Docket. No. SMM920220273-WO-PCT 1 EQUAL POWER PRECODER MAPPINGS FOR TWO-DIMENSIONAL ANTENNA ARRAYS WITH MUTUAL COUPLING PRIORITY APPLICATION [0001] This application claims priority to U.S. Provisional Application No. 63/491,617, filed March 22, 2023, the contents of which is fully incorporated herein by reference. TECHNICAL FIELD [0002] The present disclosure relates to wireless communications, and more specifically to precoding transmissions via an antenna array. BACKGROUND [0003] A wireless communications system may include one or multiple network communication devices, including base stations, which may be otherwise known as an eNodeB (eNB), a next-generation NodeB (gNB), or other suitable terminology. Each network communication device, such as a base station, may support wireless communications for one or multiple user communication devices, which may be otherwise known as user equipment (UE), or other suitable terminology. The wireless communications system may support wireless communications with one or multiple user communication devices by utilizing resources of the wireless communications system (e.g., time resources (e.g., symbols, slots, subframes, frames, or the like) or frequency resources (e.g., subcarriers, carriers). Additionally, the wireless communications system may support wireless communications across various radio access technologies including third generation (3G) radio access technology, fourth generation (4G) radio access technology, fifth generation (5G) radio access technology, and other suitable radio access technologies beyond 5G (e.g., sixth generation (6G)). [0004] To support these radio access technologies, many wireless communication devices have multiple antennas that can transmit more focused signals to a receiving device using antenna beamforming. Precoding is a generalization of beamforming to support multi-stream (or multi-layer) transmission in multi-antenna wireless communications. In conventional
Docket. No. SMM920220273-WO-PCT 2 single-stream beamforming, the same signal is emitted from each of the transmit antennas with appropriate complex amplitude weighting (i.e., phase and gain) such that the signal power is maximized at the receiver output. Multiple telecommunication standards define antenna precoder codebooks to support antenna beamforming and or multiple input/multiple output (MIMO) transmission with feedback from a receiver in the receiving device. SUMMARY [0005] The present disclosure relates to methods, apparatuses, and systems that support wireless communications by efficiently precoding transmissions made via a two-dimensional antenna array with significant mutual coupling between antenna elements of the antenna array. Some implementations of the method and apparatuses described herein may include a method for wireless communication at a communication device such as a user device or a network device. In one or more embodiments, the method may include receiving a two- dimensional precoder comprised of a vertical precoder of vertical dimension M and a horizontal precoder of horizontal dimension N. The method may include representing a composite precoder as a Kronecker product of the vertical precoder and the horizontal precoder. The method may include determining a positive definite (“Q”) matrix of dimensions M x N rows by M x N (“MN x MN”) columns based on antenna array and antenna circuitry used to drive the antenna array. The antenna array is of a first integer number “M” of rows and a second integer number “N” of columns. Each of numbers M and N are greater than 1. The method may include determining a transformed precoder based on the Q matrix. BRIEF DESCRIPTION OF THE DRAWINGS [0006] FIG. 1 is an example of a wireless communications system enabling wireless communication between base stations and user devices by efficiently precoding transmissions made via a two-dimensional antenna array with significant mutual coupling between antenna elements of the antenna array, in accordance with aspects of the present disclosure.
Docket. No. SMM920220273-WO-PCT 3 [0007] FIG.2 is an antenna array modeled as a K-port device, in accordance with aspects of the present disclosure. [0008] FIG. 3 is a two-dimensional antenna array with M rows and N columns, in accordance with aspects of the present disclosure. [0009] FIG. 4 is a diagram of a precoder implementation for a two-dimensional antenna array, in accordance with aspects of the present disclosure. [0010] FIG. 5 illustrates a block diagram of a communication device that supports efficient precoding transmissions made via a two-dimensional antenna array with significant mutual coupling between antenna elements of the antenna array, in accordance with aspects of the present disclosure. [0011] FIG.6 illustrates a flowchart of a method for wireless communication performed by a communication device for efficiently precoding transmissions made via a two- dimensional antenna array with significant mutual coupling between antenna elements of the antenna array, in accordance with aspects of the present disclosure. DETAILED DESCRIPTION [0012] In the Third Generation Partnership Project (3GPP), precoder and beam optimization relies on the assumption that the precoders map to equal power antenna patterns. However, due to the mutual coupling that occurs with antenna arrays with element spacings on the order of 0.5 to 0.7 wavelengths, this assumption is not satisfied, as power variations of multiple decibels (dB) can occur over the set of precoders. This problem has not been addressed in 3GPP presumably due to the difficulty of making accurate radiated measurements. This problem has been considered previously for one-dimensional antenna arrays, but with limitation on the types of precoder implementations in terms of voltage/current sources, transmission lines, and other implementation details. In this disclosure, equal power antenna precoder mappings are considered for the case of two- dimensional arrays with independent row and column precoders, and with the only limitation being that the implementation is linear with respect to the precoder.
Docket. No. SMM920220273-WO-PCT 4 [0013] In one aspect, a problem solved by the present disclosure is how to map precoders for two-dimensional antenna arrays with mutual coupling to equal power antenna patterns in a manner in which the modified precoder can still be implemented with separate horizontal and vertical precoders. The present disclosure provides a method for mapping antenna precoders for two-dimensional arrays with mutual coupling to equal power antenna patterns. The mapping depends on the precoder implementation, with the only limitation being that the mapping is linear with respect to the precoder so that the power can be represented as a quadratic form. By factoring this quadratic form, a linear transformation can be applied to the set of precoders such that they have equal power. By approximating the precoder with the nearest Kronecker product, the precoder can be implemented as separate vertical and horizontal precoders. [0014] In one or more embodiments, the two-dimensional antenna precoder is represented as a Kronecker product and the transmitted power can be represented as a quadratic form in terms of the Kronecker product and a positive definite matrix Q, where the positive definite matrix depends on the impedance parameters (or scattering parameters) of the antenna array and the precoder implementation. The precoder is then mapped to a new precoder using a linear transformation based on factorization of the matrix Q such that the transformed precoders all have equal power. The nearest Kronecker product is then used to map the transformed precoder to separate vertical and horizontal precoders. [0015] In one or more embodiments, a method is provided by a transmitter with a two- dimensional antenna array of dimension M x N. The method includes receiving a precoder comprised of a vertical precoder and a horizontal precoder. The method includes representing a composite precoder as a Kronecker product of the vertical precoder and the horizontal precoder. The method includes determining a positive definite matrix Q of dimension MN x MN based on the antenna array. The method includes determining a transformation matrix of dimension MN x MN based on the Q matrix. The method includes determining a transformed precoder by multiplying the composite precoder by the transformation matrix. In one or more particular embodiments, the method further includes determining the nearest
