0097] Thus, the embodiments discussed herein address the channel estimation (Figure 6) and the phase shift selection (Figure 5) of an RIS assisted system 400 in which a single antenna BS communicates through an RIS 110 with a user of interest 208 in the presence of an undesired user 212. The methods discussed herein select/calculate/design the phase shift of the RIS so as to ensure simultaneous maximum signal power at the user of interest and leakage cancellation at the undesired user as well. A low-complexity yet efficient selection method based on decomposing the channel paths and the phase shift beamforming vector into Kronecker product of factors has been proposed for this goal. By leveraging the properties of the Kronecker products, the method allocates each factor of the phase shift vector to either cancel out the leakage over the channel paths of the undesired user or coherently combine the useful signals over the channel paths of the user of interest. Additionally, a channel estimation approach is introduced which estimates the channel parameters that are required for the phase shift selection. The proposed approach relies on using an efficient 2-dimensional line spectrum optimization technique based on atomic norm minimization. The shift design vector technique along with the channel estimation approach define a cohesive phase shift framework that can be applied to many interesting applications, including the improvement of physical layer security or the leakage mitigation in heterogeneous networks and underlay cognitive radio networks. All of these applications have been considered in the simulation embodiments discussed above, in which the performance of the proposed approach has been assessed and compared with other competing techniques. [0098] A method for reconfiguring a reconfigurable intelligent surface, RIS, 110 that reflects communication signals from a base station 202 toward plural communication devices 208, 212 is now discussed with regard to Figure 16. The method includes a step 1600 of calculating frequencies associated with a communication channel between the RIS and an undesired wireless communication device, a step 1602 of calculating frequencies and gains associated with a communication channel between the RIS and a desired wireless communication device, a step 1604 of calculating, at the base station, phase shifts ^ of a reflecting beamforming vector ^ at the RIS using a Kronecker decomposition so that a first signal reflected from the RIS toward the desired wireless communication device is enhanced and a second signal reflected from the RIS toward the undesired wireless communication device is cancelled, and a step 1606 of reconfiguring a surface of the RIS based on the phase shifts [0099] The method may further include a step of decomposing a number of elements of the RIS into a product of integers, wherein the product has a number of integers, a step of performing a Kronecker decomposition of a leakage path vector between the base station and the undesired wireless communication device, based on the frequencies associated with the communication channel between the RIS and the undesired wireless communication device, wherein represents a channel vector between the RIS and the undesired wireless communication device, and ¢ represents a channel vector between the base station and the RIS, a step of performing a Kronecker decomposition of an enhanced path vector between the base station and the desired wireless communication device, based on the frequencies and gains associated with a channel vector between the RIS and the desired wireless communication device, and a step of performing a Kronecker decomposition of the reflecting beamforming vector ^, wherein the Kronecker decomposition has a number of factors equal to the number of integers. [0100] The method may also include a step of selecting and calculating Kronecker decomposed factors of the reflecting beamforming vector ^ to cancel the second signal, wherein is smaller than the number of integers, and a step of selecting and calculating a remaining number of decomposed factors of the reflecting beamforming vector ^ to enhance the first signal. [0101] In one embodiment, the method may also include a step of calculating a final reflecting beamforming vector ^ by combining the ³_^ Kronecker decomposed factors and the remaining decomposed factors, and a step of transmitting the final reflecting beamforming vector ^ to the RIS for implementing the phase shifts [0102] The above discussed method may be implemented in the base station 202. For example, the base station 202 may include, as shown in Figure 5, a transceiver 205 configured to exchange information with a desired wireless communication device 208 and with an undesired wireless communication device 212, and a processor 203 connected to the transceiver 205 and configured to calculate phase shifts of a reflecting beamforming vector to be applied at a reconfigurable intelligent surface, RIS, 110. The phase shifts are calculated using a Kronecker decomposition of the reflecting beamforming vector so that a first signal 402 emitted by the transceiver 205 and reflected from the RIS 110 toward the desired wireless communication device 208 is enhanced and a second signal 404 emitted by the transceiver 205 and reflected from the RIS 110 toward the undesired wireless communication device 212 is cancelled. [0103] The processor may be further configured to calculate (1) frequencies associated with a communication channel between the RIS and the undesired wireless communication device, and (2) frequencies and gains associated with a communication channel between the RIS and the desired wireless communication device. Additionally, the processor may be further configured to decompose a number of elements of the RIS into a product of integers, wherein the product has a number of integers. [0104] In one application, the processor is further configured to perform a Kronecker decomposition of a leakage channel vector between the base station and the undesired wireless communication device, based on the (1) frequencies associated with the communication channel between the RIS and the undesired wireless communication