Patent Publication Number: US-10334454-B2

Title: Multi-finger beamforming and array pattern synthesis

Description:
TECHNICAL FIELD 
     Various embodiments relate generally to methods and devices for multi-finger beamforming and array pattern synthesis. 
     BACKGROUND 
     Antenna-based communication systems may utilize beamforming in order to create steered antenna beams with an antenna array. Beamforming systems may adjust the phase and/or gain of each of the signals transmitted by (or received with in the receive direction) the elements of an antenna array in order to create patterns of constructive and destructive inference at certain angular directions. Through precise selection of the delays and gains of each antenna element, a beamforming architecture may control the resulting interference pattern in order to realize a steerable and adaptable radiation pattern providing different beamgain in different directions. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       In the drawings, like reference characters generally refer to the same parts throughout the different views. The drawings are not necessarily to scale, emphasis instead generally being placed upon illustrating the principles of the invention. In the following description, various embodiments of the invention are described with reference to the following drawings, in which: 
         FIGS. 1A and 1B  show simplified devices for baseband and RF beamforming according to some aspects; 
         FIG. 2  shows a communication system for beamforming according to some aspects; 
         FIG. 3  shows an exemplary radiation pattern for multi-finger beamforming according to some aspects; 
         FIGS. 4A and 4B  show an exemplary inner loop for performing multi-finger beamforming according to some aspects; 
         FIGS. 5A and 5B  shows an exemplary outer loop for performing multi-finger beamforming according to some aspects; 
         FIG. 6  shows an illustrative example of a target radiation pattern defined by relative power levels of main fingers according to some aspects; 
         FIGS. 7A and 7B  show an illustrative example of solving a least-squares problem for phase-only complex beamforming weights according to some aspects; 
         FIG. 8  shows an illustrative example of updating a maximum main finger power level and a maximum sidelobe power level according to some aspects; 
         FIG. 9  shows an exemplary radiation pattern for array pattern synthesis according to some aspects; 
         FIG. 10  shows an illustrative example of a target radiation pattern for array pattern synthesis according to some aspects; 
         FIGS. 11A and 11B  shows an exemplary method of array pattern synthesis for phase-only beamforming according to some aspects; 
         FIGS. 12A-12C  show an illustrative example of iteratively solving for phase-only complex beamforming weights for array pattern synthesis according to some aspects; 
         FIGS. 13A and 13B  show an exemplary method of array pattern synthesis for phase control with amplitude tapering beamforming according to some aspects; 
         FIGS. 14A and 14B  show an illustrative example of iteratively solving for phase control with amplitude tapering complex beamforming weights for array pattern synthesis according to some aspects; 
         FIGS. 15A and 15B  show an exemplary method of array pattern synthesis for phase and amplitude control beamforming according to some aspects; 
         FIG. 16  shows an exemplary method of performing beamforming according to some aspects; and 
         FIG. 17  shows an exemplary method of synthesizing antenna array radiation patterns according to some aspects. 
     
    
    
