Devices, methods and computer programs for energy-efficient beamforming in a holographic radio transmitter

Abstract

Devices, methods and computer programs for energy-efficient beamforming in a holographic radio transmitter are disclosed. The invention allows a structure for an antenna array of antenna elements configured to act as a holographic surface and for an associated transmit circuit in a radio transmitter device that enables energy-efficient beamforming to be performed by the radio transmitter device.

Description

[0001] DEVICES, METHODS AND COMPUTER PROGRAMS FOR ENERGY-EFFICIENT BEAMFORMING IN A HOLOGRAPHIC RADIO TRANSMITTER TECHNICAL FIELD The present disclosure relates to the field of communications, and, more particularly, to energy-efficientbeamforming in a holographic radio transmitter, and related devices, methods and computer programs. BACKGROUND Aholographic radio has emerged as a promising technology for future wireless communications. It is ex-pected to be able to arbitrarily shape electromagnetic waves generated or sensed by antennas and provide an opportunity toincrease the capacity limit of a wireless channel.However, mutual interactions (such as a mutual coupling effect) and power loss may be generated amongantenna elements in a holographic radio system. For example, such power loss may be caused by an impedance mismatchbetween an antenna array and a transmit circuit, or by a load impedance of each antenna element.Accordingly, at least in some situations, there may be a need for solutions that are capable of decreasing thepower loss and taking the mutual interactions into consideration.SUMMARY This summary is provided to introduce a selection of concepts in a simplified form that are further described below in the detailed description. This summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used to limit the scope of the claimed subject matter. It is an object of the invention to allow energy-efficient beamforming in a holographic radio transmitter. The foregoing and other objects are achieved by the features of the independent claims. Further implementation forms are apparent from the dependent claims, the description and the figures. According to a first aspect, a radio transmitter device is provided. The radio transmitter device comprises anantenna array of antenna elements. The antenna array is configured to act as a holographic surface. The antenna array has an^ × ^ mutual impedance matrix of mutual impedances between antenna elements, where each antenna element has a corre-sponding load impedance, in which N denotes a number of the antenna elements. The radio transmitter device further comprisesa transmit circuit configured to drive the holographic surface. The transmit circuit comprises at least one pair of transmissionlines per each antenna element. Each pair of transmission lines is connectable to a respective antenna element and having arespective characteristic impedance. The transmit circuit further comprises a signal generator per each antenna element. Eachsignal generator is configured to drive a respective antenna element via a respective pair of transmission lines. The radio trans-mitter device is configured to obtain a first beamforming vector. The radio transmitter device is further configured to obtain atransfer matrix based on at least one of the mutual impedance matrix, load impedances of the antenna elements, or characteristicimpedances of the pairs of transmission lines. The radio transmitter device is further configured to obtain a second beamformingvector based on a product of the obtained transfer matrix and the obtained first beamforming vector. The radio transmitterdevice is further configured to use the transmit circuit to generate a signal stream beamformed with the obtained second beam-forming vector. The radio transmitter device is further configured to transmit the generated signal stream via the antenna array.The present disclosure allows a structure for an antenna array of antenna elements configured to act as a holographic surfaceand for an associated transmit circuit in a radio transmitter device that enables energy-efficient beamforming to be performed by the radio transmitter device. In an implementation form of the first aspect, the first beamforming vector is based on a wireless channelvector of a wireless channel between the holographic surface and a receive antenna of a receiver. The wireless channel vectoris constructed based at least on positions of the antenna elements within the holographic surface, and further based on spatialdirection -based radiation power patterns of the antenna elements within the holographic surface. Alternatively, the wirelesschannel vector is estimated from pilot signals received from the receiver. Alternatively, the wireless channel vector is fed backfrom the receiver. This implementation form allows for the transmitter to capture the information of the wireless channel vectorsin alternative ways. In a case when the mutual coupling effect among the N antenna elements is weak and ignorable, and theobtained first beamforming vector is directly used to transmit a signal stream, the obtained first beamforming vector is able tofocus the signal power towards the intended receiver and so leads to a maximum received signal power.In an implementation form of the first aspect, the transfer matrix is based on a coupling transfer matrix anda mutual coupling matrix, such that ^ = ^^^^^^^, where ^ denotes the transfer matrix, ^ denotes the coupling transfer matrix,^ denotes the mutual coupling matrix, ^ denotes a scaling factor, and the coupling transfer matrix is based at least on the mutualimpedance matrix. Alternatively, the transfer matrix is based on the mutual impedance matrix, the load impedances, and thecharacteristic impedances, such that: ^^^ / ^^^( )^ / ^ ^=^(^ + ^ + ^ )^Re(^)^^Re^^^^^ , 2^ ^ ^where ^ = {^^,^} denotes the mutual impedance matrix, ^^^,^|^ = 1,2, ⋯ ^^ denotes the load impedances,^^^,^|^ = 1,2, ⋯ ^^ denotes the characteristic impedances, n denotes a first antenna element index, m denotes a second antennaelement index, ^ denotes a scaling factor, ^^ denotes an ^ × ^ diagonal load impedance matrix with ^^,^ as its ^-th diagonalentry, ^^ denotes an ^ × ^ diagonal characteristic impedance matrix with ^^,^ as its ^-th diagonal entry, ^(^)denotes an^ × ^ diagonal matrix with ^^,^ as its ^-th diagonal entry, and Re(⋅) returns a real part of its argument. This implementationform allows maximizing a directivity of the holographic surface.In an implementation form of the first aspect, the mutual coupling matrix is based on the mutual impedancematrix satisfying: where ^ = matrix is based onthe spatial direction -based radiation power pattern and a position of each antenna element satisfying:1 ^^,^= 4^^ ^^^ / ^(^)^^^ / ^(^)^^^^^ ^^^(^^^^^)^^, ∀^, ^ = 1,2, ⋯ ^,^∈^^^^where ^^(^) denotes the radiation power pattern, ^ denotes the spatial direction, and ^^ denotes the posi-tion of an antenna element ^. This implementation form allows for an accurate characterization of the mutual coupling effectamong the N antenna elements when the energy efficiency issue of the radio transmitter device is not considered. In a casewhen the radio transmitter device has perfect impedance matching between the transmit circuit and the antenna array to avoidreflection loss, and has zero real part of a load impedance per antenna element to avoid heat loss, this implementation formallows for an accurate characterization of the mutual coupling effect of the whole radio transmitter device.In an form of the first the transfer matrix is based on a coupling transfer matrix, suchthat ^ = ^^^, where ^ denotes the H denotes a conjugate transposeoperator, ^ denotes a scaling factor, at on the mutual impedance matrix. Alter-natively, the transfer matrix is based on the mutual impedance matrix, the load impedances, and the characteristic impedances, such that: ^ / ^ , == 1,2, ⋯ ^^ denotes the load impedances,^^^,^|^ = 1,2, ⋯ ^^ denotes the characteristic impedances, n denotes a first antenna element index, m denotes a second antennaelement index, ^ denotes a scaling factor, ^^denotes an ^ × ^ diagonal load impedance matrix with ^^,^as its ^-th diagonalentry, ^^ denotes an ^ × ^ diagonal characteristic impedance matrix with ^^,^ as its ^-th diagonal entry, ^(^)denotes an^ × ^ diagonal matrix with ^^,^ as its ^-th diagonal entry, and Re(⋅) returns a real part of its argument. This implementationform allows maximizing a realized beamforming gain of the holographic surface.In an implementation form of the first aspect, the coupling transfer matrix is based on the mutual impedancematrix, the characteristic impedances, and the load impedances, such that:^ / ^ ^= 2 ^Re^^(^)^^ (^ + ^^ + ^^)^^^ ^ / ^ ^ . Alternatively, the coupling transfer matrix is based on the mutual impedance matrix and a current generatormatrix associated with the radio transmitter device, such that:^ ^= ^^^^^(^)^^ ^ ^ ^, where ^ denotes allows for an accurate characteri-zation of the mutual coupling effect among the N antenna elements including the energy efficiency issue of the radio transmitterdevice. In an implementation form of the first aspect, the current generator matrix is based on the mutual impedancematrix, the characteristic impedances, and the load impedances, such that: ^= 2√2(^ + ^^ + ^^)^^^ ^ / ^ ^ . This implementation form allows for an accurate characterization of the relationship between the beam-formed signal stream generated by the transmit circuit and the resultant current vector that are fed to the N antenna elements ofthe antenna array. In an implementation form of the first aspect, the radio transmitter device further comprises a group of con-figurable external load impedances. Each configurable external load impedance is serially connected to a respective antennaelement in order form allows for online configuration of the values of the load on needs by configuring the value of a respective configu-rable external load impedance. That is, the value of the load impedance of each antenna element may be the sum of the valuesof its external load impedance and internal load impedance, thus serially connecting an external load impedance to an antennaallows value by configuring its external load impedance value. form of the first aspect, the characteristic impedances are configured based on at leastone of transmission lines of the pairs of transmission lines. This implementation form allowsoptimizing the characteristic impedances with different geometries and materials.In an implementation form of the first aspect, an amount of the pairs of transmission lines exceeds an amountof the antenna elements, and a connection between an antenna element and its respective signal generator is switchable amongthe pairs of transmission lines with different characteristic impedance values. This implementation form allows for an onlineconfiguration of the characteristic impedance for the connection between an antenna element and its respective signal generatordepending on needs by switching the connection among the transmission lines.According to a second aspect, a method for