device, where represents a channel vector between the RIS and the undesired wireless communication device, and represents a channel vector between the base station and the RIS, perform a Kronecker decomposition of an enhanced channel vector between the base station and the desired wireless communication device, based the (2) frequencies and gains associated with a channel vector between the RIS and the desired wireless communication device, and perform a Kronecker decomposition of the reflecting beamforming vector , wherein the Kronecker decomposition has a number of factors equal to the number of integers. [0105] The processor may be further configured to select and calculate Kronecker decomposed factors of the reflecting beamforming vector ^ to cancel the second signal, wherein is smaller than the number of integers, select and calculate a remaining number of decomposed factors of the reflecting beamforming vector to enhance the desired signal, calculate a final reflecting beamforming vector by combining the Kronecker decomposed factors and the remaining decomposed factors, and transmit the final reflecting beamforming vector to the RIS for implementing the phase shifts . [0106] In yet another application, the processor is further configured to reshape a training signal vector received from the desired and undesired wireless communication devices using an atomic norm minimization process, to obtain a matrix where k is 0 for the desired wireless communication device and 1 for the undesired wireless communication device; solve a semi-definite programming problem of the matrix to obtain a channel matrix and two Toeplitz matrices and extract all frequencies by calculating an annihilating filter 9(æ) and roots of the annihilating filter 9(æ) based on the Toeplitz matrices and and pair the extracted frequencies by identifying elements of a matrix which reshapes the channel matrix so that elements that are closest to 1 represent the frequencies and the frequencies and calculating the associated gains [0107] The above-discussed procedures and methods may be implemented in a computing device 203 as illustrated in Figure 17. Hardware, firmware, software or a combination thereof may be used to perform the various steps and operations described herein. Computing device 203 of Figure 17 is an exemplary computing structure that may be used in connection with such a system. [0108] Exemplary computing device 203 suitable for performing the activities described in the exemplary embodiments may include a server 1701. Such a server 1701 may include a central processor (CPU) 1702 coupled to a random access memory (RAM) 1704 and to a read-only memory (ROM) 1706. ROM 1706 may also be other types of storage media to store programs, such as programmable ROM (PROM), erasable PROM (EPROM), etc. Processor 1702 may communicate with other internal and external components through input/output (I/O) circuitry 1708 and bussing 1710 to provide control signals and the like. Processor 1702 carries out a variety of functions as are known in the art, as dictated by software and/or firmware instructions. [0109] Server 1701 may also include one or more data storage devices, including hard drives 1712, CD-ROM drives 1714 and other hardware capable of reading and/or storing information, such as DVD, etc. In one embodiment, software for carrying out the above-discussed steps may be stored and distributed on a CD- ROM or DVD 1716, a USB storage device 1718 or other form of media capable of portably storing information. These storage media may be inserted into, and read by, devices such as CD-ROM drive 1714, disk drive 1712, etc. Server 1701 may be coupled to a display 1720, which may be any type of known display or presentation screen, such as LCD, plasma display, cathode ray tube (CRT), etc. A user input interface 1722 is provided, including one or more user interface mechanisms such as a mouse, keyboard, microphone, touchpad, touch screen, voice-recognition system, etc. [0110] Server 1701 may be coupled to other devices, such as transceiver 205, etc. The server may be part of a larger network configuration as in a global area network (GAN) such as the Internet 1728, which allows ultimate connection to various landline and/or mobile computing devices. [0111] The disclosed embodiments provide a communication device that uses RIS for enhancing the transmitted signal to an intended user and suppressing the leakage to an unintended user. It should be understood that this description is not intended to limit the invention. On the contrary, the embodiments are intended to cover alternatives, modifications and equivalents, which are included in the spirit and scope of the invention as defined by the appended claims. Further, in the detailed description of the embodiments, numerous specific details are set forth in order to provide a comprehensive understanding of the claimed invention. However, one skilled in the art would understand that various embodiments may be practiced without such specific details. [0112] Although the features and elements of the present embodiments are described in the embodiments in particular combinations, each feature or element can be used alone without the other features and elements of the embodiments or in various combinations with or without other features and elements disclosed herein. [0113] This written description uses examples of the subject matter disclosed to enable any person skilled in the art to practice the same, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the subject matter is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims. References The entire content of all the publications listed herein is incorporated by reference in this patent application. [1] M. Di Renzo, M. Debbah, D.-T. Phan-Huy, A. Zappone, M.-S. Alouini, C. Yuen, V. Sciancalepore, G. C. Alexandropoulos, J. Hoydis, H. Gacanin et al., “Smart radio environments empowered by reconfigurable AI meta-surfaces: An idea whose time has come,” EURASIP Journal on Wireless Communications and Networking, vol. 2019, no.1, pp.1–20, 2019. [2] E. Basar, M. Di Renzo, J. De Rosny, M. Debbah, M.-S. Alouini, and R. 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Claims