     DESCRIPTION 
     The following detailed description refers to the accompanying drawings that show, by way of illustration, specific details and embodiments in which the invention may be practiced. 
     The word “exemplary” is used herein to mean “serving as an example, instance, or illustration”. Any embodiment or design described herein as “exemplary” is not necessarily to be construed as preferred or advantageous over other embodiments or designs. 
     The words “plurality” and “multiple” in the description or the claims expressly refer to a quantity greater than one. The terms “group (of)”, “set [of]”, “collection (of)”, “series (of)”, “sequence (of)”, “grouping (of)”, etc., and the like in the description or in the claims refer to a quantity equal to or greater than one, i.e. one or more. Any term expressed in plural form that does not expressly state “plurality” or “multiple” likewise refers to a quantity equal to or greater than one. The terms “proper subset”, “reduced subset”, and “lesser subset” refer to a subset of a set that is not equal to the set, i.e. a subset of a set that contains less elements than the set. 
     It is appreciated that any vector and/or matrix notation utilized herein is exemplary in nature and is employed solely for purposes of explanation. Accordingly, it is understood that the approaches detailed in this disclosure are not limited to being implemented solely using vectors and/or matrices, and that the associated processes and computations may be equivalently performed with respect to sets, sequences, groups, etc., of data, observations, information, signals, samples, symbols, elements, etc. Furthermore, it is appreciated that references to a “vector” may refer to a vector of any size or orientation, e.g. including a 1×1 vector (e.g. a scalar), a 1×M vector (e.g. a row vector), and an M×1 vector (e.g. a column vector). Similarly, it is appreciated that references to a “matrix” may refer to matrix of any size or orientation, e.g. including a 1×1 matrix (e.g. a scalar), a 1×M matrix (e.g. a row vector), and an M×1 matrix (e.g. a column vector). 
     A “circuit” as used herein is understood as any kind of logic-implementing entity, which may include special-purpose hardware or a processor executing software. A circuit may thus be an analog circuit, digital circuit, mixed-signal circuit, logic circuit, processor, microprocessor, Central Processing Unit (CPU), Graphics Processing Unit (GPU), Digital Signal Processor (DSP), Field Programmable Gate Array (FPGA), integrated circuit, Application Specific Integrated Circuit (ASIC), etc., or any combination thereof. Any other kind of implementation of the respective functions which will be described below in further detail may also be understood as a “circuit”. It is understood that any two (or more) of the circuits detailed herein may be realized as a single circuit with substantially equivalent functionality, and conversely that any single circuit detailed herein may be realized as two (or more) separate circuits with substantially equivalent functionality. Additionally, references to a “circuit” may refer to two or more circuits that collectively form a single circuit. The term “circuit arrangement” may refer to a single circuit, a collection of circuits, and/or an electronic device composed of one or more circuits. 
     As used herein, “memory” may be understood as a non-transitory computer-readable medium in which data or information can be stored for retrieval. References to “memory” included herein may thus be understood as referring to volatile or non-volatile memory, including random access memory (RAM), read-only memory (ROM), flash memory, solid-state storage, magnetic tape, hard disk drive, optical drive, etc., or any combination thereof. Furthermore, it is appreciated that registers, shift registers, processor registers, data buffers, etc., are also embraced herein by the term memory. It is appreciated that a single component referred to as “memory” or “a memory” may be composed of more than one different type of memory, and thus may refer to a collective component comprising one or more types of memory. It is readily understood that any single memory component may be separated into multiple collectively equivalent memory components, and vice versa. Furthermore, while memory may be depicted as separate from one or more other components (such as in the drawings), it is understood that memory may be integrated within another component, such as on a common integrated chip. 
     The term “base station” used in reference to an access point of a mobile communication network may be understood as a macro base station, micro base station, Node B, evolved NodeB (eNB), Home eNodeB, Remote Radio Head (RRH), relay point, etc. As used herein, a “cell” in the context of telecommunications may be understood as a sector served by a base station. Accordingly, a cell may be a set of geographically co-located antennas that correspond to a particular sectorization of a base station. A base station may thus serve one or more cells (or sectors), where each cell is characterized by a distinct communication channel. Furthermore, the term “cell” may be utilized to refer to any of a macrocell, microcell, femtocell, picocell, etc. 
     For purposes of this disclosure, radio communication technologies may be classified as one of a Short Range radio communication technology or Cellular Wide Area radio communication technology. Short Range radio communication technologies include Bluetooth, WLAN (e.g. according to any IEEE 802.11 standard), and other similar radio communication technologies. Cellular Wide Area radio communication technologies include Global System for Mobile Communications (GSM), Code Division Multiple Access 2002000 (CDMA2002000), Universal Mobile Telecommunications System (UMTS), Long Term Evolution (LTE), General Packet Radio Service (GPRS), Evolution-Data Optimized (EV-DO). Enhanced Data Rates for GSM Evolution (EDGE), High Speed Packet Access (HSPA; including High Speed Downlink Packet Access (HSDPA), High Speed Uplink Packet Access (HSUPA), HSDPA Plus (HSDPA+), and HSUPA Plus (HSUPA+)), Worldwide Interoperability for Microwave Access (WiMax) (e.g. according to an IEEE 802.16 radio communication standard, e.g. WiMax fixed or WiMax mobile), etc., and other similar radio communication technologies. Cellular Wide Area radio communication technologies also include “small cells” of such technologies, such as microcells, femtocells, and picocells. Cellular Wide Area radio communication technologies may be generally referred to herein as “cellular” communication technologies. It is understood that exemplary scenarios detailed herein are demonstrative in nature, and accordingly may be similarly applied to various other mobile communication technologies, both existing and not yet formulated, particularly in cases where such mobile communication technologies share similar features as disclosed regarding the following examples. Furthermore, as used herein the term GSM refers to both circuit- and packet-switched GSM, i.e. including GPRS, EDGE, and any other related GSM technologies. Likewise, the term UMTS refers to both circuit- and packet-switched GSM, i.e. including HSPA, HSDPA/HSUPA, HSDPA+/HSUPA+, and any other related UMTS technologies. 
     The term “network” as utilized herein, e.g. in reference to a communication network such as a radio communication network, encompasses both an access section of a network (e.g. a radio access network (RAN) section) and a core section of a network (e.g. a core network section). The term “radio idle mode” or “radio idle state” used herein in reference to a mobile terminal refers to a radio control state in which the mobile terminal is not allocated at least one dedicated communication channel of a mobile communication network. The term “radio connected mode” or “radio connected state” used in reference to a mobile terminal refers to a radio control state in which the mobile terminal is allocated at least one dedicated uplink communication channel of a mobile communication network. 
     Unless explicitly specified, the term “transmit” encompasses both direct (point-to-point) and indirect transmission (via one or more intermediary points). Similarly, the term “receive” encompasses both direct and indirect reception. The term “communicate” encompasses one or both of transmitting and receiving, i.e. unidirectional or bidirectional communication in one or both of the incoming and outgoing directions. The term “calculate” encompass both ‘direct’ calculations via a mathematical expression/formula/relationship and ‘indirect’ calculations via lookup or hash tables and other array indexing or searching operations. 
     Next-generation communication systems, such as those based on multiple-input multiple-output (MIMO) and millimeter wave (mmWave) technologies, are expected to have large-scale antenna arrays. These next-generation communication systems may utilize such large-scale antenna arrays to realize steerable and shapeable antenna radiation patterns that can provide varying degrees of beamgain in different directions. 
     Beamforming systems may perform processing at baseband and/or RF frequencies to shape the radiation pattern of an antenna array.  FIGS. 1A and 1B  show two simplified beamforming approaches as deployed for an exemplary four-element linear antenna array. Although examples in the following description may focus on a transmit beamforming context, skilled persons will appreciate that these descriptions can be analogously applied for receive beamforming, which may include combining the signals received at the antenna elements according to a complex beamforming weight array in order to adjust the received beam pattern. 
       FIG. 1A  illustrates a simplified digital beamforming architecture that digitally applies complex beamforming weights (composed of both a gain and phase factor) in the baseband domain. As shown in  FIG. 1A , digital baseband controller  102  may receive baseband symbol s and (in addition to other digital baseband operations) subsequently apply a complex beamforming weight vector w=[w 0  w 1  w 2  w 3 ] T  to s to generate weighted symbols ws, where each element w i , i=0, 1, 2, 3, is a complex beamforming weight. Accordingly, each resulting element [w 0 s w 1 s w 2 s w 3 s] T  may be a baseband symbol s multiplied by some complex beamforming weight w i . Digital baseband controller  102  may then map each element of ws to a respective RF chain of RF chain array  104 , which may each perform digital-to-analog conversion (DAC), radio carrier modulation, and amplification on the received weighted symbols before providing the resulting RF symbols to a respective element of antenna array  106 . Antenna array  106  may then wirelessly transmit each RF symbol. 
     By manipulating the beamforming weights of w, digital baseband controller  102  may be able to utilize each of the four antenna elements of antenna array  106  to produce a steered beam that has a greater beamgain compared to a single antenna element. The radio signals emitted by each element of antenna array  106  may combine to realize a combined waveform that exhibits a pattern of constructive and destructive interference that varies over distances and direction from antenna array  106 . Depending on a number of factors (including e.g. antenna array spacing and alignment, radiation patterns, carrier frequency, etc.), the various points of constructive and destructive interference of the combined waveform may create a focused beam lobe that can be “steered” in direction via adjustment of the phase and gain factors w i  of w.  FIG. 1A  shows several exemplary steered beams, or ‘lobes’, emitted by antenna array  106 , which digital baseband controller  102  may directly control by adjusting w. Adjustment of w may enable digital baseband controller  102  to manipulate the radiation pattern produced by antenna array  106  to form various different patterns, including direction and beamgain of main lobes, direction and suppression of sidelobes, and null steering. 
     In so-called adaptive beamforming approaches, digital baseband controller  102  may dynamically change the beamforming weights in order to adjust the direction and strength of the main lobe in addition to nulls and sidelobes. Such adaptive approaches may allow digital baseband controller  102  to steer the beam in different directions over time, which may be useful to track the location of a moving target point (e.g. a moving receiver or transmitter). In a mobile communication context, digital baseband controller  102  may identify the location of a target terminal device  108  (e.g. the direction or angle of terminal device  108  relative to antenna array  106 ) and subsequently adjust w in order to generate a beam pattern with a main lobe pointing towards terminal device  108 , thus improving the array gain at terminal device  108  and consequently improving the receiver performance. Through adaptive beamforming, digital baseband controller  102  may be able to dynamically adjust or “steer” the beam pattern as terminal device  108  moves in order to continuously provide focused transmissions to terminal device  108  (or conversely focused reception). 
       FIG. 1B  illustrates an exemplary RF beamforming architecture that applies complex beamforming weights in the RF domain. As shown in  FIG. 1B , digital baseband controller  102  may similarly receive baseband symbol s and (in addition to other digital baseband operations) split s into four duplicate symbols s. Digital baseband controller  102  may then provide symbols s to each of the RF chains of RF chain array  104 , which may each perform DAC, radio carrier modulation, and amplification on the symbols s. The RF chains of RF chain array  104  may then provide the resulting symbols s (for which the notation s is maintained for simplicity) to RF beamforming circuits  110 , which may each apply a respective complex beamforming weight w i , i=0, 1, 2, 3, to s to obtain weighted symbols w i s. RF beamforming circuits  110  may then provide the resulting symbols w i s to antenna array  106  for transmission. 
     Due to the complexity related to the implementation of RF circuitry, RF beamforming circuits  110  may be constrained with respect to the possible complex beamforming weights w i . For example, some implementations may utilize a phase-only approach in which RF beamforming circuits  110  are RF phase shifters that are capable of phase-shifting but not gain adjustment. Accordingly, in these cases w i  would be bound to |w i |=1. In other cases, RF beamforming circuits  110  may be implemented that are capable of phase and amplitude control, although the degree of amplitude control may be limited compared to the digital beamforming case. 
     While the examples shown in  FIGS. 1A and 1B  depict the use of four antennas, many next-generation communication systems are expected to deploy massive antenna arrays having 64, 128, 256, or higher antennas in a given array. These massive antenna arrays may offer a large degree of control over formulation of antenna radiation patterns. In addition to providing a high beamgain in a specific direction (e.g., to serve a single target user), another goal of these next-generation communication systems is to provide multi-finger beamforming, where multiple main lobes, or ‘fingers’, can be respectively directed to in different directions. Next-generation communication systems may therefore utilize multi-finger beamforming setups for different single-stream use cases (where the same baseband data is transmitted through all elements of the antenna array), such as for multicast transmissions to multiple users (e.g., for control channel transmission) or to provide a reliable connection through beam diversity (e.g., providing multiple beams at different directions for a single user, such as if a line-of-sight (LOS) path is obstructed). 
     Existing multi-finger beamforming solutions, however, currently suffer from several deficiencies. Many existing multi-finger beamforming solutions produce the different fingers in the radiation pattern by splitting up the antenna array into multiple sub-arrays, and then using the elements of each sub-array to form separate beams steered in different directions (e.g., by using a separate set of complex beamforming weights for each sub-array). However, these solutions are limited by the fact that only a single beam can be generated per sub-array (e.g., one per RF-chain). Additionally, there may be high beam steering error and little control over the beam gains, while the ability to control and suppress sidelobes may also be limited. 
     In addition to multi-finger beamforming, next-generation communication systems may also target array pattern synthesis. While multi-finger beamforming mainly addresses the formation of multiple fingers steered in specific directions, array pattern synthesis focuses on realizing a specific radiation pattern including main finger direction and gain control, sidelobe reduction, interference suppression, null-steering, beam broadening, etc. This array pattern synthesis is considered a much more complex problem than multi-finger beamforming, as a set of complex beamforming weights needs to be derived that realizes an entire radiation pattern (e.g., defined by desired gains across a comprehensive set of angles). Existing solutions for array pattern synthesis may use a “trial and error”-type solution, in which a large array of initial points is tested and evaluated to determine which produces the best match to a desired radiation pattern. This, however, is inefficient and can often produce sub-optimal results. 
     Accordingly, advantageous aspects of this disclosure present improved techniques for multi-finger beamforming and array pattern synthesis. These aspects can use weighted least-squares optimization to derive a set of complex beamforming weights w that, when applied at an antenna array (e.g., either at baseband or RF), realizes a desired radiation pattern. Various aspects detailed herein may also reformulate generalized least-squares problems into convex problems that can be solved with reduced complexity, thus also providing the possibility to use such techniques in adaptive beamforming. 
       FIG. 2  shows  200 communication device  200  including antenna array  202 , RF transceiver  204 , baseband modem  206 , application processor  212 , and beamforming controller  214 .  200 Communication device  200  may be implemented into and/or be a component of any radio communication device, such as any type of terminal device or network access node. Accordingly,  200 communication device  200  may implement the beamforming techniques detailed herein in the uplink direction or the downlink direction 
     Although not explicitly shown in  FIG. 2 , in some aspects communication device  200  may include one or more additional hardware and/or software components, such as processors/microprocessors, controllers/microcontrollers, other specialty or generic hardware/processors/circuits, peripheral device(s), memory, power supply, external device interface(s), subscriber identity module(s) (SIMs), user input/output devices (display(s), keypad(s), touchscreen(s), speaker(s), external button(s), camera(s), microphone(s), etc.), or other related components. 
     Communication device  200  may transmit and receive radio signals on one or more radio access networks. Baseband modem  206  may direct such communication functionality of communication device  200  according to the communication protocols associated with each radio access network, and may execute control over antenna system  202  and RF transceiver  204  in order to transmit and receive radio signals according to the formatting and scheduling parameters defined by each communication protocol. Although various practical designs may include separate communication components for each supported radio communication technology (e.g., a separate antenna, RF transceiver, digital signal processor, and controller), for purposes of conciseness the configuration of communication device  200  shown in  FIG. 2  depicts only a single instance of such components. 
     Communication device  200  may transmit and receive wireless signals with antenna system  202 , which may be a single antenna or an antenna array that includes multiple antennas. In some aspects, antenna system  202  may additionally include analog antenna combination and/or beamforming circuitry. In the receive (RX) path, RF transceiver  204  may receive analog radio frequency signals from antenna system  202  and perform analog and digital RF front-end processing on the analog radio frequency signals to produce digital baseband samples (e.g., In-Phase/Quadrature (IQ) samples) to provide to baseband modem  206 . RF transceiver  204  may include analog and digital reception components including amplifiers (e.g., Low Noise Amplifiers (LNAs)), filters, RF demodulators (e.g., RF IQ demodulators)), and analog-to-digital converters (ADCs), which RF transceiver  204  may utilize to convert the received radio frequency signals to digital baseband samples. In the transmit (TX) path, RF transceiver  204  may receive digital baseband samples from baseband modem  206  and perform analog and digital RF front-end processing on the digital baseband samples to produce analog radio frequency signals to provide to antenna system  202  for wireless transmission. RF transceiver  204  may thus include analog and digital transmission components including amplifiers (e.g., Power Amplifiers (PAs), filters, RF modulators (e.g., RF IQ modulators), and digital-to-analog converters (DACs), which RF transceiver  204  may utilize to mix the digital baseband samples received from baseband modem  206  and produce the analog radio frequency signals for wireless transmission by antenna system  202 . In some aspects baseband modem  206  may control the RF transmission and reception of RF transceiver  204 , including specifying the transmit and receive radio frequencies for operation of RF transceiver  204 . 
     As shown in  FIG. 2 , baseband modem  206  may include digital signal processor  208 , which may perform physical layer (PHY, Layer 1) transmission and reception processing to, in the transmit path, prepare outgoing transmit data provided by controller  210  for transmission via RF transceiver  204 , and, in the receive path, prepare incoming received data provided by RF transceiver  204  for processing by controller  210 . Digital signal processor  208  may be configured to perform one or more of error detection, forward error correction encoding/decoding, channel coding and interleaving, channel modulation/demodulation, physical channel mapping, radio measurement and search, frequency and time synchronization, antenna diversity processing, power control and weighting, rate matching/de-matching, retransmission processing, interference cancellation, and any other physical layer processing functions. Digital signal processor  208  may be structurally realized as hardware components (e.g., as one or more digitally-configured hardware circuits or FPGAs), software-defined components (e.g., one or more processors configured to execute program code defining arithmetic, control, and I/O instructions (e.g., software and/or firmware) stored in a non-transitory computer-readable storage medium), or as a combination of hardware and software components. In some aspects, digital signal processor  208  may include one or more processors configured to retrieve and execute program code that defines control and processing logic for physical layer processing operations. In some aspects, digital signal processor  208  may execute processing functions with software via the execution of executable instructions. In some aspects, digital signal processor  208  may include one or more dedicated hardware circuits (e.g., ASICs, FPGAs, and other hardware) that are digitally configured to specific execute processing functions, where the one or more processors of digital signal processor  208  may offload certain processing tasks to these dedicated hardware circuits, which are known as hardware accelerators. Exemplary hardware accelerators can include Fast Fourier Transform (FFT) circuits and encoder/decoder circuits. In some aspects, the processor and hardware accelerator components of digital signal processor  208  may be realized as a coupled integrated circuit. 
     Communication device  200  may be configured to operate according to one or more radio communication technologies. Digital signal processor  208  may be responsible for lower-layer processing functions of the radio communication technologies, while controller  210  may be responsible for upper-layer protocol stack functions. Controller  210  may thus be responsible for controlling the radio communication components of communication device  200  (antenna system  202 , RF transceiver  204 , and digital signal processor  208 ) in accordance with the communication protocols of each supported radio communication technology, and accordingly may represent the Access Stratum and Non-Access Stratum (NAS) (also encompassing Layer 2 and Layer 3) of each supported radio communication technology. Controller  210  may be structurally embodied as a protocol processor configured to execute protocol software (retrieved from a controller memory) and subsequently control the radio communication components of communication device  200  in order to transmit and receive communication signals in accordance with the corresponding protocol control logic defined in the protocol software. Controller  210  may include one or more processors configured to retrieve and execute program code that defines the upper-layer protocol stack logic for one or more radio communication technologies, which can include Data Link Layer/Layer 2 and Network Layer/Layer 3 functions. Controller  210  may be configured to perform both user-plane and control-plane functions to facilitate the transfer of application layer data to and from radio communication device  200  according to the specific protocols of the supported radio communication technology. User-plane functions can include header compression and encapsulation, security, error checking and correction, channel multiplexing, scheduling and priority, while control-plane functions may include setup and maintenance of radio bearers. The program code retrieved and executed by controller  210  may include executable instructions that define the logic of such functions. 
     In some aspects, communication device  200  may be configured to transmit and receive data according to multiple radio communication technologies. Accordingly, in some aspects one or more of antenna system  202 , RF transceiver  204 , digital signal processor  208 , and controller  210  may include separate components or instances dedicated to different radio communication technologies and/or unified components that are shared between different radio communication technologies. For example, in some aspects controller  210  may be configured to execute multiple protocol stacks, each dedicated to a different radio communication technology and either at the same processor or different processors. In some aspects, digital signal processor  208  may include separate processors and/or hardware accelerators that are dedicated to different respective radio communication technologies, and/or one or more processors and/or hardware accelerators that are shared between multiple radio communication technologies. In some aspects, RF transceiver  204  may include separate RF circuitry sections dedicated to different respective radio communication technologies, and/or RF circuitry sections shared between multiple radio communication technologies. In some aspects, antenna system  202  may include separate antennas dedicated to different respective radio communication technologies, and/or antennas shared between multiple radio communication technologies. Accordingly, while antenna system  202 , RF transceiver  204 , digital signal processor  208 , and controller  210  are shown as individual components in  FIG. 3 , in some aspects antenna system  202 , RF transceiver  204 , digital signal processor  208 , and/or controller  210  can encompass separate components dedicated to different radio communication technologies. 
     Application processor  212  may be a CPU, and may be configured to handle the layers above the protocol stack, including the transport and application layers. Application processor  212  may be configured to execute various applications and/or programs of communication device  200  at an application layer of communication device  200 , such as an operating system (OS), a user interface (UI) for supporting user interaction with communication device  200 , and/or various user applications. The application processor may interface with baseband modem  206  and act as a source (in the transmit path) and a sink (in the receive path) for user data, such as voice data, audio/video/image data, messaging data, application data, basic Internet/web access data, etc. In the transmit path, controller  210  may therefore receive and process outgoing data provided by application processor  212  according to the layer-specific functions of the protocol stack, and provide the resulting data to digital signal processor  208 . Digital signal processor  208  may then perform physical layer processing on the received data to produce digital baseband samples, which digital signal processor may provide to RF transceiver  204 . RF transceiver  204  may then process the digital baseband samples to convert the digital baseband samples to analog RF signals, which RF transceiver  204  may wirelessly transmit via antenna system  202 . In the receive path, RF transceiver  204  may receive analog RF signals from antenna system  202  and process the analog RF signals to obtain digital baseband samples. RF transceiver  204  may provide the digital baseband samples to digital signal processor  208 , which may perform physical layer processing on the digital baseband samples. Digital signal processor  208  may then provide the resulting data to controller  210 , which may process the resulting data according to the layer-specific functions of the protocol stack and provide the resulting incoming data to application processor  212 . Application processor  212  may then handle the incoming data at the application layer, which can include execution of one or more application programs with the data and/or presentation of the data to a user via a user interface. 
     As shown in  FIG. 2 , antenna array  202  may include N antennas. In some aspects, antenna array  202  may be a uniform linear array, while in some aspects antenna array  202  may be a rectangular array. Although various multi-finger beamforming aspects may be detailed herein with respect to linear arrays, these aspects can be extended to rectangular arrays via the two-dimensional Kroenecker product. 
     Beamforming controller  214  include hardware and/or software components. For example, in some aspects beamforming controller  214  may include one or more processors configured to retrieve and execute program code that defines the control and algorithmic functionalities of beamforming controller  214  as shown and described for  FIGS. 4, 5, 11, and 13 . In some aspects, beamforming controller  214  may include digitally-configured hardware (e.g., dedicated digital logic designed for specific functionalities) that is configured to perform control and/or algorithmic operations. In some aspects, beamforming controller  214  may include a combination of processors and digitally-configured hardware. Given the logical equivalence between hardware-defined and software-defined implementations, beamforming controller  214  is therefore not particularly limited to any particular hardware-defined or software-defined configuration. It is thus appreciated that the control and algorithmic logic described below may be embodied with many different configurations of hardware-defined and/or software-defined circuitry. Although shown separately in  FIG. 2 , in some aspects beamforming controller  208  may be included as part of another component of communication device  200 , such as a component of baseband modem  206 . The depiction of beamforming controller  208  as a separate component therefore reflects the functional separation of beamforming controller  208 , although beamforming controller  208  may be physically integrated into another component of communication device  200 . 
     Beamforming controller  214  may be configured to determine complex beamforming weights (e.g., in the manner of w as introduced above) and implement the beamforming weights at  200 communication device  200 . In some aspects,  200 communication device  200  may utilize digital beamforming, in which case beamforming controller  214  may provide the complex beamforming weights to digital signal processor  208  of baseband modem  206  (e.g., in the manner of digital baseband controller  102  of  FIG. 1A ), which may digitally apply the complex beamforming weights to outgoing baseband data. In the transmit direction, RF transceiver  204  may then process the weighted data according to transmit-path processing operations (e.g., amplification, mixing, ADC/DAC) and provide the resulting RF data to antenna array  202  for transmission. In some aspects,  200 communication device  200  may utilize RF beamforming, in which case beamforming controller  214  may provide the complex beamforming weights to RF beamforming circuits of RF transceiver  204  (e.g., in the manner of RF beamforming circuits  110  of  FIG. 1B ). In the transmit direction, the RF beamforming circuits may apply the complex beamforming weights to outgoing RF data provide the resulting weighted RF data to antenna array  202  for transmission. Beamforming controller  214  may similarly provide the complex beamforming weights (which may be the same in both the transmit and receive directions) to RF transceiver  204  in the receive direction. 
     As denoted in  FIG. 2 , the i-th antenna of antenna array  202  may have a complex beamforming weight of w i , where the complex beamforming weight of the entirety of antenna array  202  can be represented as w=[w 0  w 1  w 2  w 3 ] T . As indicated above, RF transceiver  204  may apply the complex beamforming weights w i  in a digital or RF manner. 
     As previously indicated, multi-finger beamforming may target the realization of a radiation pattern with high-beamgain fingers steered in multiple directions. Multi-finger beamforming may also target a radiation pattern having sidelobe levels below a certain power level. Beamforming controller  214  may therefore be configured to calculate the complex beamforming weight vector w that, when implemented at RF transceiver  204 , generates a radiation pattern at antenna array  202  having the desired fingers and sidelobe power upper-bound.  FIG. 3  shows an exemplary radiation pattern  300  having two main fingers at different angular directions and sidelobes that are all constrained below a particular power upper-bound. 
     In order to calculate complex beamforming weights w that produce the desired radiation pattern, beamforming controller  214  may start with input parameters that numerically define the desired radiation pattern, including the angular direction of the fingers, power at each finger, and upper-bound power for the sidelobes. Given these parameters as inputs, beamforming controller  214  may execute algorithmic logic (e.g., as program code or at digitally-configured hardware circuitry) that calculates the complex beamforming weights w. As will be detailed, in some aspects beamforming controller  214  may execute algorithmic logic based on least-squares minimization to iteratively derive complex beamforming weights w that produce a radiation pattern that approaches the desired radiation pattern defined by the input parameters. 
       FIGS. 4A-4B and 5A-5B  show flow charts illustrating the process by which beamforming controller  214  determines the complex beamforming weights w according to some aspects, where  FIGS. 4A-4B  show inner loop  400  and  FIGS. 5A-5B  show outer loop  500  of the process. As will be detailed, beamforming controller  214  may execute inner loop  400  to determine complex beamforming weights w that produce a radiation pattern with L fingers having specific power levels relative to one another (defined by vector s). Beamforming controller  214  may execute outer loop  500  to calculate the maximum finger power level d Max  (the power level of the finger with the highest power level) and the maximum sidelobe power level s Max  (the upper-bound on all of the sidelobes). As previously described, in some aspects beamforming controller  214  may include one or more processors configured to retrieve and execute program code that defines the logic of inner loop  400  and outer loop  500  as executable instructions. In some aspects, beamforming controller  214  may include one or more hardware circuits that are configured with digital logic constituting that of inner loop  400  and outer loop  500 . 
       FIG. 4A  expresses inner loop  400  algorithmically while  FIG. 4B  expresses inner loop  400  in prose, where the underlying logic at each stage in both expressions is equivalent. Likewise,  FIG. 5A  expresses outer loop  500  algorithmically while  FIG. 5B  expresses outer loop  500  in prose, where the underlying logic at each stage in both expressions is equivalent. Stages  402  (of inner loop  400 ) and  502  (of outer loop  500 ) show the input parameters to multi-finger beamforming process. As shown in stage  402 , inner loop  400  may receive the following inputs:
         N number of antennas   θ s  azimuth angles of sidelobes   θ m  azimuth angles of main fingers   s relative gains of main fingers)   w (0)  initial complex beamforming weights   C (0)  initial least-squares weights   A l  DFT sub-matrix       

     N therefore gives the number of antennas in antenna array  202 . Sidelobe azimuth angle vector θ s =[θ s,1  . . . θ s,K ], where each θ s,k , k=1, . . . , k gives the desired azimuth angles of the k-th sidelobe (e.g., expressed in radians or degrees), while main finger azimuth angle vector θ m =[θ m,1  . . . θ m,L ], where each θ m,l , l=1, . . . , L gives the desired azimuth angles of the l-th main finger. In some aspects, the azimuth angles corresponding to the transient parts of the fingers (as identified in  FIG. 3 ) can be omitted from θ s  and θ m . 
     Relative power level vector s=[s 1  . . . s L ] gives the relative power levels of each l-th main finger to the main finger with the highest power level, where the highest main finger is identified with i=1 and therefore s 1 =1 and s l ≤1, l=2, . . . , L.  FIG. 6  shows radiation pattern  600  illustrating s 1  and s 2 , where s 2  is given by the ratio of the power level of the main finger with the highest power level to the desired power level of the l=2 finger. 
     Initial complex beamforming weight vector w (0) ∈   N×1  gives the initial complex beamforming weights, which may be arbitrarily defined (where [·] (n)  denotes the value at the n-th iteration). Initial least-squares weights C (0) =I may be defined as the identity matrix I. 
     Discrete Fourier Transform (DFT) sub-matrix A l  may be composed of rows selected from the DFT matrix A, with DFT matrix A is given as 
                     A   ∈     ℂ     NO   ×   N         =       [           a   1   T             ⋮             a   k   T             ⋮             a   NO   T           ]     =     [         1         e     j   ⁢       2   ⁢           ⁢   π     NO             …         e     j   ⁢       2   ⁢           ⁢     π   ⁡     (     N   -   1     )         NO                 ⋮       ⋮       ⋮       ⋮           1         e     j   ⁢       2   ⁢           ⁢   π   ⁢           ⁢   k     NO             …         e     j   ⁢       2   ⁢           ⁢     π   ⁡     (     N   -   1     )       ⁢           ⁢   k     NO                 ⋮       ⋮       ⋮       ⋮           1         e     j   ⁢       2   ⁢           ⁢   π   ⁢           ⁢   NO     NO             …         e     j   ⁢       2   ⁢           ⁢   π   ⁢           ⁢     (     N   -   1     )     ⁢   NO     NO               ]               (   1   )               
where O is the oversampling ratio and each row k of A is a DFT codeword
 
     
       
         
           
             
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               k   d     =     ⌊       NO   λ     ⁢     d   a     ⁢     cos   ⁡     (     θ   d     )         ⌋           
identifies the row of A that can be applied as complex beamforming weights to steer an antenna pattern in angular direction θ d , where └·┘ is the operator to find the closest integer, d a  is the inter-elemental distance for a uniform linear array in terms of wavelength A. Accordingly, each row k of A can be utilized to steer the radiation pattern in a different direction, where oversampling ratio O may increase the number of rows in A and thus increase the number of available steering directions by a factor of O.
 