a radio transmitter device is provided. The radio transmitterdevice comprises an antenna array of antenna elements. The antenna array is configured to act as a holographic surface. Theantenna array has an ^ × ^ mutual impedance matrix of mutual impedances between antenna elements, where each antennaelement has a corresponding load impedance, in which N denotes a number of the antenna elements. The radio transmitterdevice further comprises a transmit circuit configured to drive the holographic surface. The transmit circuit comprises at leastone pair of transmission lines per each antenna element. Each pair of transmission lines is connectable to a respective antennaelement and having a respective characteristic impedance. The transmit circuit further comprises a signal generator per eachantenna element. Each signal generator is configured to drive a respective antenna element via a respective pair of transmissionlines. The method comprises obtaining, by the radio transmitter device, a first beamforming vector. The method further com-prises obtaining, by the radio transmitter device, a transfer matrix based on at least one of the mutual impedance matrix, loadimpedances of the antenna elements, or characteristic impedances of the pairs of transmission lines. The method further com-prises obtaining, by the radio transmitter device, a second beamforming vector based on a product of the obtained transfermatrix and the obtained first beamforming vector. The method further comprises using, by the radio transmitter device, thetransmit circuit to generate a signal stream beamformed with the obtained second beamforming vector. The method furthercomprises transmitting, by the radio transmitter device, the generated signal stream via the antenna array. The present disclosureallows a structure for an antenna array of antenna elements configured to act as a holographic surface and for an associatedtransmit circuit in a radio transmitter device that enables energy-efficient beamforming to be performed by the radio transmitterdevice. In an implementation form of the second aspect, the first beamforming vector is based on a wireless channelvector of a wireless channel between the holographic surface and a receive antenna of a receiver. The wireless channel vectoris constructed based at least on positions of the antenna elements within the holographic surface, and further based on spatialdirection -based radiation power patterns of the antenna elements within the holographic surface. Alternatively, the wirelesschannel vector is estimated from pilot signals received from the receiver. Alternatively, the wireless channel vector is fed backfrom the receiver. This implementation form allows for the transmitter to capture the information of the wireless channel vectorsin alternative ways. In a case when the mutual coupling effect among the N antenna elements is weak and ignorable, and theobtained first beamforming vector is directly used to transmit a signal stream, the obtained first beamforming vector is able tofocus the signal power towards the intended receiver and so leads to a maximum received signal power. In an implementation form of the second aspect, the transfer matrix is based on a coupling transfer matrixand a mutual coupling matrix, such that ^ = ^^^^^^^, where ^ denotes the transfer matrix, ^ denotes the coupling transfermatrix, ^ denotes the mutual coupling matrix, ^ denotes a scaling factor, and the coupling transfer matrix is based at least onthe mutual impedance matrix. Alternatively, the transfer matrix is based on the mutual impedance matrix, the load impedances,and the characteristic impedances, such that: ^=^ ^^^ / ^(^ + ^^^ ^+ ^ )^Re(^)^^Re^^(^ / ^ ^)^^ , 2^ ^where ^ = {^^,^} denotes the mutual impedance matrix, ^^^,^|^ = 1,2, ⋯ ^^ denotes the load impedances,^^^,^|^ = 1,2, ⋯ ^^ denotes the characteristic impedances, n denotes a first antenna element index, m denotes a second antennaelement index, ^ denotes a scaling factor, ^^ denotes an ^ × ^ diagonal load impedance matrix with ^^,^ as its ^-th diagonalentry, ^^ denotes an ^ × ^ diagonal characteristic impedance matrix with ^^,^ as its ^-th diagonal entry, ^(^)denotes an^ × ^ diagonal matrix with ^^,^ as its ^-th diagonal entry, and Re(⋅) returns a real part of its argument. This implementationform allows maximizing a directivity of the holographic surface.In an implementation form of the second aspect, the mutual coupling matrix is based on the mutual impedance matrix satisfying: ^^ / ^ ^^ / ^^(^)^^ Re ^ ^Re^^(^)^ ^= ^Re ( ) ^^, where ^ = ^^^,^^ denotes the mutual coupling matrix. Alternatively, the mutual coupling matrix is based onthe spatial direction -based radiation power pattern and a position of each antenna element satisfying:1 ^^,^=^ ^^^ / ^(^)^^^ / ^(^)^^^^^ ^^^(^^^^^)^^ , ∀^, ^ = 1,2, ⋯ ^, 4^^∈^^^^where ^^(^) denotes the radiation power pattern, ^ denotes the spatial direction, and ^^ denotes the posi-tion of an antenna element ^. This implementation form allows for an accurate characterization of the mutual coupling effectamong the N antenna elements when the energy efficiency issue of the radio transmitter device is not considered. In a casewhen the radio transmitter device has perfect impedance matching between the transmit circuit and the antenna array to avoidreflection loss, and has zero real part of a load impedance per antenna element to avoid heat loss, this implementation form allows for an accurate characterization of the mutual coupling effect of the whole radio transmitter device.In an implementation form of the second aspect, the transfer matrix is based on a coupling transfer matrix,such that ^ = ^^^, where ^ denotes the transfer matrix, ^ denotes the coupling transfer matrix, H denotes a conjugate trans-pose operator, ^ denotes a scaling factor, and the coupling transfer matrix is based at least on the mutual impedance matrix.Alternatively, the transfer matrix is based on the mutual impedance matrix, the load impedances, and the characteristic imped-ances, such that: ^ / ^ ^^ ^^^where ^ = ^^ denotes the load impedances,^^^,^|^ = 1,2, ⋯ ^^ denotes the characteristic impedances, n denotes a first antenna element index, m denotes a second antennaelement index, ^ denotes a scaling factor, ^^ denotes an ^ × ^ diagonal load impedance matrix with ^^,^ as its ^-th diagonalentry, ^^ denotes an ^ × ^ diagonal characteristic impedance matrix with ^^,^ as its ^-th diagonal entry, ^(^)denotes an^ × ^ diagonal matrix with ^^,^ as its ^-th diagonal entry, and Re(⋅) returns a real part of its argument. This implementationform allows maximizing a realized beamforming gain of the holographic surface.In an implementation form of the second aspect, the coupling transfer matrix is based on the mutual imped-ance matrix, the characteristic impedances, and the load impedances, such that:^ = 2 ^Re^^(^)^ / ^ ^^ (^ + ^^ + ^^)^^^ ^ / ^ ^ . Alternatively, the coupling transfer matrix is based on the mutual impedance matrix and a current generatormatrix associated with the radio^^(^)^ / ^ ^= ^ ^^ ^ ^ ^ ^, where ^ denotes the current generator matrix. This implementation form allows for an accurate characteri-zation of the mutual coupling effect the N antenna elements the issue of the radio transmitterdevice. In an implementation aspect, current generator on the mutual imped-ance matrix, the characteristic impedances, and the load impedances, such that: ^= 2√2(^ + ^^ + ^^)^^^ ^ / ^ ^ . This implementation form allows for an accurate characterization of the relationship between the beam-formed signal stream generated by the transmit circuit and the resultant current vector that are fed to the N antenna elements ofthe antenna array. In an implementation form of the second aspect, the radio transmitter device further comprises a group ofconfigurable external load impedances. Each configurable external load impedance is serially connected to a respective antennaelement in order to configure a respective load impedance. This implementation form allows for online configuration of the values of the load impedance per each antenna element depending on needs by configuring the value of a respective configu-rable external load impedance. That is, the value of the load impedance of each antenna element may be the sum of the valuesof its external load impedance and internal load impedance, thus serially connecting an external load impedance to an antennaallows configuring its load impedance value by configuring its external load impedance value.In an implementation form of the second aspect, the characteristic impedances are configured based on atleast one of geometries or materials of This implementation form allowsoptimizing the characteristic In an implementation form of the second aspect, an amount of the pairs of transmission lines exceeds anamount of the antenna elements, and a connection between an antenna element and its respective signal generator is switchableamong the pairs of transmission lines with different characteristic impedance values. This implementation form allows for anonline configuration of the characteristic impedance for the connection between an antenna element and its respective signalgenerator depending on needs by switching the connection among the transmission lines. According to a third aspect, a computer program product is provided. The computer program product com-prises program code configured to perform a method according to the second aspect, when the program code is executed on aradio transmitter device. The present disclosure allows a structure for an antenna array of antenna elements configured to actas a holographic surface and for an associated transmit circuit in a radio transmitter device that enables energy-efficientbeamforming to The apparatus comprises means for carrying out amethod according to the second aspect. The present disclosure allows a structure for an antenna array of antenna elementsconfigured to act as a holographic surface and for an associated transmit circuit in a radio transmitter device that enables energy-efficient the radio transmitter device. a network node device is provided. The network node device comprises the radiotransmitter to aspect. The present disclosure allows a structure for an antenna array of antenna ele-ments configured to act as a holographic surface and for an associated transmit circuit in a radio transmitter device that enablesenergy-efficient beamforming to be performed by the radio transmitter device.Many of the attendant features will be more readily appreciated as they become better understood by refer-ence to the following detailed description considered in connection with the accompanying drawings.DESCRIPTION OF THE DRAWINGS In the following, example embodiments are described in more detail with reference to the attached figures and drawings, in which: Fig.1 is a diagram illustrating an example system, where various embodiments of the present disclosure may be implemented; Fig. 2 is a block diagram illustrating a radio tranmitter device;Fig. 3 is a flow chart illustrating a method;Fig. 4 is a diagram illustrating a model of the disclosed transmit circuit as well as the disclosed holographicsurface implemented as an array of densely deployed antenna elements; Fig. 5 is a diagram illustrating a mutual coupling model for the disclosed holographic surface;Fig. 6 is a diagram illustrating beamforming design with optional characteristic and / or load impedanceoptimization for a general lossy holographic surface; Fig.7 is a diagram illustrating a holographic surface with an array of dipole antennas in its surface aperture;andFig. 