Docket. No. SMM920220273-WO-PCT 5 Kronecker product for the transformed precoder in terms of a vertical precoder and a horizontal precoder. [0016] FIG. 1 illustrates an example of a wireless communications system 100 enabling wireless communication between base stations and user devices while mitigating cross-link interference between base stations, in accordance with aspects of the present disclosure. The wireless communications system 100 may include one or more network devices 102, one or more UEs 104, a core network 106, and a packet data network 109. The wireless communications system 100 may support various radio access technologies. In some implementations, the wireless communications system 100 may be a 4G network, such as an LTE network or an LTE-Advanced (LTE-A) network. In some other implementations, the wireless communications system 100 may be a 5G network, such as a New Radio (NR) network. In other implementations, the wireless communications system 100 may be a combination of a 4G network and a 5G network. The wireless communications system 100 may support radio access technologies beyond 5G, or other suitable radio access technology including Institute of Electrical and Electronics Engineers (IEEE) 802.11 (Wi-Fi), IEEE 802.16 (WiMAX), IEEE 802.20. Additionally, the wireless communications system 100 may support technologies, such as time division multiple access (TDMA), frequency division multiple access (FDMA), or code division multiple access (CDMA), etc. [0017] The one or more network devices 102 may be dispersed throughout a geographic region to form the wireless communications system 100. One or more of the network devices 102 described herein may be, may include, or may be referred to as a network node, a base station, a network element, a radio access network (RAN), a base transceiver station, an access point, a NodeB, an eNodeB (eNB), a next-generation NodeB (gNB), a network device, or other suitable terminology. A network device 102 and a UE 104 may communicate via a communication link 108, which may be a wireless or wired connection. For example, a network device 102 and a UE 104 may wirelessly communicate (e.g., receive signaling, transmit signaling) over a user to user (Uu) interface. [0018] A network device 102 may provide a geographic coverage area 110 for which the network device 102 may support services (e.g., voice, video, packet data, messaging,
Docket. No. SMM920220273-WO-PCT 6 broadcast, etc.) for one or more UEs 104 within the geographic coverage area 110. For example, a network device 102 and a UE 104 may support wireless communication of signals related to services (e.g., voice, video, packet data, messaging, broadcast, etc.) according to one or multiple radio access technologies. In some implementations, a network device 102 may be moveable, for example, a satellite 107 associated with a non-terrestrial network and communicating via a satellite link 111. In some implementations, different geographic coverage areas 110 associated with the same or different radio access technologies may overlap, but the different geographic coverage areas 110 may be associated with different network devices 102. Information and signals described herein may be represented using any of a variety of different technologies and techniques. For example, data, instructions, commands, information, signals, bits, symbols, and chips that may be referenced throughout the description may be represented by voltages, currents, electromagnetic waves, magnetic fields or particles, optical fields or particles, or any combination thereof. [0019] The one or more UEs 104 may be dispersed throughout a geographic region of the wireless communications system 100. A UE 104 may include or may be referred to as a mobile device, a wireless device, a remote device, a remote unit, a handheld device, or a subscriber device, or some other suitable terminology. In some implementations, the UE 104 may be referred to as a unit, a station, a terminal, or a client, among other examples. Additionally, or alternatively, the UE 104 may be referred to as an Internet-of-Things (IoT) device, an Internet-of-Everything (IoE) device, or machine-type communication (MTC) device, among other examples. In some implementations, a UE 104 may be stationary in the wireless communications system 100. In some other implementations, a UE 104 may be mobile in the wireless communications system 100. [0020] The one or more UEs 104 may be devices in different forms or having different capabilities. Some examples of UEs 104 are illustrated in FIG.1. A UE 104 may be capable of communicating with various types of devices, such as the network devices 102, other UEs 104, or network equipment (e.g., the core network 106, the packet data network 109, a relay device, an integrated access and backhaul (IAB) node, or another network equipment), as shown in FIG.1. Additionally, or alternatively, a UE 104 may support communication with
Docket. No. SMM920220273-WO-PCT 7 other network devices 102 or UEs 104, which may act as relays in the wireless communications system 100. [0021] A UE 104a may also be able to support wireless communication directly with other UEs 104b over a communication link 112. For example, a UE 104 may support wireless communication directly with another UE 104 over a device-to-device (D2D) communication link. In some implementations, such as vehicle-to-vehicle (V2V) deployments, vehicle-to- everything (V2X) deployments, or cellular-V2X deployments, the communication link 112 may be referred to as a sidelink. For example, a UE 104a may support wireless communication directly with another UE 104b over a PC5 interface. PC5 refers to a reference point where the UE 104a directly communicates with another UE 104b over a direct channel without requiring communication with the network device 102a. [0022] A network device 102 may support communications with the core network 106, or with another network device 102, or both. For example, a network device 102 may interface with the core network 106 through one or more backhaul links 114 (e.g., via an S1, N2, or another network interface). The network devices 102 may communicate with each other over the backhaul links 114 (e.g., via an X2, Xn, or another network interface). In some implementations, the network devices 102 may communicate with each other directly (e.g., between the network devices 102). In some other implementations, the network devices 102 may communicate with each other indirectly (e.g., via the core network 106). In some implementations, one or more network devices 102 may include subcomponents, such as an access network entity, which may be an example of an access node controller (ANC). An ANC may communicate with the one or more UEs 104 through one or more other access network transmission entities, which may be referred to as a radio heads, smart radio heads, or transmission and reception points (TRPs). [0023] In some implementations, a network entity or network device 102 may be configured in a disaggregated architecture, which may be configured to utilize a protocol stack physically or logically distributed among two or more network entities or network devices 102, such as an integrated access backhaul (IAB) network, an open RAN (O-RAN) (e.g., a network configuration sponsored by the O-RAN Alliance), or a virtualized RAN
Docket. No. SMM920220273-WO-PCT 8 (vRAN) (e.g., a cloud RAN (C-RAN)). For example, a network entity or network device 102 may include one or more of a central unit (CU), a distributed unit (DU), a radio unit (RU), a RAN Intelligent Controller (RIC) (e.g., a Near-Real Time RIC (Near-RT RIC), a Non-Real Time RIC (Non-RT RIC)), a Service Management and Orchestration (SMO) system, or any combination thereof. [0024] An RU may also be referred to as a radio head, a smart radio head, a remote radio head (RRH), a remote radio unit (RRU), or a transmission and reception point (TRP). One or more components of the network entities or network devices 102 in a disaggregated RAN architecture may be co-located, or one or more components of the network entities or network devices 102 may be located in distributed locations (e.g., separate physical locations). In some implementations, one or more network entities or network devices 102 of a disaggregated RAN architecture may be implemented as virtual units (e.g., a virtual CU (VCU), a virtual DU (VDU), a virtual RU (VRU)). [0025] Split of functionality between a CU, a DU, and an RU may be flexible and may support different functionalities depending upon which functions (e.g., network layer functions, protocol layer functions, baseband functions, radio frequency functions, and any combinations thereof) are performed at a CU, a DU, or an RU. For example, a functional split of a protocol stack may be employed between a CU and a DU such that the CU may support one or more layers of the protocol stack and the DU may support one or more different layers of the protocol stack. In some implementations, the CU may host upper protocol layer (e.g., a layer 3 (L3), a layer 2 (L2)) functionality and signaling (e.g., Radio Resource Control (RRC), service data adaptation protocol (SDAP), Packet Data Convergence Protocol (PDCP). The CU may be connected to one or more DUs or RUs, and the one or more DUs or RUs may host lower protocol layers, such as a layer 1 (L1) (e.g., physical (PHY) layer) or an L2 (e.g., radio link control (RLC) layer, medium access control (MAC) layer) functionality and signaling, and may each be at least partially controlled by the CU. [0026] Additionally, or alternatively, a functional split of the protocol stack may be employed between a DU and an RU such that the DU may support one or more layers of the