     Given a DFT codeword a k   T  and a complex beamforming weight vector w, the far-field radiation pattern at azimuth angle θ k  can be calculated as
 
 d (θ k )= a   k   T   w   (2)
 
     Accordingly, after the desired main finger and sidelobe azimuth angles θ m  and θ s  are defined, beamforming controller  214  can obtain DFT sub-matrix A l  by identifying the codebook index k for each element of θ m  and θ s  and concatenating the corresponding DFT codewords (i.e., rows of A) to form A l . As DFT matrix A may be pre-calculated for a given N, beamforming controller  214  may avoid the calculation of steering codewords for each precise angle of θ m  and θ s  and may instead select the rows of A having indices that most closely match the angles of θ m  and θ s . 
     With reference back to  FIGS. 4A and 4B , beamforming controller  214  may obtain these inputs at stage  402  in preparation for the calculation of w of inner loop  400 , where beamforming controller  214  may be configured to calculate w to obtain the desired power level and direction of the while minimizing the sidelobes. In particular, beamforming controller  214  may iteratively adapt w according to least-squares optimization to converge w towards the optimal complex beamforming weights for the desired radiation pattern. 
     After obtaining the input parameters at stage  402 , beamforming controller  214  may update the iteration count n at stage  404 , where n is initially set to n=−1. Beamforming controller  214  may then begin the iterative update to w according to least-squares optimization by calculating d m   (n) , d s   (n) , and d (n) , where 
     
       
         
           
             
               
                 
                   
                     
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     Vectors d m   (n)  and d s   (n)  therefore give the desired radiation pattern for each of the main fingers and sidelobes, respectively, where each l-th element of d m   (n)  gives the complex amplitude and phase of the l-th main finger according to the current maximum main finger power level d Max , and each k-th element of d s   (n)  gives the complex amplitude and phase of the k-th sidelobe according to the current maximum sidelobe power level s Max . After calculating d m   (n)  and d s   (n) , beamforming controller  214  may then concatenate d m   (n)  and d s   (n)  to obtain matrix d (n) . 
     Matrix d (n)  therefore represents the power levels of the entire desired radiation pattern over all main finger and sidelobe azimuth angles. Beamforming controller  214  may then utilize least-squares optimization to update w such that the actual radiation pattern (i.e., the radiation pattern that would be produced if w was used at antenna array  202 ; this does not necessarily include physically using w to actually produce the actual radiation pattern) converges to d (n) . In particular, beamforming controller  214  may then calculate L (n) , R (n) , and q (n)  in stage  406  to define the least-square problem for the current iteration n, where 
     
       
         
           
             
               
                 
                   
                     
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     After calculation of L (n) , R (n) , and q (n)  in stage  408 , beamforming controller  214  may in stage  410  solve for x in the matrix system 
                     [         x           z         ]     =         [           R     (   n   )             L       (   n   )     T                 L     (   n   )           0         ]     †     ⁡     [           q     (   n   )               1         ]               (   8   )               
where [·] †  gives the pseudo-inverse. Equation (8) therefore defines the least-squares problem which beamforming controller  214  can solve to obtain x and subsequently generate w.
 
     The length  2 N vector x calculated by beamforming controller  214  via solution of Equation (8) in stage  410  represents the real and imaginary parts of w (n+1)  (the updated complex beamforming weights). Accordingly, beamforming controller  214  may calculate w (n+1)  from x in stage  412  by calculating
 
 w   (n+1)   =e   j({x}     1       N     +j{x}     N+1       2N     )   (9)
 
     As expressed in Equation (8), beamforming controller  214  may therefore generate a new length N vector by taking the first N elements of x as the real part of each element and the second N elements of x as the imaginary part of each element, and subsequently generating each of the N elements of w (n+1)  by taking the complex exponential of each element of the new vector. 
     Accordingly, the resulting elements of w (n+1)  may be phase-only, or, in other words, each element w n  of w (n+1)  is constrained by |w n |=1. Equivalently, this means that each point w n  is on the unit circle, and thus that beamforming controller  214  may solve for w (n+1)  by calculating the optimal points on the unit circle for each w n . However, as this problem is non-convex and thus complex to solve, in some aspects beamforming controller  214  may simplify the problem via a linear approximation as defined by the matrix L (n)  in Equation (5). In particular, as opposed to solving for each w n  along the unit circle, beamforming controller  214  may utilize the domain specified by L (n)  to define the tangent line along each point w n  on the unit circle, and then solve the least-squares problem along this tangent line (e.g., a linear solution as opposed to solving on the unit circle). After finding the solution along the tangent line (e.g., a point along the tangent line), beamforming controller  214  may then find the closest point on the unit circle and take this point as the new w n . This may enable beamforming controller  214  to solve the least-squares problem in polynomial time, thus rendering certain aspects suitable for adaptive beamforming uses. 
       FIGS. 7A and 7B  illustrate this computation, in which  FIG. 7A  shows a given w n   (0)  (the initial w n ). In calculating L (n) , beamforming controller  214  may define the tangent line along the unit circle at w n   (0) . As shown in  FIG. 7B , beamforming controller  214  may then solve for w n   (1)  by finding the least-squares solution along the tangent line and then finding w n   (1)  as the closest point on the unit circle to the point found along the tangent line. The solution mathematically expressed for stages  406 - 412  in Equations (3)-(9) therefore defines such a solution, where beamforming controller  214  may solve for x in Equation (8) to find the linear solution on the tangent line and then calculate w (n+1)  as the complex exponential (in other words, on the unit circle) term of x in Equation (9) to find the closest point on the unit circle. Beamforming controller  214  may store the obtained w (n+1)  as the solution w* in stage  414 . 
     As the matrix system of Equation (8) is a matrix representation of the least-squares problem, in solving for x and calculating w (n+1) , beamforming controller  214  may obtain complex beamforming weights w (n+1)  that, when applied at antenna array  202 , approaches the optimal w for the desired multi-finger radiation pattern. As previously indicated, in some aspects beamforming controller  214  may solve for w in an iterative manner, where beamforming controller  214  calculates iterative updates to w until w converges sufficiently close to the optimal w. Accordingly, following calculation of w (n+1)  in stage  412  beamforming controller  214  may proceed to calculate the least-squares weights C (n+1)  for the next iteration in stages  416 - 420  and to check whether a convergence criterion for w has been satisfied in stages  422 - 424 . 
     In particular, beamforming controller  214  may calculate the least-square weights C ll   (n+1)  for the main fingers in stage  416  and calculate the least-square weights C L+k,L+k   (n+1)  for the sidelobes in stage  418 , where
 
 C   ll   (n+1)   ∥a   H (θ m,l ) w   (n+1)   |−d   m,l   (n)   |, l= 1 , . . . ,L   (10)
 
 C   L+k,L+k   (n+1) =max{∥ a   H (θ s,k ) w   (n+1)   |−d   s,k   (n) |,0},  k= 1 , . . . ,K   (11)
 
     Beamforming controller  214  may the normalize the least-square weights C (n+1)  in stage  420  as 
     
       
         
           
             
               
                 
                   
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     Beamforming controller  214  may utilize the least-square weights C (n)  in the calculation of q (n)  and R (n)  in stage  408  and in calculating the error in stage  422 , where the least-square weights C (n)  is a diagonal matrix that dictates how much emphasis is placed on each main finger or sidelobe when adapting w (n)  during each iteration. For example, if the actual radiation pattern a H (θ m,l )w (n+1)  for a given main finger l is close to the desired radiation pattern d m,l   (n) , the corresponding weight of C (n)  for the main finger l will be low-valued and thus the update to w (n)  in the next iteration will not place a large emphasis on adapting w (n)  to shape the radiation pattern for the main finger l. Conversely, if the actual radiation pattern a H (θ m,l )w (n+1)  for a given main finger l is far from the desired radiation pattern d m,l   (n) , the corresponding weight of C (n)  for the main finger l will be high-valued and the update to w (n)  will place a large emphasis on adapting w (n)  to shape the radiation pattern for the main finger l. Accordingly, as shown by Equation (10), beamforming controller  214  may calculate the least-square weights C ll   (n+1)  for the main fingers based on the difference between the actual radiation pattern a H (θ m,l )w (n+1)  and the desired main finger radiation pattern d m,l   (n) , where larger differences will produce larger least-square weights and thus prompt larger adaptations targeting certain main fingers in the next iteration. In terms of sidelobes, beamforming controller  214  may calculate the least-square weights C L+k,L+k   (n+1)  for the sidelobes based on the maximum of either a) the difference between the actual radiation pattern a H (θ s,k )w (n+1)  and the desired sidelobe radiation pattern d s,k   (n) , zero. As beamforming controller  214  is targeting minimization of the sidelobe power level below the maximum sidelobe power level s Max , beamforming controller  214  may therefore be configured to set the least-square weights of C (n)  to zero for sidelobes that already have a power level below the maximum sidelobe power level s Max , meaning that the update to w (n)  in the next iteration will not place an emphasis on adapting w (n)  to shape the radiation pattern for these sidelobes. Beamforming controller  214  may then utilize the resulting least-square weights C (n+1) , after normalization in stage  408 , in solving for w in the next iteration. 
     In some aspects, beamforming controller  214  may be configured to execute inner loop  400  until w satisfies a convergence criterion. Accordingly, in stage  422  beamforming controller  214  may calculate the error term e (n+1) for the current iteration as
 
 e ( n+ 1)=( w   (n+1) ) H   R   (n)   w   (n+1) −2( q   (n) ) H   w   (n+1) +( d   (n) ) H   C   (n)   d   (n)   (13)
 
where e(0)=∞.
 
     After calculating e(n+1) in stage  422 , beamforming controller  214  may compare e(n+1) with e(n) in stage  424  as
 
| e ( n )− e ( n− 1)|&gt;ϵ  (14)
 
where ϵ is a predefined convergence parameter.
 
     Accordingly, beamforming controller  214  may determine whether difference between the error term e(n+1) (for the current iteration) and e(n) (for the previous iteration) is greater than ϵ. If yes, this indicates that w has not yet converged sufficiently close to the optimal complex beamforming weights for the desired radiation pattern d (n) , and beamforming controller  214  may proceed to stage  404  to perform another iteration of inner loop  400 . If no, this indicates that w has converged sufficiently close to the optimal complex beamforming weights, and beamforming controller  214  may proceed to stage  426  to conclude execution of inner loop  400 . As beamforming controller  214  stores W (n+1)  for the current iteration as w* at stage  414 , w* gives the converged complex beamforming weights at the completion of inner loop  400 . Alternatively, in some aspects beamforming controller  214  may run inner loop  400  for a predetermined number of iterations and take the value of W (n+1)  in the final iteration as w*, which may avoid excessively long runtimes and endless loops. 
     Completion inner loop  400  until the convergence condition of stage  424  is reached therefore generates complex beamforming weights w that are optimized according to least-squares to produce a desired multi-finger radiation pattern represented by d (n) . As can be seen from Equations (3) and (4), d (n) , which is derived directly from input parameters θ s  (sidelobe azimuth angles), θ d  (main finger azimuth angles), and s (relative main finger power levels), gives the azimuth angles and relative power levels of the main fingers but does not specify the maximum main finger power level d Max  and the maximum sidelobe power level s Max . As the best achievable maximum main finger power level d Max  (e.g., the highest possible maximum main finger power level d Max ) and maximum sidelobe power level s Max  (e.g., the lowest possible maximum sidelobe power level s Max ) are not initially known for the desired multi-finger radiation pattern, beamforming controller  214  may also iteratively calculate d Max  and s Max  according to outer loop  500  of  FIGS. 5A and 5B . As shown in  FIGS. 5A and 5B , beamforming controller  502  may solve inner loop  400  at stage  504 , thus obtaining complex beamforming weights w*. As different values for d Max  and s Max  are achievable for different complex beamforming weights w*, beamforming controller  214  may utilize w* as obtained from inner loop  400  to calculate an update to d Max  or s Max . 
     In particular, in stage  502  beamforming controller  214  initialize d Max  as √{square root over (N)}, which is the theoretical maximum gain level for N antennas and thus represents the highest achievable main finger gain level. Beamforming controller  214  may initialize s Max  to some arbitrary sidelobe level in stage  502 , such as −17 dB below d Max . 
     Beamforming controller  214  may then solve inner loop  400  in stage  504  to obtain w*. After obtaining w*, beamforming controller  214  may calculate error terms Δd Max  and Δs Max  as
 
Δ d   Max =max{∥ a   H (θ m,l ) w*|−d*   m,l   |, l= 1 , . . . ,L} 
 
Δ s   Max =max{max{| a   H (θ s,k ) w*|−d*   s,k ,0 }, k= 1 , . . . ,K}   (15)
 
where Δd Max  and Δs Max  are accordingly based on the difference between the actual radiation pattern and the desired radiation pattern represented by d* m  and d* s  (which depend on d Max  and s Max  as specified in Equation (3).
 
     Higher values for Δd Max  indicate that the actual radiation pattern according to w* for the main fingers is much less than the desired radiation pattern represented by d* m . Similarly, higher values for Δs Max  indicate that the actual radiation pattern according to w* for the sidelobes is much greater than the desired radiation pattern represented by d* s . Beamforming controller  214  may therefore determine whether both Δd Max  and Δs Max  are less than ε in stage  508 , where ε is a predefined convergence parameter (or, alternatively, beamforming controller  214  may run outer loop  500  for a predefined number of iterations). If Δd Max  and Δs Max  are not both less than ε, beamforming controller  214  may proceed to stage  512  to determine whether Δd Max  is greater than Δs Max . If Δd Max  is greater than Δs Max , beamforming controller  214  may subtract δ from d Max  in stage  514 , where δ is a predefined update parameter. Conversely, if Δd Max  is not greater than Δs Max , beamforming controller  214  may add δ to s Max  in stage  514  (although in some aspects, two different update parameters δ may be used to respectively update d Max  and s Max ). In some aspects, beamforming controller  214  may numerically add δ to d Max  or s Max , while in other aspects beamforming controller  214  may utilize a lookup table having a predefined set of values for d Max  and s Max  (e.g., spaced apart by δ or another fixed spacing scheme) which beamforming controller  214  may increment and decrement through when d Max  or s Max  is to be updated in stage  514  or  516 . 
     After updating one of d Max  or s Max  in stage  514  or  516 , beamforming controller  214  may return to stage  504  to solve inner loop  400  again using the newly updated d Max  or s Max  value. Solution of inner loop  400  will then produce a new w*, which beamforming controller  214  may utilize again to evaluate Δd Max , and Δs Max , as part of outer loop  500 . 
     Δd Max  and Δs Max  may eventually converge to the point that both Δd Max  and Δs Max  are less than ε. Beamforming controller  214  may identify this condition in stage  508  (i.e., that the maximum of Δd Max  and Δs Max  is less than ε), and may then end outer loop  500  at stage  510 . 
     Accordingly, beamforming controller  214  may update either d Max  or s Max  during each iteration of outer loop  500 , and may determine which of d Max  or s Max  to update based on whether a) the maximum difference between the actual and desired main finger radiation patterns, or b) the maximum difference between the actual and desired sidelobe radiation patterns, is larger.  FIG. 8  shows radiation pattern  800  illustrating the outer loop updates to d Max  and s Max . As previously indicated, different values of f Max  and s Mac  are possible depending on the complex beamforming weights w*. Accordingly, some instances of w* will allow for higher maximum main finger power levels than other instances of w*, while some instances of w* will allow for lower maximum sidelobe power levels than other instances of w*. If beamforming controller  214  calculates a Δd Max  that is high in stage  506 , this can indicate that the current selection of d Max  is too high for the current w*, and thus that d Max  should be decreased. Similarly, if beamforming controller  214  calculates a Δs Max  that is high in stage  506 , this can indicate that the current selection of s Max  is too low for the current w*, and thus that s Max  should be increased. Radiation pattern  800  thus shows exemplary updates where beamforming controller  214  decreases d Max  by δ and increases s Max  by δ. While decreasing d Max  by δ will decrease the power level of the main finger with the highest power level, the relative main finger power levels s remain the same, and accordingly the ratio between the main fingers defined by s remains constant. 
     Beamforming controller  214  may therefore continually execute inner loop  400  and outer loop  500  in such a manner, where beamforming controller iteratively solves inner loop  400  until convergence during each iteration of outer loop  500 . Beamforming controller  214  then uses the complex beamforming weights w* obtained as the output of inner loop  400  to update d Max  and s Max  as part of outer loop  500 , and repeats execution of inner loop  400  with the resulting d Max  and s Max . Beamforming controller  214  may then continue in this manner until d Max  and s Max  converge. Beamforming controller  214  may thus obtain complex beamforming weights w* that produce the desired multi-finger radiation pattern when implemented at antenna array  202 . 
     In addition to the algorithmic flow illustrated in  FIGS. 4 and 5 , in some aspects beamforming controller  214  may execute the multi-finger beamforming techniques (as program code or as digitally configured hardware) according to the following pseudocode: 
     