8 illustrates realized beamforming gains in different horizontal spatial directions of a holographic surfacewith different antenna array settings, achieved by a disclosed gain-based beamforming approach with characteristic / load im- pedance optimization and a conventional beamforming approach without characteristic / load impedance optimization. In the following, identical reference signs refer to identical or at least functionally equivalent features. DETAILED DESCRIPTION In the following description, reference is made to the accompanying drawings, which form part of the dis- closure, and in which are shown, by way of illustration, specific aspects in which the invention may be placed. It is understoodthat other aspects may be utilized, and structural or logical changes may be made without departing from the scope of theinvention. The following detailed description, therefore, is not to be taken in a limiting sense, as the scope of the invention is defined in the appended claims. For instance, it is understood that a disclosure in connection with a described method may also hold true for a corresponding device or system configured to perform the method and vice versa. For example, if a specific method step is described, a corresponding device may include a unit to perform the described method step, even if such unit is not explicitly described or illustrated in the figures. On the other hand, for example, if a specific apparatus is described based on functional units, a corresponding method may include a step performing the described functionality, even if such step is not explicitlydescribed or illustrated in the figures. Further, it is understood that the features of the various example aspects described hereinmay be combined with each other, unless specifically noted otherwise. Fig. 1 illustrates example system 100, where various embodiments of the present disclosure may be imple-mented. System 100 may comprise radio access network cell 110 of a fifth generation (5G) new radio (NR) network or of anetwork beyond 5G wireless networks. An example representation of system 100 is shown depicting receiver 130 (e.g., a userdevice), wireless channel 140 and network node device 120.User device 130 may include, e.g., a mobile phone, a smartphone, a tablet computer, a smart watch, or anyhand-held, portable and / or wearable device. User device 130 may also be referred to as a user equipment (UE). Network nodedevice 120 may comprise, e.g., a base station or a transmission and reception point (TRP). The base station or TRP may include,e.g., any device suitable for providing an air interface for user devices to connect to a wireless network via wireless transmis-sions. Furthermore, network node device 120 may comprise radio transmitter device 200 of Fig. 2.In the following, various concepts and terms that may be relevant to at least some example embodiments willbe discussed.At least in some situations, continuous surfaces may enable holographic radio links. For example, line-of-sight (LoS) holographic radio channels between two continuous surfaces may exist. An example approach for realizing close-to-continuous holographic surfaces is to densify conventional antenna arrays, e.g., by packing more antenna elements (alsoknown as radiating elements) into given apertures of transmit antenna arrays or receive antenna arrays. Such a densified antennaarray may also be referred to as a holographic surface. These surfaces may be made of a low-cost transformative wireless planarstructure comprising sub-wavelength metallic or dielectric scattering particles. When an antenna array is densified, more an-tenna elements may be placed in a same aperture size of the antenna array with a smaller spacing among them, which maycreate mutual interactions and power loss among the antenna elements. Such mutual interactions and power loss need be takeninto account during system design and analysis. Next, as an example, a holographic surface which (without loss of generality) is assumed to be centered atan origin of a three-dimensional (3D) coordinate system is discussed. Within holographic surface aperture, an array of ^ an-tennas may be deployed with an ^-th (^ = 1, 2, ⋯ ^) antenna element centered at a position ^^ ^= ^^^,^ ^^,^ ^^,^^occupyinga distinct sub-area of holographic surface. ^̅ ∈ ^ denotes a unit-power signal stream (i.e., ^(|^̅ |^) = 1 with ^(⋅) being anexpectation operator and ^ being a set of all complex numbers) to be transmitted, and ^ = (^^ ^^ ⋯ ^^)^ ∈ ^^×^ denotes anarbitrary beamforming vector with ^^×^denoting a complex space of a dimension ^ × ^. A transmit signal vector may be written as: ^= ^^̅ = (^^ ^^ ⋯ ^^)^ , (1)and a total power fed to a transmitter (referred to as a transmit power of holographic surface in this disclosure)is given by: ^^ = ^(‖^‖^^) = ^(‖^^‖̅^^) = ‖^‖^^, (2) where ‖⋅‖^ is a 2-norm operator.When the above holographic surface is used to transmit the transmit signal vector ^, a vector of currents ^ =(^^ ^^ ⋯ ^^)^ may be generated at input ports of all the antenna elements. According to the linearity of the system, the rela-tionship between them may be expressed as:^ = ^^, (3) where ^ ∈ ^^×^ (referred to as a current generator matrix in this disclosure) may model impacts of themutual coupling effect and energy efficiency of the antenna array in holographic surface (if any). When moving toward densi-fied arrays for which the spacing among the antenna elements may be arbitrarily small, interaction among the elements mayneed to be accounted for at least in some situations, and their current generator matrices ^ may generally contain non-zeroelements on their off-diagonal entries. For convenience, ^^(^|^^) ∈ ^^×^ may be defined as an electric field (E-field) intensity observed at an ar-bitrary far-field point ^ ∈ ℛ^×^ when antenna element ^ is deployed alone to be centered at the origin and activated by current^^, where ℛ^×^denotes a real space of dimension ^ × ^. Then, in a free space propagation environment, antenna ^ may be modelled as: ^^^^ ^ ^ ^ ^ ^ ^(^|^^) = ^ (^^^ ^), (4)where ^ = ‖^‖^ and ^ = ^ / ^ are, respectively, the distance and spatial direction of point ^ with respect tothe origin, ^ is the signal wavelength, and ^^(^) ∈ ^^×^is referred to as the normalized angular E-field intensity of antenna ^ in the spatial direction ^, which is an inherent characteristic of antenna ^ being independent of fed current ^^and distance^. Therefore, in the free space, the E-field intensity of the whole holographic surface at point ^ may be expressed as:^ ^ ^^^^ ^‖^^^^‖^^^^^^(^|^) = ^ ^^(^ − ^^|^^) = ^^^^^ − ^^ ^^^ ^^^ ^^^‖^ − ^^‖ ^^ ‖^ − ^^‖^^ ^^^^ ^ ^‖^‖^^^^^^^^^^^ ^ ^ ^^ ≈^^^^ ^ ^ ^^(^) =^ ^^(^)^^ ^^^^^ ⋅ ^^, ^^^ ‖^‖^^ ^^^ ^^^^^ ^ ^ ^^^^ ^ ^ = ^^^ =^ ^^^^, (5)where the approximation holds following the far-field assumption of point ^ and ^ =^^ ^^^ ^(^)^ ^^^^^^ ^^^^^^^^^^^^^(^)^ ^^ ⋯ ^^(^)^^^ ^ ∈ ^^×^. As a consequence, a radiation power density of the whole holo-graphic surface at point ^ may be given by^^^^^(^|^) =1 ‖^ (^1 2^^^^^^|^)‖^= 2^^^^^^^^^^^^ (6)with ^ = 120^ Ohm being an intrinsic impedance of the free space. Hence, a total power that is radiated outfrom holographic surface for transmitting the transmit signal vector ^ (referred to as a total radiated power of holographicsurface) may be given by:^^^^(^) = ^ ^^^^^(^|^)^^1 =^^^^ ^^ ^^^^^ ^ ^^, (7)^∈^^^^2^^∈^^^^where ^^^^ = {^ ∈ ℛ^×^|‖^‖^ = ^ } represents the set of all points on a sphere with radius ^ and centerlocation at the origin in the 3D space. From a circuit theory point of view, the total radiated power may be regarded to beequivalently consumed by a matrix of radiation impedances (referred to as a mutual impedance matrix), denoted by ^ =^^^,^^ ∈ ^^×^, that satisfies: ^^^^(^) =1 2Re(^^^^) =1 2^^^^Re(^)^^, (8)where Re(⋅) returns the real part of its argument. following may hold: Re(^) =1 ^^ ^ ^^^(9) ^^∈^^^^or equivalently:Re^^^,^^ =1 ^^^(^)^ (^)^^^^^ ^^^(^^^^^)^^ ^^ ^ , ∀^, ^ = 1, 2, ⋯ , ^.^∈^^^^(10) For any given beamforming vector ^, the radiation power density at point ^ and the total radiated power ofthe holographic surface may be averaged over a randomness of ^̅, yielding ^^^^^(^|^) = ^^^^^^^(^|^^)̅^ and ^^^^(^) =^^^^^^(^^)̅^, respectively. Accordingly, the following metrics may be considered for evaluating the performance of the holo-graphic surface: -energy efficiency: the energy efficiency of holographic surface under beamforming vector ^ may be definedas ^ ^ (^) 1 ^^^^Re(^)^^ ^^(^) ≝ ^^^^^= ⋅ , (11) ^^2 ‖^‖^^which may be upper bounded by 100%, and may be strictly less than 100% when any of the antennas inholographic surface is lossy;- directivity: the directivity of holographic surface at an observation point ^, denoted by ^^^^^(^|^), may bedefined as a ratio between a radiation power density of holographic surface at point ^, i.e., ^^^^^(^|^), and its spatial averageover a sphere centered at the origin with point ^ on it, where the latter may be expressed as^^^^(^)^^^^. Accordingly, the following may hold: ^^ ^ ^ ^4^ ^^^^^^^^^ ^^^^(^|^) ≝ ^^^( | )^ (^) = ⋅ ^ ^^^^R. (12)^^^e(^ ^^ ^)4^^ When the wireless channel is line-of-sight and ^ exhausts all the points on a 3D sphere ^^^^, the term^^^^^(^|^) may also offer a radiation power pattern of holographic surface under the beamforming vector ^;- realized beamforming gain: the^, denoted by ^^^^^(^|^), may be defined as a ratio ^^^^^(^|^), and that achieved by transmitting the same signal stream ^̅ using an idealistic isotropic antenna located at theorigin with the same transmit power ^ , whe^^^ re the latter may be expressed as ^^^^. Accordingly, the following may hold:^^^^^^^(^|^) ≝^^^^^(^|^) 4^ ^ ^ ^^^^^ ^^= ⋅ 2^ ‖. (13)^‖^^^4^^ When the wireless channel is line-of-sight and ^ exhausts all the points on the 3D sphere ^^^^, the term^ ^ ^). (14)embodiments may allow energyas an array of realistic of a mutual coupling effectis also disclosed. apparatus and method for apractical holographic surface made of an array of lossy antenna elements densely deployed within a surface aperture. Thelossy antenna elements in holographic surface aperture. disclosure may allow configuring the beamforming vector by beingtransferred from a conventional vector where the transformation operation may be characterized by a transfermatrix ^ which may exploit the elements in the holographic surface and the energyefficiency issue of holographic At least some embodiments of the present disclosure may allow constructing the transfer matrix ^ based ona coupling transfer matrix ^ and a mutual coupling matrix ^ as ^ = ^^^^^^^, with ^ being a scaling factor, to achieve a max-imum directivity of holographic surface. At least somethe coupling transfer matrix ^ as ^ = holographic