WHAT IS CLAIMED IS:

1 . A base station (202) comprising: a transceiver (205) configured to exchange information with a desired wireless communication device (208) and with an undesired wireless communication device

(212); and a processor (203) connected to the transceiver (205) and configured to calculate phase shifts of a reflecting beamforming vector to be applied at a

reconfigurable intelligent surface, RIS, (110), wherein the phase shifts

are calculated using a Kronecker decomposition of the reflecting beamforming vector so that a first signal (402) emitted by the

transceiver (205) and reflected from the RIS (110) toward the desired wireless communication device (208) is enhanced and a second signal (404) emitted by the transceiver (205) and reflected from the RIS (110) toward the undesired wireless communication device (212) is cancelled.

2. The base station of Claim 1 , wherein the processor is further configured to: calculate (1) frequencies associated with a communication

channel between the RIS and the undesired wireless communication device, and (2) frequencies

and gains

associated with a communication channel between the RIS and the desired wireless communication device.

3. The base station of Claim 2, wherein the processor is further configured to: decompose a numbe

_y of elements of the RIS into a product of integers, wherein the product has a numbe of integers.

4. The base station of Claim 3, wherein the processor is further configured to: perform a Kronecker decomposition of a leakage channel vector

between the base station and the undesired wireless communication device, based on the (1 ) frequencies associated with the communication channel

between the RIS and the undesired wireless communication device, wherein h" represents a channel vector between the RIS and the undesired wireless communication device, and g represents a channel vector between the base station and the RIS; perform a Kronecker decomposition of an enhanced channel vector

between the base station and the desired wireless communication device, based the

(2) frequencies m and gains associated with a channel vector between

the RIS and the desired wireless communication device; and perform a Kronecker decomposition of the reflecting beamforming vector

wherein the Kronecker decomposition has a number of factors equal to the number of integers.

5. The base station of Claim 4, wherein the processor is further configured to: select and calculate Kronecker decomposed factors of the reflecting beamforming vector to cancel the second signal, wherein M is smaller than the

number D_x + D_x of integers.

6. The base station of Claim 5, wherein the processor is further configured to: select and calculate a remaining number D_x + D_x - of decomposed factors

of the reflecting beamforming vector to enhance the first signal.

7. The base station of Claim 6, wherein the processor is further configured to: calculate a final reflecting beamforming vector by combining the

Kronecker decomposed factors and the remaining D_x + D_x - decomposed factors; and transmit the final reflecting beamforming vector to the RIS for implementing

the phase shifts

8. The base station of Claim 2, wherein the processor is further configured to: reshape a training signal vector received from the desired and undesired

wireless communication devices using an atomic norm minimization process, to obtain a matrix

where k is 0 for the desired wireless communication device and

1 for the undesired wireless communication device; solve a semi-definite programming problem of the matrix to obtain a

channel matrix and two Toeplitz matrices

extract all frequencies by calculating an annihilating filter H(z) and roots of the annihilating filter H(z) based on the Toeplitz matrices T and

pair the extracted frequencies by identifying elements of a matrix which

reshapes the channel matrix so that elements that are closest to 1 represent the

frequencies and the frequencies and calculating the

associated gain

9. A communication system (400) comprising: a base station (202) having a processor (203) and a transceiver (205) configured to exchange information with a desired wireless communication device

(208) and with an undesired wireless communication device (212); and a reconfigurable intelligent surface, RIS, (110) configured to reflect signals from the base station (202) toward the desired wireless communication device (208), wherein the processor (203) is connected to the transceiver (205) and is configured to calculate phase shifts xp of a reflecting beamforming vector at the

RIS (1 10) using a Kronecker decomposition so that a first signal (402) emitted by the transceiver (205) and reflected from the RIS (110) toward the desired wireless communication device (208) is enhanced and a second signal (404) emitted by the transceiver (110) and reflected from the RIS (110) toward the undesired wireless communication device (212) is cancelled.

10. The communication system of Claim 9, wherein the processor is further configured to: calculate (1 ) frequencie associated with a communication

channel between the RIS and the undesired wireless communication device, and (2) frequencies and gains associated with a communication channel

between the RIS and the desired wireless communication device.