       
         
           
               
             
               
                   
               
               
                 Algorithm for Phase-only Multi-finger Synthesis: 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
            
               
                 Input: Total number of antennas N, azimuth angles for sidelobes θ s,k , k = 1, . . . , K, 
               
               
                 azimuth angles for main beams (fingers) θ m,l , l = 1, . . . , L, relative main finger 
               
               
                 power levels s = [s 1  . . . s L ] with s 1  = 1, d Max  = {square root over (N)}, and w (0)  ϵ    N×1  initial 
               
               
                 phase values (arbitrary), least-square weights C (0)  = I, e(0) = ∞, A l  ← 
               
               
                 [a(θ m,1 ), . . . , a(θ m,L ), a(θ s,1 ), . . . , a(θ s,K )] 
               
               
                 do { 
               
               
                  if Δd Max  &gt; Δs Max ; 
               
               
                   d Max  ← d Max  − δ 
               
               
                  else; 
               
               
                   s Max  ← s Max  + δ 
               
               
                  C (0)  = I, n = 0; 
               
               
                  do{ 
               
               
                   
               
               
                   
           d   m     (   n   )       =       [         d   Max     ⁢     s   1     ⁢     e     jarg   ⁡     (         a   H     ⁡     (     θ     m   ,   1       )       ⁢     w     (   n   )         )           ,   …   ⁢           ,       d   Max     ⁢     s   L     ⁢     e     jarg   ⁡     (         a   H     ⁡     (     θ     m   ,   L       )       ⁢     w     (   n   )         )             ]     T       ,       
 
               
               
                   
               
               
                   
           d   s     (   n   )       =         s   Max     ⁡     [       e     jarg   ⁡     (         a   H     ⁡     (     θ     s   ,   1       )       ⁢     w     (   n   )         )         ,   …   ⁢           ,     e     jarg   ⁡     (         a   H     ⁡     (     θ     s   ,   K       )       ⁢     w     (   n   )         )           ]       T       ,       
 
               
               
                   
               
               
                   
           d     (   n   )       =       [         (     d   m     (   n   )       )     T     ,       (     d   s     (   n   )       )     T       ]     T       ,       
 
               
               
                   
               
               
                   
           L     (   n   )       ⁡     (     i   ,   j     )       =     {             cos   ⁡     (     arg   ⁢           ⁢     w   k     (   n   )         )       ,           i   =     j   =   k                   sin   ⁡     (     arg   ⁢           ⁢     w   k     (   n   )         )       ,             i   =   k     ,     j   =     k   +     N   ⁢     ;                       0   ,         otherwise               
 
               
               
                   
               
               
                    
         R     (   n   )       =     [           Re   ⁢     {       A   l     ⁢     C     (   n   )       ⁢     A   l   H       }               -   Im     ⁢     {       A   l     ⁢     C     (   n   )       ⁢     A   l   H       }                 Im   ⁢     {       A   l     ⁢     C     (   n   )       ⁢     A   l   H       }             Re   ⁢     {       A   l     ⁢     C     (   n   )       ⁢     A   l   H       }             ]         
 
               
               
                   
               
               
                    q (n)  = [Re{A l C (n) d (n) } Im{A l C (n) d (n) }] T   
               
               
                   
               
               
                    
       x   =         [           R     (   n   )             L       (   n   )     T                 L     (   n   )           0         ]     †     ⁡     [           q     (   n   )               1         ]           
 
               
               
                   
               
               
                    w (n+1)  = e j({x}     1       N     +j{x}     N+1       2N     ) , 
               
               
                   
               
               
                   
           C     (     n   +   1     )       ⁡     (     i   ,   i     )       =     {               max   ⁢     {       |         a   H     ⁡     (     θ     s   ,   k       )       ⁢     w     (     n   +   1     )         |     -     d     s   ,   k       (   n   )           ,   0     }       ,             i   =     L   +   1       ,   …   ⁢           ,     L   +   K                        |         a   H     ⁡     (     θ     m   ,   l       )       ⁢     w     (     n   +   1     )         |     -     d     m   ,   l       (   n   )                ,             i   =   1     ,   …   ⁢           ,   L           ,           
 
               
               
                   
               
               
                  e(n + 1) ← (w (n+1) ) H R (n) w (n+1)  − 2(q (n) ) H w (n+1)  + (d (n) ) H C (n) d (n)   
               
               
                   
               
               
                  
         C     (     n   +   1     )       ←       c     (     n   +   1     )              c     (     n   +   1     )                  
 
               
               
                   
               
               
                  n ← n + 1 
               
               
                  }while {|e(n + 1) − e(n)| &gt; ϵ} 
               
               
                   Δd Max  = max{||a H (θ m,l )w (n+1) | − d m,l   (n) |, l = 1, . . . , L}, 
               
               
                   Δs Max  = max{max{|a H (θ s,k )w (n+1) | − d s,k   (n) ,0}, k = 1, . . . , K}, 
               
               
                 }while {max{Δd Max , ΔS Max } &gt; ϵ} 
               
               
                   
               
            
           
         
       
     
     Accordingly, as previously indicated in some aspects beamforming controller  214  may be implemented as one or more processors configured to retrieve (e.g., from a non-transitory computer readable medium) and execute program code that defines the algorithmic logic detailed herein for multi-finger beamforming. In some aspects, beamforming controller  214  may include digital hardware circuitry that is digitally configured with equivalent arithmetic logic with which beamforming controller  214  can execute part or all of the algorithmic logic detailed herein. 
     In various aspects, beamforming controller  214  (or another offline processor or processing component) may be configured to perform multi-finger beamforming when operating in an offline manner, e.g., when  200 communication device  200  is not actively transmitting or receiving with any users. For instance, beamforming controller  214  may generate a predefined set of complex beamforming weight vectors (e.g., forming a codebook  ∈{w 1 , w 2  . . . , w P }, where P is the codebook size and each w p , p=1, . . . , P, is a different codeword of length N), each based on a different desired multi-finger radiation pattern, while operating offline. Once operating online (e.g., during runtime and/or when  200 communication device  200  is actively transmitting or receiving with users), beamforming controller  214  may select a complex beamforming weight vector w from   having an appropriate radiation pattern (e.g., based on user positioning/angular direction relative to antenna array  202 ) and apply the complex beamforming weight vector to transmit and/or receive at antenna array  202 . Accordingly, while beamforming controller  200200  (or another offline processor or processing component) may generate the complex beamforming weight vectors using multi-finger beamforming while offline, beamforming controller  200200  may utilize the resulting complex beamforming weight vectors when operating online. 
     In some aspects, beamforming controller  214  may be configured to perform multi-finger beamforming online, e.g., when  200 communication device  200  is actively transmitting or receiving with one or more users. For example, beamforming controller  214  may identify the angular direction of L users from antenna array  202  (e.g., based on Angle-of-Arrival or similar processing techniques) and may generate main finger azimuth angles θ m  to include θ m,l =1, . . . , L, where each θ m,l  gives the angular direction of one of the L users. In some aspects, beamforming controller  214  may also identify the distance of the L users from antenna array  202  (e.g., based on received signal strength or transmission timing processing) and generate relative main finger power levels s to include s l , l=1, . . . , L, where each s l  is directly proportional to how far each user is from antenna array  202 . Beamforming controller  214  may also generate sidelobe azimuth angles θ s  based on the remaining azimuth angles around those in θ m  (excluding transients of the main fingers). Beamforming controller  214  may then execute multi-finger beamforming according to inner loop  400  and outer loop  500  to obtain complex beamforming weights w* that approach the optimal complex beamforming weights for the multi-finger radiation pattern defined by θ s , θ m , and s. 
     In some aspects, beamforming controller  214  may repeatedly trigger multi-finger beamforming to obtain repeated updated complex beamforming weights w* when operating online, such as in part of an adaptive beamforming approach. For example, beamforming controller  214  may periodically determine the angular directions and distances of L users (where L may also increase and decrease based on how many users are within the coverage area of  200 communication device  200 ), such as by determining angular directions and distances of the users during each scheduling interval. Beamforming controller  214  may then calculate complex beamforming weights w* to use during each scheduling interval based on the current angular direction and distances of the users, where beamforming controller  214  may re-calculate w* each scheduling interval to dynamically adapt to any changes in the angular direction, distance, or number of users. 
     The multi-finger beamforming techniques can be well-suited for synthesis of radiation patterns having a number of main fingers and a general upper-bound for sidelobes, such as shown in  FIGS. 3, 6, and 8 . Aspects of this disclosure for array pattern synthesis may enable the generation of more complex patterns having not only main fingers and upper-sidelobe bounds but also null steering, interference suppression, beam broadening, and more precise sidelobe reduction.  FIG. 9  shows radiation pattern  900  illustrating an exemplary radiation pattern as generated by aspects of this disclosure for array pattern synthesis. As can be seen from  FIG. 9 , array pattern synthesis may provide greater control over the radiation pattern and enable the formulation of more complex radiation patterns. 
     In various aspects, the array pattern synthesis approaches of this disclosure can utilize phase-only control, phase control with amplitude tapering, and phase and amplitude control to generate the desired radiation patterns. As previously detailed, phase-only control approaches involve the use of complex beamforming weights that only invoke phase shifts, for example, complex beamforming weights w constrained by |w|=1, such as of the form w=e jθ . Selection of θ for w=e jθ  thus enables control over phase shifts while the constraint |w|=1 prohibits amplitude adjustment. 
     Phase control with amplitude tapering similarly allows for phase control but constrains the range of amplitude control between an upper-bound and a lower band, for example, complex beamforming weights w constrained by α≤|w|≤1, such as in the form w=βe jθ  where α≤β≤1. Accordingly, the amplitude of each complex beamforming weight can be adjusted between some lower bound α and unity gain. Phase and amplitude control enables for both phase control and amplitude control (although there may still be some constraints on the possible range of amplitude control), for example, complex beamforming weights w constrained by 0≤|w|≤1, such as in the form w=βe jθ  where θ≤β≤1. 
     Each of these approaches can be implemented either at baseband or RF. In the baseband case, a beamforming controller such as beamforming controller  214  may specify the appropriate amplitude β (with β=1 for phase-only, α≤β≤1 for phase control with amplitude tapering, and 0≤β≤1 for phase and amplitude control) and phase θ for each complex beamforming weight w n , n=1, . . . , N of complex beamforming weight vector w=[w 1  . . . w N ]. Digital signal processor  208  may receive the complex beamforming weights w and apply the complex beamforming weights w to baseband data (e.g., in the case of baseband symbol s in  FIG. 1A , e.g. with IQ multiplication) to produce weighted data, which RF transceiver  204  may subsequently transmit with antenna array  202  (in the transmit case) or process to receive transmitted data (in the receive case). 
     In the RF case, beamforming controller  214  may specify the appropriate amplitude and phase θ for each complex beamforming weight of w. A set of RF beamforming circuits of RF transceiver  204  (e.g., RF beamforming circuits  110  of  FIG. 1B ), which may include, for example, analog RF frequency shifters, gain elements, and/or tapering circuits, may receive the complex beamforming weights w and apply the complex beamforming weights w to RF data to produce weighted data, which RF transceiver  204  may subsequently transmit with antenna array  202  (in the transmit case) or process to receive transmitted data (in the receive case). 
     Similar to θ m , θ s , and s in multi-finger beamforming, array pattern synthesis may utilize input parameters that define a desired radiation pattern (e.g., radiation pattern  900 ) to derive complex beamforming weights w that realize the desired radiation pattern. Additionally, in various aspects the array pattern synthesis approaches of this disclosure may also use least-squares minimization to iteratively solve for an optimal solution for w. 
     However, the array pattern synthesis approaches may generally utilize a more rigorous algorithm to derive w from the input parameters than as done for multi-finger beamforming as described above. As detailed above regarding  FIGS. 5A and 5B , outer loop  500  of the multi-finger beamforming approaches may decide whether to decrease the maximum main finger power level d Max  or increase the maximum sidelobe power level s Max  during each iteration based on the relative levels of the main fingers and d Max  and the relative levels of the sidelobes and s Max . Accordingly, the multi-finger beamforming approaches detailed herein may allow the sidelobes to exceed the maximum sidelobe power level, in response to which outer loop  500  may trigger an increase in s Max . 
     However, various aspects of the array pattern synthesis approaches of this disclosure may place constraints on the sidelobes that can prevent the sidelobes from exceeding certain upper-bounds. In other words, array pattern synthesis approaches can iteratively update w while aiming to ensure that any update to w does not allow the sidelobes to exceed certain upper-bounds. 
     These techniques can allow for greater control over the radiation pattern. However, as the iterative updates to w may face stricter constraints in aiming to keep the sidelobes below upper-bounds, some aspects of the array pattern synthesis approaches may call for more complex algorithmic solutions. In particular, given the constraints on updates to w, the matrix inversion (e.g., pseudoinverse) of Equation (8) and stage  410  may not be possible as the 
             matrix   ⁢           [           R     (   n   )             L       (   n   )     T                 L     (   n   )           0         ]         
may not be invertible.
 
     Accordingly, instead of applying least-squares optimization over the azimuth angles for all main fingers and sidelobes as in the case of multi-finger beamforming, aspects of array pattern synthesis in this disclosure may separate the azimuth angles into two groups: a first group θ l  with azimuth angles that are subject to the least-squares constraint, and a second group θ p  with azimuth angles that are subject to the upper-bound constraint. As the upper-bound constraint may only be directed to sidelobes (as sidelobes are intended to be minimized while main fingers are intended to be maximized), the second group θ p  may only contain sidelobe azimuth angles, while the first group θ l  may contain main finger and/or sidelobe azimuth angles. Depending on the relative power level of each sidelobe in each iteration, the array pattern synthesis approaches may switch azimuth angles for certain sidelobes from the first group θ l  to the second group θ p . 
     The desired radiation pattern may be defined as d=[d 1  . . . d K ] (amplitude only) for azimuth angles θ l,i , i=1, . . . , K, where d(δ l,i ), i=1, . . . , K. The least-square problem for the azimuth angles θ l,i i=1, . . . , K can therefore be generalized as finding w where 
                     arg   ⁢           ⁢       min   w     ⁢       ∑     k   =   1     K     ⁢       c   k     ⁢                d   k     ⁢     e     j   ⁢           ⁢     φ   k           -       f   ⁡     (     θ     l   ,   k       )       ⁢       a   H     ⁡     (     θ     i   ,   k       )       ⁢   w            2     ⁢           ⁢               ⁢   such           ⁢           ⁢     
     ⁢         that   ⁢             ⁢             ⁢            f   ⁢     (     θ     l   ,   k       )     ⁢       a   H     ⁡     (     θ     i   ,   k       )       ⁢   w          2       ≤     P   l       ,     l   =   1     ,     .           .           .     ⁢           ,   L     ⁢           ⁢     
     ⁢              w   n          ≤   1     ,     n   =   0     ,     .           .           .     ⁢           ,           ⁢     N   -   1               (   16   )               
where φ k  gives the phase at the k-th least-squares constraint azimuth angle θ l,k  as φ k =arg(a H (θ l,k )w) (because the desired radiation pattern d only gives amplitude, the phase term can also be considered in the least-squares optimization), f(θ) gives the radiation pattern of the antennas of antenna array  202  (where omnidirectional antennas would have f(θ)=1), and P l , l=1, . . . , L gives the upper-bound constraints on the sidelobe azimuth angles in θ l .
 