surface.At least some embodiments of the present disclosure may allow an analytical characterization / definition ofthe coupling transfer matrix ^ which may be computed based on the mutual impedance matrix ^ and load impedance matrix^^of the antenna array, as well as the characteristic impedance matrix ^^of the transmission lines connecting the antenna elements to their corresponding signal generators. At least somethe mutual coupling matrix ^, which may radiation power pattern of each antenna At least some embodiments of the present disclosure may allow connecting each antenna element on theholographic surface to its signal generator via a setsuch that the connection between them may be At least some embodiments of the present disclosure may allow serially connecting each antenna element onthe holographic surface to aAt least somegain in a certain spatial direction, Next, example embodiments of radio transmitter device 200 are described based on Fig. 2. Some of thefeatures of the described devices are optional features which may provide further advantages. radio transmitter device 200 according to an may be included in network node device 120.Radio transmitter device 200 may comprise at least one processor or processing unit 202 and at least onememory 204 coupled to at least one processor 202, which may be used to implement the functionalities described below inmore detail. Radio transmitter device 200 further comprises transmit circuit 410 and antenna array 420, as described in moredetail below. Radio transmitter device 200transmitter device 200 may transmit protocol. Radio transmitter device 200 may be configured to provide at least one wireless radio connection, such as for examplea 3GPP mobile broadband connection (e.g., 5G or 6G). Radio transmitter device 200 may also include other elements not shown in Fig.2. At least one processor 202 may include, e.g., one or more of various processing devices, such as a coproces-a controller, a digital signal processor (DSP), a processing circuitry with or without an accompanying DSP, or various other processing devices including integrated circuits such as, for example, an application specific integratedcircuit (ASIC), programmable gate array (FPGA), a microcontroller unit (MCU), a hardware accelerator, a special-purpose computer a neural processing unit (TPU),a neural processing unit (NPU), or theMemory 204 may be 204 may include oneor more volatile memory devices, one or more non-volatile memory devices, and / or a combination of one or more volatile memory devices and non-volatile memory devices. For example, memory 204 may be embodied as semiconductor memories (such as mask ROM, PROM (programmable ROM), EPROM (erasable PROM), flash ROM, RAM (random access memory), etc.). Radio transmitter device 200 comprises antenna array 420 of antenna elements. The antenna array 420 isconfigured to act as holographic surface 430. The antenna array 420 has an ^ × ^ mutual impedance matrix of mutual imped-ances between antenna elements, where each antenna element has a corresponding load impedance. Herein, N denotes a number elements. In other words, holographic surface 430 may be implemented as antenna array 420 of ^ antennaelements deployed within a surfaceof each antenna element ^ (^ = 1, 2, ⋯ Radio transmitter device 200 further comprises transmit circuit 410 configured to drive holographic surface430. Transmit circuit 410 comprises at least one pair of transmission lines 411 per each antenna element. Each pair of trans-mission lines 411 is connectable to a respective antenna element and having a respective characteristic impedance. Transmitcircuit 410 further comprises signal generator 412 per each antenna element. Each signal generator 412 is configured to drivea respective antenna element via a respective pair of transmission lines 411.In other words, and as illustrated in diagram 400 of Fig. 4, to characterize / define the current generator matrix^ for holographic surface 430, the structure of the transmit circuit that drives the holographic surface may need to be takeninto account. As shown in Fig. 4, each antenna element ^ (^ = 1, 2, ⋯ ^) is driven by a signal generator via a pair of trans-mission lines with a real and positive characteristic impedance ^^,^. To transmit the transmit signal vector ^ in (1), the ^ signalgenerators may generate a vector of information-carrying electromagnetic waves characterized by a voltage vector ^^=(^^^^^^ ⋯ ^^^)^ ∈ ^^×^ and its corresponding current vector ^^ = (^^^^^ ^^ ⋯ ^^)^ ∈ ^^×^, which propagate forwardly in-side their transmission lines to activate the ^ antennas on holographic surface 430. When the vector of forward electromagneticwaves arrives at input ports of the antenna elements, a portion of the power carried by them may be reflected back along thetransmission lines and yield a vector of backward electromagnetic waves characterized by a voltage vector ^^= (^^^^^^ ⋯ ^^^)^ ∈ ^^×^ and its corresponding current vector ^^ = (^^^^^^ ⋯ ^^^)^∈ ^^×^. The cause of this wave reflec-tion phenomenon is an impedance mismatch between the transmit circuit and the antenna array on the two sides of the antennainput ports. Thus, a total voltage vector and a total current vector at the input ports of the ^ antenna elements may be given by:^ = ^^ + ^^ and ^ = ^^ + ^^ , (15)where ^ is the generated current vector defined in (3).The reflected electromagnetic waves may travel away from the antenna elements in the transmission linesand may no longer bounce back to the antenna array when each transmission line is supposed to be closed on a matched load.This constitutes one source of the power loss of the holographic surface, and is usually referred to as a reflection loss or a returnloss. The remaining electromagnetic waves may then penetrate through the input ports and flow into the antennaelements. When the load impedance of each antenna element ^, denoted by ^^,^, has a positive real part, a portion of the powercarried by the penetrated electromagnetic waves may be consumed by these load resistances ^Re^^^,^^^. This constitutes an-other source of the power loss of the holographic surface, and is usually referred to as a heat loss or an Ohmic loss.The residual power carried by the penetrated electromagnetic waves may be converted to a radiated powerand conveyed by a generated electromagnetic field that propagates into the surrounding 3D space. As mentioned earlier, thisradiated power may be equivalently regarded to be consumed by the mutual impedance matrix ^. Accordingly, the following may hold: ^= (^ + ^^)^, (16)where ^^ = diag^^^,^ ^^,^ ⋯ ^^,^^ ∈ ^^×^ is a diagonal matrix with its ^-th diagonal entry being ^^,^.The lemma 1 below summarizes the relationships among the transmit signal vector ^ and the forward / back-ward electromagnetic wave vectors. Lemma 1: for the holographic surface implemented as in Fig. 4, the following may hold:^^ = ^^^^ and ^^ = ^^^^ , (17)where ^^ = diag^^^,^ ^^,^ ⋯ ^^,^^ ∈ ^^×^, ^^ = ^(^)^^, (18)where ^(^) = ^^×^ − 2^^(^ + ^^ + ^^)^^(19) may be referred to as a voltage scattering matrix of the holographic surface with ^^×^ being an ^ × ^ iden-tity matrix, and ^^ = √2^^ / ^ ^^. (20)Radio transmitter device 200 is configured to obtain first beamforming vector. At least in some embodiments,first beamforming vector may be based on a wireless channel vector of wireless channel 140 between holographic surface 430and a receive antenna of receiver 130. For example, the receive antenna of receiver 130 may include a physical receive antennaor a logical antenna port. Radio transmitter device 200 is further configured to obtain a transfer matrix based on at least one of themutual impedance matrix, load impedances of the antenna elements, or characteristic impedances of the pairs of transmissionlines 411. Radio transmitter device 200 is further configured to obtain a second beamforming vector based on a productof the obtained transfer matrix and the obtained first beamforming vector.Radio transmitter device 200 is further configured to use transmit circuit 410 to generate a signal stream 401beamformed with the obtained second beamforming vector.For example, 402 in Fig. 4 represents an arbitrary beamforming vector, e.g., it may be ^(^^^^) in (46),^(^^^^^^^^) in (48), ^(^^^^) in (54), or ^(^^^^) in (57). At least in some embodiments, ^(^^^^) may be regarded as the first beam-forming vector, while ^(^^^^) and / or ^(^^^^) may be regarded as the second beamforming vector.Radio transmitter device 200 is further configured to transmit the generated signal stream via antenna array420. At least in some embodiments, the current generator matrix may be based on the mutual impedance matrix,the characteristic impedances, and the load impedances, such that ^ = 2√2(^ + ^^ + ^^)^^^ ^ / ^ ^ . In other words, and as illustrated in diagram 500 of Fig. 5 (showing mutual coupling model 510 for disclosedholographic surface 430), on the basis of Lemma 1 above, an analytical expression for the current generator matrix ^ may bedeveloped, as follows. Theorem 1: for holographic surface 430 implemented as in Fig. 4, its current generator matrix 511 may begiven by ^= 2√2(^ + ^^ + ^^)^^^ ^ / ^ ^. (21)Theorem 1 provides an analytical characterization of the mutual coupling effect and energy efficiency ofholographic surface 430, with which a complete analytical description for the transmission behavior of a general holographicsurface under an arbitrary beamforming vector ^ may be achieved, as illustrated in Fig. 5. It may be seen that, due to the mutualcoupling effect, the E-field intensity of whole holographic surface 430, and in turn the realized beamforming gain of holo-graphic surface 430 to be discussed later, may be dependent on not only the beamforming vector ^, but also on antenna array420 parameters, such as impedance matrices 512 including the mutual / load impedance matrices ^ and ^^as well as the char-acteristic impedance matrix ^^ of all the transmission lines connecting to the antenna elements 513 in antenna array 420.With the mutual coupling model described above, the energy efficiency of the holographic surface under anarbitrary beamforming vector ^ may be described next. Transmit power ^^ may be carried by the forward electromagneticwaves, which during the signal transmission may be converted to, respectively, the reflection loss denoted by ^^^^(^), the heatloss denoted by ^^^^^(^), and the radiated power ^^^^(^), i.e.,^^ = ^^^^(^) + ^^^^(^) + ^^^^^(^). (22)By substituting (21) into (11), the energy efficiency of holographic surface 430 under an arbitrary beamform-ing vector ^ may be analytically expressed as:^^^^^ ^) =^^( ^^ ‖^‖^ , (23)^where^=1 ^^Re(^)^ 2 =4^ ^ / ^( )^^ ( )( )^^^ / ^ ^^ + ^^ + ^^ Re ^ ^ + ^^ + ^^^^(24)may be referred to as an energy efficiency matrix of holographic surface 430. It can be seen that the energyefficiency ^^^^^(^) is dependent on beamforming vector ^.a holographic surface with a high energy efficiency, preferably close to 100%, may be desirable.To this end, the surface may be designed to feature both a seamless impedance matching network between thetransmit circuit and antenna array (to prevent