Docket. No. SMM920220273-WO-PCT 9 protocol stack and the RU may support one or more different layers of the protocol stack. The DU may support one or multiple different cells (e.g., via one or more RUs). In some implementations, a functional split between a CU and a DU, or between a DU and an RU may be within a protocol layer (e.g., some functions for a protocol layer may be performed by one of a CU, a DU, or an RU, while other functions of the protocol layer are performed by a different one of the CU, the DU, or the RU). [0027] A CU may be functionally split further into CU control plane (CU-CP) and CU user plane (CU-UP) functions. A CU may be connected to one or more DUs via a midhaul communication link (e.g., F1, F1-c, F1-u), and a DU may be connected to one or more RUs via a fronthaul communication link (e.g., open fronthaul (FH) interface). In some implementations, a midhaul communication link or a fronthaul communication link may be implemented in accordance with an interface (e.g., a channel) between layers of a protocol stack supported by respective network entities or network devices 102 that are in communication via such communication links. [0028] The core network 106 may support user authentication, access authorization, tracking, connectivity, and other access, routing, or mobility functions. The core network 106 may be an evolved packet core (EPC), or a 5G core (5GC), which may include a control plane entity that manages access and mobility (e.g., a mobility management entity (MME), an access and mobility management functions (AMF)) and a user plane entity that routes packets or interconnects to external networks (e.g., a serving gateway (S-GW), a Packet Data Network (PDN) gateway (P-GW), or a user plane function (UPF)). In some implementations, the control plane entity may manage non-access stratum (NAS) functions, such as mobility, authentication, and bearer management for the one or more UEs 104 served by the one or more network devices 102 associated with the core network 106. [0029] The core network 106 may communicate with the packet data network 109 over one or more backhaul links 116 (e.g., via an S1, N2, N2, or another network interface). The packet data network 109 may include an application server 118. In some implementations, one or more UEs 104 may communicate with the application server 118. A UE 104 may establish a session (e.g., a protocol data unit (PDU) session, or the like) with the core network
Docket. No. SMM920220273-WO-PCT 10 106 via a network entity or network device 102. The core network 106 may route traffic (e.g., control information, data, and the like) between the UE 104 and the application server 118 using the established session (e.g., the established PDU session). The PDU session may be an example of a logical connection between the UE 104 and the core network 106 (e.g., one or more network functions of the core network 106). [0030] In the wireless communications system 100, the network entities or network devices 102 and the UEs 104 may use resources of the wireless communications system 100 (e.g., time resources (e.g., symbols, slots, subframes, frames, or the like) or frequency resources (e.g., subcarriers, carriers)) to perform various operations (e.g., wireless communications). In some implementations, the network entities or network devices 102 and the UEs 104 may support different resource structures. For example, the network entities or network devices 102 and the UEs 104 may support different frame structures. In some implementations, such as in 4G, the network entities or network devices 102 and the UEs 104 may support a single frame structure. In some other implementations, such as in 5G and among other suitable radio access technologies, the network entities or network devices or network devices 102 and the UEs 104 may support various frame structures (i.e., multiple frame structures). The network entities or network devices 102 and the UEs 104 may support various frame structures based on one or more numerologies. [0031] One or more numerologies may be supported in the wireless communications system 100, and a numerology may include a subcarrier spacing and a cyclic prefix. A first numerology (e.g., ^=0) may be associated with a first subcarrier spacing (e.g., 15 kHz) and a normal cyclic prefix. In some implementations, the first numerology (e.g., ^=0) associated with the first subcarrier spacing (e.g., 15 kHz) may utilize one slot per subframe. A second numerology (e.g., ^=1) may be associated with a second subcarrier spacing (e.g., 30 kHz) and a normal cyclic prefix. A third numerology (e.g., ^=2) may be associated with a third subcarrier spacing (e.g., 60 kHz) and a normal cyclic prefix or an extended cyclic prefix. A fourth numerology (e.g., ^=3) may be associated with a fourth subcarrier spacing (e.g., 120 kHz) and a normal cyclic prefix. A fifth numerology (e.g., ^=4) may be associated with a fifth subcarrier spacing (e.g., 240 kHz) and a normal cyclic prefix.
Docket. No. SMM920220273-WO-PCT 11 [0032] A time interval of a resource (e.g., a communication resource) may be organized according to frames (also referred to as radio frames). Each frame may have a duration, for example, a 10 millisecond (ms) duration. In some implementations, each frame may include multiple subframes. For example, each frame may include 10 subframes, and each subframe may have a duration, for example, a 1 ms duration. In some implementations, each frame may have the same duration. In some implementations, each subframe of a frame may have the same duration. [0033] Additionally, or alternatively, a time interval of a resource (e.g., a communication resource) may be organized according to slots. For example, a subframe may include a number (e.g., quantity) of slots. The number of slots in each subframe may also depend on the one or more numerologies supported in the wireless communications system 100. For instance, the first, second, third, fourth, and fifth numerologies (i.e., ^=0, ^=1, ^=2, ^=3, ^=4) associated with respective subcarrier spacings of 15 kHz, 30 kHz, 60 kHz, 120 kHz, and 240 kHz may utilize a single slot per subframe, two slots per subframe, four slots per subframe, eight slots per subframe, and 16 slots per subframe, respectively. Each slot may include a number (e.g., quantity) of symbols (e.g., OFDM symbols). In some implementations, the number (e.g., quantity) of slots for a subframe may depend on a numerology. For a normal cyclic prefix, a slot may include 14 symbols. For an extended cyclic prefix (e.g., applicable for 60 kHz subcarrier spacing), a slot may include 12 symbols. The relationship between the number of symbols per slot, the number of slots per subframe, and the number of slots per frame for a normal cyclic prefix and an extended cyclic prefix may depend on a numerology. It should be understood that reference to a first numerology (e.g., ^=0) associated with a first subcarrier spacing (e.g., 15 kHz) may be used interchangeably between subframes and slots. [0034] In the wireless communications system 100, an electromagnetic (EM) spectrum may be split, based on frequency or wavelength, into various classes, frequency bands, frequency channels, etc. By way of example, the wireless communications system 100 may support one or multiple operating frequency bands, such as frequency range designations FR1 (410 MHz – 7.125 GHz), FR2 (24.25 GHz – 52.6 GHz), FR3 (7.125 GHz – 24.25 GHz),