1 1. The communication system of Claim 10, wherein the processor is further configured to: decompose a number N_x x N_y of elements of the RIS into a product of integers, wherein the product has a number D_x + D_x of integers.

12. The communication system of Claim 11 , wherein the processor is further configured to: perform a Kronecker decomposition of a leakage channel vector

between the base station and the undesired wireless communication device, based on the (1 ) frequencies m associated with the communication channel

between the RIS and the undesired wireless communication device, wherein

represents a channel vector between the RIS and the undesired wireless communication device, and g represents a channel vector between the base station and the RIS; perform a Kronecker decomposition of an enhanced channel vector

between the base station and the desired wireless communication device, based the

(2) frequencies associated with a channel vector between

the RIS and the desired wireless communication device; and perform a Kronecker decomposition of the reflecting beamforming vector </>, wherein the Kronecker decomposition has a number of factors equal to the number

D_x + D_x of integers.

13. The communication system of Claim 12, wherein the processor is further configured to: select and calculate Kronecker decomposed factors of the reflecting

beamforming vector to cancel the second signal, wherein is smaller than the

number D_x + D_x of integers.

14. The communication system of Claim 13, wherein the processor is further configured to: select and calculate a remaining number D_x + D_x - of decomposed factors

of the reflecting beamforming vector o enhance the first signal.

15. The communication system of Claim 14, wherein the processor is further configured to: calculate a final reflecting beamforming vector by combining the

Kronecker decomposed factors and the remaining D_x + D_x - decomposed

factors; and transmit the final reflecting beamforming vector o the RIS for implementing

the phase shifts

16. The communication system of Claim 10, wherein the processor is further configured to: reshape a training signal vecto received from to the desired and

undesired wireless communication devices using an atomic norm minimization process, to obtain a matrix where k is 0 for the desired wireless communication

device and 1 for the undesired wireless communication device; solve a semi-definite programming problem of the matrix o obtain a

channel matrix and two Toeplitz matrices

extract all frequencies by calculating an annihilating filter H(z) and roots of the annihilating filter H(z) based on the Toeplitz matrices and

pair the extracted frequencies by identifying elements of a matrix which

reshapes the channel matrix

so that elements that are closest to 1 represent the frequencies and the frequencie and calculating the

associated gains

17. A method for reconfiguring a reconfigurable intelligent surface, RIS, (110) that reflects communication signals from a base station (202) toward plural communication devices (208, 212), the method comprising: calculating (1600) frequencie associated with a communication

channel between the RIS (110) and an undesired wireless communication device

(212); calculating (1602) frequencies associated with a

communication channel between the RIS (110) and a desired wireless communication device (208); calculating (1604), at the base station (202), phase shifts ip of a reflecting beamforming vector at the RIS (110) using a Kronecker decomposition so that a

first signal (402) reflected from the RIS (110) toward the desired wireless communication device (208) is enhanced and a second signal (404) reflected from the RIS (1 10) toward the undesired wireless communication device (212) is cancelled; and reconfiguring (1606) a surface of the RIS (1 10) based on the phase shifts xp.

18. The method of Claim 17, further comprising: decomposing a number N_x x N_y of elements of the RIS into a product of integers, wherein the product has a number D_x + D_x of integers; performing a Kronecker decomposition of a leakage path vector

between the base station and the undesired wireless communication device, based on the frequencies ssociated with the communication channel between

the RIS and the undesired wireless communication device, wherein epresents a channel vector between the RIS and the undesired

wireless communication device, and g represents a channel vector between the base station and the RIS; performing a Kronecker decomposition of an enhanced path vector

between the base station and the desired wireless communication device, based the frequencie associated with a channel vector between the

RIS and the desired wireless communication device; and performing a Kronecker decomposition of the reflecting beamforming vector

</>, wherein the Kronecker decomposition has a number of factors equal to the number D_x + D_x of integers.

19. The method of Claim 18, further comprising: selecting and calculating Kronecker decomposed factors of the reflecting

beamforming vector to cancel the second signal, wherein is smaller than the

number D_x + D_x of integers; and selecting and calculating a remaining number D_x + D_x - M_T of decomposed factors of the reflecting beamforming vector o enhance the first signal.

20. The method of Claim 19, further comprising: calculating a final reflecting beamforming vector by combining the

Kronecker decomposed factors and the remaining D_x + D_x - decomposed

factors; and transmitting the final reflecting beamforming vector to the RIS for

implementing the phase shifts