reflection loss) and a minimal load resistance per antenna element (to mitigateheat loss), which implies that the energy efficiency matrix ^ may need to be as close to an identity matrix as possible. Mathe-matically, a necessary and sufficient condition to avoid heat loss may be to let: Re(^^) = ^^×^ , (25)where ^^×^ is an ^ × ^ all-zero matrix. In addition, a sufficient condition to prevent reflection loss may beto let ^(^)= ^^×^, which according to the definition in (19) is guaranteed when: Im(^^) = −Im(^) and ^^ = Re(^ + ^^), (26)where Im(⋅) returns the imaginary part of its argument. By substituting (25) and (26) into (24), it may beverified that: ^= 4^Re(^)^^ / ^ ^^ ^^ ^ / ^ ^2Re(^)^ Re(^)^2Re(^)^ ^Re(^)^= ^^×^ . (27)Hence, this may result in ^^^^^(^) =^^^^×^^ ‖^‖^^= 100% for any beamforming vector ^. It should be noted that for a holographic surface implemented as a densely deployed antenna array, its mutualimpedance matrix ^ may be a full matrix. Consequently, conditions (25) and (26) imply that to achieve a lossless transmissionunder any beamforming vector ^, both the load impedance matrix ^^ and characteristic impedance matrix ^^ may need to befull matrices. At least in some embodiments, the coupling transfer matrix may be based on the mutual impedance matrix, ^ / ^the characteristic impedances, and the load impedances, such that ^ = 2 ^Re^^(^)^^ (^ + ^^ + ^^)^^^ ^ / ^ ^. Alternatively,the coupling transfer matrix may be based on the mutual impedance matrix and a current generator matrix associated with radio, such that ^ = ^^^(^)^ / ^transmitter device 200 ^^ ^^ ^^, where ^ denotes the current generator matrix.In other words, the mutual coupling model in (21) is applicable to a holographic surface in a general com-munication system as long as there is no scatterer in a near-field region of the holographic surface to alter its mutual coupling effect and energy efficiency. In the following, a general point-to-point communication system model based on the mutualcoupling model in (21) is described, which will facilitate the later discussion about the beamforming design of the system.In a LoS communication is equipped with a holographic surface discussed above totransmit the signal stream ^̅, and a receive antenna with a normalized angular E-field intensity ^^(^) ∈ ^^×^ (^ ∈ ^^^^) isdeployed to be centered at the far-field observation point ^ = ^^ to receive an attenuated version of ^̅. Then from (5), thereceived signal, denoted by ^ ∈ ^, may be expressed as:^^^^ ^^ = ^ ⋅ ^^^ ^ ^(−^)^^^^^(^|^) = ^^⋅ ^^^(−^)^^^^ ^^ =^^^^^‖^^(−^)‖^^ ⋅^^^^^ ^‖^^(^)‖^^^^^^^^ ^^^^^^^‖^^(^)‖^^ ^^ ⋯ ^^^^^^^^‖^^(^)‖^^^^ ^ ^^,(28) where ^ is a constant that is irrelevant to the holographic surface and dependent only on the reception capa-^ bility of the receive antenna, and ^^=^^(^^)^^(^)‖^^(^^)‖^⋅‖^^(^)‖^. The received noise is ignored in (28) for brevity. For convenience, ^^(^) denotes the directivity of antenna element ^ in the spatial direction ^, which maybe defined similarly to (12) as: ^^⋅ 1‖^( )‖^ 1‖ ( )‖^ 2^^^|^^ ^^^^^^ ( )^ ^^ = 4^ ⋅|^= 4^^|^⋅2^ ∫^^∈^^^^‖^^(^′)‖^1 ^^^ ′^ ∫^^∈^^^^‖^^(^′)‖^^^^′ 4^ ‖^( ^ = ⋅^^)‖^^, (29)Re^^^,^^ where ^^ is an arbitrary current fed to antenna ^ and the last equality follows from (10). Then, the followingmay hold: ^ ^ ^ / ^ ‖^(^)‖^ = ^ ^ (^) ⋅ Re^^ ^^. (30) 4^^ ^,^^^^On the basis of (30), the term ^^‖^^(^)‖ ^ ^^ ^^ ^^ (^ = 1, 2 ⋯ ^) in (28) may be rewritten as^ (2^)^ / ^ / ^ ^^^^^^^Re^^^,^^ ^^‖^^(^)‖^^^ ^^ ^^=^^^^^^ / ^(^)^^^^^ ⋅ ^^ 2^ / ^2. (31)Substituting this into (28), the following is obtained: ^^^^ = ^(2^)^ / ^^^^^‖^^(−^)‖^2^^ / ^^ ^^ ⋅^^^^^^ / ^(^)^^^^^^^^^^^^ / ^(^)^^^^ ^^^^^ ⋯ ^^^^^ / ^(^)^^^^ ^^^^^^ (^)^ / ^ ⋅^Re^^ ^ ^ ^^ 2 ^^ ^ / ^ ( )^ / ^ ^^^^‖ ( )‖(^)=^2^ ^ ^^−^^2^^ / ^^⋅ ^^ ⋅ ^Re^^ ^ ^ 2^^ = ^^^, (32)^ where ^ = ^(^^)^ / ^^^^ ^ ^^‖^^(^^)‖^^^^ / ^^ ⋅ ^^ is the channel vector between the holographic surface and the receiveantenna, ^^ = (ℎ^^ ℎ^^ ⋯ ℎ^^) is a scaled version of ^ with its entries given byℎ^^ / ^ ^= ^^^^(^)^^^^ ^^^^^ , ∀^ = 1, 2, ⋯ ^, (33)^(^) = diag^^^,^ ^^,^ ⋯ ^^,^^ ∈ ^^×^is a diagonal matrix containing only the diagonal entries of matrix ^, and (^)^ / ^ ^ / ^ ^≝ ^Re^^ ^ ^^ = 2 ^Re^^(^)^^ (^ + ^^ + ^ )^^^ ^ / ^ (34) 2^ ^may be referred to as a coupling transfer matrix.At least in some embodiments, the mutual coupling matrix may be based on the mutual impedance matrix( ^^ / ^ ^^ / ^satisfying ^ = ^Re^^^)^^ Re(^) ^Re^^(^)^^ , where ^ = ^^^,^^ denotes the mutual coupling matrix. Alternatively,the mutual coupling matrix may be based on the spatial direction -based radiation power pattern and a position of each antennaelement satisfying ^^,^ =^ ^^∫^∈^^^^ ^^^ / ^(^)^^^ / ^(^)^^^^^^^^ ^^^^^^^, ∀^, ^ = 1,2, ⋯ ^, where ^^(^) denotes the radiationpower pattern, ^ denotes the spatial direction, and ^^ denotes the position of an antenna element ^.In other words, equation (32) models a general point-to-point LoS communication system for a transmission between a holographic surface and a single receive antenna. In a case when all the antennas in the holographic surface areidentical to each other with the same polarization direction, it follows that ^^ = ^^ , ∀^, ^ = 1, 2, ⋯ , ^. In addition, if theholographic surface was lossless, further satisfying the lossless conditions (25) and (26), the following Corollary 1 may beobtained. Corollary 1: for a holographic surface implemented as an array of identical antennas and satisfying the loss- less conditions (25) and (26), its coupling transfer matrix may reduce to: ^= ^^^ / ^, (35), = , , ⋯ . (3 )In practice, the receive antenna may be deployed to have its polarization direction match the polarizationdirection of the received E-field intensity 514 at point ^, i.e.,^^^^(^^)^^^^^^^^^^^‖^^(^^)‖^⋅^^^^^^^^^^^^= 1, to maximize the received signal^power. In this case, it may be seen from (33) that the received signal power is given by^(|^|^) = ^(|^ ⋅ ^^^(−^)^^^^^(^|^)|^) =2^|^|^‖^^^(−^)‖^1 ^ ⋅ ‖^ (^|^)‖^2^^^^^ ^= 2^|^|^‖^^^(−^)‖^^ ⋅ ^^^^^(^|^). (38)That is to say, the received signal power ^(|^|^) may be proportional to an average radiation power densityat point ^, ^^^^^(^|^), upon a constant scaling factor, and so may also be proportional to the directivity and realized beam-forming gain defined in (12) and (13). Hence, these two performance metrics may still be used to reflect the beamformingperformance of such a LoS communication system. From (38) and (32), the radiation power density ^^^^^(^|^) involved in(12) and (13) may be rewritten as: ^^^^ ^(| | )^^^^^ ^(^|^) =^ ^^^=. (39)2^|^|^‖^^^(−^)‖^^4^^^In addition, the total radiated power ^^^^(^) may be rewritten from (34) and (36) as:^^^^(^) =1^ ^ ( ) ^ ^ 2^ ^ Re ^ ^^ = ^ ^= (40) Hence, by substituting (39) and (40) into (12) and (13), the directivity and realized beamforming gain of theholographic surface under an arbitrary beamforming vector ^ may be explicitly expressed as, respectively:^^^^^^ 4^ ^^ ( | )^^^^^^^(^|^) = ^^^^^ ^ = (41) ^^^^(^) ^^^^^^^ and ^^^^ ^ (^|^)^^^^ ( | ) ^^^^^^^^ ^ ^ ≝^^= ‖^‖^. (42)^^4^ ^ The LoS communication system model in (32) may be straightforwardly extended to a more general systemwith a multi-path channel between the transmitter and receiver. Specifically, the between the transmitterand receiver may be assumed to comprise a number of ^ propagation paths located in the far-fieldregions of both the transmitter and receiver. For each path ^, ^^ ∈ ℛ^×^ (‖^^‖^ = 1) and ^^ ∈ ℛ^×^ (‖^^‖^ = 1) may beused to represent its departure and arrival direction, ^^may be used to represent the propagation distance along each path ^,^^,^ ∈ ℛ^×^ (^^^,^^^ = 1) may be defined as a rotation matrix charactering the polarization direction change of the electro-magnetic wave during the propagation / reflection along the ^-th channel path, and ^^ ∈ ^ may be defined to account for theextra loss and phase change caused by the absorption / diffraction of the scatterer(s) generating the ^-th channel path. Then, thecorresponding multi-path communication system may be modelled into the same form as (32) with its channel vector ^ rede-fined as:^^ ^ / ^ ^^^^^^^ = ^(2^) ^∑^^^^ ‖^^(^^ )‖^2^^ / ^^⋅ ^^ (43) where ^ ^^ = ^‖^^(^^)‖^^^^^ . still proportional to the average radiation power density at point ^ upon a constant scaling factor, and so the performancemetrics in (41) and (42) may also be adopted by updating the with (44), to evaluate the performance of a generalmulti-path communication system. At least in some embodiments, the wireless channel vector may be constructed based at least on positions ofthe antenna elements within holographic surface 430, and further based on spatial direction -based radiation power patterns ofthe antenna elements within holographic surface 430. Alternatively, the wireless channel vector may be estimated from pilotsignals receivedIngraphic surface 430.terpart ^^, where ^^ ismay be used as an radiation power a , orthe pilot signals received from the receive antenna of the receiver, or fed back from the receiver. At least in some situations, a beamforming vector for a receive antenna at point ^ could be designed to onlymatch the channel vector (or equivalently its scaled version), i.e.: ^= ^(^^^^) ≝ ^^^^^^=^^‖^‖^^^ . (46)^ ^^^^^ This approach is ^ ^However, in the disclosure a different linear transformation ^ is performed on the conventional beamformingvector ^(^^^^)to generate a new beamforming vector ^, such that it takes into account the antenna coupling effect and energyefficiency of the holographic surface in a^ are disclosed.In a first alternative coupling matrix, such that ^ = ^^^^^^^, where ^ denotes the transfer matrix, ^ denotes the coupling transfer matrix, ^ de-notes the mutual coupling matrix, ^ denotes a scaling factor, and the coupling transfer matrix is based at least on the mutualimpedance matrix. the mutual impedance matrix, the load impedances, and the characteristic impedances, such that ^ =^^^ (^ + ^^ + ^^)^Re(^)^^^ ^Re^^(^ / ^ ^)^^, where ^ = {^^,^} denotes themutual impedance matrix, ^^^,^|^ = 1,2, ⋯ ^^ denotes theimpedances, n denotes a first antenna element index, m factor, ^^denotes an ^ × ^ diagonal load impedance matrix with ^^,^ as its ^-th diagonal entry, ^^ denotes an ^ × ^ diagonal charac-teristic impedance matrix with ^^,^ as its ^-th diagonal entry, ^(^) denotes an ^ × ^ diagonal matrix with ^^,^ as its ^-th di-agonal entry, and Re(⋅) returns a real part of its argument.In other words, the transformation ^ may be selected to maximize the directivity of holographic surface 430.Specifically, it can be seen that the directivity in (41) may be upper bounded by:^ ^^ / ^^ ^^^^^^ / ^ ⋅ ^^ / ^^^ ^^^^^^ / ^^ ⋅ ^^ ^^^^ (^| ^^) =≤ ^ ^ ^^^^‖^^ / ^^^‖^^‖^^ / ^^^‖^ ^= ^^^^^^^^ ≝ ^ (^^^^)^^^^ (^), (52)where the equality holds when: ^^ / ^^^ ∝ ^^^^^^ / ^^^ (53) or equivalently: ^ This approach is referred to as a directivity-based beamforming, and the corresponding transformation ^ maybe determined as:^ = ^^ = ^ = ^^^ / ^^ (^ + ^^ + ^^)^Re(^)^^^ ^Re^^(^ / ^ ^)^^ (55) 2 ^^^^^with ^ =^ ^^^^^^^^^^^ being a scaling factor to satisfy the power some situations, this^embodiment may allow the transformation ^^ in (55) to fully take into account the energy efficiencyof the lossy holographic surface, and thereby the resultant directivity may be maximized.In a second alternative embodiment, the transfer matrix may be based on a coupling transfer matrix, such that^ = ^^^, where ^ denotes the transfer matrix, ^ denotes the coupling transfer matrix, H denotes a conjugate transpose opera-tor, ^ denotes a scaling factor, and the coupling transfer matrix is based at least on the mutual impedance matrix. Alternatively,the transfer matrix may be based on the mutual impedance matrix, the load impedances, and the characteristic impedances,^ / ^ / ^such that ^ = 2^^ ^^ (^ + ^^ + ^^)^^^Re^^(^)^^, where ^ = {^^,^} denotes the mutual impedance matrix, ^^^,^|^ =1,2, ⋯ ^^ denotes the load impedances, ^^^,^|^ = 1,2, ⋯ ^^ denotes the characteristic impedances, n denotes a first antennaelement index, m denotes a second antenna element index, ^ denotes a scaling factor, ^^ denotes an ^ × ^ diagonal load im-pedance matrix with ^^,^ as its ^-th diagonal entry, ^^ denotes an ^ × ^ diagonal characteristic impedance matrix with ^^,^as its ^-th diagonal entry, ^(^) denotes an ^ × ^ diagonal matrix with ^^,^ as its ^-th diagonal entry, and Re(⋅) returns a realpart of its argument.In other words, the transfgain of the holographic surface. Specificall^^^^^^ ⋅‖^‖^ ^^^^^(^|^) ≤^^= ^^^^^^ ^≝ ^ (^^^^)^^^^ (^), (56) ^^^^^^^.