Docket. No. SMM920220273-WO-PCT 12 FR4 (52.6 GHz – 114.25 GHz), FR4a or FR4-1 (52.6 GHz – 71 GHz), and FR5 (114.25 GHz – 300 GHz). In some implementations, the network entities or network devices 102 and the UEs 104 may perform wireless communications over one or more of the operating frequency bands. In some implementations, FR1 may be used by the network entities or network devices 102 and the UEs 104, among other equipment or devices for cellular communications traffic (e.g., control information, data). In some implementations, FR2 may be used by the network entities or network devices 102 and the UEs 104, among other equipment or devices for short- range, high data rate capabilities. [0035] FR1 may be associated with one or multiple numerologies (e.g., at least three numerologies). For example, FR1 may be associated with a first numerology (e.g., ^=0), which includes 15 kHz subcarrier spacing; a second numerology (e.g., ^=1), which includes 30 kHz subcarrier spacing; and a third numerology (e.g., ^=2), which includes 60 kHz subcarrier spacing. FR2 may be associated with one or multiple numerologies (e.g., at least 2 numerologies). For example, FR2 may be associated with a third numerology (e.g., ^=2), which includes 60 kHz subcarrier spacing; and a fourth numerology (e.g., ^=3), which includes 120 kHz subcarrier spacing. [0036] In one or more embodiments, UE 104c uses a two-dimensional “MxN” antenna array 130a to transmit an uplink 131a to network device 102b. According to aspects of the present disclosure, UE 104c is optimized for two-dimensional precoder optimization for MxN antenna array 130a to transmit uplink 131a. Similarly, in one or more embodiments, network device 102b uses a two-dimensional “MxN” antenna array 130b to transmit a downlink 131b to UE 104c. According to aspects of the present disclosure, network device 102b is optimized for two-dimensional precoder optimization for MxN antenna array 130b to transmit downlink 131b. [0037] One aspect of the disclosure provides equal power precoder mappings for two- dimensional antenna arrays with mutual coupling. In 3GPP, precoder and beam optimization relies on the assumption that the precoders map to equal power antenna patterns. Generally, the objective is to select the precoder and/or beam which achieves the highest signal-to-noise ratio or highest user throughput, subject to a power constraint, since the result will: (i)
Docket. No. SMM920220273-WO-PCT 13 Maximize the user signal-to-noise ratio or user throughput in a power limited scenario; and (ii) Minimize the power needed to achieve a given target user signal-to-noise ratio or user throughput and thus minimize the interference into other users in an interference limited scenario. [0038] In 3GPP, all of the precoders have unit norm so that single layer precoders ^ for an antenna array with K antenna elements have the property that: [0039] ‖^‖^ = ∑^ ^^^ |^^|^ = 1 [0040] and multi-layer precoders ^ with L layers have the property that: [0041] ‖^‖^ = ∑^ ^^^ ∑^ ^^^ ^^^,^^^ = 1. [0042] There is an implicit assumption that precoders having equal norm will result in antenna patterns with equal power. As an example, let ^^ ^^, ^^ denote the antenna pattern (in watts per steradian) that results when precoder ^ is applied by the transmitter, where ^ is the azimuth angle and ^ is the elevation angle. The radiated power is then given by: 0043] p ^ = ^ ^^ ^ [ ^ ^ ^^ ^ "^! ^ ^! ^^^^, ^^ sin ^ &^ &^ . [0044] The assumption in the design of the precoders is that if ‖^^ ‖ = ‖^^ ‖, then p^^^ ^ = p^^^ ^. However, this assumption is not satisfied in the case that there is mutual coupling between the elements of the antenna array. [0045] Mutual coupling occurs when antenna elements are closely spaced relative to the wavelength of the carrier frequency that is to be transmitted or received by the array. Significant mutual coupling may occur when the antenna elements are spaced by less than a few wavelengths. Minimal mutual coupling will occur when the antenna elements are separated by ten or more wavelengths. With mutual coupling, when one antenna element in the array is driven with a voltage or current source, the radiation emitted from this element will induce currents on the neighboring antenna elements which will cause these neighboring antenna elements to radiate. Therefore, the antenna elements generally behave differently within the array than if the antenna element is removed from the array.
Docket. No. SMM920220273-WO-PCT 14 [0046] FIG. 2 is an antenna array modeled as a K-port device. In general, the mutual coupling can be represented by a K-port model as illustrated in FIG.2. Here the relationship between the voltage vector ' and current vector ^ is given by [0047] ' = ( ^ [0048] where ' is a vector of length K of voltages across each port terminal, ^ is the vector of currents into each of the ports, and ( is the impedance matrix. For an M x N matrix, the number of antenna ports is ) = * +. In the absence of mutual coupling, the matrix ( is diagonal and can be expressed as ( = diag/0!, 0^,…,0234 where 0^ denotes the self-impedance of the i-th antenna element. The matrix ( is non-diagonal if there is coupling between the antenna elements in the array. [0049] This issue has been considered previously in the case of one-dimensional antenna arrays. The present disclosure considers the case of a two-dimensional array. The disclosure assumes a two-dimensional antenna array of antenna elements with M rows and N columns. To begin, an assumption is made that the spacing between the elements in the horizontal dimension are equal (i.e., the spacing between the columns). Similarly, the spacing between the antenna elements in each column are assumed to be equal (i.e., the spacing between the rows). The spacing between the rows may or may not be equal to the spacing between the columns. For example, the spacing between the columns may be equal to 0.5 λ while the spacing between the rows is 0.7 λ, where λ is the wavelength of the carrier used for transmission or reception. [0050] With a few exceptions, most of the following description will also apply to the case where the columns are not equally spaced, and the rows are not equally spaced. In one example, the M rows can lie in the horizontal dimension and the N columns can lie in the vertical dimension. In another example, the M rows can lie in the vertical dimension while the N columns can lie in the horizontal dimension. More generally, the two-dimensional array can have any orientation with respect to a three-dimensional coordinate system. The three- dimensional coordinate system can be a Euclidean coordinate system or a spherical coordinate system.
Docket. No. SMM920220273-WO-PCT 15 [0051] FIG. 3 is a two-dimensional antenna array with M rows and N columns. For the example considered here, the M rows will lie in the horizontal dimension and the N columns 2 3 will lie in the vertical dimension as in FIG.2. In the example, let the matrix 5 = /A^,^4^^^,^^^ denote the M x N array of antenna elements. In 3GPP, the precoders for a two-dimensional 2 array are defined as two one-dimensional vectors each having unit norm. Let 78 = /w8,^4^^^ 3 denote the length-M vector applied in the vertical dimension and let 7: = /w:,^ 4 ^^^ denote the length-N vector applied in the horizontal dimension. Typically, these vectors are Discrete Fourier Transform vectors of the form [0052] 7: = ;1/√+>?exp^B 2DE^F − 1^/+^H3 ^^^ for E ∈ ?0, … , + − 1H, and similarly, [0053] 78 = ;1/√*>?exp^B 2DE^F − 1^/*^H2 ^^^ for E ∈ ?0, … , * − 1H. [0054] This set of precoders will yield N and M beams that cover the range [−90, 90] degrees in increments of 180/M and 180/N degrees in azimuth and elevation, respectively, if the element spacing in each dimension is equal to λ/2. More generally, if the elements have a spacing of d/λ and K beams are required from −90 degrees to 90 degrees, the precoders can be given by [0055] 7: = 1/√+> Mexp NB 2D ^O 3 ; P Q E^F − 1^/)RS ^^^ for E ∈ ?0, … , ) − 1H and 2 [0056] 78 = ;1/√*> Mexp NB 2D ^OT Q E^F − 1^/)RS ^^^ for E ∈ ?0, … , ) − 1H, [0057] where &: and &8 denote the horizontal and vertical spacing of the elements, respectively. [0058] In order to define the precoding operation as an inner product, the elements of the 2 3 two-dimensional M x N antenna array 5 = /a^,^4^^^,^^^ can be reordered as the MN x 1 vector U = vec^X^ in which the columns of the matrix A are stacked with the first column at the top and the N-th column at the bottom. Furthermore, let the combined precoder 7Y be defined as the Kronecker product:
Docket. No. SMM920220273-WO-PCT 16 [0059] 7Y = 78 ⊗ 7: [0060] which also has dimension MN x 1. The transmitter then applies the weights of the combined precoding vector 7Y to the vector of antenna elements U. As in the case of a linear array, a general assumption is that if the precoders 7Y have equal norm, then the power of the resulting antenna patterns ^7[^^, ^^ will have equal power. However, this assumption does not hold if there is mutual coupling between the antenna elements so that the impedance matrix Z is non-diagonal. [0061] FIG. 4 is a diagram of a precoder implementation for a two-dimensional antenna array. The combined precoder is indicated on the left side of FIG. 4 with MN outputs. The impedance matrix for the antenna array is shown on the right with MN inputs. While the 3GPP specification indicates the precoders to be used, the specification does not say anything about how the precoders should be applied. For example, the precoders could be implemented as a voltage source with a series impedance or as a current source with a shunt impedance. Additionally, there will generally be transmission lines between the voltage or current source and the antenna load. There may be an isolator at the source and matching circuitry at the antenna load. All of these implementation details will affect the mapping between the precoder 7Y and the antenna pattern ^7[ ^^, ^^. [0062] For a given precoder 7Y, the resulting antenna pattern, and thus the radiated power will depend on the precoder implementation. If the antenna pattern for a given implementation of the precoder 7Y is given by ^7[ ^^, ^^, where ^ is the azimuth angle and ^ is the elevation angle, then the power radiated from the antenna array can be expressed as ^ ^^ ^ [0063] p^7Y ^ = ^^ ^ "^! ^ ^! ^7[ ^^, ^^ sin ^ &^ &^ . [0064] Alternatively, the power received by the antenna array can be expressed in terms of the voltage vector ' and the current vector ^ at the input into the * ∙ + port device with impedance Z, which is given by: [0065] Re ^ ' ^ ^ ^ = Re ^ ' ^ ( _^ ' ^ = Re ^ ^ ^ ( ^ ^ ^ . [0066] The power into the array must equal the power out of the array so that
Docket. No. SMM920220273-WO-PCT 17 [0067] ^ ^^ ^ ^^ "^! ^ ^ ^ ^! ^7[ ^^, ^^ sin ^ &^ &^ = Re^' ^^. [0068] There are many possible implementations of a precoder, all of which yield a different relationship between the precoder 7Y, the antenna pattern ^7[ ^^, ^^ and the transmitted power p^7Y ^. In particular, the following possibilities can be considered: (i) Thevenin sources; (ii) Norton sources; (iii) Thevenin sources with transmission lines; and (iv) Thevenin source with transmission lines and source isolators. For all four of these cases, the transmitted power can be expressed as the quadratic form [0069] p^7Y ^ = 7`^ a 7Y [0070] where the matrix a is positive definite. [0071] For Thevenin sources, the matrix a is given by: [0072] a = ;( + (>_^ Re ( ;( + (>_^ b_d:e8 ^ ^ b_d:e8 [0073] where (b_d:e8 is the diagonal matrix of the series source impedances for the Thevenin sources and ( is the impedance matrix for the antenna array. [0074] For Norton sources, the matrix a is given by: [0075] a = (^ ;( + (>_^ Re ( ;( + (>_^ b_ghi b_ghi ^ ^ b_ghi (b_ghi [0076] where (b_ghi is the diagonal matrix of the shunt impedances for the Norton sources and ( is the impedance matrix for the antenna array. [0077] For Thevenin sources with transmission lines between the sources and the antenna elements, the matrix a is given by: a = N( + _^ _^ [0078] b_d:e8 (jk ^ l ^ R Re ; (jk ^ l ^> N(b_d:e8 + (jk ^ l ^ R [0079] where [0080] ( l = Z ;( + B Z n tan 2Dl >; Z n >_^ jk ^ ^ ! ! og ^ ^ ! og + B ( tan^2Dl^ , [0081] Z! is the impedance of the transmission line, l is the length of the transmission line in wavelengths, nog is the square identity matrix of dimension *+, (b_d:e8 is the
Docket. No. SMM920220273-WO-PCT 18 diagonal matrix of the series source impedances for the Thevenin sources, and ( is the impedance matrix for the antenna array. [0082] For a Thevenin source with transmission lines between the sources and the antenna elements and isolators at the source, the matrix a is given by: [0083] a = Z! (b _ _^ d:e8 ;nog − Re^q^q^> (b _ _^ d:e8 [0084] where q denotes the scattering matrix, Z! is the impedance of the transmission line, nog is the square identity matrix of dimension *+, (b_d:e8 is the diagonal matrix of the series source impedances for the Thevenin sources, and ( is the impedance matrix for the antenna array. The scattering matrix can be expressed in terms of the impedance matrix as [0085] q = ^( + Z!nog ^_^^( − Z!nog ^ . [0086] In order to modify the set of precoders 7Y so that the resulting antenna patterns have equal power, the matrix Q in the expression must be known to the transmitter, and this requires knowledge of the impedance matrix ( of the antenna array or equivalent parameters from which the impedance matrix can be determined. The impedance matrix ( can be expressed in terms of the scattering parameters q as [0087] ( = Z!^nog − q^_^^S + nog^ , [0088] where q can be measured using a network analyzer. Alternatively, the UE or gNB may be capable of measuring the reflected power from its antenna array and thus the S parameters and the impedance matrix ( can be determined. Another set of parameters equivalent to the impedance parameters are the admittance parameters Y from which the matrix ( can be expressed as ( = s_^. [0089] More generally, consideration can be provided for the case that the system is linear with respect to the precoder 7Y so that the voltage vector ' at the input to the MN-port antenna array can be expressed as ' = t 7Y , where t is an MN x MN matrix. Then the current vector ^ can be expressed as [0090] ^ = (_^' = (_^t 7Y ,
Docket. No. SMM920220273-WO-PCT 19 [0091] and the power is given by: [0092] p^7Y ^ = Re^'^^^ = Re^^t 7Y ^^(_^t 7Y ^ [0093] = Re^7u^ t^ (_^t 7Y^ [0094] = ^ ^ ^7u^ t^ (_^t 7Y + 7u^ t^ (_^t 7Y ^ [0095] = 7u^ v 7Y [0096] where [0097] v = t^wx^(^t. [0098] Thus, as long as the system driving the antenna array is linear with respect to the precoder 7Y, the transmitted power can always be represented as a quadratic form in terms of the real part of the impedance matrix. [0099] Given that the transmit power can always be expressed in this quadratic form for a linear system, consideration can be given on how to map the antenna precoders to equal power antenna patterns. According to aspects of the disclosure, there are essentially two approaches that can be considered: [00100] Approach 1: The first approach is to scale the existing set of precoders ?7YH by the inverse square root of p^7Y ^, so that [00101] 7Y y = 7[ , z7u{ v 7[ [00102] and [00103] p^7Y y^ = 7y`^ a 7Y y = 7u{ v 7[ 7u{ v 7[ = 1 [00104] so that the set of scaled precoders ?7Y yH all have equal power. The scaled precoders should be used for the data symbol precoders and the demodulation reference symbol precoders (DMRS). It should be noted, however, that the scaling factor is precoder dependent and so can be different for each precoder 7Y. There can be a drawback with this approach
Docket. No. SMM920220273-WO-PCT 20 when channel state information reference symbols (CSI-RS) are used by the receiving device to evaluate channel quality for each precoder. Unless the scaling factor ^7`^ v 7Y ^_^/^ used to scale each precoder 7Y is known to the receiver, the receiver cannot determine the channel quality that would result from the application of the scaled precoders ?7Y yH to the channel state information reference symbols. [00105] The scaling factors used at the transmitter can be signaled to the receiver. Depending on the number of precoders in the set ?7Y yH, one of the following two methods may be preferable: [00106] (i) Signal the scaling factor associated with each precoder. With this method, the complexity corresponds to the number of precoders in the set ?7Y yH; and [00107] (ii) Signal the upper diagonal (or lower diagonal) portion of the matrix v to the receiver. Here the number of coefficients is given by MN(MN+1)/2, where the antenna array has dimension M x N. [00108] The scaling values in method (i) are real-valued and the non-diagonal values in the matrix are in general complex-valued, so that the number of real valued coefficients is given by MN(MN+(1/2)). [00109] In implementation, Approach 1 does not change the mapping between the precoder 7Y and the shape of the antenna pattern ^7[^^, ^^; Rather, the approach only scales the pattern by a real constant. Thus, the shape of the antenna patterns ^7[ ^^, ^^ and ^7[ | ^^, ^^ are the same. [00110] Also, with this approach, the precoder can be implemented as separate vertical and horizontal precoders 78 and 7: with multiplication of one of the precoders by the scalar ^7`^ v 7Y^_^/^. This implementation may be beneficial in cases where the transmitter hardware cannot implement arbitrary precoders and can only implement precoders which can be represented as a Kronecker product.