     Solution of the least-squares problem defined by Equation (16) is non-convex and NP-hard; however, aspects provided below for phase-only and phase control with amplitude tapering can present convex solutions. Furthermore, aspects provided below also specify how to select least-square weights c k  and φ k , and feasible constraints. Therefore, the following aspects of array pattern synthesis can reformulate the above optimization problem for array pattern synthesis to solve it in polynomial time. 
       FIG. 10  shows an exemplary radiation pattern  1000  illustrating the least-squares constraints on the azimuth angles of θ l  according to d and the upper-bound constraints on the azimuth angles of θ p  according to P l . In particular, radiation pattern  1000  illustrates two least-square constraints for azimuth angles θ l,k  defined by d i  and upper-bound constraints on the power level of some azimuth angels θ p,k  for side-lobe reduction. 
       FIGS. 11A and 11B  show an exemplary process  1100  for a phase-only array pattern synthesis approach according to some aspects.  FIG. 11A  expresses process  1100  algorithmically while  FIG. 11B  expresses process  1100  in prose, where the underlying logic at each stage in both expressions is equivalent. In the manner as detailed above for aspects of multi-finger beamforming, beamforming controller  214  may execute the algorithmic logic of process  1100  as one or more processors executing program code that defines the algorithmic logic and/or one or more dedicated hardware circuits that are digitally configured according to the algorithmic logic. 
     As shown in  FIGS. 11A and 11B , beamforming controller  214  may begin process  1100  with the following inputs at stage  1102 :
         N number of antennas   θ l  azimuth angles with least-squared constraint   θ p  azimuth angles with upper-bound constraint   w (0)  initial complex beamforming weights   d initial desired radiation pattern   P l  upper-bound sidelobe gain   C (0)  initial least-squares weights   e(0) initial error term
 
where θ l  is a vector θ l =[θ l,1  . . . θ l,K ] containing the azimuth angles θ l,i =1, . . . , K, θ p  is a vector θ p =[θ p,1  . . . θ p,L ] containing the azimuth angles θ p,i , i=1, . . . , L, w (0)  is a vector w=[w 0  w 1  . . . w N−1 ] T  containing the initial complex beamforming weights w n , n=0, . . . , N−1 (can be arbitrarily chosen), d is a vector d=[d i  . . . d k ] containing the desired radiation pattern for the least-squared constraint azimuth angles θ l,i , i=1, . . . , K as d i =d(θ l,i ), i=1, . . . , K, P l , l=1, . . . , size (θ p ), are the maximum sidelobe power levels for each of the azimuth angles in θ p  (which are all sidelobe azimuth angles), C (0)  is a diagonal matrix initialized to the identity matrix I that gives the initial least-square weights, and e(0)=∞ gives the initial error term used for the convergence check.
       

     As denoted in stage  1102 , process  1100  may initialize the second group θ p  (the upper-bound constraint azimuth angles) as a null set Ø, and accordingly may initialize the first group θ l  (the least-squares constraint azimuth angles) as containing all of the azimuth angles. Each of the azimuth angles may be pre-identified as either a main finger azimuth angle or a sidelobe azimuth angle. Desired radiation pattern vector d defines the desired power level at each main finger and sidelobe azimuth angle, and therefore acts as the input parameter defining the overall radiation pattern. 
     Beamforming controller  214  may selectively switch certain sidelobe azimuth angles from the first group θ l  to the second group θ p  throughout the course of execution of process  1100 . In particular, as soon beamforming controller  214  calculates an iterative update to w (n)  that causes the actual radiation pattern for a given sidelobe azimuth angle of θ l  to fall below its desired radiation pattern, beamforming controller  214  may switch the sidelobe azimuth angle from θ l  to θ p , thus changing the sidelobe azimuth angle from being subject to a least-squared constraint to being subject to an upper-bound constraint. Each azimuth angle in θ p  may therefore be a sidelobe azimuth angle for which the current w (n)  keeps the radiation pattern below a certain upper bound, while each azimuth angle in θ l  may either be a main finger azimuth angle (which may always remain in θ l  and thus subject to a least-squares constraint) or a sidelobe azimuth angle that w (n)  does not yet contain below its upper bound. 
     As beamforming controller  214  may aim to determine complex beamforming weights w that keep the radiation pattern at sidelobe azimuth angles below certain upper bounds (while realizing main fingers that reach certain maximum power levels), beamforming controller  214  may then calculate subsequent iterative updates to w (n)  while ensuring that any update to w (n)  does not cause the radiation pattern at any of the azimuth angles in θ p  (which are all sidelobe azimuth angles) to exceed their assigned upper bounds. Beamforming controller  214  may therefore iteratively solve the least-squares problem for the azimuth angles of θ l  to determine a w (n)  while constraining all updates to w (n)  to keep the azimuth angles of θ p  below their assigned upper bounds. 
     With reference back to  FIGS. 11A and 11B , after obtaining the initial parameters at stage  1102 , beamforming controller  214  may increment iteration counter n to n+1 in stage  1104 . Beamforming controller  214  may then calculate L (n) , d (n) , x (n) , Q p,l   (n) , A l   (n) , R r , and q r  in stage  1106 , which define the least-squares problem in the current iteration. 
     As previously indicated, beamforming controller  214  may execute process  1100  in accordance with a phase-only approach, where each w n  of w is bound by |w n |=1. Beamforming controller  214  may redefine the least-squares problem generalized above in Equation (8) according to a phase-only approach for process  1100 . 
     In particular, as previously described regarding  FIGS. 7A and 7B , each w n  may be a point on the unit circle, and in solving the least-squares problem beamforming controller  214  may aim to find w as a set of points on the unit circle that minimize the least-square error between the actual radiation pattern produced by w and the desired radiation pattern d. Beamforming controller  214  may therefore solve the least-squares problem with process  1100  in the manner as described for multi-finger beamforming in inner loop  400  and outer loop  500 , namely by, for each w n  of w (n) , plotting the tangent line on the unit circle, solving the least-squares problem in a linear fashion along the tangent line, finding the closest point on the unit circle to the solution on the tangent line, and taking this closest point as the new w n  for w (n+1) . 
     Accordingly, beamforming controller  214  may calculate L (n)  in stage  1104  to define the tangent lines for the current w (n) , in particular as 
     
       
         
           
             
               
                 
                   
                     
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     Beamforming controller  214  may calculate the desired radiation pattern d (n)  for the azimuth angles in θ l  in the current iteration (e.g., for the least-squared constraint azimuth angles) based on the phase values of the radiation pattern produced by the current w (n) , in particular as 
                     d     (   n   )       =       [       d   1     ⁢     e     j   ⁢           ⁢     φ   1         ⁢   …   ⁢           ⁢     d     size   ⁡     (     θ   l     )         ⁢     e     j   ⁢           ⁢     φ     size   ⁡     (     θ   l     )               ]     T             (   18   )               
where each element d i , i=1, . . . , K is the desired radiation pattern for the i-th azimuth angle in θ l  and φ i =arg(a H (θ l,i )w n ) gives the phase at the i-th azimuth angle of θ l  given w (n)  as described above regarding Equation (31).
 
     Beamforming controller  214  may calculate x (n)  based on the real and imaginary parts of w (n)  as
 
 x   (n) =[Re{ w   (n) }Im{ w   (n) }] T   (19)
 
     Beamforming controller  214  may calculate Q p,l   (n)  as 
                           Q     p   ,   l         (   n   )       =     [           Re   ⁢     {       a   ⁡     (     θ     p   ,   l       )       ⁢       a   H     (     θ       p   ,   l     )       }                   -   Im     ⁢     {       a   ⁡     (     θ     p   ,   l       )       ⁢       a   H     (     θ       p   ,   l     )       }                     Im   ⁢     {       a   ⁡     (     θ     p   ,   l       )       ⁢       a   H     (     θ       p   ,   l     )       }                 Re   ⁢     {       a   ⁡     (     θ     p   ,   l       )       ⁢       a   H     (     θ       p   ,   l     )       }                 ]       ,     
     ⁢     l   =   1     ,           ⁢     .           .           .     ⁢           ,     size   ⁡     (     θ   p     )         ⁢                   (   20   )               
where Q p,l   (n)  represents real matrix of the matrix of azimuth angles with upper-bound constraint for iteration n and constraint l, This is used to solve the problem in real domain.
 
     Beamforming controller  214  may calculate A l   (n)  as
 
 A   l   (n) ←[ a (θ l,1 ), . . . , a (θ l,size(θ     l     ) )]  (21)
 
where, similar to the multi-finger beamforming approach, A is the DFT matrix and A l   (n)  therefore gives the DFT-submatrix containing the rows of A having indices that correspond to the azimuth angles of θ l .
 
     Beamforming controller  214  may calculate R r  as 
                       R   r     =     [           Re   ⁢     {       A   l     (   n   )       ⁢         C     (   n   )       ⁡     (     A   l     (   n   )       )       H       }               -   Im     ⁢     {       A   l     (   n   )       ⁢         C     (   n   )       ⁡     (     A   l     (   n   )       )       H       }                 Im   ⁢     {       A   l     (   n   )       ⁢         C     (   n   )       ⁡     (     A   l     (   n   )       )       H       }             Re   ⁢     {       A   l     (   n   )       ⁢         C     (   n   )       ⁡     (     A   l     (   n   )       )       H       }             ]       ⁢                   (   22   )               
where R r  is a matrix in real domain obtained by expanding the weighted least square objective function in (16). It is used to solve problem in real domain in a convex optimization.
 
     Beamforming controller  214  may calculate q r  as
 
 q   r =[Re{ A   l   (n)   C   (n)   d   (n) }Im{ A   l   (n)   C   (n)   d   (n) }] T   (23)
 
where q r  is a vector in real domain obtained by expanding the weighted least square objective function in (16). It is used to solve problem in real domain in a convex optimization.
 
     Accordingly, given L (n) , d (n) , x (n) , Q p,l   (n) , A l   (n) , R r , q r , and C (n)  (either C (0) =I or C (n&gt;0)  as obtained in the previous iteration), beamforming controller  214  may solve the least-squares problem in stage  1108  using phase-only beamforming while considering the upper-bound constraint on the azimuth angles of θ p . As previously indicated, in order to make the optimization complex, the phase-only constraint |w n |=1 is replaced by a linear constraint as in  FIGS. 7A and 7B , where beamforming controller  214  performs the optimization on the tangent line at point w n   (0)  for each antenna element of antenna array  202 . After solving along the tangent line, beamforming controller  214  finds w n   (1)  as the closest point on the unit circle to the solution found on the tangent line. Expressed mathematically, beamforming controller  214  may calculate 
     
       
         
           
             
               
                 
                   
                     
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     Beamforming controller  214  may therefore solve the least-squares problem with phase-only beamforming by finding the vector x that minimizes the least-squares expression d (n)     H   C (n) d (n) +x (n)     H   R r x (n) −2x (n)     H   q r  while also keeping the radiation pattern at the upper-bound constraint azimuth angles of θ p  below their respective upper bounds given by P l  and constraining the solution to points located on the unit circle. Beamforming controller  214  may also set the error term e(n) as the value of d (n)     H   C (n) d (n) +x (n)     H   R r x (n) −2x (n)     H   q r  given the identified x, or in other words, the minimum value of d (n)     H    C (n) d (n) +x (n)     H   R r x (n) −2x (n)     H   q r  that also satisfies the relevant constraints on x. 
     Following stage  1108 , beamforming controller  214  may calculate w (n+1)  from x in stage  1110  as
 
 w   (n+1)   e   j({x}     1       N     +j{x}     N+1       N     )   (25)
 
which makes w (n+1)  have a constant modulus and thus.
 
     Beamforming controller  214  may then set w (n+1)  for the current solution as w* in stage  1112 . Beamforming controller  214  may then update the least-square weights C (n)  and, if necessary, select azimuth angles of θ l  to switch to θ p . In particular, beamforming controller  214  may update the least-square weights for the sidelobe azimuth angles in stage  1118  by calculating
 
 C   ii   (n+1) =max{| a   H (θ l,i ) w   (n+1)   |−d   i ,0}  (26)
 
or in other words, by setting C ii   (n+1)  for each i-th azimuth angle of θ l  that is a sidelobe as either a) the difference between the magnitude of the actual radiation pattern |a H (θ l,i )w (n+1) | and the desired radiation pattern d i  for the i-th azimuth angle if the magnitude of the actual radiation pattern is greater than the desired radiation pattern, or b) zero if the magnitude of the actual radiation pattern is not greater than the desired radiation pattern.
 
     Accordingly, the least-square weights C ii   (n+1)  will be non-zero for sidelobe azimuth angles for which the actual radiation pattern given by w (n+1)  is greater than the desired radiation pattern, and will be zero for sidelobe azimuth angles for which the actual radiation pattern given by w (n+1)  is greater than the desired radiation pattern. As beamforming controller  214  may aim to keep the radiation pattern for sidelobe azimuth angles below their respective desired radiation pattern, beamforming controller  214  may switch the sidelobe azimuth angles in θ l  having zero-valued least-square weights C ii   (n+1)  to θ p , and constrain subsequent updates to w to keeping the radiation pattern for these sidelobe azimuth angles to below their desired radiation pattern. 
     Accordingly, beamforming controller  214  may check in stage  1120  whether any of the least-square weights C ii   (n+1)  for sidelobe azimuth angles are equal to zero and, if so, move the sidelobe azimuth angles with zero-valued least-square weights C ii   (n+1)  from θ l  to θ p  in stage  1122 . Beamforming controller  214  may keep the sidelobe azimuth angles with non-zero least-square weights C ii   (n+1)  in θ l . 
     Movement of sidelobe azimuth angles from θ l  to θ p  will change the size of the dependent vectors and matrices calculated by beamforming controller  214  in stage  1106 , as the number of azimuth angles in θ l  will decrease and the number of azimuth angles in θ p  will increase. Beamforming controller  214  may then store current θ l  and θ p  for subsequent use in stage  1106  of the next iteration. Beamforming controller  214  will then perform subsequent iterations with different vector and matrix sizes according to the current size of θ l  and θ p . 
     Following stage  1110 , beamforming controller  214  may also update the least-square weights C ii   (n+1)  for the azimuth angles in θ l , which includes main finger azimuth angles and sidelobe azimuth angles for which w n  does not yet reduce the actual radiation pattern below their desired radiation pattern. Beamforming controller  214  may update the least-square weights C ii   (n+1)  for the azimuth angles in θ l  in stage  1124  as
 
 C   ii   (n+1)   =∥a   H (θ p,i ) w   (n+1)   |−d   i |  (27)
 
     Beamforming controller  214  may then normalize all of the least-square weights of C (n+1)  (for both main finger and sidelobe azimuth angles) in stage  1126  as 
                     C     (     n   +   1     )       ←       C     (     n   +   1     )           ||     ⁢     C     (     n   +   1     )       ⁢     ||                 (   28   )               
and subsequently store the least-square weights C (n+1)  for subsequent use in stage  1106  of the next iteration.
 
     Beamforming controller  214  may continue to iteratively update w according to process  1100  in this manner. At the end of each iteration n, beamforming controller  214  may check for convergence by comparing the error term e(n) of the current iteration to the error term e(n−1) of the previous iteration, and determining whether the magnitude of the difference is greater than convergence parameter ϵ in stage  1114 . If the magnitude of the difference |e(n)−e(n−1)| is greater than E, beamforming controller  214  may conclude that w (n)  has not yet converged to the optimal solution and may proceed to the next iteration n+1 at stage  1104 . If the magnitude of the difference |e(n)−e(n−1)| is greater than ϵ, beamforming controller  214  may conclude that w (n)  has converged and may end process  1100  at stage  1116 , thus taking the current w (n)  as the final complex beamforming weights w*. Alternatively, beamforming controller  214  may execute process  1100  for a predefined number of iterations, and take the value of w (n+1)  at the final iteration as w*. 
     Equivalent to the arithmetic logic described regarding  FIGS. 11A and 11B , beamforming controller  214  may execute process  1100  (as program code or as digitally configured hardware) according to the following pseudocode: 
     
       
         
           
               
             
               
                   
               
               
                 Algorithm for Phase-only Array Pattern Synthesis: 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
            
               
                 Input: Total number of antennas N, azimuth angles θ l,k  ϵ θ l   (0) , θ p,l  ϵ θ p   (0)  = ø, and 
               
               
                 w (0)  ϵ    N×1  initial phase values (arbitrary), desired raditation pattern d i  for each 
               
               
                 azimuth angles, power upper bound P l , least-square weights C (0)  = I. e(0) = ∞. 
               