^ This approach is referred to as a gain-based beamforming, and the corresponding transformation ^ may bedetermined as:^ ^ ^ / ^^ / ^ ^= ^ = ^^ = 2^^^ (^ + ^^ + ^^)^^^Re^^(^)^^ (58) ^^th ^ =^^^wi^ ^^^^^^^ . At least in some situations, this embodiment may allow the transformation ^^ in (58) to^ fully take into account the coupling effect andbeamforming gain may be maximized. Inconventional beamforming vector other than ^ By comparing (52) with (56), it follows that: ^(^^^^)^^^^ (^) = ^ (^^^^^^^^ ^^^^)^ ≤ ^ (^^^^^^^^ ^^^^)^ ≤ ^^^^^^^^^(^^^^)^ =^(^^^^^^^^)(^) = ^^^^^^^^. (59) Equation (59) offers an upper bound for the maximum realized beamforming gain that may be achieved byan arbitrary holographic surface. According to Corollary 1, this upper bound may be achievable when the holographic surfacesatisfies the lossless conditions (25) and (26). In addition, this upper bound may be independent of the characteristic and loadimpedance matrices ^^ and ^^, and may only depend on the mutual coupling matrix ^ that is in turn related to the mutualimpedance matrix ^ of antenna array 420.Next, optimization of the array / circuit parameters is described to further enhance the realized beamforminggain of the system. Diagram 600 of Fig. 6 summarizees this overall beamforming design with optional characteristic and / orload impedance optimization for a general lossy holographic surface. The maximum realized beamforming gain in (56) may be explicitly rewritten as^ where ^^ = 2^^ ^Re^^(^) tially be further boosted by optimizing ^^and ^^. At least in some embodiments, be configured based on geometries and / ormaterials of the transmission lines of the pairs ofAt least in some lines 411 may exceed an amount ofthe antenna elements, and a connectionamong the pairs of transmission lines 411 Nsignal generators. In an example, each antenna ele-ment and its of transmission lines, i.e., there may be at least N pairsof transmission lines. However, this does not mean that there has to be exactly N pairs of in the transmitcircuit. That is, there may be M (M>N) pairs of transmission lines in the transmit circuit, such that the connection between eachantenna and its corresponding signal generator may be achieved using either a given pair of transmission lines or another pairof transmission lines among the M pairs of transmission lines.Thus, in another example, there may be M = NK pairs of transmission lines that are divided into N sets ofpairs of transmission lines with each set having K pairs of transmission lines. Thus, the connection between the n-th antenna and the n-th signal generator may be achieved by a pair of transmission lines within the n-th set of pairs of transmission lines,and such a connection may be switched among the K pairs of transmission lines in the n-th set.In yet other words, the characteristic impedances in the diagonals of ^^ may theoretically take any positivevalues by manufacturing transmission lines with different geometries and materials. In an implementation example, it may beoptimized in advance for a fixed communication link before manufacturing the involved transmission lines. In another imple-mentation example, a set of transmission lines with different characteristic impedances may be manufactured and equipped forthe connection between each antenna element and its signal generator, and the transmitter may adaptively switch among thesetransmission lines to establish the connection depending on the need. In this disclosure, a lower bound ^(^^)^ > 0 may be setto the characteristic impedances for numerical stability, and a feasible region of ^^ may be considered to be ^^^ =^^^|^^,^ ≥ ^^(^^), ^ = 1,2 ⋯ , ^^. At least in some embodiments, radio transmitter device 200 may further comprise a group of configurableexternal load impedances 403. Each configurable external load impedance 403 may be serially connected to a respective an-tenna element in order to configure a respective load impedance.In other words, the load impedance matrix ^^ may be adjusted by serially connecting an external impedance^ (^) ^,^ to each antenna ^, i.e., ^^,^ = ^(^) ^,^ + ^(^) ^,^with ^ (^) ^,^being the original internal load impedance of antenna ^. In practice, these external impedances ^^ (^) ^,^^ may be madedepending on the need. Since a positive ,the external impedances may be assumed to be purely imaginary, and so the feasible region of ^^ may be ^^^ =^^(^) ^|Re^^^,^^ = Re ^^^,^^ , ^ = 1,2, ⋯ , ^^.Thus, the following ^ where: ^^(^^, ^^) = ^^(^ + ^^ + ^^)^^^^(^ + ^^ + ^^)^^^^^ . (62)Problem P1 may be non-sub-problems and alternatively optimize the cifically, both ^^ and ^^ may be assumed to be properly initialized, and ^ = ^ + ^^ + ^^ and ^^ may be defined as a length-^ column vector with “1” on its ^-th entry and “0” elsewhere for convenience. Firstly, to increase ^^(^^, ^^) by updating ^^,^to ^^,^ + ^, where ^ ∈ [−^^,^ + ^(^^)^ , +∞), the optimal value of ^ may be obtained by solving the following sub-P1 − 1: max(^^^^(^),^∈[^^^^) ^,^^^^,^^) where: ^^^^(^) = ^^(^ + ^^^^^^)^^(^^ + ^^^^^^)(^ + ^^^^^^)^^^^^. (64)Secondly, to increase ^^(^^, ^^) by updating the imaginary part of ^^,^ to Im^ ^^,^^ + ^ , where ^ ∈[^, +∞), with ^ ≪ 0 being a ^problem: where: ^^^^(^) = ^^(^ + ^^^^^^^)^^^^(^ + ^^^^^^^)^^^^^. (66)By adopting a Sherman-Morrison formula, i.e., (^ + ^^^)^^ = ^^^ − ^^^^^^^^^^^^^^^^^for any invertible squarematrix ^ ∈ ^^×^ and vectors ^, ^ ∈ ^^×^, and after some mathematical derivations, both subproblems P1-1 and P1-2 may becast into a more general and common form as: ^^^^ + ^^^ + ^P2: max ^ ( ) ^^∈[^,^^)^ ^ =(^^ + 1)(^∗^ + 1), (67)with ^, ^^ , ^^ , ^^ ∈ ℛ, b ∈ ^ and ^^ > 0. Therefore, both P1-1 and P1-2 may be solved in the same way.Theorem 2: the solution to problem P2 is given by:any value,ì^^ = ^^ = 0;ïï max^^, −Re(^)^ , ^^ = 0, ^^ > 0;^^ = 0, ^^ < 0;^ =+∞, |^ − ^^ |ax ^^,^ (68) ím ^ ^−^ ^^ > 0 orïï|^|^^ ,^^^ ^^ < 0, ^^^ + ^^ ≥ 0;î+∞, ^^ < 0, ^^^ + ^^ < 0,where ^^ = ^^|^|^ − (^ + ^∗)^^ and ^^ = ^^|^|^− ^^. In each update of ^^,^ or ^^,^, the obtained value of ^^(^^ , ^^) may be non-decreasing. Further consideringthat the realized beamforming gain is a finite value, it may be concluded that the alternative optimization process converges.In addition, in practice both the characteristic and load impedances may either be jointly optimized, or only optimize the char-acteristic impedances by skipping the steps of the load impedance optimization, or only optimize the load impedances by skipping the steps of the characteristic impedance optimization. Fig. 3 shows an example flow chart that illustrates method 300 for radio transmitter device 200, according toan embodiment of the Aselements. Antenna arraypedance matrix of mutual pedance, in which N denotes a number of the antenna elements. Radio transmitter device 200 further comprises a transmitcircuit 410 configured to drive holographic surface 430.411 per each antenna element. Each pair of transmissionrespective characteristic impedance.Each signal generator 412 is configured toAt operation 301, radio At operation 302, radio transmitter device 200 obtains the transfer matrix based on the mutual impedancematrix, the load impedances of the antenna elements, and / or the characteristic impedances of the pairs of transmission lines411. At operation 303, radio of the obtained transfer matrix and the obtained At operation transmitter device 200 uses transmit circuit 410 to generate the signal stream beam-formed with the obtained second beamforming vector. At operation 305, radio transmitter device 200 transmits the generated signal stream via antenna array 420.Method 300 may be carried out by radio transmitter device 200. Operations 301-305 may, for example, becarried out by at least one processor 202 and memory 204. Further features of method 300 directly resulting from thefunctionalities and parameters of radio transmitter device 200 are not repeated here. Method 300 can be carried out by computerprograms or portions thereof.Below, some computer simulation results are presented to further demonstrate the advantages of the disclosedbeamforming solution with optional characteristic / load impedance optimization by taking into account the mutual couplingeffect between the antenna elements and energy efficiency issue of theDiagram 700 of Fig. 7 illustrates a rectangular graphic surface has length ^ = ^ and width ^ = ^, and is deployed on the ^ − ^ plane having its long and wide sides parallelto the ^ − and ^ −axes, respectively, and its center located at the origin. An array in the holographic surface is made of identicaland ^ −axis directed dipole antennas, where each dipole antenna is cylinder-shaped with length ^, diameter ^ = 0.01^, andcenter-fed with gap ^ = 0.01^ in the middle. The carrier frequency is fixed at 0.75 GHz. During the impedance optimization,the characteristic and load impedances are initialized at, respectively, ^^,^ = 50 Ohm and ^^,^ = 0 Ohm, ∀^ = 1, 2, ⋯ , ^, andthe lower bound of {^^,^} is set at at ^(^^)^ = 1 Ohm. E.g., the following array settings may be considered:- ConvURA: a conventional uniform rectangular antenna array (URA) of ^^ = 3 columns and ^^ = 2rows with horizontal spacing ^^= ^ / 2 and vertical spacing ^_2 = ^ / 2, where the length of all dipoles is set at ^ = ^ / 2 − ^to avoid overlapping between colinear dipoles; -DensURA: a densified URA with ^^ = 5, ^^ = 2, ^^ = ^ / 4, ^^ = ^ / 2 and ^ = ^ / 2 − ^;- DensUCA: a densified uniform cubic antenna array (UCA) formed by ^^ = 2 layers of DensURAsabove placed in parallel with spacing ^^ = ^ / 4, i.e., a thickness of ^ = ^ / 4 is introduced to the holographic surface.As shown inin different horizontal target directions, illustrated in Fig.7 satisfying ^ = (sin^cos^ sin^sin^ cos^ , and a horizontal target direction refers to that with ^ = ^ / 2.The gain-based beamforming