Docket. No. SMM920220273-WO-PCT 21 [00111] Approach 2: The second approach is to transform the set of precoders ?7YH into a second set of precoders ?7Y yH by multiplying each precoder by a common transformation matrix } such that [00112] 7Y y = } 7Y [00113] and [00114] p^7Y y^ = 7y`^ a 7Y y = 7u^ 7Y = |7Y|^, [00115] and note that all of the precoders in the set ?7Y H have equal norm. [00116] To see how } can be defined, note that because the matrix v is Hermitian and positive definite, matrix v has an eigendecomposition given by [00117] v = ~^~^ [00118] where the columns of U are the eigenvectors of Q, and the elements of the diagonal matrix ^ are the corresponding eigenvalues, where eigenvalues are both real and non- negative. The approach defines [00119] ^ = ^^/^~^ [00120] where ^^/^ is the square root of ^ and notes that [00121] v = ^^^. [00122] According to aspect, the matrix ^1/2 is non-unique since each eigenvalue of Q has both a positive and negative square root. Here, ^1/2 is chosen to be the matrix comprised of the positive square roots. However, depending on the specifics of the implementation and the antenna patterns that result, defining ^1/2 to incorporate the negative square root for some eigenvalues may be beneficial in some cases. [00123] As next step in the process, define } = ^_^ so that [00124] ^^7Y y^ = 7y`^ a 7Y y = 7`^ }^ a } 7Y [00125] = 7`^ ^_^ ^^^ ^_^ 7Y y [00126] = 7`^ 7Y = |7Y|^ .
Docket. No. SMM920220273-WO-PCT 22 [00127] Since all of the precoders ?7YH have the same norm, all of the transformed precoders ?7Y yH will produce antenna patterns with the same power. [00128] The method of Approach 2 has the advantage that no scaling factors need to be signaled from the transmitter to the receiver. As long as the same transformation } is applied to the reference symbol precoders as is applied to the data symbol precoders ?7YH, the receiver can apply the original precoders ?7YH to determine the channel quality for the precoders ?7Y yH. [00129] With Approach 2, the shape of the antenna patterns ^7[ | ^^, ^^ for the transformed precoders ?7Y yH will be different than the shape of the antenna patterns ^7[ ^^, ^^ for the original precoders ?7YH since the transformation is not a simple scaling of the power. [00130] Unlike Approach 1, with Approach 2 the precoder cannot be implemented as separate vertical and horizontal precoders 78 and 7:. The reason for this is that the precoder 7Y y given by [00131] 7Y y = } 7Y [00132] cannot, in general, be expressed as a Kronecker product of a length-M vector and a length-N vector. As a result, the precoder 7Y y cannot be implemented as independent column and row precoders. [00133] If it is required that the precoder 7Y y be implemented as independent column and row precoders, then the nearest Kronecker product for 7Y y can be used. The nearest Kronecker product can be found by first unstacking the vector 7Y y into a matrix ^ of dimension M x N. Accordingly, let the singular value decomposition of ^ be given by [00134] ^ = ^q^^ . [00135] The nearest Kronecker product can then be expressed as [00136] ^^^ ^^ ⊗ '^
Docket. No. SMM920220273-WO-PCT 23 [00137] where ^^ and '^ are the first columns of U and V, and S11 is the element in the first row and first column of S. With this, the column precoder is given by ^^^^ ^^ and the row precoder is given by ^^^^ '^. [00138] As a summary of precoder mapping procedures, consideration is given for a two- 2 3 dimensional array with M rows and N columns. Let the matrix 5 = /A^,^ 4 ^^^,^^^ denote the M x N array of antenna elements. Define the MN x 1 vector U = vec^X^ in which the columns of the matrix A are stacked with the first column at the top and the N-th column at the bottom. [00139] An assumption can be made that the precoders for the vertical and horizontal dimensions are defined independently. 2 [00140] Let the set ?78H denote the set of length-M precoders 78 = /w8,^4^^^ applied in 3 the vertical dimension and let the set ?7: H denote the set of length-N vectors 7: = /w:,^4^^^ applied in the horizontal dimension. Also, an assumption is made that the same vertical precoder is applied across all N columns and the same horizontal precoder is applied across all M rows. [00141] Let the set ?7Y H denote the set of combined precoders defined as the Kronecker product of the vertical beamformer and the horizontal beamformer [00142] 7Y = 78 ⊗ 7: [00143] where 78 is an element of ?78H and 7: is an element of ?7:H. The dimension of the combined precoder 7Y is MN x 1. [00144] As was shown above, as long at the transmitter implementation of the precoder is linear with respect to the precoder 7Y, the transmitter power corresponding to precoder 7Y can be expressed as [00145] p^7Y ^ = 7u^ v 7Y , [00146] where the MN x MN matrix Q is positive definite. The matrix Q is known to the transmitter and has eigendecomposition given by
Docket. No. SMM920220273-WO-PCT 24 [00147] v = ~^~^ [00148] The matrix P is then defined as [00149] ^ = ^^/^~^ [00150] where ^^/^ is the positive square root of ^ (with all positive values). Finally, define } = ^_^. [00151] Approach 1: For Approach 1, the equal energy precoders ?7Y yH are given by [00152] 7Y y = 7[ { . z7u v 7[ [00153] The equal energy precoders are applied to the data symbols and the demodulation reference symbols, but not to the channel state information reference symbols. The transmitter then signals one of the following to the receiver: (i) the upper diagonal portion of the matrix v, or (ii) The scaling factor ^7`^ v 7Y ^_^/^ for all of the precoders in the set ?78 H. [00154] [00155] Approach 2: For Approach 2, the equal energy precoders ?7Y yH are given by [00156] 7Y y = } 7Y. [00157] As long as the same transformation } is applied to the reference symbol precoders as is applied to the data symbol precoders ?7Y H, the receiver can apply the original precoders ?7Y H to determine the channel quality for the precoders ?7Y yH. Thus, the transformation } must be applied to both the channel state information reference symbols (CSI-RS) and to the demodulation reference symbols. [00158] If the precoder must be implemented as a Kronecker product, then the nearest Kronecker product can be used, which can be expressed in terms of the singular value decomposition of the matrix ^ that is defined as the unstacked vector 7Y y . If ^ has the singular value decomposition [00159] ^ = ^q^^ , [00160] then the nearest Kronecker product is given by
Docket. No. SMM920220273-WO-PCT 25 [00161] ^^^ ^^ ⊗ '^ [00162] where ^^ and '^ are the first columns of U and V, and S11 is the element in the first row and first column of S. With this, the column precoder is given by ^^^^ ^^ and the row precoder is given by ^^^^ '^. [00163] FIG.5 illustrates an example of a block diagram 500 of a communication device 502 that performs wireless communication on an uplink or a downlink by efficiently precoding transmissions made via a two-dimensional antenna array with significant mutual coupling between antenna elements of the antenna array. The communication device 502 may be an example of a network device 102 or base station (102) or a UE 104 (FIG. 1), as described herein, that support wireless communication. The communication device 502 may include components for bi-directional communications including components for transmitting and receiving communications, such as a processor 504, a memory 506, and a transceiver 508. These components may be in electronic communication or otherwise coupled (e.g., operatively, communicatively, functionally, electronically, electrically) via one or more interfaces (e.g., buses). [00164] The processor 504, the memory 506, the transceiver 508, or various combinations thereof or various components thereof may be examples of means for performing various aspects of the present disclosure as described herein. For example, the processor 504, the memory 506, the transceiver 508, or various combinations or components thereof may support a method for performing one or more of the operations described herein. [00165] In some implementations, the processor 504, the memory 506, the transceiver 508, or various combinations or components thereof may be implemented in hardware (e.g., in communications management circuitry). The hardware may include a processor, a digital signal processor (DSP), an application-specific integrated circuit (ASIC), a field- programmable gate array (FPGA) or other programmable logic device, a discrete gate or transistor logic, discrete hardware components, or any combination thereof configured as or otherwise supporting a means for performing the functions described in the present disclosure. A controller 507 includes the processor 504 that configures the communication