               
                 do { 
               
               
                   
               
               
                    
           L     (   n   )       ⁡     (     i   ,   j     )       =     {             cos   ⁡     (     arg   ⁢           ⁢     w   k     (   n   )         )       ,           i   =     j   =   k                   sin   ⁡     (     arg   ⁢           ⁢     w   i     (   n   )         )       ,             i   =   k     ,     j   =     k   +   N                   0   ,         otherwise               
 
               
               
                   
               
               
                    
           Find   ⁢     :     ⁢           ⁢     d     (   n   )         =       [         d   1     ⁢     e     jarg   ⁡     (         a   H     ⁡     (     θ     l   ,   1       )       ⁢     w     (   n   )         )           ,   …   ⁢           ,       d     |     θ   l     |       ⁢     e     jarg   ⁡     (         a   H     ⁡     (     θ     l   ,     |     θ   l     |         )       ⁢     w     (   n   )         )             ]     T       ,       
 
               
               
                   
               
               
                     A l   (n)  ← [a(θ l,1 ), . . . , a(θ l,|θ     l     | )], 
               
               
                   
               
               
                   
           Q     p   ,   l       (   n   )       =     [           Re   ⁢     {       a   ⁡     (     θ     p   ,   l       )       ⁢       a   H     ⁡     (     θ     p   ,   l       )         }               -   Im     ⁢     {       a   ⁡     (     θ     p   ,   l       )       ⁢       a   H     ⁡     (     θ     p   ,   l       )         }                 Im   ⁢     {       a   ⁡     (     θ     p   ,   l       )       ⁢       a   H     ⁡     (     θ     p   ,   l       )         }             Re   ⁢     {       a   ⁡     (     θ     p   ,   l       )       ⁢       a   H     ⁡     (     θ     p   ,   l       )         }             ]       ,     l   =   1     ,   …   ⁢           ,     |     θ   p     |         
 
               
               
                   
               
               
                    
         R   r     =     [           Re   ⁢     {       A   l     (   n   )       ⁢         C     (   n   )       ⁡     (     A   l     (   n   )       )       H       }               -   Im     ⁢     {       A   l     (   n   )       ⁢         C     (   n   )       ⁡     (     A   l     (   n   )       )       H       }                 Im   ⁢     {       A   l     (   n   )       ⁢         C     (   n   )       ⁡     (     A   l     (   n   )       )       H       }             Re   ⁢     {       A   l     (   n   )       ⁢         C     (   n   )       ⁡     (     A   l     (   n   )       )       H       }             ]         
 
               
               
                   
               
               
                     q r  = [Re{A l   (n) C (n) d (n) } Im{A l   (n) C (n) d (n) }] T   
               
               
                      x (n)  = [Re{w (n) } Im{w (n) }] T   
               
               
                   Solve: 
               
               
                   
               
               
                    
           e   ⁡     (   n   )       ⁢     :     ⁢           ⁢       min   x     ⁢           ⁢       d       (   n   )     H       ⁢     C     (   n   )       ⁢     d     (   n   )             +       x       (   n   )     H       ⁢     R   r     ⁢     x     (   n   )         -     2   ⁢     x       (   n   )     H       ⁢     q   r           
             s   .   t   .           ⁢     L     (   n   )         ⁢     x     (   n   )         =   1     ,     
     ⁢                  Q     p   ,   l       (   n   )         ⁢     x     (   n   )              2     ≤       P   l         ,     l   =   1     ,   …   ⁢           ,     |     θ   p     |         
 
               
               
                   
               
               
                   Find: 
               
               
                      w (n+1)  = e j({x}     1       N     +j{x}     N+1       2N     )   
               
               
                   
               
               
                   
         C   ll     (     n   +   1     )       =     {               max   ⁢     {       |         a   H     ⁡     (     θ     p   ,   l       )       ⁢     w     (     n   +   1     )         |     -     d   l         ,   0     }       ,           if   ⁢           ⁢   sidelobe                      |         a   H     ⁡     (     θ     p   ,   l       )       ⁢     w     (     n   +   1     )         |     -     d   l              ,           if   ⁢           ⁢   main   ⁢           ⁢   beam           ,     l   =   1     ,   …   ⁢           ,     |     θ   l     |             
 
               
               
                   
               
               
                  if C ll   (n+1)  = = 0, do θ l   (n+1)  ← θ l   (n) \θ l,l  and θ p   (n+1)  ← θ p   (n)  ∪ θ l,l  ∀l 
               
               
                    Normalize C (n+1)   
               
               
                 n ← n + 1} while (|e(n) − e(n − 1)| ≥ ϵ) 
               
               
                   
               
            
           
         
       
     
       FIGS. 12A-12B  show the iterative progression of process  1100  according to some aspects, where upward-pointing arrows indicate least-squares constraint azimuth angles in θ l  and downward-pointing arrows indicate upper-bound constraint azimuth angles in θ p . As shown in  FIG. 12A , the initial radiation pattern produced by w (n)  yields two sidelobe azimuth angles that are above their desired radiation pattern given by d i  while the remaining sidelobe azimuth angles are below their desired radiation pattern. Accordingly, upon evaluating the least-square constraints C (n)  in stages  1118  and  1120 , beamforming controller  214  may move the remaining sidelobe azimuth angles from θ l  to θ p  and leave the two sidelobe azimuth angles in θ l  along with the main finger azimuth angle. Beamforming controller  214  may then solve the least-squares problem for w (n+1)  in the next iteration using phase-only beamforming while constraining the azimuth angles of θ p  to remain below their respective upper-bound power levels P l . Accordingly, as shown in  FIG. 12B , beamforming controller  214  may obtain w (n+1)  that keeps the radiation pattern for the upper-bound constraint azimuth angles in θ l  below their respective P 1 . Beamforming controller  214  may again solve the least-squares problem for w (n+2)  to arrive at the radiation pattern shown in  FIG. 12C , where all sidelobe azimuth angles are below their respective desired power levels P l  and are thus in θ p . As beamforming controller  214  may decide that w has converged based on the error term e(n), beamforming controller  214  may continue to iteratively update w after moving all sidelobe azimuth angles to θ p . 
       FIGS. 13A and 13B  show an exemplary process  1300 , which describes array pattern synthesis using phase control with amplitude tapering according to some aspects.  FIG. 13A  expresses process  1300  algorithmically while  FIG. 13B  expresses process  1300  in prose, where the underlying logic at each stage in both expressions is equivalent. Beamforming controller  214  may similarly iteratively calculate complex beamforming weights w using least-squares optimization for a first group θ l  of azimuth angles while using an upper-bound constraint for a second group θ p  of azimuth angles. However, instead of constraining complex beamforming weights w n  of w to |w n |=1, beamforming controller  214  may constrain w n  to α≤|w n |≤1 according to phase control with amplitude tapering. 
     Accordingly, beamforming controller  214  may solve the least-squares problem in a similar manner to that of the phase-only approach, with the primary difference being in the definition of the search space constraints for each w n  of w.  FIGS. 14A and 14B  illustrate the search space constraining w n  to α≤|w n |≤1 that beamforming controller  214  may apply to transform the least-squares problem into a convex problem. Accordingly, instead of searching the entire two-dimensional ring defined by α≤|w n |≤1 for updates to w n , beamforming controller  214  may instead define a region of the unit circle as shown in  FIG. 14A  as the search space for solving the least-squares problem. In particular, given w n   (k)  from iteration k, the region is defined as being between a line perpendicular to the line e jw     n       (k)   , where the minimum distance between the origin and this line is α, and the unit circle. For each iteration k, beamforming controller  214  may rotate this region according to the solution of the previous iteration w n   (k−1)  and define a new region satisfying these conditions with which to solve for w n   (k)  Beamforming controller  214  may iteratively solve the least-squares for w (k)  in this region based on the azimuth angles of θ l  while similarly constraining the azimuth angles of θ p  to remain below their respective upper bounds P l . 
     Process  1300  describes the algorithmic logic which beamforming controller  214  may execute (as program code or as digitally-configured hardware circuitry) to solve for w using phase control with amplitude tapering. As can be seen from  FIGS. 13A and 13B , beamforming controller  214  may solve for w using similar algorithmic logic, with the exception being in the definition of the constraint conditions for the least-squares solution in stage  1308  and the initialization and calculation of the terms related to these constraints in stages  1302  and  1306 , respectively. In particular, beamforming controller  214  may initialize L k  in stage  1302  as 
                       L     (   n   )       ⁡     (     i   ,   j     )       =     {             1   ,           i   =     j   =   k               0       otherwise         ,     k   =   1     ,           ⁢     .           .           .     ⁢           ,   N               (   29   )               
which defines the unit circle as shown in  FIGS. 14A and 14B .
 
     Beamforming controller  214  may then increment the iteration count at stage  1304 , and proceed to stage  1306  to calculate Z k   (n) , d (n) , x (n) , Q p,l   (n) , A l   (n) , R r , and q r , which are the parameters used by beamforming controller  214  to define and solve the least-squares problem. In particular, beamforming controller  214  may calculate d (n) , x (n) , Q p,l   (n) , A l   (n) , R r , and q r  in the manner expressed above in Equations (18)-(23), respectively, which may rely on C (n)  from the previous iteration (either) C (0) =1 as initialized or C (n&gt;0)  as calculated in stages  1318 ,  1324 , and  1326 ). Beamforming controller  214  may calculate Z k   (n)  as 
                       Z   k     (   n   )       ⁡     (     i   ,   j     )       =     {                 cos   ⁡     (     arg   ⁢           ⁢     w   n     (   n   )         )       a     ,           i   =     j   =   k                     sin   ⁡     (     arg   ⁢           ⁢     w   n     (   n   )         )       a     ,             i   =   k     ,     j   =     k   +   N                   0   ,         otherwise         ,     k   =   1     ,           ⁢     .           .           .     ⁢           ,   N               (   30   )               
which defines the line shown in  FIGS. 14A and 14B  (perpendicular to the line formed between the origin and w n ) that intersects the unit circle and thus defines the other boundary of the region used for the search space.
 
     After obtaining the parameters in stage  1306 , beamforming controller  214  may solve the least-squares problem in stage  1308  for the azimuth angles of θ l  according to the relevant constraints for the upper bound constraint azimuth angles of θ p  and the complex beamforming weight constraints on w n  according to phase control with amplitude tapering. As shown in  FIGS. 13A and 13B , beamforming controller  214  may evaluate the same expression d (n)     H   C (n) d (n) +x (n)     H   R r x (n) −2x (n)     H   q r  as in the phase-only case to obtain x (e.g., the vector x that minimizes this expression) and e(n) (e.g., the value of this expression given the identified solution x) while using the same upper-bound constraint ∥√{square root over (Q p,l   (n) )}x (n) ∥ 2 ≤√{square root over (P l )}, l=1, . . . , size(θ p ) for the azimuth angles of θ p . However, as the search space is different than the phase-only case, beamforming controller  214  may also constrain the solution x
 
∥√{square root over ( L   k )} x   (n) ∥ 2 ≤1,  k= 1, . . . , N  
 
such that
 
 Z   k   (n)   x   (n) ≥1,  k= 1, . . . , N   (31)
 
which limits the search space for x to restrict the complex beamforming weights w n  of w (n)  to be within the region defined in  FIGS. 14A and 14B .
 
     Beamforming controller  214  may then calculate w (n+1)  from x in stage  1310  as w (n+1) ={x} 1   N +j{x} N+1   2N  which, unlike in Equation (9), does not have a constant modulus as w (n+1)  can have any amplitude between α and 1 for phase control with amplitude tapering). Beamforming controller  214  may then perform the remaining process of process  1300  in the same manner as in process  1100 , including storing w (n+1)  as the current solution w (*)  in stage  1312 , updating the least-square weights C (n)  for the sidelobe azimuth angles of θ l  and switching certain sidelobe azimuth angles from θ l  to θ p  in stages  1318 - 1322 , updating the least-square weights C (n)  for the main finger azimuth angles of θ l  and normalizing the least-square weights C (n)  in stages  1324  and  1326 , and checking for convergence with e(n) in stages  1314  and  1316 . Beamforming controller  214  may therefore iteratively find a solution w* that produces a radiation pattern approaching the desired radiation pattern using phase control with amplitude tapering with process  1300 . 
     Equivalent to the algorithmic logic described above for process  1300 , beamforming controller  214  may execute process  1300  (as program code or as digitally configured hardware) according to the following pseudocode: 
     
       
         
           
               
             
               
                   
               
               
                 Algorithm for Phase control and Amplitude Tapering Array Pattern Synthesis: 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
            
               
                 Input: Total number of antennas N, azimuth angles θ l,k  ϵ θ l   (0) , θ p,l  ϵ θ p   (0)  = 
               
               
                 w (0)  ϵ    N×1  initial phase values (arbitrary), desired radiation pattern d i , minimum 
               
               
                 amplitude level a, power upper bound P l , least-square weights C (0)  = I |θ     l     | . e(0) = 
               
               
                 ∞. 
               
               
                 Do { 
               
               
                   
               
               
                    
           L   k     ⁡     (     i   ,   j     )       =     {             1   ,           i   =     j   =   k                 0   ,         otherwise         ,     k   =   1     ,   …   ⁢           ,   N           
 
               
               
                   
               
               
                    
           Z   k     (   n   )       ⁡     (     i   ,   j     )       =     {                 cos   (     arg   ⁢           ⁢     w   n     (   n   )           a     ,           i   =     j   =   k                     sin   ⁡     (     arg   ⁢           ⁢     w   n     (   n   )         )       a     ,             i   =   k     ,     j   =     k   +   N                   0   ,         otherwise         ,     k   =   1     ,   …   ⁢           ,   N           
 
               
               
                   
               
               
                    
           Find   ⁢     :     ⁢           ⁢     d     (   n   )         =       [         d   1     ⁢     e     jarg   ⁡     (         a   H     ⁡     (     θ     l   ,   1       )       ⁢     w     (   n   )         )           ,   …   ⁢           ,       d     |     θ   l     |       ⁢     e     jarg   ⁡     (         a   H     ⁡     (     θ     l   ,     |     θ   l     |         )       ⁢     w     (   n   )         )             ]     T       ,       
 
               
               
                   
               
               
                     A l   (n)  ← [a(θ l,1 ), . . . , a(θ l,|θ     l   |)], 
               
               
                   
               
               
                   
           Q     p   ,   l       (   n   )       =     [           Re   ⁢     {       a   ⁡     (     θ     p   ,   l       )       ⁢       a   H     ⁡     (     θ     p   ,   l       )         }               -   Im     ⁢     {       a   ⁡     (     θ     p   ,   l       )       ⁢       a   H     ⁡     (     θ     p   ,   l       )         }                 Im   ⁢     {       a   ⁡     (     θ     p   ,   l       )       ⁢       a   H     ⁡     (     θ     p   ,   l       )         }             Re   ⁢     {       a   ⁡     (     θ     p   ,   l       )       ⁢       a   H     ⁡     (     θ     p   ,   l       )         }             ]       ,     l   =   1     ,   …   ⁢           ,     |     θ   p     |         
 
               
               
                   
               
               
                    
         R   r     =     [           Re   ⁢     {       A   l     (   n   )       ⁢         C     (   n   )       ⁡     (     A   l     (   n   )       )       H       }               -   Im     ⁢     {       A   l     (   n   )       ⁢         C     (   n   )       ⁡     (     A   l     (   n   )       )       H       }                 Im   ⁢     {       A   l     (   n   )       ⁢         C     (   n   )       ⁡     (     A   l     (   n   )       )       H       }             Re   ⁢     {       A   l     (   n   )       ⁢         C     (   n   )       ⁡     (     A   l     (   n   )       )       H       }             ]         
 
               
               
                   
               
               
                     q r  = [Re{A l   (n) C (n) d (n) } Im{A l   (n) C (n) d (n) }] T   
               
               
                      x (n)  = [Re{w (n) } Im{w (n) }] T   
               
               
                   Solve: 
               
               
                   
               
               
                    
           e   ⁡     (   n   )       ⁢     :     ⁢           ⁢       min   x     ⁢           ⁢       d       (   n   )     H       ⁢     C     (   n   )       ⁢     d     (   n   )             +       x       (   n   )     H       ⁢     R   r     ⁢     x     (   n   )         -     2   ⁢     x       (   n   )     H       ⁢     q   r           
           s   .   t   .           ⁢                L   k       ⁢     x     (   n   )              2       ≤   1     ,     k   =   1     ,   …   ⁢           ,   N       
             Z   k     (   n   )       ⁢     x     (   n   )         ≥   1     ,     k   =   1     ,   …   ⁢           ,   N       
                      Q     p   ,   l       (   n   )         ⁢     x     (   n   )              2     ≤       P   l         ,     l   =   1     ,   …   ⁢           ,     |     θ   p     |         
 
               
               
                   
               
               
                   Find: 
               
               
                      w (n+1)  = {x} 1   N  + j{x} N+1   2N   
               
               
                   
               
               
                   
         C   ll     (     n   +   1     )       =     {               max   ⁢     {       |         a   H     ⁡     (     θ     p   ,   l       )       ⁢     w     (     n   +   1     )         |     -     d   l         ,   0     }       ,           if   ⁢           ⁢   sidelobe                      |         a   H     ⁡     (     θ     p   ,   l       )       ⁢     w     (     n   +   1     )         |     -     d   l              ,           if   ⁢           ⁢   main   ⁢           ⁢   beam           ,     l   =   1     ,   …   ⁢           ,     |     θ   l     |             
 
               
               
                   
               
               
                  if C ll   (n+1)  = = 0, do θ l   (n+1)  ← θ l   (n) \θ l,l  and θ p   (n+1)  ← θ p   (n)  ∪ θ l,l  ∀l 
               
               
                    Normalize C (n+1)   
               
               
                 n ← n + 1} 
               
               
                 while (|e(n) − e(n − 1)| ≥ ϵ) 
               
               
                   
               
            
           
         
       
     
       FIGS. 15A and 15B  show an exemplary process  1500 , which beamforming controller  214  may execute to perform array pattern synthesis with phase and amplitude control beamforming.  FIG. 15A  expresses process  1500  algorithmically while  FIG. 15B  expresses process  1500  in prose, where the underlying logic at each stage in both expressions is equivalent. Similar to the differences between processes  1100  and  1300 , beamforming controller  214  may utilize a different constraint on the complex beamforming weights w n  in accordance with phase and amplitude control beamforming. In particular, as opposed to limiting complex beamforming weights w n  to points along the unit circle as |w n |=1 or to a certain ring-shaped region within the unit circle as α≤|w n |≤1, beamforming controller  214  may constrain complex beamforming weights w n  to anywhere within the unit circle as 0≤|w n |≤1. 
     Accordingly, as the constraint related to Z k   (n)  is unnecessary (as it relates to the amplitude tapering lower bound α) for the phase and amplitude control case, beamforming controller  214  may perform generally the same process in process  1500  as in process  1300  with the omission of the calculation of Z k   (n)  in stage  1506  and the omission of the application of the related constraint Z k   (n) x (n) ≥1, k=1, . . . , N in stage  1508 . Beamforming controller  214  may therefore perform stages  1502 - 1526  of process  1500  in the same manner as stages  1302 - 1326  of process  1300  with the exception of these omissions. 
     Beamforming controller  214  may therefore execute process  1500  with program code or with digitally-configured hardware circuitry to solve for complex beamforming weights w* using least-squares optimization where each w n  of w* satisfies 0≤|w n |≤1. Equivalent to the algorithmic logic described above for process  1500 , beamforming controller  214  may execute process  1500  (as program code or as digitally configured hardware) according to the following pseudocode: 
     
       
         
           
               
            
               
                   
               
               
                 Algorithm for Phase and Amplitude Control Array Pattern Synthesis: 
               
            
           
           
               
               
            
               
                   
                 Input: Total number of antennas N, azimuth angles θ l,k  ϵ θ l   (0) , θ p,l  ϵ θ p   (0)  = ϕ, and 
               
               
                   
                 w (0)  ϵ       Nx1    initial phase values (arbitrary), desired radiation pattern d i , power 
               
               
                   
                 upper bound P l , least-square weights C (0)  = I |θ     l     |  · e(0) = ∞. 
               