approach withconventional beamforming approach withoutvURA for reference. From Fig. 8 the following observations can be made:- for ConvURA, the conventional beamforming approach without impedance optimization achieves itsmaximum realized beamforming gain of 8.86 dB in the normal direction of the URA (i.e., ^ = ^ / 2 and ^ = 0), which gradu-ally reduces to 3.46 dB when the target direction moves from the normal direction to the end-fire directions (i.e., ^ = ^ / 2 and^ = ±^ / 2) of the surface; -the beamforming approach with impedance optimization can significantly increase the re-alized beamforming gain of the ConvURA. The realized beamforming gain is still the highest in the normal direction of thesurface, i.e., 9.75 dB, and gradually reduces to 6.95 dB when the target direction moves to the end- which arehigher by about 0.89~3.49 dB than those achieved by the conventional beamforming approach without impedance optimiza-tion; -array densification can significantly enhance the beamforming performance of the holographic surfaceunder the gain-basedfire direction of theend-fire direction of forming approach without impedance optimization. Even if both the ConvURA and DensURA are allowed to be arbitrarilyoriented to serve a common spatial direction, this gain is still 3.37 dB higher than the later achieved in the normal direction. Inaddition, if the array is further densified in the normal direction of the surface, i.e., from DensURA to DensUCA, the realizedbeamforming gain in the normal direction is also increased to 13.29 dB and becomes the highest in all horizontal target direc-tions, which is 4.43 dB higher than that achieved by ConvURA using the conventional beamforming approach without imped-ance optimization. In summary, with the disclosed gain-based beamforming design and characteristic / load impedance optimi-zation, array densification can significantly enhance the realized beamforming gain of the holographic surface in the direction that the array is densified. Radio transmitter device 200 may comprise means for performing at least one method described herein. Inone example, the means may comprisefigured to, when executed by the at least The functionality described herein can be performed, at least in part, by one or more computer programproduct components such as software components. According to an embodiment, radio transmitter device 200 may comprise aprocessor or processor circuitry, such as for example a microcontroller, configured by the program code when executed to execute the embodiments of the operations and functionality described. Alternatively, or in addition, the functionality described herein can be performed, at least in part, by one or more hardware logic components. For example, and without limitation, illustrative types of hardware logic components that can be used include Field-programmable Gate Arrays (FPGAs), Applica- tion-specific Integrated Circuits (ASICs), Application-specific Standard Products (ASSPs), System-on-a-chip systems (SOCs),Complex Programmable Logic Devices (CPLDs), Tensor Processing Units (TPUs), and Graphics Processing Units (GPUs).Any range or device value given herein may be extended or altered without losing the effect sought. Also, any embodiment may be combined with another embodiment unless explicitly disallowed. Although the subject matter has been described in language specific to structural features and / or acts, it is to be understood that the subject matter defined in the appended claims is not necessarily limited to the specific features or acts described above. Rather, the specific features and acts described above are disclosed as examples of implementing the claims and other equivalent features and acts are intended to be within the scope of the claims. It will be understood that the benefits and advantages described above may relate to one embodiment or may relate to several embodiments. The embodiments are not limited to those that solve any or all of the stated problems or those that have any or all of the stated benefits and advantages. It will further be understood that reference to 'an' item mayrefer to one or more of those items.The steps of the methods described herein may be carried out in any suitable order, or simultaneously where appropriate. Additionally, individual blocks may be deleted from any of the methods without departing from the spirit and scope of the subject matter described herein. Aspects of any of the embodiments described above may be combined with aspects of any of the other embodiments described to form further embodiments without losing the effect sought. The term 'comprising' is used herein to mean including the method, blocks or elements identified, but that such blocks or elements do not comprise an exclusive list and a method or apparatus may contain additional blocks or elements. It will be understood that the above description is given by way of example only and that various modifica- tions may be made by those skilled in the art. The above specification, examples and data provide a complete description of the structure and use of exemplary embodiments. Although various embodiments have been described above with a certain degree of particularity, or with reference to one or more individual embodiments, those skilled in the art could make numerous alterations to the disclosed embodiments without departing from the spirit or scope of this specification. CLAIMS 1. A radio transmitter device (200), comprising: an antenna array (420) of antenna elements, the antenna array (420) configured to act as a holographic surface(430), the antenna array (420) having an ^ × ^ mutual impedance matrix of mutual impedances between antenna elements,where each antenna element has a corresponding load impedance, in which N denotes a number of the antenna elements; anda transmit circuit (410) configured to drive the holographic surface (430), the transmit circuit (410) compris- ing: at least one mission lines (411) connectable to a respective antenna aasignal generator (412) per each antenna element, each signal generator (412) configuredto drive a respective antenna element via a respective pair of transmission lines (411),wherein the radio transmitter device (200) is configured to: obtain a first beamforming vector; obtain a transfer matrix based on at least one of the mutual impedance matrix, load impedances of the antenna elements, or characteristic impedances of the pairs of obtain a second beamforming beamforming vector; use the transmit circuit (410) to generate a signal stream beamformed with the obtained second beamformingvector; and transmit the generated signal stream via the antenna array (420).2. The radio transmitter device (200) according to claim 1, wherein the first beamforming vector is basedon a wireless channel vector of a wireless receiver (130), the wireless channel vector constructed based at least on positions of the antenna elements within the holographic surface (430), andfurther based on spatial direction -based radiation power patterns of the antenna elements within the holographic surface (430);estimated from pilot signals received from the receiver (130); or fed back from the receiver (130). 3. The radio acoupling transferfer matrix, ^ denotes the coupling coupling transfer matrix is based at least on the mutual impedance matrix; orthe mutual impedance matrix, the load impedances, and the characteristic impedances, such that:^ ^^^^^ ^ / ^ ^= / ^(^ + ^^ + ^^)^Re(^)^^Re^^(^)^^ , 2^where ^ = {^^,^} denotes the mutual impedance matrix, ^^^,^|^ = 1,2, ⋯ ^^ denotes the load impedances,^^^,^|^ = 1,2, ⋯ ^^ denotes the characteristic impedances, n denotes a first antenna element index, m denotes a second antennaelement index, ^ denotes a scaling factor, ^^ denotes an ^ × ^ diagonal load impedance matrix with ^^,^ as its ^-th diagonalentry, ^^ denotes an ^ × ^ diagonal characteristic impedance matrix with ^^,^ as its ^-th diagonal entry, ^(^)denotes an^ × ^ diagonal matrix with ^^,^as its ^-th diagonal entry, and Re(⋅) returns a real part of its argument.4. The radio transmitter device (200) according to claim 3, wherein the mutual coupling matrix is basedon: the mutual impedance matrix satisfying: ^ Re^^( )^ / ^ ^^ / ^ ^= ^ ^ ^^ Re(^) ^Re^^(^)^^, where ^ = ^^^,^^ denotes the mutual coupling matrix; orthe spatial direction -based radiation power pattern and a position of each antenna element satisfying:1 ^^,^=^ ^^^ / ^(^)^^ / ^(^)^^^^^ ^^^(^^^^^)^^ , ∀^, ^ = 1,2, ⋯ ^, 4^^^∈^^^^where ^^(^) denotes the radiation power pattern, ^ denotes the spatial direction, and ^^ denotes the posi-tion of an antenna element ^.5. The radio transmitter device (200) according to claim 1 or 2, wherein the transfer matrix is based on:a coupling transfer matrix, such that ^ = ^^^, where ^ denotes the transfer matrix, ^ denotes the couplingtransfer matrix, H denotes a conjugate transpose operator, ^ denotes a scaling factor, and the coupling transfer matrix is basedat least on the mutual impedance matrix; or the mutual impedance matrix, the load impedances, and the characteristic impedances, such that:^ / ^^ ^ ^^( )^ / ^ ^= 2^^ (^ + ^ + ^ )^^^Re^^^^ , where ^ = {^^,^} denotes the mutual impedance matrix, ^^^,^|^ = 1,2, ⋯ ^^ denotes the load impedances,^^^,^|^ = 1,2, ⋯ ^^ denotes the characteristic impedances, n denotes a first antenna element index, m denotes a second antennaelement index, ^ denotes a scaling factor, ^^ denotes an ^ × ^ diagonal load impedance matrix with ^^,^ as its ^-th diagonalentry, ^^ denotes an ^ × ^ diagonal characteristic impedance matrix with ^^,^ as its ^-th diagonal entry, ^(^)denotes an^ × ^ diagonal matrix with ^^,^ as its ^-th diagonal entry, and Re(⋅) returns a real part of its argument.6. The radio transmitter device (200) according to claim 3 or 5, wherein:the coupling transfer matrix is based on the mutual impedance matrix, the characteristic impedances, and theload impedances, such that: ^= 2 ^Re^^(^)^ / ^ ^^ (^ + ^^ + ^^)^^^ ^ / ^ ^ ; or the coupling transfer matrix is based on the mutual impedance matrix and a current generator matrix associ-ated with the radio transmitter device (200), such that: ^^^(^)^ / ^ ^= ^ ^ ^^ ^ ^, where ^ denotes the current generator matrix.7. The radio transmitter device (200) according to claim 6, wherein the current generator matrix is based on the mutual impedance matrix, the characteristic impedances, and the load impedances, such that: ^= 2√2(^ + ^ + ^ )^^^ / ^ ^^^^. 8. The radio transmitter device (200) according to any of claims 1 to 7, further comprising a group ofconfigurable external load impedances (403), each configurable external load impedance (403) serially connected to a respec-tive antenna element in order to configure a respective load impedance.9. The radio transmitter device (200) according to any of claims 1 to 8, wherein the characteristic imped-ances are configured based on at least one of geometries or materials of the transmission lines of the pairs of transmission lines(411).10. The radio transmitter device (200) according to any of claims 1 to 9, wherein an amount of the pairs oftransmission lines (411) exceeds an amount of the antenna elements, and a connection between an antenna element and itsrespective signal generator (412) is switchable among the pairs of transmission lines (411) with different characteristic imped-ance values. 