Docket. No. SMM920220273-WO-PCT 26 device 502 to perform the functionality of the present disclosure. The controller 507 is communicatively coupled to the memory 506 to execute program code. Controller 507 may include dedicated memory accessible solely by the processor 504, that is a portion of memory 506. In some implementations, the processor 504 and the memory 506 coupled with the processor 504 may be configured to perform one or more of the functions as a controller 507 described herein (e.g., executing, by the processor 504, instructions stored in the memory 506). In an example, the processor 504 of a device controller 514 executes a communication application 509 to generate a signal for transmission by wireless communication via a two- dimensional antenna array 535. The processor 504 of a device controller 514 further executes a two-dimensional precoder mapping optimizer 540 to configure communication device 502 to efficiently precode the signal for transmission via the two-dimension antenna array 535. [00166] The processor 504 may include an intelligent hardware device (e.g., a general- purpose processor, a DSP, a CPU, a graphics processing unit (GPU), a microcontroller, an ASIC, an FPGA, a programmable logic device, a discrete gate or transistor logic component, a discrete hardware component, or any combination thereof). In some implementations, the processor 504 may be configured to operate a memory array using a memory controller. In some other implementations, a memory controller may be integrated into the processor 504. The processor 504 may be configured to execute computer-readable instructions stored in a memory (e.g., the memory 506) to cause the communication device 502 to perform various functions of the present disclosure. [00167] The memory 506 may include random access memory (RAM) and read-only memory (ROM). The memory 506 may store computer-readable, computer-executable code including instructions that, when executed by the processor 504 cause the communication device 502 to perform various functions described herein. The code may be stored in a non- transitory computer-readable medium such as system memory or another type of memory. In some implementations, the code may not be directly executable by the processor 504 but may cause a computer (e.g., when compiled and executed) to perform functions described herein. In some implementations, the memory 506 may include, among other things, a basic
Docket. No. SMM920220273-WO-PCT 27 input/output (I/O) system (BIOS) which may control basic hardware or software operation such as the interaction with peripheral components or devices. [00168] The communication device 502 may have an antenna subsystem 512 that includes the two-dimensional array 535. In some implementations, the antenna subsystem 512 of the communication device 502 may also include a single antenna for certain wireless communications. However, in some other implementations, the antenna subsystem 512 of the communication device 502 may have more than one antenna (i.e., multiple antennas), including multiple antenna panels or antenna arrays such as two-dimensional antenna array 535, which may be capable of concurrently transmitting or receiving multiple wireless transmissions. The transceiver 508 may communicate bi-directionally using one or more receivers 515 and one or more transmitters 517, via the antenna subsystem 512, wired, or wireless links, as described herein. For example, the transceiver 508 may represent a wireless transceiver and may communicate bi-directionally with another wireless transceiver. The transceiver 508 may also include a modem to modulate the packets, to provide the modulated packets to the antenna subsystem 512 for transmission, and to demodulate packets received from the antenna subsystem 512. The communication device 502 has the at least one transceiver 508 that includes at least one receiver 515 and at least one transmitter 517 that enable the communication device 502 to communicate with a network entity or network device 102a and/or alternatively to a user device, such as UE 104a (FIG.1). Specifically, in one embodiment, the communication device 502 can be a network device 102a such as a base station and transmits signals to a UE 104a. However, in an alternate embodiment, the communication device 502 can be a UE 104a and transmits signals to a network device 102a such as a base station. Thus, in the embodiments in which the communication device 502 is a base station, the controller can configure the base station to perform the various processes described herein. Alternatively, in the embodiments in which the communication device 502 is a UE, the controller can configure the UE to perform the various processes described herein. [00169] The communication device 502 may include a communication module 519 that is communicatively coupled to the controller 514. In some implementations, the
Docket. No. SMM920220273-WO-PCT 28 communication module 519 may be configured to perform various operations (e.g., receiving, monitoring, transmitting) using or otherwise in cooperation with the receiver 515, the transmitter 517, or both. For example, the communication module 519 may receive information from the receiver 515, send information to the transmitter 517, or be integrated in combination with the receiver 515, the transmitter 517, or both to receive information, transmit information, or perform various other operations as described herein. Although the communication module 519 is illustrated as a separate component, in some implementations, one or more functions described with reference to the communication module 519 may be supported by or performed by a processing subsystem such as controller 514, the memory 506, or any combination thereof. For example, the memory 506 may store code, which may include instructions executable by the controller 514 to cause/configure the communication device 502 to perform various aspects of the present disclosure as described herein. Alternatively, or in addition, the controller 514 and the memory 506 may be otherwise configured to perform or support such operations. [00170] FIG. 6 illustrates a flowchart of a method 600 for wireless communication at a communication device of performing wireless communication on an uplink or a downlink by efficiently precoding transmissions made via a two-dimensional antenna array with significant mutual coupling between antenna elements of the antenna array. The operations of the method 600 may be implemented by a device or its components as described herein. For example, the operations of the method 600 may be performed by a user device such as UE 104, by network device 102 (e.g., base station) (FIG. 1), or more generally by communication device 502 (FIG. 5). In some implementations, the communication device 502 may execute a set of instructions to control the function elements of the user device or network device to perform the described functions. Additionally, or alternatively, the communication device 502 may perform aspects of the described functions using special- purpose hardware. [00171] At 605, the method 600 may include receiving a two-dimensional precoder comprised of a vertical precoder of vertical dimension M and a horizontal precoder of horizontal dimension N. The operations of 605 may be performed in accordance with
Docket. No. SMM920220273-WO-PCT 29 examples as described herein. In some implementations, aspects of the operations of 605 may be performed by a device as described with reference to FIGs.1 and 5. [00172] At 610, the method 600 may include representing a composite precoder as a Kronecker product of the vertical precoder and the horizontal precoder. The operations of 610 may be performed in accordance with examples as described herein. In some implementations, aspects of the operations of 610 may be performed by a device as described with reference to FIGs.1 and 5. [00173] At 615, the method 600 may include determining a positive definite (“Q”) matrix of dimensions M x N rows by M x N (“MN x MN”) columns based on antenna array and antenna circuitry used to drive the antenna array. The antenna array is of a first integer number “M” of rows and a second integer number “N” of columns. Each of numbers M and N is greater than 1. The operations of 615 may be performed in accordance with examples as described herein. In some implementations, aspects of the operations of 615 may be performed by a device as described with reference to FIGs.1 and 5. [00174] At 620, the method 600 may include determining a transformed precoder based on the Q matrix. The operations of 620 may be performed in accordance with examples as described herein. In some implementations, aspects of the operations of 620 may be performed by a device as described with reference to FIGs.1 and 5. [00175] According to one or more aspects of the present disclosure, the method 600 may further include receiving a signal for transmission. The method 600 may further include applying the transformed precoder to the signal to generate a precoded signal. The method 600 may further include transmitting, by the transceiver, the precoded signal over a physical channel. In one or more embodiments, the method 600 may further include transmitting the precoded signal over a downlink physical channel from a network device (base station) to a user device (UE). In one or more alternate embodiments, where the operating device executing the various described processes is a UE, the method 600 may further include transmitting the precoded signal over an uplink physical channel from a user device (the UE) to a network device (e.g., the base station).
Docket. No. SMM920220273-WO-PCT 30 [00176] In one or more embodiments, the Q matrix depends at least in part on one or more of impedance parameters and scattering parameters of the antenna array. In one or more embodiments, the transceiver is configured to transmit a precoded signal in a transmission band that corresponds to a first range of wavelength, and the antenna array includes antenna elements subject to mutual coupling by being spaced in less than the first range of wavelength. [00177] According to one or more aspects of a first solution of the present disclosure, the method 600 may further include determining a scaling factor based on the Q matrix. The method 600 may further include determining the transformed precoder by multiplying the composite precoder by the scaling factor. In one or more embodiments, the transformed precoder is a data symbol precoder. In one or more embodiments, the transformed precoder is a demodulation reference symbol precoder. In one or more embodiments, method 600 may further include transmitting at least one of the scaling factors for each scaled precoder. In one or more embodiments, method 600 may further include transmitting the coefficients of the upper diagonal portion of the Q matrix. In one or more embodiments, method 600 may further include transmitting at least one of the scaling factors for each scaled precoder and the coefficients of the upper diagonal portion of the Q matrix. According to one or more aspects of a second solution of the present disclosure, the method 600 may further include determining a transformation matrix of dimension MN x MN based on the Q matrix. The method 600 may further include determining the transformed precoder by multiplying the composite precoder by the transformation matrix. In one or more embodiments, the transformed precoder is a data symbol precoder. In one or more embodiments, the transformed precoder is a reference symbol precoder. In one or more embodiments, the reference symbols precoder is a demodulation symbol precoder. In one or more embodiments, the reference symbol precoder is a channel state information reference symbol precoder. [00178] In one or more embodiments, the method 600 may further include determining the Kronecker product as a nearest Kronecker product, which enables implementation of the transformed precoder in terms of the vertical precoder and the horizontal precoder. In one or
Docket. No. SMM920220273-WO-PCT 31 more embodiments, the method 600 may further include linearly mapping the two- dimensional precoder to the input of the array, enabling transmit power to be represented as a quadratic form that can be factored. The factoring of the quadratic form is used to define a linear transformation that transforms the precoders into a set of transformed precoders corresponding to equal transmit power. [00179] The various illustrative blocks and components described in connection with the disclosure herein may be implemented or performed with a general-purpose processor, a DSP, an ASIC, a CPU, an FPGA or other programmable logic device, discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein. A general-purpose processor may be a microprocessor, but in the alternative, the processor may be any processor, controller, microcontroller, or state machine. A processor may also be implemented as a combination of computing devices (e.g., a combination of a DSP and a microprocessor, multiple microprocessors, one or more microprocessors in conjunction with a DSP core, or any other such configuration. [00180] The functions described herein may be implemented in hardware, software executed by a processor, firmware, or any combination thereof. If implemented in software executed by a processor, the functions may be stored on or transmitted over as one or more instructions or code on a computer-readable medium. Other examples and implementations are within the scope of the disclosure and appended claims. For example, due to the nature of software, functions described herein may be implemented using software executed by a processor, hardware, firmware, hardwiring, or combinations of any of these. Features implementing functions may also be physically located at various positions, including being distributed such that portions of functions are implemented at different physical locations. [00181] Computer-readable media includes both non-transitory computer storage media and communication media including any medium that facilitates transfer of a computer program from one place to another. A non-transitory storage medium may be any available medium that may be accessed by a general-purpose or special-purpose computer. By way of example, and not limitation, non-transitory computer-readable media may include RAM,
Docket. No. SMM920220273-WO-PCT 32 ROM, electrically erasable programmable ROM (EEPROM), flash memory, compact disk (CD) ROM or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other non-transitory medium that may be used to carry or store desired program code means in the form of instructions or data structures and that may be accessed by a general-purpose or special-purpose computer, or a general-purpose or special-purpose processor. [00182] Any connection may be properly termed a computer-readable medium. For example, if the software is transmitted from a website, server, or other remote source using a coaxial cable, fiber optic cable, twisted pair, digital subscriber line (DSL), or wireless technologies such as infrared, radio, and microwave, then the coaxial cable, fiber optic cable, twisted pair, DSL, or wireless technologies such as infrared, radio, and microwave are included in the definition of computer-readable medium. Disk and disc, as used herein, include CD, laser disc, optical disc, digital versatile disc (DVD), floppy disk and Blu-ray disc where disks usually reproduce data magnetically, while discs reproduce data optically with lasers. Combinations of the above are also included within the scope of computer-readable media. [00183] As used herein, including in the claims, “or” as used in a list of items (e.g., a list of items prefaced by a phrase such as “at least one of” or “one or more of”) indicates an inclusive list such that, for example, a list of at least one of A, B, or C means A or B or C or AB or AC or BC or ABC (i.e., A and B and C). Also, as used herein, the phrase “based on” shall not be construed as a reference to a closed set of conditions. For example, an example step that is described as “based on condition A” may be based on both a condition A and a condition B without departing from the scope of the present disclosure. In other words, as used herein, the phrase “based on” shall be construed in the same manner as the phrase “based at least in part on. Further, as used herein, including in the claims, a “set” may include one or more elements. [00184] The terms “transmitting,” “receiving,” or “communicating,” when referring to a network entity, may refer to any portion of a network entity (e.g., a base station, a CU, a DU,
Docket. No. SMM920220273-WO-PCT 33 a RU) of a RAN communicating with another device (e.g., directly or via one or more other network entities). [00185] The description set forth herein, in connection with the appended drawings, describes example configurations and does not represent all the examples that may be implemented or that are within the scope of the claims. The term “example” used herein means “serving as an example, instance, or illustration,” and not “preferred” or “advantageous over other examples.” The detailed description includes specific details for the purpose of providing an understanding of the described techniques. These techniques, however, may be practiced without these specific details. In some instances, known structures and devices are shown in block diagram form to avoid obscuring the concepts of the described example. [00186] The description herein is provided to enable a person having ordinary skill in the art to make or use the disclosure. Various modifications to the disclosure will be apparent to a person having ordinary skill in the art, and the generic principles defined herein may be applied to other variations without departing from the scope of the disclosure. Thus, the disclosure is not limited to the examples and designs described herein but is to be accorded the broadest scope consistent with the principles and novel features disclosed herein.