               
                   
               
            
           
           
               
               
               
            
               
                   
                 Do{ 
                   
               
               
                   
               
               
                   
                   
                  
         •   ⁢           ⁢       L   k     ⁡     (     i   ,   j     )         =     {             1   ,           i   =     j   =   k                 0   ,         otherwise         ,           ⁢     k   =   1     ,           ⁢   …   ⁢           ,   N           
 
               
               
                   
               
               
                   
                   
                  
           •   ⁢           ⁢     Find:     ⁢           ⁢     d     (   n   )         =       [         d   1     ⁢     e     jarg   ⁡     (         a   H     ⁡     (     θ     l   ,   1       )       ⁢     w     (   n   )         )           ,   …   ⁢           ,       d          θ   l            ⁢     e     jarg   ⁡     (         a   H     ⁡     (     θ     l   ,          θ   l              )       ⁢     w     (   n   )         )             ]     T       ,     
     ⁢           ⁢       A   l     (   n   )       ←     [       a   ⁡     (     θ     l   ,   1       )       ,   …   ⁢           ,     a   ⁡     (     θ     l   ,          θ   l              )         ]       ,       
 
               
               
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                           
                             
                               Q 
                               
                                 p 
                                 , 
                                 l 
                               
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                             = 
                             
                               [ 
                               
                                 
                                   
                                     
                                       Re 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         { 
                                         
                                           
                                             a 
                                             ⁡ 
                                             
                                               ( 
                                               
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                                           ⁢ 
                                           
                                             
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                                         - 
                                         Im 
                                       
                                       ⁢ 
                                       
                                         { 
                                         
                                           
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                                             ⁡ 
                                             
                                               ( 
                                               
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                                       ⁢ 
                                       
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                                               ( 
                                               
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                                       ⁢ 
                                       
                                           
                                       
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                                         } 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           , 
                           
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                             = 
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                           , 
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                           ⁢ 
                           
                               
                           
                           , 
                           
                              
                             
                               θ 
                               p 
                             
                              
                           
                         
                         ⁢ 
                         
                             
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                           R 
                           r 
                         
                         = 
                         
                           [ 
                           
                             
                               
                                 
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                                   ⁢ 
                                   
                                     { 
                                     
                                       
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                                         l 
                                         
                                           ( 
                                           n 
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                                                 ) 
                                               
                                             
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                                       ⁢ 
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                                         ⁢ 
                                         m 
                                       
                                       ⁢ 
                                       
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                                             A 
                                             l 
                                             
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                                               n 
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                                               n 
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                         ⁢ 
                         
                           
                             x 
                             
                               ( 
                               n 
                               ) 
                             
                           
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                                       ⁢ 
                                       
                                         { 
                                         
                                           w 
                                           
                                             ( 
                                             n 
                                             ) 
                                           
                                         
                                         } 
                                       
                                     
                                   
                                   
                                     
                                       
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                                         ⁢ 
                                         m 
                                       
                                       ⁢ 
                                       
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                                             ( 
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                               ] 
                             
                             T 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                   
                   
                 • Solve: 
               
               
                   
               
               
                   
                   
                    
           e(n):     ⁢           ⁢       min   x     ⁢           ⁢       d       (   n   )     H       ⁢     C     (   n   )       ⁢     d     (   n   )             +       x       (   n   )     H       ⁢     R   r     ⁢     x     (   n   )         -     2   ⁢     x       (   n   )     H       ⁢     q   r           
               ⁢         s   .   t   .           ⁢                L   k       ⁢     x     (   n   )              2       ≤   1     ,     k   =   1     ,   …   ⁢           ,   N         
               ⁢                    Q     p   ,   l       (   n   )         ⁢     x     (   n   )              2     ≤       P   l         ,     l   =   1     ,   …   ⁢           ,          θ   p            ⁢               
 
               
               
                   
               
               
                   
                   
                 • Find: 
               
               
                   
               
               
                   
                   
                 
                   
                     
                       
                         
                             
                         
                         ⁢ 
                         
                           
                             w 
                             
                               ( 
                               
                                 n 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           = 
                           
                             
                               
                                 { 
                                 x 
                                 } 
                               
                               1 
                               N 
                             
                             + 
                             
                               j 
                               ⁢ 
                               
                                 
                                   { 
                                   x 
                                   } 
                                 
                                 
                                   N 
                                   + 
                                   1 
                                 
                                 
                                   2 
                                   ⁢ 
                                   N 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                             
                         
                         ⁢ 
                         
                           
                             C 
                             u 
                             
                               ( 
                               
                                 n 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                           = 
                           
                             { 
                             
                               
                                 
                                   
                                     
                                       
                                         max 
                                         ⁢ 
                                         
                                           { 
                                           
                                             
                                               
                                                  
                                                 
                                                   
                                                     
                                                       a 
                                                       H 
                                                     
                                                     ⁡ 
                                                     
                                                       ( 
                                                       
                                                         θ 
                                                         
                                                           p 
                                                           , 
                                                           l 
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                   ⁢ 
                                                   
                                                     w 
                                                     
                                                       ( 
                                                       
                                                         n 
                                                         + 
                                                         1 
                                                       
                                                       ) 
                                                     
                                                   
                                                 
                                                  
                                               
                                               - 
                                               
                                                 d 
                                                 l 
                                               
                                             
                                             , 
                                             0 
                                           
                                           } 
                                         
                                       
                                       , 
                                     
                                   
                                   
                                     
                                       if 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       sidelobe 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                          
                                         
                                           
                                              
                                             
                                               
                                                 
                                                   a 
                                                   H 
                                                 
                                                 ⁡ 
                                                 
                                                   ( 
                                                   
                                                     θ 
                                                     
                                                       p 
                                                       , 
                                                       l 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               ⁢ 
                                               
                                                 w 
                                                 
                                                   ( 
                                                   
                                                     n 
                                                     + 
                                                     1 
                                                   
                                                   ) 
                                                 
                                               
                                             
                                              
                                           
                                           - 
                                           
                                             d 
                                             l 
                                           
                                         
                                          
                                       
                                       , 
                                     
                                   
                                   