11. A method (300) for a radio transmitter device (200) comprising: an antenna array (420) of antenna elements, the antenna array (420) configured to act as a holographic surface(430), the antenna array (420) having an ^ × ^ mutual impedance matrix of mutual impedances between antenna elements,where each antenna element has a corresponding load impedance, in which N denotes a number of the antenna elements; andatransmit circuit (410) configured to drive the holographic surface (430), the transmit circuit (410) compris-ing: at least one pair of transmission lines (411) per each antenna element, each pair of trans-mission lines (411) connectable to a respective antenna element and having a respective characteristic impedance; anda signal generator (412) per each antenna element, each signal generator (412) configured to drive a respective antenna element via a respective pair of transmission lines (411), wherein the method (300) comprises: obtaining (301), by the radio transmitter device (200), a first beamforming vector;obtaining (302), by the radio transmitter device (200), a transfer matrix based on at least one of the mutualimpedance matrix, load impedances of the antenna elements, or characteristic impedances of the pairs of transmission lines(411);obtaining (303), by the radio transmitter device (200), a second beamforming vector based on a product ofthe obtained transfer matrix and the obtained first beamforming vector; using (304), by the radio transmitter device (200), the transmit circuit (410) to generate a signal stream beam-formed with the obtained second beamforming vector; and transmitting (305), by the radio transmitter device (200), the generated signal stream via the antenna array (420). 12. A computer program product comprising program code configured to perform a method according to claim 11, when the program code is executed on a radio transmitter device. 13. An apparatus, comprising means for carrying out the method (300) according to claim 11. 14. A network node device (120), comprising the radio transmitter device (200) according to any of claims1 to 10.

[0002] ABSTRACTDevices, methods and computer programs for energy-efficient beamforming in a holographic radiotransmitter are disclosed. The invention allows a structure for an antenna array of antenna elementsconfigured to act as a holographic surface and for an associated transmit circuit in a radio trans-mitter device that enables energy-efficient beamforming to be performed by the radio transmitterdevice. (Fig.4)

Claims

CLAIMS1 . A radio transmitter device (200), comprising: an antenna array (420) of antenna elements, the antenna array (420) configured to act as a holographic surface (430), the antenna array (420) having an IV x N mutual impedance matrix of mutual impedances between antenna elements, where each antenna element has a corresponding load impedance, in which N denotes a number of the antenna elements; and a transmit circuit (410) configured to drive the holographic surface (430), the transmit circuit (410) comprising: at least one pair of transmission lines (411) per each antenna element, each pair of transmission lines (411) connectable to a respective antenna element and having a respective characteristic impedance; and a signal generator (412) per each antenna element, each signal generator (412) configured to drive a respective antenna element via a respective pair of transmission lines (411), wherein the radio transmitter device (200) is configured to: obtain a first beamforming vector; obtain a transfer matrix based on at least one of the mutual impedance matrix, load impedances of the antenna elements, or characteristic impedances of the pairs of transmission lines (411); obtain a second beamforming vector based on a product of the obtained transfer matrix and the obtained first beamforming vector; use the transmit circuit (410) to generate a signal stream beamformed with the obtained second beamforming vector; and transmit the generated signal stream via the antenna array (420).2 . The radio transmitter device (200) according to claim 1 , wherein the first beamforming vector is based on a wireless channel vector of a wireless channel (140) between the holographic surface (430) and a receive antenna of a receiver (130), the wireless channel vector being: constructed based at least on positions of the antenna elements within the holographic surface (430), and further based on spatial direction -based radiation power patterns of the antenna elements within the holographic surface (430); estimated from pilot signals received from the receiver (130); or fed back from the receiver (130).3 . The radio transmitter device (200) according to claim 2, wherein the transfer matrix is based on: a coupling transfer matrix and a mutual coupling matrix, such that T = EA-1C-1, where T denotes the transfer matrix, A denotes the coupling transfer matrix, C denotes the mutual coupling matrix, s denotes a scaling factor, and the coupling transfer matrix is based at least on the mutual impedance matrix; or the mutual impedance matrix, the load impedances, and the characteristic impedances, such that:where Z = {Zm n} denotes the mutual impedance matrix, {zL n|n=1,2, ••• 1V} denotes the load impedances, {zO n|n=1,2, ••• N] denotes the characteristic impedances, n denotes a first antenna element index, m denotes a second antenna element index, s denotes a scaling factor, ZLdenotes an IV x IV diagonal load impedance matrix with ZL nas its n-tli diagonal entry, Zodenotes an IV x IV diagonal characteristic impedance matrix with Zo nas its n-th diagonal entry, Z(d)denotes an IV x IV diagonal matrix with Zn nas its n-tli diagonal entry, and Re(-) returns a real part of its argument.4 . The radio transmitter device (200) according to claim 3, wherein the mutual coupling matrix is based on: the mutual impedance matrix satisfying:where C = {cm n} denotes the mutual coupling matrix; or the spatial direction -based radiation power pattern and a position of each antenna element satisfying: Vm, n = 1,2, ••• N,where Rn(u) denotes the radiation power pattern, u denotes the spatial direction, and tndenotes the position of an antenna element n.5 . The radio transmitter device (200) according to claim 1 or 2, wherein the transfer matrix is based on: a coupling transfer matrix, such that T = EAH, where T denotes the transfer matrix, A denotes the coupling transfer matrix, H denotes a conjugate transpose operator, s denotes a scaling factor, and the coupling transfer matrix is based at least on the mutual impedance matrix; or the mutual impedance matrix, the load impedances, and the characteristic impedances, such that:where Z = {Zm n} denotes the mutual impedance matrix, {zL n|n = 1,2, ••• 1V} denotes the load impedances, {zO n|n = 1,2, ••• N] denotes the characteristic impedances, n denotes a first antenna element index, m denotes a second antenna element index, s denotes a scaling factor, ZLdenotes an IV x IV diagonal load impedance matrix with ZL nas its n-th diagonal entry, Zodenotes an IV x IV diagonal characteristic impedance matrix with Zo nas its n-th diagonal entry, Z(d)denotes an IV x IV diagonal matrix with Znnas its n-th diagonal entry, and Re(-) returns a real part of its argument.6 . The radio transmitter device (200) according to claim 3 or 5, wherein: the coupling transfer matrix is based on the mutual impedance matrix, the characteristic impedances, and the load impedances, such that:the coupling transfer matrix is based on the mutual impedance matrix and a current generator matrix associated with the radio transmitter device (200), such that:where B denotes the current generator matrix.7 . The radio transmitter device (200) according to claim 6, wherein the current generator matrix is based on the mutual impedance matrix, the characteristic impedances, and the load impedances, such that:B = 2V2(Z + ZL+ Z0)-1ZQ / 2.8 . The radio transmitter device (200) according to any of claims 1 to 7, further comprising a group of configurable external load impedances (403), each configurable external load impedance (403) serially connected to a respective antenna element in order to configure a respective load impedance.9 . The radio transmitter device (200) according to any of claims 1 to 8, wherein the characteristic impedances are configured based on at least one of geometries or materials of the transmission lines of the pairs of transmission lines (411).10 . The radio transmitter device (200) according to any of claims 1 to 9, wherein an amount of the pairs of transmission lines (411) exceeds an amount of the antenna elements, and a connection between an antenna element and its respective signal generator (412) is switchable among the pairs of transmission lines (411) with different characteristic impedance values.11 . A method (300) for a radio transmitter device (200) comprising: an antenna array (420) of antenna elements, the antenna array (420) configured to act as a holographic surface (430), the antenna array (420) having an IV x IV mutual impedance matrix of mutual impedances between antenna elements, where each antenna element has a corresponding load impedance, in which N denotes a number of the antenna elements; and atransmit circuit (410) configured to drive the holographic surface (430), the transmit circuit (410) comprising: at least one pair of transmission lines (411) per each antenna element, each pair of transmission lines (411) connectable to a respective antenna element and having a respective characteristic impedance; and a signal generator (412) per each antenna element, each signal generator (412) configured to drive a respective antenna element via a respective pair of transmission lines (411), wherein the method (300) comprises: obtaining (301), by the radio transmitter device (200), a first beamforming vector; obtaining (302), by the radio transmitter device (200), a transfer matrix based on at least one of the mutual impedance matrix, load impedances of the antenna elements, or characteristic impedances of the pairs of transmission lines (411);obtaining (303), by the radio transmitter device (200), a second beamforming vector based on a product of the obtained transfer matrix and the obtained first beamforming vector; using (304), by the radio transmitter device (200), the transmit circuit (410) to generate a signal stream beamformed with the obtained second beamforming vector; and transmitting (305), by the radio transmitter device (200), the generated signal stream via the antenna array (420).12 . A computer program product comprising program code configured to perform a method according to claim 11, when the program code is executed on a radio transmitter device.13 . An apparatus, comprising means for carrying out the method (300) according to claim 11.14 . A network node device (120), comprising the radio transmitter device (200) according to any of claims 1 to 10.

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