                                     
                                       if 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
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     Aspects of array pattern synthesis described herein may be advantageous in codebook design for massive MIMO and mmWave systems, in particular with respect to lower sidelobes (e.g., for interference rejection), broad beams (e.g., for fast sector sweeps, diversity and multiplexing gain), null steering (e.g., for directional blocker/interference mitigation), and multi-finger beam design (e.g., for single-stream applications such as control channel transmission). Aspects of array pattern synthesis may be particular applicable for beam broadening patterns, which may yield higher main beam power and less ripple at the main beam compared to existing solutions. 
     Similar to as described above for multi-finger beamforming, various aspects of the array pattern synthesis approach (according to any of process  1100 ,  1300 , or  1500  may be performed offline or online. Accordingly, in various aspects beamforming controller  214  (or another offline processor or processing component) may be configured to perform array pattern synthesis beamforming when operating in an offline manner, e.g., when  200 communication device  200  is not actively transmitting or receiving with any users. For instance, beamforming controller  214  may generate a predefined set of complex beamforming weight vectors (e.g., forming a codebook  ∈{w 1 , w 2  . . . w P }, where P is the codebook size and each w p , p=1, . . . , P, is a different codeword of length N), each based on a different desired radiation pattern, while operating offline. Once operating online (e.g., during runtime and/or when  200 communication device  200  is actively transmitting or receiving with users), beamforming controller  214  may select a complex beamforming weight vector w from   having an appropriate radiation pattern (e.g., based on user positioning/angular direction relative to antenna array  202 ) and apply the complex beamforming weight vector to transmit and/or receive at antenna array  202 . Accordingly, while beamforming controller  200200  (or another offline processor or processing component) may generate the complex beamforming weight vectors using array pattern synthesis beamforming while offline, beamforming controller  200200  may utilize the resulting complex beamforming weight vectors when operating online. 
     In some aspects, beamforming controller  214  may be configured to perform array pattern synthesis beamforming online, e.g., when  200 communication device  200  is actively transmitting or receiving with one or more users. For example, beamforming controller  214  may identify the angular direction of L users from antenna array  202  (e.g., based on Angle-of-Arrival or similar processing techniques) and may generate a desired radiation pattern defined by d, θ l , and P l , where certain azimuth angles of θ l  may be main finger azimuth angles with desired power levels defined by d and other azimuth angles of θ l  may be sidelobe azimuth angles defined by P. In some aspects, beamforming controller  214  may also identify the distance of the L users from antenna array  202  (e.g., based on received signal strength or transmission timing processing) and generate the power levels of d based on how far each user is from antenna array  202 . In some aspects, beamforming controller  214  may also identify certain angular directions that are targeted for null steering or interference suppression, such as based on the locations of other users that  200 communication device  200  is not actively transmitting or receiving with. Beamforming controller  214  may then execute multi-finger beamforming according to any of processes  1100 ,  1300 , or  1500  to obtain complex beamforming weights w* that approach the optimal complex beamforming weights for the desired radiation pattern defined by d, θ l , and P l . 
     In some aspects, beamforming controller  214  may repeatedly trigger array pattern synthesis beamforming to obtain repeated updated complex beamforming weights w* when operating online, such as in part of an adaptive beamforming approach. For example, beamforming controller  214  may periodically determine the angular directions and distances of L users (where L may also increase and decrease based on how many users are within the coverage area of  200 communication device  200 ), such as by determining angular directions and distances of the users during each scheduling interval. Beamforming controller  214  may then calculate complex beamforming weights w* to use during each scheduling interval based on the current angular direction and distances of the users, where beamforming controller  214  may re-calculate w* each scheduling interval to dynamically adapt to any changes in the angular direction, distance, or number of users. 
       FIG. 16  shows an exemplary process  1600  of performing beamforming according to some aspects. As shown in  FIG. 16 , process  1600  includes determining a set of beamforming weights for an antenna array based on a target radiation pattern having a plurality of main fingers by an iteration of identifying a search space of beamforming weights for a plurality of elements of the antenna array, and determining, based on contribution of one or more of the plurality of elements of the antenna array to multiple of the plurality of main fingers, an updated set of beamforming weights in the search space to reduce a difference between an actual radiation pattern and the target radiation pattern ( 1610 ), the method further including transmitting or receiving radio signals with the antenna array based on the updated set of beamforming weights ( 1620 ). 
       FIG. 17  shows an exemplary process  1700  of performing beamforming according to some aspects. As shown in  FIG. 17 , process  1700  includes iteratively determining a set of beamforming weights for an antenna array based on a target radiation pattern with an iteration of identifying a search space for an updated set of beamforming weights for a plurality of elements of the antenna array, and determining, based on reducing a difference between an actual radiation pattern and the target radiation pattern, the updated set of beamforming weights in the search space based on a first set of angles of the target radiation pattern having an optimization constraint and a second set of angles of the target radiation pattern having an upper-bound power level constraint ( 1710 ), the method further including selecting the updated set of beamforming weights as the set of beamforming weights ( 1720 ). 
     In one or more further exemplary aspects of the disclosure, one or more of the features described above in reference to  FIGS. 1-15  may be further incorporated into process  1600  and/or  1700 . In particular, process  1600  and/or  1700  may be configured to perform further and/or alternate processes as detailed regarding beamforming controller  214  and/or  200 communication device  200 . 
     The terms “user equipment”, “UE”, “mobile terminal”, “user terminal”, “terminal device”, etc., may apply to any wireless communication device, including cellular phones, tablets, laptops, personal computers, wearables, multimedia playback and other handheld electronic devices, consumer/home/office/commercial appliances, vehicles, and any number of additional electronic devices capable of wireless communications. 
     While the above descriptions and connected figures may depict electronic device components as separate elements, skilled persons will appreciate the various possibilities to combine or integrate discrete elements into a single element. Such may include combining two or more circuits for form a single circuit, mounting two or more circuits onto a common chip or chassis to form an integrated element, executing discrete software components on a common processor core, etc. Conversely, skilled persons will recognize the possibility to separate a single element into two or more discrete elements, such as splitting a single circuit into two or more separate circuits, separating a chip or chassis into discrete elements originally provided thereon, separating a software component into two or more sections and executing each on a separate processor core, etc. 
     It is appreciated that implementations of methods detailed herein are demonstrative in nature, and are thus understood as capable of being implemented in a corresponding device. Likewise, it is appreciated that implementations of devices detailed herein are understood as capable of being implemented as a corresponding method. It is thus understood that a device corresponding to a method detailed herein may include one or more components configured to perform each aspect of the related method. 
     All acronyms defined in the above description additionally hold in all claims included herein. 
     The following examples pertain to further aspects of this disclosure: 
     Example 1 is a method of performing beamforming, the method including determining a set of beamforming weights for an antenna array based on a target radiation pattern having a plurality of main fingers by an iteration of identifying a search space of beamforming weights for a plurality of elements of the antenna array, and determining, based on contribution of one or more of the plurality of elements of the antenna array to multiple of the plurality of main fingers, an updated set of beamforming weights in the search space to reduce a difference between an actual radiation pattern and the target radiation pattern, the method further including transmitting or receiving radio signals with the antenna array based on the updated set of beamforming weights. 
     In Example 2, the subject matter of Example 1 can optionally include wherein the actual radiation pattern characterizes a radiation pattern from applying the updated set of beamforming weights at an antenna array. 
     In Example 3, the subject matter of Example 1 or 2 can optionally include wherein determining the updated set of beamforming weights in the search space includes determining the updated set of beamforming weights based on a least-squares problem that generates a result with minimized difference between the target radiation pattern and the actual radiation pattern. 
     In Example 4, the subject matter of Example 3 can optionally include wherein the least-squares problem includes contribution from each of the plurality of elements of the antenna array to each of the plurality of main fingers of the target radiation pattern. 
     In Example 5, the subject matter of any one of Examples 1 to 4 can optionally include wherein the search space for the updated set of beamforming weights constrains the updated set of beamforming weights to have unity gain. 
     In Example 6, the subject matter of any one of Examples 1 to 4 can optionally include wherein the set of beamforming weights are phase-only beamforming weights located on the unit circle. 
     In Example 7, the subject matter of any one of Examples 1 to 6 can optionally include wherein, for a first element of the plurality of elements, the search space for the updated beamforming weight is a tangent line of the unit circle at a current beamforming weight of the first element, and wherein determining the updated set of beamforming weights in the search space includes determining a solution to a least-squares problem along the tangent line for the first element of the antenna array, locating the closest point on the unit circle to the solution on the tangent line for the first element of the antenna array, and using the closest point on the unit circle as the updated beamforming weight for the first element of the antenna array in the updated set of beamforming weights. 
     In Example 8, the subject matter of any one of Examples 1 to 7 can optionally include wherein determining the set of beamforming weights for the antenna array includes iteratively identifying the search space and determining the updated set of beamforming weights until the updated set of beamforming weights satisfies a convergence condition. 
     In Example 9, the subject matter of Example 8 can optionally include wherein the iteration further includes determining the actual radiation pattern based on the updated set of beamforming weights obtained in the iteration, determining an error term based on the difference between the actual radiation pattern and the desired radiation pattern, determining that the convergence condition is satisfied if the error term is less than a predefined convergence parameter. 
     In Example 10, the subject matter of Example 8 or 9 can optionally include the method further including after the updated set of beamforming weights satisfies the convergence condition, updating a maximum main finger power level or updating a maximum sidelobe power level based on the actual radiation pattern produced by the updated set of complex beamforming weights, and iteratively re-determining the set of beamforming weights for the antenna array by identifying the search space and determining the updated set of beamforming parameters based on the maximum main finger power level and the maximum sidelobe power level. 
     In Example 11, the subject matter of Example 10 can optionally include wherein updating the maximum main finger power level or updating the maximum sidelobe power level includes incrementing the maximum sidelobe power level or decrementing the maximum main finger power level. 
     In Example 12, the subject matter of any one of Examples 1 to 11 can optionally include wherein the target radiation pattern is defined over one or more main finger angles and one or more sidelobe angles, and wherein determining the updated set of beamforming weights includes determining an updated set of beamforming weights for which the actual radiation pattern has predetermined ratios between the power levels of the one or more main finger angles. 
     In Example 13, the subject matter of any one of Examples 1 to 12 can optionally further include identifying a new target radiation pattern based on adaptive beamforming, and iteratively determining a new set of beamforming weights for the antenna array based on the new target radiation pattern. 
     In Example 14, the subject matter of Example 13 can optionally further include transmitting or receiving radio signals with the antenna array based on the new set of beamforming weights. 
     Example 15 is a non-transitory computer readable medium storing instructions that, when executed by a processor, cause the processor to perform the method of any one of Examples 1 to 14. 
     Example 16 is a non-transitory computer readable medium storing instructions that, when executed by a controller of a communication device, cause the communication device to perform the method of any one of Examples 1 to 14. 
     Example 17 is a communication device including an antenna array, a beamforming controller, and a transmit or receive path, that is configured to perform the method of any one of Examples 1 to 16. 
     Example 18 is a method of synthesizing antenna array radiation patterns, the method including iteratively determining a set of beamforming weights for an antenna array based on a target radiation pattern with an iteration of identifying a search space for an updated set of beamforming weights for a plurality of elements of the antenna array, and determining, based on reducing a difference between an actual radiation pattern and the target radiation pattern, the updated set of beamforming weights in the search space based on a first set of angles of the target radiation pattern having an optimization constraint and a second set of angles of the target radiation pattern having an upper-bound power level constraint, the method further including selecting the updated set of beamforming weights as the set of beamforming weights. 
     In Example 19, the subject matter of Example 18 can optionally include wherein the actual radiation pattern characterizes the radiation pattern produced by applying the updated set of beamforming weights at an antenna array. 
     In Example 20, the subject matter of Example 18 or 19 can optionally include wherein the target radiation pattern is defined over one or more main finger angles and one or more sidelobe angles. 
     In Example 21, the subject matter of Example 20 can optionally include wherein the iteration further includes determining if the actual radiation pattern at any of the one or more sidelobe angles in the first set of angles is below a target power level, and moving the identified sidelobe angles to the second set of angles if the actual radiation pattern is below the target power level. 
     In Example 22, the subject matter of Example 21 can optionally include wherein the first set of angles initially includes all of the one or more main finger angles and all of the one or more sidelobe angles. 
     In Example 23, the subject matter of any one of Examples 18 to 22 can optionally include wherein determining the updated set of beamforming weights includes determining a solution for a least-squares problem for the first set of angles based on the optimization constraint while constraining the updated set of beamforming weights to keep the actual radiation pattern below target power levels at the second set of angles. 
     In Example 24, the subject matter of Example 23 can optionally include wherein the optimization constraint minimizes the difference between target power levels at the first set of angles of the target radiation pattern and the power levels at the first set of angles of the actual radiation pattern. 
     In Example 25, the subject matter of any one of Examples 18 to 24 can optionally include wherein iteratively determining the set of beamforming weights for the antenna array includes iteratively identifying the search space and determining the updated set of beamforming weights until the updated set of beamforming weights satisfies a convergence condition. 
     In Example 26, the subject matter of Example 25 can optionally include wherein each iteration further includes determining the actual radiation pattern based on the updated set of beamforming weights obtained in the iteration, determining an error term based on the difference between the actual radiation pattern and the desired radiation pattern, and determining that the convergence condition is satisfied if the error term is less than a predefined convergence parameter. 
     In Example 27, the subject matter of any one of Examples 18 to 26 can optionally include wherein the set of beamforming weights are for phase-only beamforming and wherein the search space constrains the updated set of beamforming weights to locations on the unit circle. 
     In Example 28, the subject matter of any one of Examples 18 to 27 can optionally include wherein, for a first element of the plurality of elements, the search space for the updated beamforming weight is a tangent line of the unit circle at a current beamforming weight of the first element, and wherein determining the updated set of beamforming weights in the search space includes determining a solution to a least-squares problem for the first set of angles along the tangent line for the first element of the antenna array, locating the closest point on the unit circle to the solution on the tangent line for the first element, and using the closest point on the unit circle as the updated beamforming weight for the first element in the updated set of beamforming weights. 
     In Example 29, the subject matter of any one of Examples 18 to 26 can optionally include wherein the set of beamforming weights are for phase control with amplitude tapering beamforming, and wherein the search space constrains the updated set of beamforming weights locations within a two-dimensional ring including radial distances between a minimum tapering amplitude and the unit circle. 
     In Example 30, the subject matter of any one of Examples 18 to 26 or 29 can optionally include wherein, for a first element of the plurality of elements, the search space for the updated beamforming weight is a region between the unit circle and a line intersecting the unit circle that is perpendicular to the line between the origin of the unit circle and a current beamforming weight, and wherein determining the updated set of beamforming weights in the search space includes determining the updated set of beamforming weights in the region. 
     In Example 31, the subject matter of any one of Examples 18 to 26 can optionally include wherein the set of beamforming weights are phase and amplitude control beamforming and wherein the search space constrains the updated set of beamforming weights locations within the unit circle. 
     In Example 32, the subject matter of any one of Examples 18 to 26 or 31 can optionally include wherein, for a first element of the plurality of elements, the search space for the updated beamforming weight is the unit circle, and wherein determining the updated set of beamforming weights in the search space includes determining the updated set of beamforming weights in the unit circle. 
     In Example 33, the subject matter of any one of Examples 18 to 32 can optionally further include performing the method offline to generate a beamforming codebook including sets of beamforming weights for different target radiation patterns. 
     In Example 34, the subject matter of any one of Examples 18 to 32 can optionally further include transmitting or receiving radio signals with the antenna array based on the set of beamforming weights. 
     Example 35 is a non-transitory computer readable medium storing instructions that, when executed by a processor, cause the processor to perform the method of any one of Examples 18 to 35. 
     Example 36 is a non-transitory computer readable medium storing instructions that, when executed by a controller of a communication device, cause the communication device to perform the method of any one of Examples 18 to 35. 
     Example 37 is a communication device including an antenna array, a beamforming controller, and a transmit or receive path, that is configured to perform the method of any one of Examples 18 to 35. 
     Example 38 is a communication device including an antenna array, and a beamforming controller configured to determine a set of beamforming weights for the antenna array based on a target radiation pattern having a plurality of main fingers, wherein the beamforming controller is configured to, in each of a plurality of iterations identify a search space of beamforming weights for a plurality of elements of the antenna array, and determine, based on contribution of one or more of the plurality of elements of to multiple of the plurality of main fingers, an updated set of beamforming weights in the search space to reduce a difference between an actual radiation pattern and the target radiation pattern, the antenna array configured to transmit or receive radio signals based on the updated set of beamforming weights. 
     In Example 39, the subject matter of Example 38 can optionally include wherein the beamforming controller is implemented as one or more processors configured to execute program code or as digitally-configured hardware circuitry. 
     In Example 40, the subject matter of Example 38 or 39 can optionally include wherein the actual radiation pattern characterizes a radiation pattern from applying the updated set of beamforming weights at an antenna array. 
     In Example 41, the subject matter of any one of Examples 38 to 40 can optionally include wherein the beamforming controller is configured to determine the updated set of beamforming weights in the search space by determining the updated set of beamforming weights based on a least-squares problem that generates a result with minimized difference between the target radiation pattern and the actual radiation pattern. 
     In Example 42, the subject matter of Example 41 can optionally include wherein the least-squares problem includes contribution from each of the plurality of elements of the antenna array to each of the plurality of main fingers of the target radiation pattern. 
     In Example 43, the subject matter of any one of Examples 38 to 42 can optionally include wherein the search space for the updated set of beamforming weights constrains the updated set of beamforming weights to have unity gain. 
     In Example 44, the subject matter of any one of Examples 38 to 42 can optionally include wherein the set of beamforming weights are phase-only beamforming weights located on the unit circle. 
     In Example 45, the subject matter of any one of Examples 38 to 44 can optionally include wherein, for a first element of the plurality of elements, the search space for the updated beamforming weight is a tangent line of the unit circle at a current beamforming weight of the first element, and wherein the beamforming controller is configured to determine the updated set of beamforming weights in the search space by determining a solution to a least-squares problem along the tangent line for the first element of the antenna array, locating the closest point on the unit circle to the solution on the tangent line for the first element of the antenna array, and using the closest point on the unit circle as the updated beamforming weight for the first element of the antenna array in the updated set of beamforming weights. 
     In Example 46, the subject matter of any one of Examples 38 to 45 can optionally include wherein the beamforming controller is configured to determine the set of beamforming weights for the antenna array by iteratively identifying the search space and determining the updated set of beamforming weights until the updated set of beamforming weights satisfies a convergence condition. 
     In Example 47, the subject matter of Example 46 can optionally include wherein the beamforming controller is further configured to, in each of the plurality of iterations determine the actual radiation pattern based on the updated set of beamforming weights obtained in the iteration, determine an error term based on the difference between the actual radiation pattern and the desired radiation pattern, determine that the convergence condition is satisfied if the error term is less than a predefined convergence parameter. 
     In Example 48, the subject matter of Example 46 or 47 can optionally include wherein the beamforming controller is further configured to after the updated set of beamforming weights satisfies the convergence condition, update a maximum main finger power level or update a maximum sidelobe power level based on the actual radiation pattern produced by the updated set of complex beamforming weights, and iteratively re-determine the set of beamforming weights for the antenna array by identifying the search space and solve for the updated set of beamforming parameters based on the maximum main finger power level and the maximum sidelobe power level. 
     In Example 49, the subject matter of Example 48 can optionally include wherein the beamforming controller is configured to update the maximum main finger power level or update the maximum sidelobe power level by incrementing the maximum sidelobe power level or decrementing the maximum main finger power level. 
     In Example 50, the subject matter of any one of Examples 38 to 49 can optionally include wherein the target radiation pattern is defined over one or more main finger angles and one or more sidelobe angles, and wherein the beamforming controller is configured to determine the updated set of beamforming weights by determining an updated set of beamforming weights for which the actual radiation pattern has predetermined ratios between the power levels of the one or more main finger angles. 
     In Example 51, the subject matter of any one of Examples 38 to 50 can optionally include wherein the beamforming controller is further configured to identify a new target radiation pattern based on adaptive beamforming, and iteratively determine a new set of beamforming weights for the antenna array based on the new target radiation pattern. 
     In Example 52, the subject matter of Example 51 can optionally include wherein the antenna array is further configured to transmit or receive radio signals with the antenna array based on the new set of beamforming weights. 
     In Example 53, the subject matter of any one of Examples 38 to 52 can optionally further include a transmit or receive path that interfaces with the antenna array and the beamforming controller. 
     Example 54 is a communication device for synthesis of radiation patterns for antenna arrays, the communication device including a beamforming controller including one or more processors or digitally-configured hardware circuitry, the beamforming controller configured to iteratively determine a set of beamforming weights for an antenna array based on a target radiation pattern, wherein the beamforming controller is configured to, for each of a plurality of iterations identify a search space for an updated set of beamforming weights for a plurality of elements of the antenna array, and determine, based reducing a difference between an actual radiation pattern and the target radiation pattern, the updated set of beamforming weights in the search space based on a first set of angles of the target radiation pattern having an optimization constraint and a second set of angles of the target radiation pattern having an upper-bound power level constraint, the beamforming controller further configured to select the updated set of beamforming weights as the set of beamforming weights. 
     In Example 55, the subject matter of Example 54 can optionally further include the antenna array, wherein the antenna array is configured to transmit or receive radio signals based on the set of beamforming weights. 
     In Example 56, the subject matter of Example 54 or 55 can optionally further include a transmit or receive path. 
     In Example 57, the subject matter of any one of Examples 54 to 56 can optionally include wherein the actual radiation pattern characterizes the radiation pattern produced by applying the updated set of beamforming weights at an antenna array. 
     In Example 58, the subject matter of any one of Examples 54 to 57 can optionally include wherein the target radiation pattern is defined over one or more main finger angles and one or more sidelobe angles. 
     In Example 59, the subject matter of Example 58 can optionally include wherein the beamforming controller is further configured to, in each iteration determine if the actual radiation pattern at any of the one or more sidelobe angles in the first set of angles is below a target power level, and move the identified sidelobe angles to the second set of angles if the actual radiation pattern is below the target power level 
     In Example 60, the subject matter of Example 59 can optionally include wherein the first set of angles initially includes all of the one or more main finger angles and all of the one or more sidelobe angles. 
     In Example 61, the subject matter of any one of Examples 54 to 60 can optionally include wherein the beamforming controller is configured to determine the updated set of beamforming weights by determining a solution for least-squares problem for the first set of angles based on the optimization constraint while constraining the updated set of beamforming weights to keep the actual radiation pattern below target power levels at the second set of angles. 
     In Example 62, the subject matter of Example 61 can optionally include wherein the optimization constraint minimizes the difference between target power levels at the first set of angles of the target radiation pattern and the power levels at the first set of angles of the actual radiation pattern. 
     In Example 63, the subject matter of any one of Examples 54 to 62 can optionally include wherein the beamforming controller is configured to iteratively determine the set of beamforming weights for the antenna array by iteratively identifying the search space and determining the updated set of beamforming weights until the updated set of beamforming weights satisfies a convergence condition. 
     In Example 64, the subject matter of Example 63 can optionally include wherein the beamforming controller is further configured to, in each iteration determine the actual radiation pattern based on the updated set of beamforming weights obtained in the iteration, determine an error term based on the difference between the actual radiation pattern and the desired radiation pattern, and determine that the convergence condition is satisfied if the error term is less than a predefined convergence parameter. 
     In Example 65, the subject matter of any one of Examples 54 to 64 can optionally include wherein the set of beamforming weights are for phase-only beamforming and wherein the search space constrains the updated set of beamforming weights to locations on the unit circle. 
     In Example 66, the subject matter of any one of Examples 54 to 65 can optionally include wherein, for a first element of the plurality of elements, the search space for the updated beamforming weight is a tangent line of the unit circle at a current beamforming weight of the first element, and wherein the beamforming controller is configured to solve for the updated set of beamforming weights in the search space by determining a solution to a least-squares problem for the first set of angles along the tangent line for the first element, locating the closest point on the unit circle to the solution on the tangent line for the first element, and using the closest point on the unit circle as the updated beamforming weight for the first element in the updated set of beamforming weights. 
     In Example 67, the subject matter of any one of Examples 54 to 64 can optionally include wherein the set of beamforming weights are for phase control with amplitude tapering beamforming and wherein the search space constrains the updated set of beamforming weights locations within a two-dimensional ring including radial distances between a minimum tapering amplitude and the unit circle. 
     In Example 68, the subject matter of any one of Examples 54 to 64 or 67 can optionally include wherein, for a first element of the plurality of elements, the search space for the updated beamforming weight is a region between the unit circle and a line intersecting the unit circle that is perpendicular to the line between the origin of the unit circle and a current beamforming weight, and wherein the beamforming controller is configured to determine the updated set of beamforming weights in the search space by determining the updated set of beamforming weights in the region. 
     In Example 69, the subject matter of any one of Examples 54 to 64 can optionally include wherein the set of beamforming weights are phase and amplitude control beamforming and wherein the search space constrains the updated set of beamforming weights locations within the unit circle. 
     In Example 70, the subject matter of any one of Examples 54 to 64 or 69 can optionally include wherein, for a first element of the plurality of elements, the search space for the updated beamforming weight is the unit circle, and wherein the beamforming controller is configured to determine the updated set of beamforming weights in the search space by determining the updated set of beamforming weights in the unit circle. 
     In Example 71, the subject matter of any one of Examples 54 to 70 can optionally include wherein the beamforming controller is configured to iteratively calculate the set of beamforming weights offline to generate a beamforming codebook including sets of beamforming weights for different target radiation patterns. 
     While the invention has been particularly shown and described with reference to specific embodiments, it should be understood by those skilled in the art that various changes in form and detail may be made therein without departing from the spirit and scope of the invention as defined by the appended claims. The scope of the invention is thus indicated by the appended claims and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced.