Patent Publication Number: US-2022224464-A1

Title: Fractional cooperative multipoint network operation

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This patent document claims priority to and benefits of U.S. Provisional Patent Application No. 62/845,037 filed 8 May 2019 and U.S. Provisional Patent Application No. 62/852,240 filed 23 May 2019. The entire content of the before-mentioned patent applications are incorporated by reference as part of the disclosure of this patent document. 
    
    
     TECHNICAL FIELD 
     The present document relates to mobile wireless communication, and more particularly, to massive cooperative multipoint network operation. 
     BACKGROUND 
     Due to an explosive growth in the number of wireless user devices and the amount of wireless data that these devices can generate or consume, current wireless communication networks are fast running out of bandwidth to accommodate such a high growth in data traffic and provide high quality of service to users. 
     Various efforts are underway in the telecommunication industry to come up with next generation of wireless technologies that can keep up with the demand on performance of wireless devices and networks. Many of those activities involve situations in which a large number of user devices may be served by a network. 
     SUMMARY 
     This document discloses devices and techniques useful for embodiments of wireless technologies in which cooperative multipoint operation of wireless networks is achieved. 
     In one example aspect, a method of wireless communication is disclosed. The method includes determining, by a network device, a cooperative multipoint (COMP) management status of wireless devices served by the network device, and providing, by the network device, wireless connectivity to the one or more wireless devices, wherein the network device jointly manages transmission resources for a first wireless device due to the COMP management status being a joint COMP status and the network device locally manages transmission resources for a second wireless device due to the COMP management status being a local COMP status. 
     In yet another example aspect, a wireless communication system is disclosed. The system includes an arrangement of a plurality of network nodes in which each network node is configured to provide wireless connectivity to wireless devices using a mode that includes a joint cooperative multipoint (COMP) mode and a local COMP mode, wherein, in the joint COMP mode, transmission resources for wireless devices are managed cooperatively with other network nodes, and wherein, in the local COMP mode, transmission resources for wireless devices are managed locally, without explicit coordination with other network nodes. 
     In yet another example aspect, a wireless communication apparatus that implements the above-described methods is disclosed. 
     In yet another example aspect, the methods may be embodied as processor-executable code and may be stored on a computer-readable program medium. 
     These, and other, features are described in this document. 
    
    
     
       DESCRIPTION OF THE DRAWINGS 
       Drawings described herein are used to provide a further understanding and constitute a part of this application. Example embodiments and illustrations thereof are used to explain the technology rather than limiting its scope. 
         FIG. 1  shows an example of a mobile wireless network. 
         FIG. 2  shows an example of a cellular 3-sector hexagonal model. 
         FIG. 3  shows examples of interference circumferences in wireless networks. 
         FIG. 4  shows an example of distributed cooperative multipoint (COMP) clusters. 
         FIG. 5  shows examples of links, nodes and clusters in a wireless network. 
         FIG. 6  shows examples of sizing of COMP clusters. 
         FIG. 7  shows an example of staged COMP clustering. 
         FIG. 8  shows another example of staged COMP clustering. 
         FIG. 9  shows an example in which one cluster with three nodes are depicted. 
         FIG. 10  shows an example of a wireless network with one cluster and 7 nodes. 
         FIG. 11  shows an example of a wireless network with 3 clusters and 16 nodes. 
         FIG. 12  shows an example of a wireless network with 7 clusters and 31 nodes. 
         FIG. 13  shows an example of evolution of spectral efficiency from SU-MIMO to MU-MIMO with delay-Doppler channel prediction to MU-MIMO with delay-Doppler channel prediction and COMP. 
         FIG. 14  shows an example of joint downlink transmissions in the COMP framework. 
         FIGS. 15 and 16  show examples of the “zero forcing” operation used for the joint downlink transmissions. 
         FIG. 17  shows an example of joint uplink transmissions in the COMP framework. 
         FIG. 18  shows an example of the “zero forcing” operation used for the joint uplink transmissions. 
         FIG. 19  shows an example of the information flow in joint downlink and uplink transmissions. 
         FIG. 20  shows an example of a handover in the COMP zone. 
         FIG. 21  shows an example of the scalability and locality of the COMP zone. 
         FIG. 22  shows an example of the interactions between the control plane (CP) and the user plane (UP). 
         FIG. 23  shows an example of the split between the control unit (CU) and distributed unit (DU), as well as the split between the control plane (CP) and the user plane (UP). 
         FIG. 24  shows an example of the split options in the protocol stack. 
         FIG. 25  shows another example of the split options in the protocol stack. 
         FIG. 26  shows yet another example of the split options in some layers of the protocol stack. 
         FIG. 27  shows yet another example of the split options in the protocol stack. 
         FIG. 28  shows an example of the mapping of the control unit (CU) and distributed unit (DU) functions according to the split points. 
         FIG. 29  shows an example of a eNodeB (eNB) and/or gNodeB (gNB) architecture. 
         FIG. 30  shows an example of a lower layer downlink (DL) split description. 
         FIG. 31  shows an example of a lower layer uplink (UL) split description. 
         FIG. 32  show examples of the interactions between the remote unit and the central unit. 
         FIG. 33  shows an example of the interface between MU-MIMO and COMP, in accordance with embodiments of the disclosed technology. 
         FIG. 34  shows an example of a common public radio interface (CPRI). 
         FIG. 35  shows a quantitative example of the base requirements and additional bandwidth requirements required by COMP. 
         FIG. 36  shows a table of reference scenario system parameters. 
         FIG. 37  show examples comparing the SNR of quantized and unquantized signals. 
         FIG. 38  shows graphs that demonstrate benefits of the disclosed technologies. 
         FIG. 39  shows an example of wireless channels between a first wireless terminal (terminal A) and a second wireless terminal (Terminal B). 
         FIG. 40  is an illustrative example of a detection tree. 
         FIG. 41  depicts an example network configuration in which a hub services for user equipment (UE). 
         FIG. 42  depicts an example embodiment in which an orthogonal frequency division multiplexing access (OFDMA) scheme is used for communication. 
         FIG. 43  illustrates the concept of precoding in an example network configuration. 
         FIG. 44  is a spectral chart of an example of a wireless communication channel. 
         FIG. 45  illustrates examples of downlink and uplink transmission directions. 
         FIG. 46  illustrates spectral effects of an example of a channel prediction operation. 
         FIG. 47  graphically illustrates operation of an example implementation of a zero-forcing precoder (ZFP). 
         FIG. 48  graphically compares two implementations—a ZFP implementation and regularized ZFP implementation (rZFP). 
         FIG. 49  shows components of an example embodiment of a precoding system. 
         FIG. 50  is a block diagram depiction of an example of a precoding system. 
         FIG. 51  shows an example of a quadrature amplitude modulation (QAM) constellation. 
         FIG. 52  shows another example of QAM constellation. 
         FIG. 53  pictorially depicts an example of relationship between delay-Doppler domain and time-frequency domain. 
         FIG. 54  is a spectral graph of an example of an extrapolation process. 
         FIG. 55  is a spectral graph of another example of an extrapolation process. 
         FIG. 56  compares spectra of a true and a predicted channel in some precoding implementation embodiments. 
         FIG. 57  is a block diagram depiction of a process for computing prediction filter and error covariance. 
         FIG. 58  is a block diagram illustrating an example of a channel prediction process. 
         FIG. 59  is a graphical depiction of channel geometry of an example wireless channel. 
         FIG. 60A  is a graph showing an example of a precoding filter antenna pattern. 
         FIG. 60B  is a graph showing an example of an optical pre-coding filter. 
         FIG. 61  is a block diagram showing an example process of error correlation computation. 
         FIG. 62  is a block diagram showing an example process of precoding filter estimation. 
         FIG. 63  is a block diagram showing an example process of applying an optimal precoding filter. 
         FIG. 64  is a graph showing an example of a lattice and QAM symbols. 
         FIG. 65  graphically illustrates effects of perturbation examples. 
         FIG. 66  is a graph illustrating an example of hub transmission. 
         FIG. 67  is a graph showing an example of the process of a UE finding a closest coarse lattice point. 
         FIG. 68  is a graph showing an example process of UE recovering a QPSK symbol by subtraction. 
         FIG. 69  depicts an example of a channel response. 
         FIG. 70  depicts an example of an error of channel estimation. 
         FIG. 71  shows a comparison of energy distribution of an example of QAM signals and an example of perturbed QAM signals. 
         FIG. 72  is a graphical depiction of a comparison of an example error metric with an average perturbed QAM energy. 
         FIG. 73  is a block diagram illustrating an example process of computing an error metric. 
         FIG. 74  is a block diagram illustrating an example process of computing perturbation. 
         FIG. 75  is a block diagram illustrating an example of application of a precoding filter. 
         FIG. 76  is a block diagram illustrating an example process of UE removing the perturbation. 
         FIG. 77  is a block diagram illustrating an example spatial Tomlinson Harashima precoder (THP). 
         FIG. 78  is a spectral chart of the expected energy error for different exemplary pulse amplitude modulated (PAM) vectors. 
         FIG. 79  is a plot illustrating an example result of a spatial THP. 
         FIG. 80  shows an example of a wireless system including a base station with L antennas and multiple users. 
         FIG. 81  shows an example of a subframe structure that can be used to compute second-order statistics for training. 
         FIG. 82  shows an example of prediction training for channel estimation. 
         FIG. 83  shows an example of prediction for channel estimation. 
         FIG. 84  is a block diagram of an example of the prediction setup in an FDD system. 
         FIG. 85  is an example of a transmitter and receiver. 
         FIGS. 86A, 86B and 86C  show examples of different bandwidth partitions. 
         FIG. 87  shows an example of a bandwidth partition with the same time interval. 
         FIG. 88  shows an example of a bandwidth partition with a different time interval. 
         FIG. 89  shows an example of channel prediction over the same time interval. 
         FIG. 90  shows an example of channel prediction over a different time interval. 
         FIG. 91A  shows an example of overlaid radiation beam patterns for four users. 
         FIG. 91B  shows an example of overlaid angle-of-arrivals for the users in  FIG. 91A . 
         FIG. 92  shows an example of a wireless system in which fractional COMP is implemented. 
         FIG. 93  shows an example of antenna coverage using dual polarization antennas. 
         FIG. 94  is a graphical representation of benefits of layered wireless communication. 
         FIG. 95  shows an example of communication bandwidth utilization in a COMP network. 
         FIG. 96  is a block diagram of an example system that controls MAC-PHY interfaces across a scheduler interface. 
         FIG. 97  is a block diagram of another example system that shows the multi-user (MU)-MIMO system Application Programming Interfaces (APIs). 
         FIGS. 98A-98D  show example hardware platforms used for implementing some of the techniques described in the embodiments. 
         FIG. 99  shows an example of a wireless transceiver apparatus. 
         FIG. 100  is a flowchart for an example method of wireless communication. 
     
    
    
     DETAILED DESCRIPTION 
     To make the purposes, technical solutions and advantages of this disclosure more apparent, various embodiments are described in detail below with reference to the drawings. Unless otherwise noted, embodiments and features in embodiments of the present document may be combined with each other. 
     1. Brief Introduction 
     Cellular wireless service providers have begun planning and deployment of next generation networks to support deployment of denser deployments of higher bandwidth user devices. Furthermore, the ever-increasing reliance on wireless connectivity has raised users&#39; expectations of Quality of Service and seamless availability of wireless connectivity everywhere. 
     Cloud Radio Access Network (C-RAN) is one example of a network architecture in which a centralized cloud-based access network provides wireless connectivity to wireless terminals. However, C-RAN deployments rely on expensive deployments of fiber optic infrastructure to connect base stations with each other and with a central network controller. Furthermore, such an architecture requires planning, and deployments can be relatively slow due to the labor and resources required to lay down fiber. As a result, C-RAN and similar solutions are expensive, and cannot be quickly deployed (or taken down) to meet short term increase in demand of wireless services. Furthermore, when such an deployment reaches its maximum capacity, incremental deployment is often not possible without having to significantly alter the existing infrastructure. 
     The techniques described in the present document can be used in wireless network embodiments to overcome such problems. In one example aspect, network nodes may be deployed using short range, high speed millimeter wavelength (mmwave) links. Such installations have minimal footprint and power requirements and can be deployed and taken down to quickly meet time and geography-specific demand for wireless services. 
     In another beneficial aspect, the present technology may be used to deploy networks that provide short links between base stations, or network nodes, thereby providing reduced latency, jitter and fronthaul traffic loading in wireless networks. 
     In another beneficial aspect, the disclosed techniques may be used to manage a soft handover whereby a user equipment (UE) and N neighboring Base stations (typically N=3) constitute a cooperative multi-point (COMP) service zone. 
     In another beneficial aspect, embodiments may benefit from increased network performance without any change or replacement of existing antennas on towers, e.g., does require setting new mmwave links or computing platforms. The inventor&#39;s rough calculations have shown that it may be possible for embodiments to increase network capacity by at least a factor of two and at least 5db Signal to Interference and Noise Ratio (SINR) improvement. 
     Some embodiments of the disclosed distributed COMP technology may be used to address both intra-cell and inter-cell interference, or alternatively inter-sector interference and cell edge poor coverage, using a computer platform that processes jointly all three sectors of all towers in a cluster. One advantage is that the physical front end, e.g., antennas on tower, may not have to be changed, and yet the techniques may be embodied for boosting performance. 
     As further described in the present document, in some embodiments, distributed COMP may include groups of cell towers in which all cell towers carry the functionality of a Remote Radio Head (RRH) while one of them carry the computation for the cluster and is connected to the network for TCP/IP traffic. In other words, there is no need for a fronthaul to the network. Cluster formation may be performed using one of the techniques described in the present document. A cluster takes advantage of shared resource management and load balancing. 
       FIG. 1  shows an example of a mobile wireless network  100 . In this simplified drawing, a wireless terminal  102  is provided wireless connectivity by a network-side node  104 . The wireless terminal  102  may be, for example, a smartphone, a tablet, an Internet of Things (IoT) device, a smartwatch, etc. The network node  104  may be a base station that establishes and operates a cell of wireless communication. The communication channel between the wireless terminal  102  and the node  104  may include reflectors such as buildings, trees, moving objects such as vehicles that tend to distort signal transmissions to and from the wireless terminal  102 . During operation, the wireless terminal  102  may move away from the node  104  and may have to be handed over to or share connectivity with another network node (not explicitly shown in the drawing). In some cases, the network node  104  may cooperatively operate with other nodes to provide a multi-point transmission/reception to the wireless terminal  102  such that the mobility of the wireless terminal  102  does not hamper connectivity with the wireless services. 
     Embodiments of the disclosed technology provide various improvements to the operation of wireless networks and equipment, including: 
     1) Accurate geometry extraction and multipath attributes acquisition, based on instantaneous measurements over a limited band and over a short period of time. For example, the sparse channel representation technique, as described in Section 3, provides a computationally efficient was of modeling and predicting channels. The computations can be performed by a network device on behalf of multiple base stations and thus the network device can control transmissions from/to the multiple base stations so that wireless devices can move freely between coverage areas of the base stations without any interference from transmissions from/to other base stations in the distributed cooperative zone of base stations. 
     2) Accurate channel prediction on same band or on a different adjacent band based on instantaneous measurements over a limited band and over a short period of time, as described in Sections 2-5. The sparse channel measurement may be performed using very few reference signal transmissions and thus channel conditions in multiple neighboring cells can be quickly acquired and used at a network-side server that controls operation of distributed base stations in a cooperative manner. 
     3) Use of predicted channel state information for centralized &amp; distributed MU-MIMO precoding. For example, Sections 2, 4 and 5 describe certain techniques for predicting channels at different time instances, frequencies and spatial positions. 
     4) Use of predicted channel state information to determine Modulation and Coding Scheme (MCS) attributes (Resource block bit loading-modulation order and forward error correction codes). 
     5) Use of predicted channel State information to determine retransmission to meet delivery reliability criteria. For example, the network server may obtain accurate channel estimates at future times or other frequencies and, without waiting for ACK feedbacks, is able to decide retransmission strategy based on the channel conditions. For example, channel condition may be compared with a threshold and retransmission may be used when channel condition falls below the threshold. 
     6) Base Station clustering &amp; front haul network organization for defining CoMP regions &amp; Soft handoff between CoMP regions, as described in Section 2. 
     7) Pilot arrangement to minimize pilot contamination, as described in Section 6. The central awareness of channels for all base stations in a zone or a cluster allows the cluster controller on the network side to arrange pilots from different base stations to be non-overlapping in terms of their transmission resources. 
     8) Signal processing to separate pilot mixtures and contamination mitigation. 
     2. Embodiments of the Distributed COMP Architecture 
     Embodiments of the disclosed technology include distributed COMP architectures that implement a separation of a base station&#39;s functionality of transmission and reception of radio frequency (RF) signals between UEs and the functionality of channel estimation, prediction, precoding and retransmission management. Furthermore, millimeter (mm) wave links may be established between the RF functionality sites and remote or network-side computing servers for exchanging information related to ongoing operation of the cellular network. 
       FIG. 2  shows an example of a cellular 3-sector hexagonal model. In this model, a base station transceiver may be operated at center of each small circle and may provide wireless connectivity using three spatial sectors/beams that span 120 degrees surrounding the base station. The larger concentric circles show the neighboring cells in which a transition of a UE operating in a sector at the center may occur due to mobility. The concentric circles also show interference circumferences where neighboring sectors may affect signal quality in each other. 
       FIG. 3  shows an enlarged view of interference circumferences in the wireless network depicted in  FIG. 2 . 
       FIG. 4  shows an example of distributed cooperative multipoint (COMP) clusters. In this example, each cluster includes base stations 1 to 7, where one base station is at a center and the other base station are vertices of a hexagonal around the center station. The base stations may offer 3-sector coverage as described with respect to  FIG. 2 . The base stations may be connected with a wireless connection (possibly with a Line of Sight), that is depicted as the straight lines joining each base station, or node, 1 to 7 to neighboring base stations. 
       FIG. 5  shows examples of links, nodes and clusters in a wireless network. Links are labeled using lower case letters a, b, c . . . etc. Nodes are labeled using numbers. Clusters are labeled using upper case letters A, B, . . . etc. 
       FIG. 6  shows examples of sizing of COMP clusters. For example, a single network-side resource may handle the channel determination and prediction tasks for all sectors within a given cluster. 
     The following calculations may be used for resource planning in the network. 
       #Nodes=9 n{circumflex over ( )} 2−3 n+ 1
 
       #Clusters=3 n{circumflex over ( )} 2−3 n+ 1
 
       #Links=6(3 n{circumflex over ( )} 2−3 n+ 1)
 
       3 n ( n− 1)+1=( R/D ){circumflex over ( )}2
 
     The table below shows example values which may be used in some embodiments. 
     
       
         
           
               
               
               
               
               
             
               
                   
                   
               
               
                   
                 n 
                 cells 
                 Clusters 
                 R/D 
               
               
                   
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                 10 
                 871 
                 271 
                 16.46 
               
               
                   
                 20 
                 3541 
                 1141 
                 33.78 
               
               
                   
                 30 
                 8011 
                 2611 
                 51.10 
               
               
                   
                   
               
            
           
         
       
     
       FIG. 7  shows an example of staged COMP clustering. As depicted in  FIG. 7 , COMP zones may start from the top left side of an area, and may be gradually staged to become larger and larger in terms of their cooperative operation. 
       FIG. 8  shows another example of staged COMP clustering that starts from center of the area and progressively grows in an outward direction. 
       FIG. 9  shows an example in which one cluster with three nodes (base stations) are depicted. The three nodes may be communicating with each other using a RF link such as a mmwave link that operates in a low latency manner. 
       FIG. 10  shows an example of a wireless network in which one cluster and 7 nodes are depicted. 
       FIG. 11  shows an example of a wireless network depicting 3 clusters and 16 nodes. 
       FIG. 12  shows an example of a wireless network depicting 7 clusters and 31 nodes. 
     The embodiments described in the present document may be used to achieve wide scale COMP, as is described herein. For example, the clustering approach may be used on regional basis, while the entire network may include clustered and COMP-based operating cell, while some other cells may be operating in the conventional manner. 
     One limitation with present day implementations of wireless technologies is that wireless networks are not able to achieve full rank operation and manage interference among various transmission points (e.g., base stations). For example, in embodiments in which dual polarization is used for signal transmissions, only about 40% gain in efficiency is achieved over single polarization transmission due to imperfections in estimation and/or use of transmission rank of the channel. As a result, often, in practical implementations, transmissions are performed using single user MIMO (SU-MIMO) even in cases where MU-MIMO operation ins possible. As further described throughout the present document, the techniques described herein may be used to achieve the following operational advantages: 
     (1) Use MU-MIMO in a true sense—i.e., every time it is theoretically possible, then it is practically used 
     (2) Predict channel at a future time or at a different frequency accurately, using, for example, reciprocity and sparse channel computation techniques described in the present document. Using the predicted channel, scheduling operation may be improved in the selection of correct resource blocks for a UE, and also in selecting a modulation scheme for the selected resource blocks. 
     (3) Coordinate operation of towers (base stations) to minimize interference and improve signal to noise ratio (SNR) per layer of transmission. 
       FIG. 13  shows an example of evolution of spectral efficiency from SU-MIMO to MU-MIMO with delay-Doppler channel prediction to MU-MIMO with delay-Doppler channel prediction and COMP. The three practical advantages that can be achieved by embodiments that use the disclosed techniques are highlighted in  FIG. 13 . 
       FIG. 14  shows an example of a three-tower downlink transmission (DL) scheme in which one of the base stations B 3  operates as a COMP server (e.g., cluster server). As shown therein, the network node (or base station or base tower) B 1  generates a transmission T 1  that is communicated to UEs (or mobile devices) U 1 , U 2  and U 3  through channels h 11 , h 12  and h 13 , respectively. Similarly, the network nodes B 2  and B 3  generate transmissions T 2  and T 3  that are also communicated with U 1 , U 2  and U 3  through the channels {h 21 , h 22 , h 23 } and {h 31 , h 32 , h 33 }, respectively. 
     One advantage of the depicted embodiment is that the connection between base stations and the mobile core network does not need any changes and can continue to operate as before. The base stations are communicating with each other through a separate link for exchanging information regarding channel conditions and UE related information such as the h coefficients that can be used to generate the pre-coding coefficients. 
     In the example shown in  FIG. 14 , B 1  transmits scheduling information (denoted (s 1 )Txd 1 ) and channel condition information (denoted h 1x ) to B 3  (which is operating as the COMP server) over the separate link. Similarly, B 2  transmits {(s 1 )Txd 2 , h 2x } to B 3  over the separate link. B 3  uses the scheduling and channel condition information to generate weighting coefficients, which are transmitted back to B 1  and B 2  along with scheduling information (and denoted {(s 1 )Txd 3 , w 1x } and {(s 1 )Txd 3 , w 2x }, respectively) over the separate link. 
     In some embodiments, and as shown in CoMP server computations in  FIG. 14 , B 3  computes the weights (w ij ) based on the schedule (u i ) and the channels (h ij ). The subsequent transmissions from the base stations are based on these weights and modulated symbols, and in an example, are a weight average as shown in  FIG. 14 . 
     In some embodiments, and as shown in  FIG. 14 , a mobile core can be configured to generate the transmission schedules ((s 1 )Txd 1 , (s 1 )Txd 2  and (s 1 )Txd 3  for B 1 , B 2  and B 3 , respectively) for the base stations. 
     In some embodiments, the information exchanged by the base stations over the separate link is on the order of 400 Mbps/20 MHz/layer and can be easily accommodated by a wireless transmission link between base stations using, for example, orthogonal time frequency space (OTFS) modulated signals. In other embodiments, the separate link may be a mmwave link or an 10G optical fiber link. 
     One advantage of the depicted system is that each base station or tower only needs to connect to a first-hop neighbor, thereby simplifying operation of the b2b link between base station. Furthermore, the COMP server base station only has to be able to coordinate downstream and upstream traffic for its nearest neighbors, a task that is within reasonable limits of computation to make it practical. 
     In some embodiments, traffic on the downlink may be characterized by the following steps:
         Network sends data as usual to the different towers   Each tower senses the coupling between its antenna and the UE (channel [c ij ])   Each tower schedules users as usual “distributed and decoupled” (could be done centrally)   Towers sends to a designated aggregation tower the channel-coupling data for each UE, copy of data, and the scheduling information   The aggregation tower could be tower that has fiber connection or hosts the computation engine   The computation engine derive weighing coefficients from channel sensing and schedule information and distributes the weights back to the different towers   Towers in addition exchange data as opposed to IQ samples (at least eight times less/antenna port traffic between towers) to generate the transmit signal (based, for example, on the split options shown in  FIG. 24 or 25 )       

       FIG. 15  shows mathematical details of how the h coefficients can be used to achieve full rank of operation for the wireless channels in the case of joint downlink transmission to two mobile devices. In  FIG. 15 , [T 1  T 2 ] T  represents the transmission which is a product of the weights derived by the COMP server (w ij ) and the modulated symbols (m i ), [U 1  U 2 ] T  represents the received symbols through the channels (h ij ), and the derivation of the weights using a “zero-forcing” solution are shown therein. 
       FIG. 16  shows mathematical details in the case of three mobile devices, where H is the matrix of h coefficients representing the various layers of the channel characteristics, and follows a similar “zero-forcing” derivation as described in the context of  FIG. 15 . The weights w (ideally) are inverse of the H matrix to completely eliminate effects of the channel. 
       FIG. 17  shows operation of the uplink portion of the COMP portion, in a similar manner as described with respect to  FIG. 14  (for DL). In  FIG. 17 , R i  represents the received signal at base station (or network node) B i , which is also transmitted by B 1  and B 2  to B 3  (which is operating as the COMP server) using, for example, the 3GPP split option 7.1 (further detailed in the context of  FIG. 22  and denoted (1.7) R i  in  FIG. 17 ). 
     In some embodiments, the uplink traffic may be characterized by as follows:
         Each tower schedules users as usual “distributed and decoupled” (could be done centrally)   Each tower senses the coupling between its antenna and the UE (channel [c ij ])   Towers sends to a designated aggregation tower the received signal split 7.1, the channel coupling data for each UE and the scheduling information   The aggregation tower could be tower that has fiber connection or hosts the computation engine   The computation engine recovers the UL data and distributes the UL data back to the different towers   Towers in addition exchange data as opposed to IQ samples (at least eight times less/antenna port traffic between towers) to generate the transmit signal (based, for example, on the split options shown in  FIG. 24 or 25 )   Tower sends received data back to network as usual       

       FIG. 18  describes mathematical equations for the uplink channel estimation and equalization, using the same notation and terminology as used in  FIGS. 15 and 16 , and describes the “zero-forcing” implementation for the uplink. 
       FIG. 19  shows combined UL/DL operation of the COMP configurations of  FIGS. 14 and 17 , and uses the same notation and terminology as described above. 
       FIG. 20  shows an example of scaling of the COMP networks. As depicted, cluster A is being managed by the base station that is at the center. The base station controls transmissions to/from a UE, including movement of the UE from one cell to another, by communication with the other towers. A similar cluster pattern may extend across the entire geographic region, and ideally uniformly covers entire region. However, because the clustering is controlled and managed locally, some other clusters (e.g., cluster E) may operate partially COMP manner. 
     In some embodiments, and as shown in  FIG. 20 , a UE (U 1 ) moving between zones within a domain (A) is managed by one CoMP Node (A). In other embodiments, a UE (U 2 ) moving between two domains (A and E) is managed by two CoMP Nodes (A and E). 
       FIG. 21  is a further expanded version of the COMP network operation depicted in  FIG. 20 , and shows an example of the scalability and locality of the COMP zone. In some embodiments, and as shown in  FIG. 21 , each CoMP node is connected to six other non-CoMP nodes and each non-CoMP node is connected to three other CoMP nodes. 
       FIG. 22  is a block diagram showing an example base station implementation in which the COMP techniques described herein may be managed. As shown therein, the COMP techniques are implemented based on the interactions between the Control Plane, the Uplink User Plane, the Downlink User Plane, and the Radio Unit (RU). 
     As shown in  FIG. 22 , the Control Plane (CP) Network Function (NF) Radio Resource Control (RRC) implements the corresponding 3GPP protocol layer. It is mainly responsible for the establishment, maintenance and release of connections to the UEs. The required interaction with the UEs happens by generating RRC control messages, which are then forwarded to the User Plane. By handing over the generated messages to the Packet Data Convergence Protocol (PDCP) layer, they enter the User Plane processing chain and are finally transmitted through the antennas. Corresponding RRC messages generated by the UEs are processed by the Uplink User Plane chain and then forwarded to the CP NF. Thus a full communication between the CP NF RRC and the UEs is enabled through the User Plane. 
       FIG. 22  further shows specific interactions with the User Plane, which include DL buffer status ( 1 ), payload selection ( 2 ), DL resource assignment and generation of UL transmission grants ( 3 ), retransmission control ( 4 ), coding scheme ( 6 ), antenna mapping, precoding and modulation scheme ( 7 ), channel state information (CSI) from UL sounding ( 10 ), CSI from reporting and UL scheduling requests ( 11 ), and hybrid ARQ (HARQ) status ( 12 ). In addition, 3GPP functional splits in the User Plane are shown. 
       FIG. 23  shows a more detailed view of an example DU control plane interface used in the implementation. AIV are the layers (air interface variants) that ae provided with schedule from a short-term scheduler that defines resource block scheduling based on channel characterization. In particular,  FIG. 23  shows a multi-AIV deployment based on horizontal split Option 2 (as shown in  FIG. 22 ) in combination with the related Control Plane/User Plane split, therein demonstrating the needed Control Plane split between the Central Unit (CU) and the Distributed Unit (DU). 
       FIG. 24  shows an example of function split options that can be implemented within the User Plane protocol stack. The key rationale behind any choice of function split is to obtain the largest possible extent of centralization that a specific deployment architecture supports. A large extent of centralization of functionalities allows to exploit gains related to, e.g., centralized joint transmission, centralized scheduling, centralized flow control etc., but at the expense of increased fronthaul data rate requirements and increasingly stringent latency requirements. 
     The function splits shown in the example in  FIG. 24  affect the basic data rate scaling behavior. For example, using the “Split 2” option results in the required data rate scaling with system bandwidth, number of quantization bits per in-phase/quadrature (I/Q) sample, and number of antennas. In contrast, using the “Split 7-3” option results in the data volume here scaling solely with the user data rates and the selected forward error correction (FEC) code strength, and not strictly with the system bandwidth, number of antenna ports etc. 
       FIG. 25  shows another example of function splits for the DL and UL baseband signal processing through a base station. As shown therein, User processing typically comprises the PDCP, RLC, MAC, FEC, QAM and precoding on the downlink signal path, and the corresponding uplink blocks, whereas Cell processing typically comprises resource mapping, IFFT and CP insertion, parallel-to-serial conversion, and the final RF output stage. 
       FIG. 26  shows an example of MAC/PHY layer division of data processing performed in a base station. As shown therein, the transmission selection unit (TSU) is configured to operate above the PHY layer, resulting in MAC-PHY functions operating without TSU and the MAC-MAC functions operating with TSU. The TSU is typically configured to select one of the pre-calculated scheduling assignments from the CU based on results from the UL PHY and forward it further to the DL PHY to generate the next subframe accordingly. 
       FIG. 27  shows another example of possible functional splits between the central and distributed units. As shown therein, eight possible options. These include:
         Option 1 (RRC/PDCP, 1A-like split): In this split option, RRC is in the central unit while PDCP, RLC, MAC, physical layer and RF are kept in the distributed unit. Thus the entire user plane is in the distributed unit.   Option 2 (PDCP/RLC split): Option 2 may be a base for an X2-like design due to similarity on U-plane but some functionality may be different.   Option 3 (High RLC/Low RLC Split): In this option, the approach is based on Real time/Non-Real time functions split.   Option 4 (RLC-MAC split): In this split option, RRC, PDCP, and RLC are in the central unit. MAC, physical layer, and RF are in the distributed unit.   Option 5 (Intra MAC split): In this split option, RF, physical layer and lower part of the MAC layer (Low-MAC) are in the Distributed Unit, and the higher part of the MAC layer (High-MAC), RLC and PDCP are in the Central Unit.   Option 6 (MAC-PHY split): The MAC and upper layers are in the central unit (CU). PHY layer and RF are in the DU. The interface between the CU and DUs carries data, configuration, and scheduling-related information (e.g. MCS, Layer Mapping, Beamforming, Antenna Configuration, resource block allocation, etc.) and measurements.   Option 7 (Intra PHY split): Multiple realizations of this option are possible, including asymmetrical options which allow obtaining benefits of different sub-options for UL and DL independently.   Option 8 (PHY-RF split): This option allows to separate the RF and the PHY layer. This split permit centralization of processes at all protocol layer levels, resulting in very tight coordination of the RAN. This allows efficient support of functions such as CoMP, MIMO, load balancing, mobility.       

       FIG. 28  depicts the processing and keys it to the processing performed for fronthaul communication (e.g., on air interface and to/from UEs and b2b) and backhaul (e.g., to and from core network). In the example of the mapping of the control unit (CU) and distributed unit (DU) functions according to the split points shown in  FIG. 28 , the 4G split is at the CPRI interface. In another example, the 5G(a) high layer split is at the F1 interface. In yet another example, the 5G(b) lower layer split is at the Fx interface. In yet another example, 5G(c) shows an example of a cascaded split. 
       FIG. 29  shows an example block diagram of implementation of a base station (labeled as gNB) and its corresponding CU-RU division and interface to a management system. In some embodiments, the architecture includes a Lower Layer Split Central Unit (lls-CU), which is a logical node that includes the eNB/gNB functions (commonly also referred to as BBU), excepting those functions allocated exclusively to the Radio Unit (RU). The lls-CU manages real-time control and user plane functions of the RUs. In other embodiments, the architecture further includes a Radio Unit (RU), which is a logical node that includes a subset of the gNB functions as required by split option 7-2x (‘x’ here refers to xRAN). Management functions of the RUcan be controlled over the LLS-M interface by the lls-CU or a management system. 
       FIG. 30  shows an example of the function split 7.2x for the downlink (DL). As shown therein, the iFFT, CP addition, and digital beamforming functions reside in the Radio Unit (RU). The rest of the PHY functions including resource element mapping, precoding, layer mapping, modulation, scrambling, rate matching and coding reside in the lls-CU. Beamforming specific processing (expansion from layers/beams to digital transceivers) resides within the RU. 
       FIG. 31  shows an example of the function split 7.2x for the uplink (UL). As shown therein, the FFT, CP removal and digital beamforming functions reside in the RU. The rest of the PHY functions including resource element de-mapping, equalization, demodulation, de-scrambling, rate de-matching and de-coding reside in the lls-CU. Beamforming specific processing (combining inputs from multiple digital transceivers to a set of beams/layers) resides within the RU. 
       FIG. 32  shows examples of timelines of signal transmissions in the downstream and upstream direction during the operation of a base station in a COMP network and for the maximum tolerable CU-RU round trip time (RTT). 
     In some embodiments, network entry starts with the UE transmitting a random access preamble in the PRACH. The eNB is required to answer within a configurable random access response window (as shown in  FIG. 32( a ) ). This window starts three subframes after the last subframe with the respective preamble transmission (preambles may span two subframes), and has a configurable length between two and ten subframes. 
       FIG. 32( b )  shows an example of suspending a HARQ process in the DL, which results in delaying the scheduling of the retransmission by the desired amount of time, and in the UL, forcibly sending an ACK. The ACK causes the UE to refrain from further retransmissions. The drawback of forced ACK is that only every second transmission opportunity within each HARQ process is useable by the respective UE, halving the achievable per UE peak rate. 
     To mitigate the impact on UL power-limited UEs, the forced ACK may be combined with subframe bundling as shown in  FIG. 32( c ) . In subframe bundling, four consecutive subframes form a single UL transmission. 
       FIGS. 32( d ) and 32( e )  show examples of the uplink and downlink timing diagrams when the transmission selection unit (TSU) is incorporated (e.g., the TSU shown in  FIG. 26 ). 
       FIG. 33  shows an example of a controller  3300  that can be used to enhance the operation of a conventional base station based on some of the techniques described in the present document. Such a centralized base station may operate as a network-side server that controls scheduling for other base stations in a COMP wireless system. The centralized base station may also be implemented simply as a network-side server, without having associated base station functionality of providing connectivity directly to mobile devices via an air interface with these mobile devices. The enhanced base station, with a controller shown on the side, may continue to operate as usual (e.g., implementing base station functionalities prescribed by 3GPP), while the controller  3300  receives scheduling information and user to channel mapping from the base station and provides the base station with layering and association of UEs with layers to improve overall channel usage. For example, the controller  3300  will help achieve coordination between various layers of transmission by (1) associating UEs with respective layers, (2) controlling scheduling of each layer for its users in a conventional manner, and (3) grouping the UEs such that MU-MIMO transmissions that maximize the use of channel rank are achieved. The controller  3300  may process such information either for communication with UEs in a RAN established by the centralized base station and/or control communication on behalf of other base stations in a COMP arrangement, e.g., as described with respect to  FIGS. 1-21 . 
     For example, as shown in  FIG. 33 , the controller  3300  may communicate with a MAC/PHY protocol stack as follows. The PHY layer may be coupled to antennas  3302  through a signal separation and precoding (SSP) block  3304 . The SSP block may perform the function of precoding signals to be transmitted based on the coefficients 9 received from the controller (which the controller  3300  will have calculated using, e.g., sparse channel estimation and channel prediction techniques described herein). For received signals, the SSP block  3304  may perform the function of signal separation (e.g., extracting separate layers of communication from the received multi-layer MIMO communication). The N layers of communication (N is a positive integer, typically from 1 to 8), On the receiving side, upon separation into N layers, the communication may provide to the controller  3300  scheduling information 3, and/or condition information on various user channels 10 (e.g., obtained through channel sounding of uplink channels). The scheduling information 3 may include information such as payload selection, downlink resource assignment and uplink grant information. 
     On the transmit side, the controller  3300  may determine a layer association 7 for signal to be transmitted. The layer association 7 may be processed by a multiplexing block  3306  such that all outgoing traffic of a same layer is mapped together on a layer for transmission. The multiplexing block  3306  may also perform the tasks of mapping outgoing user data to antennas, specifying precoder to be used for each stream of data and a modulation scheme (bitloading) to be used for each layer or stream. The layered streams of data may then be precoded in the SSP block  3304  and transmitted via antennas  3302  (in some examples, a single antenna may be used). 
       FIG. 34  shows an example of a common public radio interface (CPRI) that may be used for the b2b connection among base stations. As shown therein, centralized processing is employed in conjunction with multiple remote radio heads (RRHs) that are connected by CPRI links via optical fiber, which advantageously enable joint reception with LTE. 
       FIG. 35  is a tabular example of bitrates used and achieved by some embodiments of COMP networks described herein. The factor N in the formulas shown in  FIG. 35  reflects the number of coordinated base stations (cluster size). In an example, N will be between 2 and 7. 
       FIG. 36  is a tabular example of parameters used in an example implementation. As shown therein, N RE,i  and N SC  may take two different values because the number of resource elements per resource block and control region is reduced if the actual subframe includes an SRS. Furthermore, the number of resource blocks carrying either data or control (sum (N RB,n )+N reg ) depends on the load (10 percent, 30 percent, 50 percent, 100 percent). 
       FIG. 37  shows graphs of signal to noise performance achieved as a function of quantization used in some example implementations of COMP systems. As shown therein, and depending on the working point, a resolution of at least 5 to 6 bits is needed. In order to gain robustness against power disparity between user transmissions and with inter-cell interference entering the game, we need to add 1 to 2 more bits per dimension per IQ sample to be able to follow a wider dynamic range and to not cripple interference suppression/cancellation. 
       FIG. 38  shows graphs that demonstrate benefits of the disclosed technologies. As shown in the left-hand side graph, using increasing the number of antenna ports (while keeping the number of antenna ports per layer the same) results in the xHaul data rate increasing across all the 3GPP split options considered. The right-hand side graph of  FIG. 38  shows the efficacy of multi-AIV implementations compared to LTE across all the 3GPP split options considered. In particular, it is seen therein that the case with greater than 6 GHz supports a higher xHaul data rate than the scenario with less than 6 GHz. 
     3. Sparse Channel Representation 
     A wireless channel, between a transmitting antenna of a device to a receiving antenna of another device, may be described by a geometric model of rays originating from the transmitting device, as shown in  FIG. 39 . Some of these rays may be directly received by the antenna of the other device, when there is a Line-of-Sight (LoS) between the antennas, and some may be reflected from objects and only then received in a Non-Line-of-Sight (NLoS) trace between the antennas. Each ray may be associated with a certain propagation delay, complex gain and angle of arrival (AoA) with respect to the receiving antenna. The two antennas and/or the reflecting objects may not be stationary, resulting in the well-known Doppler effect. The receiving device may use multiple antennas (antenna array) to receive the transmitted signal. 
     More formally, let α i , τ i , θ i  and υ i  represent the complex gain, delay, AoA and Doppler of ray i, respectively. Then, for N r  rays (or reflectors), the wireless channel response at time t, space s and frequency f is 
     
       
         
           
             
               
                 
                   
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     where, the space dimension is used for multiple receive antennas. 
     The channel response, H, may be considered to be a super-position of all received rays. This patent document describes a method for extracting the values α i , τ i , θ i  and υ i  from H(t, s, f), under the assumptions that N r  is small (for example, typical situations may have between zero to 10 reflectors). Obtaining these values, gives a sparse and compact channel representation of equation (0)), regardless of how large the frequency, time and space dimensions are. 
     For example, a channel with 3 NLoS reflectors, received over 16 antennas, 512 frequency tones and 4 time samples, will be described by 3×4=12 values (gain, delay, angle and Doppler for 3 reflectors) instead of 16×512×4=32768 values. 
     Furthermore, with the knowledge of the values of α i , τ i , θ i  and υ i , a covariance matrix for the channel, R HH , can be constructed for any desired frequency, time and space instances. This can be applied for predicting the channel response in any one of these dimensions. 
     This section describes two methods for constructing the covariance matrix under the sparse channel assumptions. Both methods use convex optimization techniques. Variations of these methods, or alternative methods may be used as well. 
     It is assumed that the channel response, H, is given for N t , N s  and N f  time, space and frequency grid points, respectively. This channel response may be obtained from known reference signals transmitted from one device to the other. 
     3.1 Method 1—Rays (Reflectors) Detection 
     The following algorithm solves the optimization problem of finding the complex values of vectors in delay, angular and Doppler dimensions, which after transformation to frequency, time and space, will give a channel response, which is the closest to the empirical measurement of the channel response H, under the assumption that the number of elements with non-negligible energy in these vectors is small (sparse). 
     More specifically, let&#39;s define grids of M τ , M θ  and M υ points over the delay, angular and Doppler dimensions, respectively. These grids represent the desired detection resolution of these dimensions. Let λ τ , λ θ  and λ υ be vectors of complex values over these grids. The constructed channel response is 
     
       
         
           
             
               
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     The general optimization problem minimizes ∥λ τ ∥ 1 , ∥λ θ ∥ 1  and ∥λ υ ∥ 1 , subject to 
     
       
         
           
             
               
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     where ∥·∥ 1  is the L1 norm and ε represents a small value (which may correspond to the SNR of the channel). 
     The above optimization problem may be too complex to solve directly. To reduce the complexity, one possible alternative is to solve the problem sequentially for the different dimensions. Any order of the dimensions may be applied. 
     For example, embodiments may start with the delay dimension and solve the optimization problem of minimizing ∥λ τ ∥ 1 , subject to 
     
       
         
           
             
               
                 
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     For the solution, we detect the delay indexes with non-negligible energy, m τ ∈T, such that |λ τ (m τ )| 2 ≥ε λ , where ε λ  represents an energy detection threshold (which may correspond to the SNR of the channel). Then, embodiments may continue to solve for the next dimension, for example, the angular dimension. For the next optimization problem, we reduce the delay dimension from M τ  indexes, to the set of indexes, T. Thus, we solve the optimization problem of minimizing ∥λ τ,θ ∥ 1 , subject to 
     
       
         
           
             
               
                 
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     Note, that the size of the optimization vector is now T·M θ  and m τ,θ  is an index to this vector, corresponding to delay indexes in T and angular indexes in M θ . For this solution, some embodiments may detect the delay-angular indexes with non-negligible energy, m τ,θ ∈TΘ, such that |λ τ,θ (m τ,θ )| 2 ≥ε λ  and continue to the final dimension, Doppler. Here, embodiments may solve the optimization problem of minimizing ∥λ τ,θ,υ ∥ 1 , subject to 
     
       
         
           
             
                 
             
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                         ∑ 
                         
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                           = 
                           0 
                         
                         
                           
                             N 
                             f 
                           
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           
                              
                             
                               
                                 H 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     , 
                                     s 
                                     , 
                                     f 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 ( 
                                 
                                   
                                     H 
                                     ^ 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       t 
                                       , 
                                       s 
                                       , 
                                       f 
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                              
                           
                           2 
                         
                         
                           
                             
                                
                               
                                 H 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     , 
                                     s 
                                     , 
                                     f 
                                   
                                   ) 
                                 
                               
                               ) 
                             
                              
                           
                           2 
                         
                       
                     
                   
                 
               
               ≤ 
               ɛ 
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             where 
           
         
       
       
         
           
             
               
                 H 
                 ^ 
               
               ⁡ 
               
                 ( 
                 
                   t 
                   , 
                   s 
                   , 
                   f 
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   
                     m 
                     
                       τ 
                       , 
                       θ 
                     
                   
                   ∈ 
                   
                     T 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Θ 
                   
                 
                 
                     
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     
                       m 
                       υ 
                     
                     = 
                     0 
                   
                   
                     
                       M 
                       υ 
                     
                     - 
                     1 
                   
                 
                 ⁢ 
                 
                   
                     
                       λ 
                       
                         t 
                         , 
                         θ 
                         , 
                         υ 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         m 
                         
                           τ 
                           , 
                           θ 
                           , 
                           υ 
                         
                       
                       ) 
                     
                   
                   · 
                   
                     e 
                     
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         jf 
                         · 
                         
                           
                             m 
                             τ 
                           
                           / 
                           
                             M 
                             τ 
                           
                         
                       
                     
                   
                   · 
                   
                     e 
                     
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         js 
                         · 
                         
                           
                             m 
                             θ 
                           
                           / 
                           
                             M 
                             θ 
                           
                         
                       
                     
                   
                   · 
                   
                     e 
                     
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         jt 
                         · 
                         
                           
                             m 
                             υ 
                           
                           / 
                           
                             M 
                             υ 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     The size of the optimization vector is now TΘ·M υ and m τ,θ,υ is an index to this vector, corresponding to delay indexes in T, angular indexes in Θ and Doppler indexes in M υ . Finally, embodiments may detect the Doppler indexes with non-negligible energy, m τ,θ,υ ∈TΘΥ, such that |λ τ,θ,υ (m τ,θ,v )| 2 ≥ε λ . The final information, representing the sparse channel, is now a small set of |TΘΥ| values. 
     Now, for any selection of time, space and frequency grids, denoted by the indexes t′, s′ and f′, we can use this representation to construct a covariance for the channel as 
     
       
         
           
             
               
                 H 
                 ^ 
               
               ⁡ 
               
                 ( 
                 
                   
                     t 
                     ′ 
                   
                   , 
                   
                     s 
                     ′ 
                   
                   , 
                   
                     f 
                     ′ 
                   
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   
                     
                       m 
                       τ 
                     
                     ∈ 
                     T 
                   
                   ⁢ 
                   
                       
                   
                 
                 
                     
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     
                       m 
                       θ 
                     
                     ∈ 
                     Θ 
                   
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       
                         m 
                         υ 
                       
                       ∈ 
                       Υ 
                     
                   
                   ⁢ 
                   
                     
                       
                         λ 
                         
                           t 
                           , 
                           θ 
                           , 
                           υ 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           m 
                           
                             τ 
                             , 
                             θ 
                             , 
                             υ 
                           
                         
                         ) 
                       
                     
                     · 
                     
                       e 
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         jf 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ′ 
                           · 
                           
                             
                               m 
                               τ 
                             
                             / 
                             
                               M 
                               τ 
                             
                           
                         
                       
                     
                     · 
                     
                       e 
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         js 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ′ 
                           · 
                           
                             
                               m 
                               θ 
                             
                             / 
                             
                               M 
                               θ 
                             
                           
                         
                       
                     
                     · 
                     
                       e 
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         jt 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ′ 
                           · 
                           
                             
                               m 
                               υ 
                             
                             / 
                             
                               M 
                               υ 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               
                 R 
                 HH 
               
               = 
               
                 
                   
                     H 
                     ^ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         t 
                         ′ 
                       
                       , 
                       
                         s 
                         ′ 
                       
                       , 
                       
                         f 
                         ′ 
                       
                     
                     ) 
                   
                 
                 · 
                 
                   
                     
                       H 
                       * 
                     
                     ^ 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         t 
                         ′ 
                       
                       , 
                       
                         s 
                         ′ 
                       
                       , 
                       
                         f 
                         ′ 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     3.2 Method 2—Maximum Likelihood 
     The following algorithm solves the optimization problem of finding the most likelihood covariance matrix, for an empirical channel measurement. Let&#39;s consider the function r(·), which translates a covariance from the delay, angular or Doppler dimensions, to frequency, space or time dimensions 
         r (α,κ, x )=sinc(κ· x )· e   j2παx  
 
     The covariance of the channel is a Toeplitz matrix generated by the function: 
     
       
         
           
             
               
                 
                   R 
                   = 
                   
                     
                       ∑ 
                       
                         τ 
                         = 
                         0 
                       
                       
                         
                           M 
                           τ 
                         
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           θ 
                           = 
                           0 
                         
                         
                           
                             M 
                             θ 
                           
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             υ 
                             = 
                             0 
                           
                           
                             
                               M 
                               υ 
                             
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             
                               λ 
                               t 
                             
                             ⁡ 
                             
                               ( 
                               τ 
                               ) 
                             
                           
                           · 
                           
                             r 
                             ⁡ 
                             
                               ( 
                               
                                 τ 
                                 , 
                                 
                                   κ 
                                   τ 
                                 
                                 , 
                                 f 
                               
                               ) 
                             
                           
                           · 
                           
                             
                               λ 
                               θ 
                             
                             ⁡ 
                             
                               ( 
                               θ 
                               ) 
                             
                           
                           · 
                           
                             r 
                             ⁡ 
                             
                               ( 
                               
                                 θ 
                                 , 
                                 
                                   κ 
                                   θ 
                                 
                                 , 
                                 s 
                               
                               ) 
                             
                           
                           · 
                           
                             
                               λ 
                               υ 
                             
                             ⁡ 
                             
                               ( 
                               υ 
                               ) 
                             
                           
                           · 
                           
                             
                               r 
                               ⁡ 
                               
                                 ( 
                                 
                                   υ 
                                   , 
                                   
                                     κ 
                                     υ 
                                   
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     In the above equation, M τ , M θ  and M υ are the desired resolutions in delay, angular and Doppler dimensions, κ τ , κ θ  and κ υ are constants and f, s and t are indexes in the frequency, space and time grids. The variables λ τ , λ θ  and λ υ are the unknown non-negative weights that needs to be determined for each element in these three dimensions. To find them, some embodiments may solve the optimization problem of finding the covariance R that maximizes the probability of getting the empirical channel response H. More formally, find 
     
       
         
           
             
               
                 
                   
                     
                       
                         R 
                         * 
                       
                       = 
                       
                         
                           argmax 
                           R 
                         
                         ⁢ 
                         
                           P 
                           ( 
                           H 
                            
                         
                         ⁢ 
                         R 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   where 
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     P 
                     ( 
                     H 
                      
                   
                   ⁢ 
                   R 
                 
               
               ) 
             
             = 
             
               
                 1 
                 
                   
                     π 
                     ⁢ 
                     
                        
                       R 
                        
                     
                   
                 
               
               · 
               
                 e 
                 
                   
                     - 
                     
                       1 
                       2 
                     
                   
                   ⁢ 
                   
                     H 
                     H 
                   
                   ⁢ 
                   
                     R 
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   H 
                 
               
             
           
         
       
     
     One possible method for solving this, is to use convex optimization techniques for an equivalent minimization problem. The assumption of a sparse channel representation is not used explicitly to formalize the optimization problem. However, the geometric physical model of the channel in delay, angular and Doppler dimensions, implicitly implies of a sparse channel representation. 
     To reduce the complexity of solving such an optimization problem, it is possible to perform the optimization sequentially over the dimensions (in any order), in a similar way to the one described for method 1. First, we solve for one of the dimensions, for example delay. We find the delay covariance 
     
       
         
           
             
               
                 R 
                 τ 
               
               ⁡ 
               
                 ( 
                 f 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   τ 
                   = 
                   0 
                 
                 
                   
                     M 
                     τ 
                   
                   - 
                   1 
                 
               
               ⁢ 
               
                 
                   
                     λ 
                     τ 
                   
                   ⁡ 
                   
                     ( 
                     τ 
                     ) 
                   
                 
                 · 
                 
                   r 
                   ⁡ 
                   
                     ( 
                     
                       τ 
                       , 
                       
                         κ 
                         τ 
                       
                       , 
                       f 
                     
                     ) 
                   
                 
               
             
           
         
       
     
     that maximizes the probability 
     
       
         
           
             
               
                 
                   P 
                   ( 
                   H 
                    
                 
                 ⁢ 
                 
                   R 
                   τ 
                 
               
               ) 
             
             = 
             
               
                 1 
                 
                   
                     π 
                     ⁢ 
                     
                        
                       
                         R 
                         τ 
                       
                        
                     
                   
                 
               
               · 
               
                 e 
                 
                   
                     - 
                     
                       1 
                       2 
                     
                   
                   ⁢ 
                   
                     H 
                     H 
                   
                   ⁢ 
                   
                     R 
                     τ 
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   H 
                 
               
             
           
         
       
     
     Then, some embodiments may detect the delay indexes with non-negligible energy, τ∈T, such that |λ τ (τ)| 2 ≥ε λ , where ε λ  represents an energy detection threshold (which may correspond to the SNR of the channel). Then, some embodiments may continue to solve for the next dimension, for example, the angular dimension. Some embodiments may find the delay-angular covariance matrix as follows: 
     
       
         
           
             
               
                 R 
                 
                   τ 
                   ⁢ 
                   θ 
                 
               
               ⁡ 
               
                 ( 
                 
                   f 
                   , 
                   s 
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   τ 
                   ∈ 
                   T 
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     θ 
                     = 
                     0 
                   
                   
                     
                       M 
                       θ 
                     
                     - 
                     1 
                   
                 
                 ⁢ 
                 
                   
                     
                       λ 
                       
                         τ 
                         ⁢ 
                         θ 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         τ 
                         , 
                         θ 
                       
                       ) 
                     
                   
                   · 
                   
                     r 
                     ⁡ 
                     
                       ( 
                       
                         τ 
                         , 
                         
                           κ 
                           r 
                         
                         , 
                         f 
                       
                       ) 
                     
                   
                   · 
                   
                     r 
                     ⁡ 
                     
                       ( 
                       
                         θ 
                         , 
                         
                           κ 
                           θ 
                         
                         , 
                         s 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     that maximizes the probability 
     
       
         
           
             
               
                 
                   P 
                   ( 
                   H 
                    
                 
                 ⁢ 
                 
                   R 
                   
                     τ 
                     ⁢ 
                     θ 
                   
                 
               
               ) 
             
             = 
             
               
                 1 
                 
                   
                     π 
                     ⁢ 
                     
                        
                       
                         R 
                         
                           τ 
                           ⁢ 
                           θ 
                         
                       
                        
                     
                   
                 
               
               · 
               
                 e 
                 
                   
                     - 
                     
                       1 
                       2 
                     
                   
                   ⁢ 
                   
                     H 
                     H 
                   
                   ⁢ 
                   
                     R 
                     τθ 
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   H 
                 
               
             
           
         
       
     
     Again, some embodiments may detect the delay-angular indexes with non-negligible energy, τ, θ∈TΘ, such that |λ τθ (τ, θ)| 2 ≥E λ  and continue to solve for the final dimension, Doppler. Some embodiments may find the delay-angular-Doppler covariance matrix 
     
       
         
           
             
               
                 R 
                 τθυ 
               
               ⁡ 
               
                 ( 
                 
                   f 
                   , 
                   s 
                   , 
                   t 
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   τ 
                   , 
                   
                     θ 
                     ∈ 
                     
                       T 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Θ 
                     
                   
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     υ 
                     = 
                     0 
                   
                   
                     
                       M 
                       υ 
                     
                     - 
                     1 
                   
                 
                 ⁢ 
                 
                   
                     
                       λ 
                       τθυ 
                     
                     ⁡ 
                     
                       ( 
                       
                         τ 
                         , 
                         θ 
                         , 
                         υ 
                       
                       ) 
                     
                   
                   · 
                   
                     r 
                     ⁡ 
                     
                       ( 
                       
                         τ 
                         , 
                         
                           κ 
                           π 
                         
                         , 
                         f 
                       
                       ) 
                     
                   
                   · 
                   
                     r 
                     ⁡ 
                     
                       ( 
                       
                         θ 
                         , 
                         
                           κ 
                           θ 
                         
                         , 
                         s 
                       
                       ) 
                     
                   
                   · 
                   
                     r 
                     ⁡ 
                     
                       ( 
                       
                         v 
                         , 
                         
                           κ 
                           υ 
                         
                         , 
                         t 
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     that maximizes the probability 
     
       
         
           
             
               
                 
                   P 
                   ( 
                   H 
                    
                 
                 ⁢ 
                 
                   R 
                   
                     τ 
                     ⁢ 
                     θ 
                     ⁢ 
                     v 
                   
                 
               
               ) 
             
             = 
             
               
                 1 
                 
                   
                     π 
                     ⁢ 
                     
                        
                       
                         R 
                         τθυ 
                       
                        
                     
                   
                 
               
               · 
               
                 e 
                 
                   
                     - 
                     
                       1 
                       2 
                     
                   
                   ⁢ 
                   
                     H 
                     H 
                   
                   ⁢ 
                   
                     R 
                     τθυ 
                     
                       - 
                       1 
                     
                   
                   ⁢ 
                   H 
                 
               
             
           
         
       
     
     Finally, some embodiments may detect the delay-angular-Doppler indexes with non-negligible energy, τ, θ, υ∈TΘΥ, such that |λ τθυ (τ, θ, υ)| 2 ≥ε λ  and use them to construct a covariance for the channel for any selection of frequency, space and time grids, denoted by the indexes f′, s′ and t′ 
     
       
         
           
             
               R 
               ⁡ 
               
                 ( 
                 
                   
                     f 
                     ′ 
                   
                   , 
                   
                     s 
                     ′ 
                   
                   , 
                   
                     t 
                     ′ 
                   
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   τ 
                   , 
                   θ 
                   , 
                   
                     υ 
                     ∈ 
                     
                       T 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Θ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Υ 
                     
                   
                 
               
               ⁢ 
               
                 
                   
                     λ 
                     τθυ 
                   
                   ⁡ 
                   
                     ( 
                     
                       τ 
                       , 
                       θ 
                       , 
                       υ 
                     
                     ) 
                   
                 
                 · 
                 
                   r 
                   ⁡ 
                   
                     ( 
                     
                       τ 
                       , 
                       
                         κ 
                         τ 
                       
                       , 
                       
                         f 
                         t 
                       
                     
                     ) 
                   
                 
                 · 
                 
                   r 
                   ⁡ 
                   
                     ( 
                     
                       θ 
                       , 
                       
                         κ 
                         θ 
                       
                       , 
                       
                         s 
                         ′ 
                       
                     
                     ) 
                   
                 
                 · 
                 
                   
                     r 
                     ⁡ 
                     
                       ( 
                       
                         υ 
                         , 
                         
                           κ 
                           υ 
                         
                         , 
                         
                           t 
                           ′ 
                         
                       
                       ) 
                     
                   
                   . 
                 
               
             
           
         
       
     
     3.3 Detection Tree for Reduced Complexity 
     The optimization problems, solved for a grid size of M points in one of the dimensions can be iteratively solved by constructing the M points in a tree structure. For example, M=8 can be constructed as a tree of 3 levels, as shown in  FIG. 40 . For each tree level, l, some embodiments may solve the optimization problem for m l ≤M. Then, some embodiments detect branches in the tree, where the total energy of the optimized vector is smaller than a threshold and eliminate them. The next level, will have a new m l  value, that does not include the removed branches. In this way, when an execution gets to the bottom levels of the tree, the size of m l  becomes smaller and smaller relative to M. Overall, this technique reduces the complexity significantly, especially when the number of detected elements (reflectors) is much smaller compare to the detection resolution M. 
       FIG. 40  shows a detection tree example for M=8. In each tree level, for every valid node, the detected energy is compared to a threshold. If it is above it, the descendant tree branches survive (solid nodes). If it is below it, the descendant tree branches are eliminated (dashed-lines nodes) and their descendant nodes are not processed anymore (marked with a cross). The first tree level processes two nodes and keeps them. The second tree level, processes 4 nodes and keep only two of them. The third tree level, processes 4 nodes and keeps two of them, corresponding to the location of the reflectors (arrows). 
     It will be appreciated by practitioners of the art that it is possible to use the detection tree for both method 1 and 2, described above. 
     3.4 Prediction Filter Examples 
     Once the reflectors are detected (method 1), or the covariance weights are determined (method 2), a covariance matrix can be constructed for any frequency, space and time grids. If we denote these grids as a set of elements, Y, and denote X as a subset of Y, and representing the grid elements for an instantaneous measurement of the channel as H X , then the prediction filter may be computed as 
         C=R   YX ·( R   XX ) −1  
 
     and the predicted channel is computed as 
     
       
      
       Ĥ 
       Y 
       =C·H 
       X  
      
     
     The matrices R YX  and R XX  are a column decimated, and a row-column decimated, versions of the channel constructed covariance matrix. These matrices are decimated to the grid resources represented by X. 
     3.5 Channel Prediction in a Wireless System 
     The described techniques may be used for predicting the wireless channels in a Time Division Duplex (TDD) or a Frequency Division Duplex (FDD) system. Such a system may include base stations (BS) and multiple user equipment (UE). This technique is suitable for both stationary and mobile UE. Generally, these techniques are used to compute a correct covariance matrix representing the wireless channels, based on a sparse multi-dimensional geometric model, from a relatively small number of observations (in frequency, time and space). From this covariance matrix, a prediction filter is computed and applied to some channel measurements, to predict the channels in some or all the frequency, space and time dimensions. The predicted channels for the UE, along with other predicted channels for other UE, may be used to generate a precoded downlink transmission from one BS to multiple UE (Multi-User MIMO, Mu-MIMO), or from several BS to multiple UE (also known as CoMP—Coordinated Multi-Point or distributed Mu-MIMO). 
     Note, that although most of the computational load, described in the following paragraphs, is attributed to the BS (or some other network-side processing unit), some of it may be performed, in alternative implementations, in the UE. 
     3.5.1 TDD Systems 
     In this scenario, the BS predicts the wireless channels from its antennas to the UE in a future time instance. This may be useful for generating a precoded downlink transmission. The UE may transmit at certain time instances reference signals to the BS, from which the BS will estimate the wireless channels response. Note, that typically, a small number of time instances should be sufficient, which makes it a method, suitable for mobile systems. Then, the estimated channel responses (whole or partial), are used with one of the described methods, to determine the covariance matrix of the channels and compute a prediction filter. This processing may take place in the base station itself, or at a remote or a network-side processing unit (also known as “cloud” processing). The prediction filter may be applied to some of the channel responses already received, or to some other estimated channel responses, to generate a prediction of the wireless channels, at a future time instance and over the desired frequency and space grids. 
     3.5.2 FDD Systems 
     In this scenario too, the BS predicts the wireless channels from its antennas to the UE in a future time instance. However, the UE to BS uplink transmissions and the BS to UE downlink transmissions are over different frequency bands. The generation of the of prediction filter is similar to TDD systems. The UE may transmit at certain time instances reference signals to the BS, from which the BS will estimate the wireless channels response. Then, the estimated channel responses (whole or partial), are used with one of the described methods, to determine the covariance matrix of the channels and compute a prediction filter. In parallel, at any time instance, the BS may transmit reference signals to the UE. The UE will feedback to the BS through its uplink, some the received reference signals (all or partial), as raw or processed information (implicit/explicit feedback). The BS will generate, if needed, an estimated channel response for the downlink channel, from the information received from the UE and apply the prediction filter to it. The result is a predicted channel at the downlink frequency band and at a future time instance. 
     3.5.3 Self-Prediction for MCS Estimation 
     It is useful for the BS to know the quality of the prediction of the channels in order to determine correctly which modulation and coding (MCS) to use for its precoded transmission. The more accurate the channels are represented by the computed covariance matrix; the higher prediction quality is achieved, and the UE will have a higher received SNR. 
     4. Multiple Access and Precoding in OTFS 
     This section covers multiple access and precoding protocols that are used in typical OTFS systems.  FIG. 41  depicts a typical example scenario in wireless communication is a hub transmitting data over a fixed time and bandwidth to several user devices (UEs). For example: a tower transmitting data to several cell phones, or a Wi-Fi router transmitting data to several devices. Such scenarios are called multiple access scenarios. 
     Orthogonal Multiple Access 
     Currently the common technique used for multiple access is orthogonal multiple access. This means that the hub breaks it&#39;s time and frequency resources into disjoint pieces and assigns them to the UEs. An example is shown in  FIG. 42 , where four UEs (UE 1 , UE 2 , UE 3  and UE 4 ) get four different frequency allocations and therefore signals are orthogonal to each other. 
     The advantage of orthogonal multiple access is that each UE experience its own private channel with no interference. The disadvantage is that each UE is only assigned a fraction of the available resource and so typically has a low data rate compared to non-orthogonal cases. 
     Precoding Multiple Access 
     Recently, a more advanced technique, precoding, has been proposed for multiple access. In precoding, the hub is equipped with multiple antennas. The hub uses the multiple antennas to create separate beams which it then uses to transmit data over the entire bandwidth to the UEs. An example is depicted in  FIG. 43 , which shows that the hub is able to form individual beams of directed RF energy to UEs based on their positions. 
     The advantage of precoding it that each UE receives data over the entire bandwidth, thus giving high data rates. The disadvantage of precoding is the complexity of implementation. Also, due to power constraints and noisy channel estimates the hub cannot create perfectly disjoint beams, so the UEs will experience some level of residual interference. 
     Introduction to Precoding 
     Precoding may be implemented in four steps: channel acquisition, channel extrapolation, filter construction, filter application. 
     Channel acquisition: To perform precoding, the hub determines how wireless signals are distorted as they travel from the hub to the UEs. The distortion can be represented mathematically as a matrix: taking as input the signal transmitted from the hubs antennas and giving as output the signal received by the UEs, this matrix is called the wireless channel. 
     Channel prediction: In practice, the hub first acquires the channel at fixed times denoted by s 1 , s 2 , . . . , s n . Based on these values, the hub then predicts what the channel will be at some future times when the pre-coded data will be transmitted, we denote these times denoted by t 1 , t 2 , . . . , t m . 
     Filter construction: The hub uses the channel predicted at t 1 , t 2 , . . . , t m  to construct precoding filters which minimize the energy of interference and noise the UEs receive. 
     Filter application: The hub applies the precoding filters to the data it wants the UEs to receive. 
     Channel Acquisition 
     This section gives a brief overview of the precise mathematical model and notation used to describe the channel. 
     Time and frequency bins: the hub transmits data to the UEs on a fixed allocation of time and frequency. This document denotes the number of frequency bins in the allocation by N f  and the number of time bins in the allocation by N t . 
     Number of antennas: the number of antennas at the hub is denoted by L h , the total number of UE antennas is denoted by L u . 
     Transmit signal: for each time and frequency bin the hub transmits a signal which we denote by φ(f, t)∈   L     h    for f=1, . . . , N f  and t=1, . . . , N t . 
     Receive signal: for each time and frequency bin the UEs receive a signal which we denote by y(f, t)∈   L     u    for f=1, . . . , N f  and t=1, . . . , N t . 
     White noise: for each time and frequency bin white noise is modeled as a vector of iid Gaussian random variables with mean zero and variance N 0 . This document denotes the noise by w(f, t)∈   L     u    for f=1, . . . , N f  and t=1, . . . , N t . 
     Channel matrix: for each time and frequency bin the wireless channel is represented as a matrix and is denoted by H(f, t)∈   L     u     ×L     h    for f=1, . . . , N f  and t=1, . . . , N t . 
     The wireless channel can be represented as a matrix which relates the transmit and receive signals through a simple linear equation: 
         y ( f,t )= H ( f,t )φ( f,t )+ w ( f,t )  (1)
 
     for f=1, . . . , N f  and t=1, . . . , N t .  FIG. 44  shows an example spectrogram of a wireless channel between a single hub antenna and a single UE antenna. The graph is plotted with time as the horizontal axis and frequency along the vertical axis. The regions are shaded to indicate where the channel is strong or weak, as denoted by the dB magnitude scale shown in  FIG. 44 . 
     Two common ways are typically used to acquire knowledge of the channel at the hub: explicit feedback and implicit feedback. 
     Explicit Feedback 
     In explicit feedback, the UEs measure the channel and then transmit the measured channel back to the hub in a packet of data. The explicit feedback may be done in three steps. 
     Pilot transmission: for each time and frequency bin the hub transmits a pilot signal denoted by p(f, t)∈   L     h    for f=1, . . . , N f  and t=1, . . . , N t . Unlike data, the pilot signal is known at both the hub and the UEs. 
     Channel acquisition: for each time and frequency bin the UEs receive the pilot signal distorted by the channel and white noise: 
         H ( f,t ) p ( f,t )+ w ( f,t ),  (2)
 
     for f=1, . . . , N f  and t=1, . . . , N t . Because the pilot signal is known by the UEs, they can use signal processing to compute an estimate of the channel, denoted by Ĥ(f, t). 
     Feedback: the UEs quantize the channel estimates Ĥ(f, t) into a packet of data. The packet is then transmitted to the hub. 
     The advantage of explicit feedback is that it is relatively easy to implement. The disadvantage is the large overhead of transmitting the channel estimates from the UEs to the hub. 
     Implicit Feedback 
     Implicit feedback is based on the principle of reciprocity which relates the uplink channel (UEs transmitting to the hub) to the downlink channel (hub transmitting to the UEs).  FIG. 45  shows an example configuration of uplink and downlink channels between a hub and multiple UEs. 
     Specifically, denote the uplink and downlink channels by H up  and H respectively, then: 
         H ( f,t )= AH   up   T ( f,t ) B,   (3)
 
     for f=1, . . . , N f  and t=1, . . . , N t . Where H up   T (f, t) denotes the matrix transpose of the uplink channel. The matrices A∈   L     u     ×L     u    and B∈   L     h     ×L     h    represent hardware non-idealities. By performing a procedure called reciprocity calibration, the effect of the hardware non-idealities can be removed, thus giving a simple relationship between the uplink and downlink channels: 
         H ( f,t )= H   up   T ( f,t )  (4)
 
     The principle of reciprocity can be used to acquire channel knowledge at the hub. The procedure is called implicit feedback and consists of three steps. 
     Reciprocity calibration: the hub and UEs calibrate their hardware so that equation (4) holds. 
     Pilot transmission: for each time and frequency bin the UEs transmits a pilot signal denoted by p(f, t)∈   L     u    for f=1, . . . , N f  and t=1, . . . , N t . Unlike data, the pilot signal is known at both the hub and the UEs. 
     Channel acquisition: for each time and frequency bin the hub receives the pilot signal distorted by the uplink channel and white noise: 
         H   up ( f,t ) p ( f,t )+ w ( f,t )  (5)
 
     for f=1, . . . , N f  and t=1, . . . , N t . Because the pilot signal is known by the hub, it can use signal processing to compute an estimate of the uplink channel, denoted by  (f, t). Because reciprocity calibration has been performed the hub can take the transpose to get an estimate of the downlink channel, denoted by Ĥ(f, t). 
     The advantage of implicit feedback is that it allows the hub to acquire channel knowledge with very little overhead; the disadvantage is that reciprocity calibration is difficult to implement. 
     Channel Prediction 
     Using either explicit or implicit feedback, the hub acquires estimates of the downlink wireless channel at certain times denoted by s 1 , s 2 , . . . , s n  using these estimates it must then predict what the channel will be at future times when the precoding will be performed, denoted by t 1 , t 2 , . . . , t m .  FIG. 46  shows this setup in which “snapshots” of channel are estimated, and based on the estimated snapshots, a prediction is made regarding the channel at a time in the future. As depicted in  FIG. 46 , channel estimates may be available across the frequency band at a fixed time slots, and based on these estimates, a predicated channel is calculated. 
     There are tradeoffs when choosing the feedback times s 1 , s 2 , . . . , s n . 
     Latency of extrapolation: Refers to the temporal distance between the last feedback time, s n , and the first prediction time, t 1 , determines how far into the future the hub needs to predict the channel. If the latency of extrapolation is large, then the hub has a good lead time to compute the pre-coding filters before it needs to apply them. On the other hand, larger latencies give a more difficult prediction problem. 
     Density: how frequent the hub receives channel measurements via feedback determines the feedback density. Greater density leads to more accurate prediction at the cost of greater overhead. 
     There are many channel prediction algorithms in the literature. They differ by what assumptions they make on the mathematical structure of the channel. The stronger the assumption, the greater the ability to extrapolate into the future if the assumption is true. However, if the assumption is false then the extrapolation will fail. For example: 
     Polynomial extrapolation: assumes the channel is smooth function. If true, can extrapolate the channel a very short time into the future ≈0.5 ms. 
     Bandlimited extrapolation: assumes the channel is a bandlimited function. If true, can extrapolated a short time into the future ≈1 ms. 
     MUSIC extrapolation: assumes the channel is a finite sum of waves. If true, can extrapolate a long time into the future ≈10 ms. 
     Precoding Filter Computation and Application 
     Using extrapolation, the hub computes an estimate of the downlink channel matrix for the times the pre-coded data will be transmitted. The estimates are then used to construct precoding filters. Precoding is performed by applying the filters on the data the hub wants the UEs to receive. Before going over details we introduce notation. 
     Channel estimate: for each time and frequency bin the hub has an estimate of the downlink channel which we denote by Ĥ(f, t)∈   L     u     ×L     h    for f=1, . . . , N f  and t=1, . . . , N t . 
     Precoding filter: for each time and frequency bin the hub uses the channel estimate to construct a precoding filter which we denote by W(f, t)∈   L     h     ×L     u    for f=1, . . . , N f  and t=1, . . . , N t . 
     Data: for each time and frequency bin the UE wants to transmit a vector of data to the UEs which we denote by x(f, t)∈   L     u    for f=1, . . . , N f  and t=1, . . . , N t . 
     Hub Energy Constraint 
     When the precoder filter is applied to data, the hub power constraint is an important consideration. We assume that the total hub transmit energy cannot exceed N f N t L h . Consider the pre-coded data: 
         W ( f,t ) x ( f,t ),  (6)
 
     for f=1, . . . , N f  and t=1, . . . , N t . To ensure that the pre-coded data meets the hub energy constraints the hub applies normalization, transmitting: 
       λ W ( f,t ) x ( f,t ),  (7)
 
     for f=1, . . . , N f  and t=1, . . . , N t . Where the normalization constant λ is given by: 
     
       
         
           
             
               
                 
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     Receiver SNR 
     The pre-coded data then passes through the downlink channel, the UEs receive the following signal: 
       λ H ( f,t ) W ( f,t ) x ( f,t )+ w ( f,t ),  (9)
 
     for f=1, . . . , N f  and t=1, . . . , N t . The UE then removes the normalization constant, giving a soft estimate of the data: 
     
       
         
           
             
               
                 
                   
                     
                       
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     for f=1, . . . , N f  and t=1, . . . , N t . The error of the estimate is given by: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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     The error of the estimate can be split into two terms. The term H(f, t)W(f, t)−x(f, t) is the interference experienced by the UEs while the term 
     
       
         
           
             
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     gives the noise experienced by the UEs. 
     When choosing a pre-coding filter there is a tradeoff between interference and noise. We now review the two most popular pre-coder filters: zero-forcing and regularized zero-forcing. 
     Zero Forcing Precoder 
     The hub constructs the zero forcing pre-coder (ZFP) by inverting its channel estimate: 
         W   ZF ( f,t )=( Ĥ *( f,t ) Ĥ ( f,t )) −1   Ĥ *( f,t ),  (12)
 
     for f=1, . . . , N f  and t=1, . . . , N t . The advantage of ZPP is that the UEs experience little interference (if the channel estimate is perfect then the UEs experience no interference). The disadvantage of ZFP is that the UEs can experience a large amount of noise. This is because at time and frequency bins where the channel estimate Ĥ(f, t) is very small the filter W ZF (f, t) will be very large, thus causing the normalization constant λ to be very small giving large noise energy.  FIG. 47  demonstrates this phenomenon for a SISO channel. 
     Regularized Zero-Forcing Pre-Coder (rZFP) 
     To mitigates the effect of channel nulls (locations where the channel has very small energy) the regularized zero forcing precoder (rZFP) is constructed be taking a regularized inverse of its channel estimate: 
         W   rZF ( f,t )=( Ĥ *( f,t ) Ĥ ( f,t )+α I ) −1   Ĥ *( f,t ),  (13)
 
     for f=1, . . . , N f  and t=1, . . . , N t . Where α&gt;0 is the normalization constant. The advantage of rZFP is that the noise energy is smaller compared to ZPF. This is because rZFP deploys less energy in channel nulls, thus the normalization constant λ is larger giving smaller noise energy. The disadvantage of rZFP is larger interference compared to ZFP. This is because the channel is not perfectly inverted (due to the normalization constant), so the UEs will experience residual interference.  FIG. 48  demonstrates this phenomenon for a SISO channel. 
     As described above, there are three components to a precoding system: a channel feedback component, a channel prediction component, and a pre-coding filter component. The relationship between the three components is displayed in  FIG. 49 . 
     OTFS Precoding System 
     Various techniques for implementing OTFS precoding system are discussed. Some disclosed techniques can be used to provide the ability to shape the energy distribution of the transmission signal. For example, energy distribution may be such that the energy of the signal will be high in regions of time frequency and space where the channel information and the channel strength are strong. Conversely, the energy of the signal will be low in regions of time frequency and space where the channel information or the channel strength are weak. 
     Some embodiments may be described with reference to three main blocks, as depicted in  FIG. 50 . 
     Channel prediction: During channel prediction, second order statistics are used to build a prediction filter along with the covariance of the prediction error. 
     Optimal precoding filter: using knowledge of the predicted channel and the covariance of the prediction error: the hub computes the optimal precoding filter. The filter shapes the spatial energy distribution of the transmission signal. 
     Vector perturbation: using knowledge of the predicted channel, precoding filter, and prediction error, the hub perturbs the transmission signal. By doing this the hub shapes the time, frequency, and spatial energy distribution of the transmission signal. 
     Review of OTFS Modulation 
     A modulation is a method to transmit a collection of finite symbols (which encode data) over a fixed allocation of time and frequency. A popular method used today is Orthogonal Frequency Division Multiplexing (OFDM) which transmits each finite symbol over a narrow region of time and frequency (e.g., using subcarriers and timeslots). In contrast, Orthogonal Time Frequency Space (OTFS) transmits each finite symbol over the entire allocation of time and frequency. Before going into details, we introduce terminology and notation. 
     We call the allocation of time and frequency a frame. We denote the number of subcarriers in the frame by N f . We denote the subcarrier spacing by df. We denote the number of OFDM symbols in the frame by N t . We denote the OFDM symbol duration by dt. We call a collection of possible finite symbols an alphabet, denoted by A. 
     A signal transmitted over the frame, denoted by φ, can be specified by the values it takes for each time and frequency bin: 
       φ( f,t )∈ ,  (14)
 
     for f=1, . . . , N f  and t=1, . . . , N t . 
       FIG. 51  shows an example of a frame along time (horizontal) axis and frequency (vertical) axis.  FIG. 52  shows an example of the most commonly used alphabet: Quadrature Amplitude Modulation (QAM). 
     OTFS Modulation 
     Suppose a transmitter has a collection of N f N t  QAM symbols that the transmitter wants to transmit over a frame, denoted by: 
         x ( f,t )∈ A,   (15)
 
     for f=1, . . . , N f  and t=1, . . . , N t . OFDM works by transmitting each QAM symbol over a single time frequency bin: 
       φ( f,t )= x ( f,t ),  (16a)
 
     for f=1, . . . , N f  and t=1, . . . , N t . The advantage of OFDM is its inherent parallelism, this makes many computational aspects of communication very easy to implement. The disadvantage of OFDM is fading, that is, the wireless channel can be very poor for certain time frequency bins. Performing pre-coding for these bins is very difficult. 
     The OTFS modulation is defined using the delay Doppler domain, which is relating to the standard time frequency domain by the two-dimensional Fourier transform. 
     The delay dimension is dual to the frequency dimension. There are N τ  delay bins with N τ =N f . The Doppler dimension is dual to the time dimension. There are N ν  Doppler bins with N ν =N t . 
     A signal in the delay Doppler domain, denoted by ϕ, is defined by the values it takes for each delay and Doppler bin: 
       ϕ(τ,ν)∈ ,  (16b)
 
     for τ=1, . . . , N τ  and ν=1, . . . , N ν . 
     Given a signal ϕ in the delay Doppler domain, some transmitter embodiments may apply the two-dimensional Fourier transform to define a signal φ in the time frequency domain: 
       φ( f,t )=( F ϕ)( f,t ),  (17)
 
     for f=1, . . . , N f  and t=1, . . . , N t . Where F denotes the two-dimensional Fourier transform. 
     Conversely, given a signal φ in the time frequency domain, transmitter embodiments could apply the inverse two-dimensional Fourier transform to define a signal ϕ in the delay Doppler domain: 
       ϕ(τ,ν)=( F   −1 φ)(τ,ν),  (18)
 
     for τ=1, . . . , N τ  and ν=1, . . . , N ν . 
       FIG. 53  depicts an example of the relationship between the delay Doppler and time frequency domains. 
     The advantage of OTFS is that each QAM symbol is spread evenly over the entire time frequency domain (by the two-two-dimensional Fourier transform), therefore each QAM symbol experience all the good and bad regions of the channel thus eliminating fading. The disadvantage of OTFS is that the QAM spreading adds computational complexity. 
     MMSE Channel Prediction 
     Channel prediction is performed at the hub by applying an optimization criterion, e.g., the Minimal Mean Square Error (MMSE) prediction filter to the hub&#39;s channel estimates (acquired by either implicit or explicit feedback). The MMSE filter is computed in two steps. First, the hub computes empirical estimates of the channel&#39;s second order statistics. Second, using standard estimation theory, the hub uses the second order statistics to compute the MMSE prediction filter. Before going into details, we introduce notation: 
     We denote the number of antennas at the hub by L h . We denote the number of UE antennas by L u . We index the UE antennas by u=1, . . . , L u . We denote the number frequency bins by N f . We denote the number of feedback times by n past . We denote the number of prediction times by n future .  FIG. 54  shows an example of an extrapolation process setup. 
     For each UE antenna, the channel estimates for all the frequencies, hub antennas, and feedback times can be combined to form a single N f L h n past  dimensional vector. We denote this by: 
         Ĥ   past ( u )∈   N     f     L     h     n     past   ,  (19)
 
     Likewise, the channel values for all the frequencies, hub antennas, and prediction times can be combined to form a single N f L h n future  dimensional vector. We denote this by: 
         H   future ( u )∈   N     f     L     h     n     future   ,  (20)
 
     In typical implementations, these are extremely high dimensional vectors and that in practice some form of compression should be used. For example, principal component compression may be one compression technique used. 
     Empirical Second Order Statistics 
     Empirical second order statistics are computed separately for each UE antenna in the following way: 
     At fixed times, the hub receives through feedback N samples of Ĥ past (u) and estimates of H future (u). We denote them by: Ĥ past (u) i  and Ĥ future (u) i  for i=1, . . . , N. 
     The hub computes an estimate of the covariance of Ĥ past (u), which we denote by {circumflex over (R)} past (u): 
     
       
         
           
             
               
                 
                   
                     
                       
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     The hub computes an estimate of the covariance of H future (u), which we denote by {circumflex over (R)} future (u): 
     
       
         
           
             
               
                 
                   
                     
                       
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     The hub computes an estimate of the correlation between H future (u) and Ĥ past (u), which we denote by {circumflex over (R)} past,future (u): 
     
       
         
           
             
               
                 
                   
                     
                       
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     In typical wireless scenarios (pedestrian to highway speeds) the second order statistics of the channel change slowly (on the order of 1-10 seconds). Therefore, they should be recomputed relatively infrequently. Also, in some instances it may be more efficient for the UEs to compute estimates of the second order statistics and feed these back to the hub. 
     MMSE Prediction Filter 
     Using standard estimation theory, the second order statistics can be used to compute the MMSE prediction filter for each UE antenna: 
         C ( u )= {circumflex over (R)}   future,past ( u ) {circumflex over (R)}   past   −1 ( u ),  (24)
 
     Where C(u) denotes the MMSE prediction filter. The hub can now predict the channel by applying feedback channel estimates into the MMSE filter: 
         Ĥ   future ( u )= C ( u ) Ĥ   past ( u ).  (25)
 
     Prediction Error Variance 
     We denote the MMSE prediction error by ΔH future (u), then: 
         H   future ( u )= Ĥ   future ( u )+Δ H   future ( u ).  (26)
 
     We denote the covariance of the MMSE prediction error by R error (u), with: 
         R   error ( u )= [Δ H   future ( u )Δ H   future ( u )*].  (27)
 
     Using standard estimation theory, the empirical second order statistics can be used to compute an estimate of R error (u): 
         {circumflex over (R)}   error ( u )= C ( u ) {circumflex over (R)}   past ( u ) C ( u )*− C ( u ) {circumflex over (R)}   future,past ( u )*− {circumflex over (R)}   future,past ( u ) C ( u )*+ R   future ( u )  (28)
 
     Simulation Results 
     We now present simulation results illustrating the use of the MMSE filter for channel prediction. Table 1 gives the simulation parameters and  FIG. 55  shows the extrapolation setup for this example. 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 1 
               
               
                   
                   
               
             
            
               
                   
                 Subcarrier spacing 
                 15 
                 kHz 
               
            
           
           
               
               
               
            
               
                   
                 Number of subcarriers 
                 512 
               
            
           
           
               
               
               
               
            
               
                   
                 Delay spread 
                 3 
                 μs 
               
               
                   
                 Doppler spread 
                 600 
                 Hz 
               
            
           
           
               
               
               
            
               
                   
                 Number of channel feedback estimates 
                  5 
               
            
           
           
               
               
               
               
            
               
                   
                 Spacing of channel feedback estimates 
                 10 
                 ms 
               
            
           
           
               
               
               
            
               
                   
                 Prediction range 
                 0-20 ms into the future 
               
               
                   
                   
               
            
           
         
       
     
     Fifty samples of Ĥ past  and Ĥ future  were used to compute empirical estimates of the second order statistics. The second order statistics were used to compute the MMSE prediction filter.  FIG. 56  shows the results of applying the filter. The results have shown that the prediction is excellent at predicting the channel, even 20 ms into the future. 
     Block Diagrams 
     In some embodiments, the prediction is performed independently for each UE antenna. The prediction can be separated into two steps: 
     1) Computation of the MMSE prediction filter and prediction error covariance: the computation can be performed infrequently (on the order of seconds). The computation is summarized in  FIG. 57 . Starting from left in  FIG. 57 , first, feedback channel estimates are collected. Next, the past, future and future/past correlation matrices are computed. Next the filter estimate C(u) and the error estimate are computed. 
     2) Channel prediction: is performed every time pre-coding is performed. The procedure is summarized in  FIG. 58 . 
     Optimal Precoding Filter 
     Using MMSE prediction, the hub computes an estimate of the downlink channel matrix for the allocation of time and frequency the pre-coded data will be transmitted. The estimates are then used to construct precoding filters. Precoding is performed by applying the filters on the data the hub wants the UEs to receive. Embodiments may derive the “optimal” precoding filters as follows. Before going over details we introduce notation. 
     Frame (as defined previously): precoding is performed on a fixed allocation of time and frequency, with N f  frequency bins and N t  time bins. We index the frequency bins by: f=1, . . . , N f . We index the time bins by t=1, . . . , N t . 
     Channel estimate: for each time and frequency bin the hub has an estimate of the downlink channel which we denote by Ĥ(f, t)∈   L     u     ×L     h   . 
     Error correlation: we denote the error of the channel estimates by ΔH(f, t), then: 
         H ( f,t )= Ĥ ( f,t )+Δ H ( f,t ),  (29)
 
     We denote the expected matrix correlation of the estimation error by R ΔH  (f, t)∈   L     h     ×L     h   , with: 
         R   ΔH ( f,t )= [Δ H ( f,t )*Δ H ( f,t )].  (30)
 
     The hub can be easily compute these using the prediction error covariance matrices computed previously: {circumflex over (R)} error (u) for u=1, . . . , L u . 
     Signal: for each time and frequency bin the UE wants to transmit a signal to the UEs which we denote by s(f, t)∈   L     u   . 
     Precoding filter: for each time and frequency bin the hub uses the channel estimate to construct a precoding filter which we denote by W(f, t)∈   L     h     ×L     u   . 
     White noise: for each time and frequency bin the UEs experience white noise which we denote by n(f, t)∈   L     u   . We assume the white noise is iid Gaussian with mean zero and variance N 0 . 
     Hub Energy Constraint 
     When the precoder filter is applied to data, the hub power constraint may be considered. We assume that the total hub transmit energy cannot exceed N f N t L h . Consider the pre-coded data: 
         W ( f,t ) s ( f,t ),  (31)
 
     To ensure that the pre-coded data meets the hub energy constraints the hub applies normalization, transmitting: 
       λ W ( f,t ) s ( f,t ),  (32)
 
     Where the normalization constant A is given by: 
     
       
         
           
             
               
                 
                   λ 
                   = 
                   
                     
                       
                         
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                   33 
                   ) 
                 
               
             
           
         
       
     
     Receiver SINR 
     The pre-coded data then passes through the downlink channel, the UEs receive the following signal: 
       λ H ( f,t ) W ( f,t ) s ( f,t )+ n ( f,t ),  (34)
 
     The UEs then removes the normalization constant, giving a soft estimate of the signal: 
     
       
         
           
             
               
                 
                   
                     
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                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
     The error of the estimate is given by: 
     
       
         
           
             
               
                 
                   
                     
                       
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                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
     The error can be decomposed into two independent terms: interference and noise. Embodiments can compute the total expected error energy: 
     
       
         
           
             
               
                 
                   
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                     ⁢ 
                     
                         
                     
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                     ⁢ 
                     
                         
                     
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                   37 
                   ) 
                 
               
             
           
         
       
     
     Optimal Precoding Filter 
     We note that the expected error energy is convex and quadratic with respect to the coefficients of the precoding filter. Therefore, calculus can be used to derive the optimal precoding filter: 
     
       
         
           
             
               
                 
                   
                     
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                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
     Accordingly, some embodiments of an OTFS precoding system use this filter (or an estimate thereof) for precoding. 
     Simulation Results 
     We now present a simulation result illustrating the use of the optimal precoding filter. The simulation scenario was a hub transmitting data to a single UE. The channel was non line of sight, with two reflector clusters: one cluster consisted of static reflectors, the other cluster consisted of moving reflectors.  FIG. 59  illustrates the channel geometry, with horizontal and vertical axis in units of distance. It is assumed that the hub has good Channel Side Information (CSI) regarding the static cluster and poor CSI regarding the dynamic cluster. The optimal precoding filter was compared to the MMSE precoding filter.  FIG. 60A  displays the antenna pattern given by the MMSE precoding filter. It can be seen that the energy is concentrated at ±45°, that is, towards the two clusters. The UE SINR is 15.9 dB, the SINR is relatively low due to the hub&#39;s poor CSI for the dynamic cluster. 
       FIG. 60B  displays the antenna pattern given by the optimal precoding filter as described above, e.g., using equation (38). In this example, the energy is concentrated at −45°, that is, toward the static cluster. The UE SINR is 45.3 dB, the SINR is high (compared to the MMSE case) due to the hub having good CSI for the static reflector. 
     The simulation results depicted in  FIGS. 34A and 34B  illustrate the advantage of the optimal pre-coding filter. The filter it is able to avoid sending energy towards spatial regions of poor channel CSI, e.g., moving regions. 
     Example Block Diagrams 
     Precoding is performed independently for each time frequency bin. The precoding can be separated into three steps: 
     [1] Computation of error correlation: the computation be performed infrequently (on the order of seconds). The computation is summarized in  FIG. 61 . 
     [2] Computation of optimal precoding filter: may be performed every time pre-coding is performed. The computation is summarized in  FIG. 62 . 
     [3] Application of the optimal precoding filter: may be performed every time pre-coding is performed. The procedure is summarized in  FIG. 63 . 
     OTFS Vector Perturbation 
     Before introducing the concept of vector perturbation, we outline the application of the optimal pre-coding filter to OTFS. 
     OTFS Optimal Precoding 
     In OTFS, the data to be transmitted to the UEs are encoded using QAMs in the delay-Doppler domain. We denote this QAM signal by x, then: 
         x (τ,ν)∈ A   L     u   ,  (39)
 
     for τ=1, . . . , N τ  and ν=1, . . . , N ν . A denotes the QAM constellation. Using the two-dimensional Fourier transform the signal can be represented in the time frequency domain. We denote this representation by X: 
         X ( f,t )=( Fx )( f,t ),  (40)
 
     for f=1, . . . , N f  and t=1, . . . , N t . F denotes the two-dimensional Fourier transform. The hub applies the optimal pre-coding filter to X and transmit the filter output over the air: 
       λ W   opt ( f,t ) X ( f,t )  (41)
 
     for f=1, . . . , N f  and t=1, . . . , N t . λ denotes the normalization constant. The UEs remove the normalization constant giving a soft estimate of X: 
     
       
         
           
             
               
                 
                   
                     
                       
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                         soft 
                       
                       ⁡ 
                       
                         ( 
                         
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                           , 
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                         ) 
                       
                     
                     = 
                     
                       
                         
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                           ⁡ 
                           
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                           ⁡ 
                           
                             ( 
                             
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                       + 
                       
                         
                           1 
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                         ⁢ 
                         
                           w 
                           ⁡ 
                           
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                               f 
                               , 
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                   , 
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
           
         
       
     
     for f=1, . . . , N f  and t=1, . . . , N t . The term w(f, t) denotes white noise. We denote the error of the soft estimate by E: 
         E ( f,t )= X   soft ( f,t )− X ( f,t ),  (43)
 
     for f=1, . . . , N f  and t=1, . . . , N t . The expected error energy was derived earlier in this document: 
     
       
         
           
             
               
                 
                   
                     expected 
                     ⁢ 
                     
                         
                     
                     ⁢ 
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                     ⁢ 
                     
                         
                     
                     ⁢ 
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                   = 
                   
                     
                       
                         ∑ 
                         
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                           = 
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                         f 
                       
                       ⁢ 
                       
                         
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                               error 
                             
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                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
     Where: 
     
       
         
           
             
               
                 
                   
                     
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                     ⁡ 
                     
                       ( 
                       
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                         , 
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                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
     We call the positive definite matrix M error (f, t) the error metric. 
     Vector Perturbation 
     In vector perturbation, the hub transmits a perturbed version of the QAM signal: 
         x (τ,ν)+ p (τ,ν),  (46)
 
     for τ=1, . . . , N τ  and ν=1, . . . , N ν . Here, p(τ,ν) denotes the perturbation signal. The perturbed QAMs can be represented in the time frequency domain: 
         X ( f,t )+ P ( f,t )=( Fx )( f,t )+( Fp )( f,t ),  (47)
 
     for f=1, . . . , N f  and t=1, . . . , N t . The hub applies the optimal pre-coding filter to the perturbed signal and transmits the result over the air. The UEs remove the normalization constant giving a soft estimate of the perturbed signal: 
         X ( f,t )+ P ( f,t )+ E ( f,t ),  (48)
 
     for f=1, . . . , N f  and t=1, . . . , N t . Where E denotes the error of the soft estimate. The expected energy of the error is given by: 
       expected error energy=Σ f=1   N     f   Σ t=1   N     t   ( X ( f,t )+ P ( f,t ))* M   error ( f,t )( X ( f,t )+ P ( f,t ))  (49)
 
     The UEs then apply an inverse two dimensional Fourier transform to convert the soft estimate to the delay Doppler domain: 
         x (τ,ν)+ p (τ,ν)+ e (τ,ν),  (50)
 
     for τ=1, . . . , N τ  and ν=1, . . . , N ν . The UEs then remove the perturbation p(τ, ν) for each delay Doppler bin to recover the QAM signal x. 
     Collection of Vector Perturbation Signals 
     One question is: what collection of perturbation signals should be allowed? When making this decision, there are two conflicting criteria: 
     1) The collection of perturbation signals should be large so that the expected error energy can be greatly reduced. 
     2) The collection of perturbation signals should be small so the UE can easily remove them (reduced computational complexity): 
         x (τ,ν)+ p (τ,ν)→ x (τ,ν)  (51)
 
     Coarse Lattice Perturbation 
     An effective family of perturbation signals in the delay-Doppler domain, which take values in a coarse lattice: 
         p (τ,ν)∈ B   L     u   ,  (52)
 
     for τ=1, . . . , N τ  and ν=1, . . . , N ν . Here, B denotes the coarse lattice. Specifically, if the QAM symbols lie in the box: [−r, r]×j[−r, r] we take as our perturbation lattice B=2r +2rj . We now illustrate coarse lattice perturbation with an example. 
     EXAMPLES 
     Consider QPSK (or 4-QAM) symbols in the box [−2,2]×j[−2,2]. The perturbation lattice is then B=4 +4j .  FIG. 64  illustrates the symbols and the lattice. Suppose the hub wants to transmit the QPSK symbol 1+1j to a UE. Then there is an infinite number of coarse perturbations of 1+1j that the hub can transmit.  FIG. 65  illustrates an example. The hub selects one of the possible perturbations and transmits it over the air.  FIG. 66  illustrates the chosen perturbed symbol, depicted with a single solid circle. 
     The UE receives the perturbed QPSK symbol. The UE then removes the perturbation to recover the QPSK symbol. To do this, the UE first searches for the coarse lattice point closest to the received signal.  FIG. 67  illustrates this. 
     The UE subtracts the closest lattice point from the received signal, thus recovering the QPSK symbol 1+1j.  FIG. 68  illustrates this process. 
     Finding Optimal Coarse Lattice Perturbation Signal 
     The optimal coarse lattice perturbation signal, p opt , is the one which minimizes the expected error energy: 
         p   opt =argmin p Σ f=1   N     f   Σ t=1   N     t   ( X ( f,t )+ P ( f,t ))* M   error ( f,t )( X ( f,t )+ P ( f,t ))  (53)
 
     The optimal coarse lattice perturbation signal can be computed using different methods. A computationally efficient method is a form of Thomlinson-Harashima precoding which involves applying a DFE filter at the hub. 
     Coarse Lattice Perturbation Example 
     We now present a simulation result illustrating the use of coarse lattice perturbation. The simulation scenario was a hub antenna transmitting to a single UE antenna. Table 2 displays the modulation parameters. Table 3 display the channel parameters for this example. 
     
       
         
           
               
               
               
             
               
                   
                 TABLE 2 
               
               
                   
                   
               
             
            
               
                   
                 Subcarrier spacing 
                 30 kHz 
               
               
                   
                 Number of subcarriers 
                 256 
               
               
                   
                 OFDM symbols per frame 
                  32 
               
               
                   
                 QAM order 
                 Infinity (uniform in the unit box) 
               
               
                   
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
             
               
                   
                 TABLE 3 
               
               
                   
                   
               
             
            
               
                   
                 Number of reflectors 
                 20 
               
            
           
           
               
               
               
               
            
               
                   
                 Delay spread 
                 2 
                 μs 
               
               
                   
                 Doppler spread 
                 1 
                 KHz 
               
               
                   
                 Noise variance 
                 −35 
                 dB 
               
               
                   
                   
               
            
           
         
       
     
       FIG. 69  displays the channel energy in the time (horizontal axis) and frequency (vertical axis) domain. 
     Because this is a SISO (single input single output) channel, the error metric M error (f, t) is a positive scaler for each time frequency bin. The expected error energy is given by integrating the product of the error metric with the perturbed signal energy: 
       expected error energy=Σ f=1   N     f   Σ t=1   N     t     M   error ( f,t )| X ( f,t )+ P ( f,t )| 2   (54)
 
       FIG. 70  displays an example of the error metric. One hundred thousand random QAM signals were generated. For each QAM signal, the corresponding optimal perturbation signal was computed using Thomlinson-Harashima precoding.  FIG. 71  compares the average energy of the QAM signals with the average energy of the perturbed QAM signals. The energy of QAM signals is white (evenly distributed) while the energy of the perturbed QAM signals is colored (strong in some time frequency regions and weak in others). The average error energy of the unperturbed QAM signal was −24.8 dB. The average error energy of the perturbed QAM signal was −30.3 dB. The improvement in error energy can be explained by comparing the energy distribution of the perturbed QAM signal with the error metric. 
       FIG. 72  shows a comparison of an example error metric with an average perturbed QAM energy. The perturbed QAM signal has high energy where the error metric is low, conversely it has low energy where the error metric is high. 
     The simulation illustrates the gain from using vector perturbation: shaping the energy of the signal to avoid time frequency regions where the error metric is high. 
     Block Diagrams 
     Vector perturbations may be performed in three steps. First, the hub perturbs the QAM signal. Next, the perturbed signal is transmitted over the air using the pre-coding filters. Finally, the UEs remove the perturbation to recover the data. 
     Computation of error metric: the computation can be performed independently for each time frequency bin. The computation is summarized in  FIG. 73 . See also Eq. (45). As shown, the error metric is calculated using channel prediction estimate, the optimal coding filter and error correlation estimate. 
     Computation of perturbation: the perturbation is performed on the entire delay Doppler signal. The computation is summarized in  FIG. 74 . As shown, the QAM signal and the error metric are used to compute the perturbation signal. The calculated perturbation signal is additively applied to the QAM input signal. 
     Application of the optimal precoding filter: the computation can be performed independently for each time frequency bin. The computation is summarized in  FIG. 75 . The perturbed QAM signal is processed through a two dimensional Fourier transform to generate a 2D transformed perturbed signal. The optimal precoding filter is applied to the 2D transformed perturbed signal. 
     UEs removes perturbation: the computation can be  FIG. 76 . At UE, the input signal received is transformed through an inverse 2D Fourier transform. The closest lattice point for the resulting transformed signal is determined and then removed from the 2D transformed perturbed signal. 
     Spatial Tomlinson Harashima Precoding 
     This section provides additional details of achieving spatial precoding and the beneficial aspects of using Tomlinson Harashima precoding algorithm in implementing spatial precoding in the delay Doppler domain. The embodiments consider a flat channel (with no frequency or time selectivity). 
     Review of Linear Precoding 
     In precoding, the hub wants to transmit a vector of QAMs to the UEs. We denote this vector by x∈   L     u   . The hub has access to the following information: 
     An estimate of the downlink channel, denoted by: Ĥ∈   L     u     ×L     h   . 
     The matrix covariance of the channel estimation error, denoted by: R ΔH ∈   L     h     ×L     h   . 
     From this information, the hub computes the “optimal” precoding filter, which minimizes the expected error energy experienced by the UEs: 
     
       
         
           
             
               W 
               opt 
             
             = 
             
               
                 
                   ( 
                   
                     
                       
                         
                           H 
                           ^ 
                         
                         * 
                       
                       ⁢ 
                       
                         H 
                         ^ 
                       
                     
                     + 
                     
                       R 
                       
                         Δ 
                         ⁢ 
                         H 
                       
                     
                     + 
                     
                       
                         
                           
                             N 
                             0 
                           
                           ⁢ 
                           
                             L 
                             u 
                           
                         
                         
                           L 
                           h 
                         
                       
                       ⁢ 
                       I 
                     
                   
                   ) 
                 
                 
                   - 
                   1 
                 
               
               ⁢ 
               
                 
                   H 
                   ^ 
                 
                 * 
               
             
           
         
       
     
     By applying the precoding filter to the QAM vector the hub constructs a signal to transmit over the air: λW opt x∈   L     h   , where λ is a constant used to enforce the transmit energy constraints. The signal passes through the downlink channel and is received by the UEs: 
       λ HW   opt   x+w,  
 
     Where w∈   L     u    denotes AWGN noise. The UEs remove the normalization constant giving a soft estimate of the QAM signal: 
     
       
      
       x+e,  
      
     
     where e∈   L     u    denotes the estimate error. The expected error energy can be computed using the error metric: 
       expected error energy= x*M   error   x    
     where M error  is a positive definite matrix computed by: 
     
       
         
           
             
               M 
               error 
             
             = 
             
               
                 
                   
                     ( 
                     
                       
                         
                           H 
                           ^ 
                         
                         ⁢ 
                         
                           W 
                           opt 
                         
                       
                       - 
                       I 
                     
                     ) 
                   
                   * 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       
                         H 
                         ^ 
                       
                       ⁢ 
                       
                         W 
                         opt 
                       
                     
                     - 
                     I 
                   
                   ) 
                 
               
               + 
               
                 
                   
                     W 
                     opt 
                     * 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         R 
                         
                           Δ 
                           ⁢ 
                           H 
                         
                       
                       + 
                       
                         
                           
                             N 
                             0 
                           
                           ⁢ 
                           
                             L 
                             u 
                           
                         
                         
                           L 
                           h 
                         
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   W 
                   opt 
                 
               
             
           
         
       
     
     Review of Vector Perturbation 
     The expected error energy can be greatly reduced by perturbing the QAM signal by a vector v∈   L     u   . The hub now transmits λW opt (x+v)∈   L     h   . After removing the normalization constant, the UEs have a soft estimate of the perturbed QAM signal: 
     
       
      
       x+v+e  
      
     
     Again, the expected error energy can be computed using the error metric: 
       expected error energy=( x+v )* M   error ( x+v ) 
     The optimal perturbation vector minimizes the expected error energy: 
         v   opt =argmin v ( x+v )* M   error ( x+v ). 
     Computing the optimal perturbation vector is in general NP-hard, therefore, in practice an approximation of the optimal perturbation is computed instead. For the remainder of the document we assume the following signal and perturbation structure: 
     The QAMs lie in the box [−1,1]×j[−1,1]. 
     The perturbation vectors lie on the coarse lattice: (2 +2j ) L     u   . 
     Spatial Tomlinson Harashima Precoding 
     In spatial THP a filter is used to compute a “good” perturbation vector. To this end, we make use of the Cholesky decomposition of the positive definite matrix M error : 
     
       
      
       M 
       error 
       =U*DU,  
      
     
     where D is a diagonal matrix with positive entries and U is unit upper triangular. Using this decomposition, the expected error energy can be expressed as: 
       expected error energy=( U ( x+v ))* D ( U ( x+v ))= z*Dz=Σ   n=1   L     u     D ( n,n )| z ( n )| 2 , 
     where z=U(x+v). We note that minimizing the expected error energy is equivalent to minimizing the energy of the z entries, where: 
         z ( L   u )= x ( L   u )+ v ( L   u ), 
         z ( n )= x ( n )+ v ( n )+Σ m=n+1   L     u     U ( n,m )( x ( m )+ v ( m )),
 
     for n=1, 2, . . . , L u −1. Spatial THP iteratively choses a perturbation vector in the following way. 
         v ( L   u )=0 
     Suppose v(n+1), v(n+2), . . . , v(L u ) have been chosen, then: 
         v ( n )=− ( x ( n )+Σ m=n+1   L     u     U ( n,m )( x ( m )+ v ( m )))
 
     where  denotes projection onto the coarse lattice. We note that by construction the coarse perturbation vector bounds the energy of the entries of z by two.  FIG. 77  displays a block diagram of spatial THP. 
     Simulation Results 
     We now present the results of a simple simulation to illustrate the use of spatial THP. Table 4 summarizes the simulation setup. 
     
       
         
           
               
             
               
                 TABLE 4 
               
               
                   
               
               
                 Simulation setup 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                   
                 Number of hub antennas 
                 2 
               
               
                   
                 Number of UEs 
                 2 (one antenna each) 
               
               
                   
                 Channel condition number 
                  10 dB 
               
               
                   
                 Modulation 
                 PAM infinity (data uniformly 
               
               
                   
                   
                 disturbed on the interval [−1, 1]) 
               
               
                   
                 Data noise variance 
                 −35 dB 
               
               
                   
                 Channel noise variance 
                 −35 dB 
               
               
                   
                   
               
            
           
         
       
     
       FIG. 78  displays the expected error energy for different PAM vectors. We note two aspects of the figure. 
     The error energy is low when the signal transmitted to UE 1  and UE 2  are similar. Conversely, the error energy is high when the signals transmitted to the UEs are dissimilar. We can expect this pattern to appear when two UEs are spatially close together; in these situations, it is advantageous to transmit the same message to both UEs. 
     The error energy has the shape of an ellipses. The axes of the ellipse are defined by the eigenvectors of M error . 
     A large number data of PAM vectors was generated and spatial THP was applied.  FIG. 79  shows the result. Note that the perturbed PAM vectors are clustered along the axis with low expected error energy. 
     5. Channel Estimation for OTFS Systems 
     This section overviews channel estimation for OTFS systems, and in particular, aspects of channel estimation and scheduling for a massive number of users. A wireless system, with a multi-antenna base-station and multiple user antennas, is shown in  FIG. 80 . Each transmission from a user antenna to one of the base-station antennas (or vice versa), experiences a different channel response (assuming the antennas are physically separated enough). For efficient communication, the base-station improves the users&#39; received Signal-to-Interference-Noise-Ratio (SINR) by means of precoding. However, to precode, the base-station needs to have an accurate estimation of the downlink channels to the users during the transmission time. 
     In some embodiments, and when the channels are not static and when the number of users is very large, some of the challenges of such a precoded system include:
         Accurately and efficiently estimating all the required channels   Predicting the changes in the channels during the downlink transmission time       

     Typical solutions in systems, which assume a low number of users and static channels, are to let each user transmit known pilot symbols (reference signals) from each one of its antennas. These pilots are received by all the base-station antennas and used to estimate the channel. It is important that these pilot symbols do not experience significant interference, so that the channel estimation quality is high. For this reason, they are typically sent in an orthogonal way to other transmissions at the same time. There are different methods for packing multiple pilots in an orthogonal (or nearly-orthogonal) way, but these methods are usually limited by the number of pilots that can be packed together (depending on the channel conditions) without causing significant interference to each other. Therefore, it becomes very difficult to have an efficient system, when the number of user antennas is high and the channels are not static. The amount of transmission resources that is needed for uplink pilots may take a considerable amount of the system&#39;s capacity or even make it unimplementable. For prediction of the channel, it is typically assumed that the channel is completely static and will not change from the time it was estimated till the end of the downlink transmission. This assumption usually causes significant degradation in non-static channels. 
     It is assumed that the downlink and uplink channels are reciprocal and after calibration it is possible to compensate for the difference in the uplink-downlink and downlink-uplink channel responses. Some example embodiments of the calibration process using reciprocity are further discussed in Section 5. 
     Embodiments of the disclosed technology include a system and a method for packing and separating multiple non-orthogonal pilots, as well as a method for channel prediction. In such a system, it is possible to pack together a considerably higher number of pilots comparing to other commonly used methods, thus allowing an accurate prediction of the channel for precoding. 
     Second-Order Training Statistics 
     The system consists of a preliminary training step, in which all users send uplink orthogonal pilots to the base-station. Although these pilots are orthogonal, they may be sent at a very low rate (such as one every second) and therefore do not overload the system too much. The base-station receives a multiple of N SOS  such transmissions of these pilots, and use them to compute the second-order statistics (covariance) of each channel. 
       FIG. 81  shows an example of such a system, where a subframe of length 1 msec consists of a downlink portion (DL), a guard period (GP) and an uplink portion (UL). Some of the uplink portion is dedicated to orthogonal pilots (OP) and non-orthogonal pilots (NOP). Each specific user is scheduled to send on these resources its pilots every 1000 subframes, which are equivalent to 1 sec. After the reception of N SOS  subframes with pilots (equivalent to N SOS  seconds), the base-station will compute the second-order statistics of this channel. 
     The computation of the second-order statistics for a user antenna u is defined as:
         For each received subframe i=1, 2, . . . , N SOS  with orthogonal pilots and for each one of the L base-station receive antennas-estimate the channel along the entire frequency band (N f  grid elements) from the pilots and store it as the i-th column of the matrix H (u)  with dimensions (N f ·L)×N SOS      Compute the covariance matrix R HH   (u) =(H (u) ) H H (u) , where (·) H  is the Hermitian operator.   For the case that the channel H (u)  is non-zero-mean, both the mean and the covariance matrix should be determined.       

     To accommodate for possible future changes in the channel response, the second-order statistics may be updated later, after the training step is completed. It may be recomputed from scratch by sending again N SOS  orthogonal pilots, or gradually updated. One possible method may be to remove the first column of H (u)  and attach a new column at the end and then re-compute the covariance matrix again. 
     The interval at which these orthogonal pilots need to be repeated depends on the stationarity time of the channel, e.g., the time during which the second-order statistics stay approximately constant. This time can be chosen either to be a system-determined constant, or can be adapted to the environment. In particular, users can determine through observation of downlink broadcast pilot symbols changes in the second-order statistics, and request resources for transmission of the uplink pilots when a significant change has been observed. In another embodiment, the base-station may use the frequency of retransmission requests from the users to detect changes in the channel, and restart the process of computing the second-order statistics of the channel. 
     To reduce the computational load, it is possible to use principal component analysis (PCA) techniques on R HH   (u) . We compute {λ (u) }, the K (u)  most dominant eigenvalues of R HH   (u) , arranged in a diagonal matrix D (u) =diag(λ 1   (u) , λ 2   (u) , . . . , λ K     (u)     (u) ) and their corresponding eigenvectors matrix V (u) . Typically, K (u)  will be in the order of the number of reflectors along the wireless path. The covariance matrix can then be approximated by R HH   (u) ≈V (u) ·D (u) ·(V (u) ) H . 
     Non-Orthogonal Pilots 
     The non-orthogonal pilots (NOP), P (u) , for user antenna u, may be defined as a pseudo-random sequence of known symbols and of size N NOP , over a set of frequency grid elements. The base-station can schedule many users to transmit their non-orthogonal pilots at the same subframe using overlapping time and frequency resources. The base-station will be able to separate these pilots and obtain a high-quality channel estimation for all the users, using the method describes below. 
     Define the vector Y of size (L·N NOP )×1, as the base-station received signal over all its antennas, at the frequency grid elements of the shared non-orthogonal pilots. Let {tilde over (V)} (u)  be the eigenvectors matrix V (u)  decimated along its first dimension (frequency-space) to the locations of the non-orthogonal pilots. 
     The base-station may apply a Minimum-Mean-Square-Error (MMSE) estimator to separate the pilots of every user antenna:
         For every user antenna u, compute       

         R   YY   (u) =[ {tilde over (V)}   (u)   ⊙P   (u) ]· D   (u) ·[ {tilde over (V)}   (u)   ⊙P   (u) ] H  
 
         R   XY   (u)   ={tilde over (V)}   (u)   ·D   (u) ·[ {tilde over (V)}   (u)   ⊙P   (u) ] H  
 
     Herein, ⊙ is defined as the element-by-element multiplication. For a matrix A and vector B, the A⊙B operation includes replicating the vector B to match the size of the matrix A before applying the element-by-element multiplication. 
     If principal component analysis (PCA) is not used, the covariance matrices can be computed directly as: 
         R   YY   (u) =( P   (u) [ P   (u) ] H )⊙ R   HH   (u)  
 
         R   XY   (u) =(1[ P   (u) ] H ) ⊙R   HH   (u)            For the set of user antennas shared on the same resources u∈U, compute       
         R   YY =Σ u∈U   R   YY   (u)  
 
     and invert it. Note that it is possible to apply PCA here as well by finding the dominant eigenvalues of R YY (D R     YY   ) and their corresponding eigenvectors matrix (V R     YY   ) and approximating the inverse with R YY   −1 ≈V R     YY   ·D R     YY     −1 ·(V R     YY   ) H .
         For each user antenna u, compute the pilot separation filter       

     
       
      
       C 
       P 
       (u) 
       =R 
       XY 
       (u) 
       ·R 
       YY 
       −1  
      
         
         
           
             For each user antenna u, separate its non-orthogonal pilots by computing 
           
         
       
    
     
       
      
       H 
       NOP 
       (u) 
       =C 
       P 
       (u) 
       ·Y  
      
     
     Note that H NOP   (u)  is the channel response over the frequency grid-elements of the non-orthogonal pilots for the L base-station received antennas. It may be also interpolated along frequency to obtain the channel response over the entire bandwidth. 
     Prediction Training 
     The method described in the previous section for separating non-orthogonal pilots is applied to train different users for prediction. In this step, a user sends uplink non-orthogonal pilots on consecutive subframes, which are divided to 3 different sections, as shown in the example in  FIG. 82 . 
     1. Past—the first N past  subframes. These subframes will later be used to predict future subframes. 
     2. Latency—the following N latency  subframes are used for the latency required for prediction and precoding computations. 
     3. Future—the last N future  subframes (typically one), where the channel at the downlink portion will be later predicted. 
     Each user, is scheduled N PR  times to send uplink non-orthogonal pilots on consecutive N past +N latency +N future  subframes. Note that in one uplink symbol in the subframe, both orthogonal and non-orthogonal pilots may be packed together (although the number of orthogonal pilots will be significantly lower than the number of non-orthogonal pilots). The base-station applies the pilot separation filter for the non-orthogonal pilots of each user and computes H NOP   (u) . To reduce storage and computation, the channel response may be compressed using the eigenvector matrix computed in the second-order statistics step 
         H   K   (u) =( {tilde over (V)}   (u) ) H   ·H   NOP   (u)    
     For subframes, which are part of the “Past” section, store H K   (u)  as columns in the matrix H past,(i)   (u) , where i=1, 2, . . . , N PR . Use all or part of the non-orthogonal pilots to interpolate the channel over the whole or part of the downlink portion of the “Future” subframes, compress it using {tilde over (V)} (u)  and store it as H future,(i)   (u) . Compute the following covariance matrices: 
         R   past,(i)   (u)   =H   past,(i)   (u) ·( H   past,(i)   (u) ) H  
 
         R   future,(i)   (u)   =H   future,(i)   (u) ·( H   future,(i)   (u) ) H  
 
         R   future_past,(i)   (u)   =H   future,(i)   (u) ·( H   past,(i)   (u) ) H  
 
     After all N PR  groups of prediction training subframes have been scheduled, compute the average covariance matrices for each user 
     
       
         
           
             
               R 
               past 
               
                 ( 
                 u 
                 ) 
               
             
             = 
             
               
                 1 
                 
                   N 
                   PR 
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   
                     N 
                     PR 
                   
                 
                 ⁢ 
                 
                   R 
                   
                     past 
                     , 
                     
                       ( 
                       i 
                       ) 
                     
                   
                   
                     ( 
                     u 
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               R 
               future 
               
                 ( 
                 u 
                 ) 
               
             
             = 
             
               
                 1 
                 
                   N 
                   PR 
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   
                     N 
                     PR 
                   
                 
                 ⁢ 
                 
                   R 
                   
                     future 
                     , 
                     
                       ( 
                       i 
                       ) 
                     
                   
                   
                     ( 
                     u 
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               R 
               future_past 
               
                 ( 
                 u 
                 ) 
               
             
             = 
             
               
                 1 
                 
                   N 
                   PR 
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   
                     N 
                     PR 
                   
                 
                 ⁢ 
                 
                   R 
                   
                     future_past 
                     , 
                     
                       ( 
                       i 
                       ) 
                     
                   
                   
                     ( 
                     u 
                     ) 
                   
                 
               
             
           
         
       
     
     Finally, for each user compute the MMSE prediction filter 
         C   PR   (u)   =R   future_past   (u) ·( R   past   (u) ) −1  
 
     and its error variance for the precoder 
         R   E   (u)   =R   future   (u)   −C   PR   (u) ·( R   future_past   (u) ) H .
 
     Scheduling a Downlink Precoded Transmission 
     For each subframe with a precoded downlink transmission, the base-station should schedule all the users of that transmission to send uplink non-orthogonal pilots for N past  consecutive subframes, starting N past +N latency  subframes before it, as shown in FIG.  83 . The base-station will separate the non-orthogonal pilots of each user, compress it and store the channel response as H K,past   (u) . Then, it will apply the prediction filter to get the compressed channel response for the future part 
     
       
      
       H 
       K,future 
       (u) 
       =C 
       PR 
       (u) 
       ·H 
       K,past 
       (u)  
      
     
     Finally, the uncompressed channel response is computed as 
     
       
      
       H 
       future 
       (u) 
       ={tilde over (V)} 
       (u) 
       ·H 
       K,future 
       (u)  
      
     
     The base-station may correct for differences in the reciprocal channel by applying a phase and amplitude correction, α(f), for each frequency grid-element 
         H   future_reciprocity   (u) ( f )=α( f )· H   future   (u) ( f )
 
     Then, use H future_reciprocity   (u)  and R E   (u)  of the participating users to compute the precoder for the downlink transmission. 
     Scheduling of the Uplink Pilots 
     If during a frame there are multiple orthogonal resources available for pilot transmission (e.g., different timeslots or different frequency grid elements), then the set of uplink pilots that needs to be transmitted can be divided into sets such that each set is transmitted on a different resource. The criterion of for the division into sets can be, e.g., the achievable pilot SINR. The transmission of non-orthogonal pilots leads to a reduction in the achievable pilot SINR, which is the more pronounced the stronger the alignment of the vector spaces containing the correlation matrices from different users is. Thus, arranging users in sets such that two pilots with very similar correlation matrices are not transmitted at the same time improves performance. However, other criteria are possible as well. For example, for users that have only a low SINR during data transmission, achieving a high pilot SINR might be wasteful; thus, achieving an optimal “matching” of the pilot SINR to the data SINR might be another possible criterion. 
     The embodiments of the disclosed technology described in this section may be characterized, but not limited, by the following features:
         A wireless system in which a network node performs precoded downlink transmissions, which support a massive number of users, consisting of channel prediction, reciprocity adjustment and precoding, based on the second-order statistics of the channels.   A system including a mix of uplink orthogonal pilots and non-orthogonal pilots.   Computing the second-order statistics of a channel based on orthogonal pilots.   Separating non-orthogonal pilots from multiple users, using second-order statistics and computation of channel estimation.   Training for prediction of channel estimates.   Scheduling non-orthogonal uplink pilots based on second-order statistics.   Compressing channel responses using PCA       

     6. Pilot Scheduling to Reduce Transmission Overhead 
     This section covers scheduling pilots to reduce transmission overhead and improve the throughput of a wireless communication system. One possible FWA system design is based on separating users based on their angular power spectra. For example, users can operate in parallel if they do not create “significant” interference in each other&#39;s “beams.” A beam may for example be a Luneburg beam. A precoding vector can also be associated with a beam pattern. However, for ease of explanation, the word “precoder pattern” is used in the present description. Consider as an example a system with 8 beams in a 90-degree sector, such that any two adjacent beams have overlapping beam patterns, while beams whose difference of indices is at least 2 are orthogonal to each other. If there is a pure line of sight (LoS), or a small angular spread around the LoS direction, then a spatial reuse factor of 2 may be possible. For example, beams 1, 3, 5, and 7 can operate in parallel (and similarly beam 2, 4, 6, 8). However, most channels provide a larger angular spread than can be handled by such a configuration, so that only beams with a wider angular separation may use the same time/frequency resources; e.g., a reuse factor on the order of 4 may be achieved. This means that only 2 users can operate on the same time-frequency resources within one sector, so that the overall performance gain compared to traditional systems is somewhat limited. 
     Considerably better spatial reuse can be achieved when the user separation is based on instantaneous channel state information, using joint receive processing of the multiple beam signals, and joint precoding, for the uplink and downlink, respectively. To take the example of the uplink, with N antenna (beam) ports, N signals can be separated, so that N users can be active at the same time (and analogously for the downlink). The simplest way to achieve this is zero-forcing, though it may suffer from poor performance in particular if users are close together (in mathematical terms, this occurs if their channel vectors are nearly linearly dependent). More sophisticated techniques, such as turbo equalization in the uplink, and Tomlinson-Harashima Precoding (THP) in the downlink can improve the performance further. Such implementations can increase signal to interference plus noise ratio (SINR) for the users, though they may not increase the degrees of freedom. 
     However, while these methods have great advantages, they rely on the knowledge of the instantaneous channel state information (CSI) for the processing, while the beam-based transmission can be performed simply by the time-averaged (for FWA) or second order (for mobile) systems CSI. The problem is aggravated by two facts: 
     1) while N users can be served in parallel (since they are separated by their different instantaneous CSI), the pilots cannot be separated this way (because the CSI is not yet known when the pilots are transmitted—it is a “chicken and egg” problem). Thus, pilots can be separated based on their average or second-order statistics. 
     2) OTFS modulation may have a higher pilot overhead compared to, e.g., OFDMA, because of the spreading of the information over the whole time-frequency plane, such that each user attempts to determine the CSI for the whole bandwidth. 
     Example System Model and Basic Analysis 
     A. Assumptions for the Analysis 
     An example system is described and for ease of explanation, the following assumptions are made: 
     1) Luneburg lens with 8 beams. Adjacent beams have overlap, beams separated by at least 1 other beam have a pattern overlap separation of better than 30 dB. However, in general, any number of beams may be used. 
     2) For the uplink, no use of continuous pilots. Channels might be estimated either based on the pilots embedded in the data packets. Alternatively, placing a packet in a queue for, say 4 ms, to allow transmission of uplink pilots before the transmission of data can improve channel estimation performance. 
     3) For the downlink, every UE observes broadcast pilots, which, in this example, are sent periodically or continuously, and extrapolates the channel for the next downlink frame. It then might send this information, in quantized form, to the BS (for the case that explicit channel state feedback is used). 
     4) The discussion here only considers the basic degrees of freedom for the pilot tones, not the details of overhead associated with delay-Doppler versus time-frequency multiplexing. In some implementations, both may give approximately the same overhead. 
     5) A frame structure with 1 ms frame duration is used. Different users may transmit in different frames. It is assumed that in the uplink and for the precoded pilots of the downlink, two pilots are transmitted per user, one at the beginning of the frame, and one at the end of the frame, so that interpolation can be done. For the broadcast pilots in the downlink, this may not be done, since it will be transmitted once per frame anyway, so that interpolation and extrapolation is implicitly possible. 
     6) A system bandwidth of 10 MHz is assumed. 
     B. Efficiency of an Example System 
     The following presents a first example calculation of the pilot overhead when the pilots in all beams are kept completely orthogonal. For the example, first compute the degrees of freedom for the pilot for each user. With 10 MHz bandwidth and 1 ms frame duration, and two polarizations, there are in general 10,000 “resolvable bins” (degrees of freedom) that can be used for either data transmission or pilot tone transmission. The propagation channel has 200 degrees of freedom (resolvable delay bin 100 ns and 5 microseconds maximum excess delay means 50 delay coefficients characterize the channel, plus two resolvable Doppler bins within each channel, on each of two polarizations). Thus, the pilot tones for each user constitute an overhead of 2% of the total transmission resources. Due to the principle of OTFS of spreading over the whole system bandwidth and frame duration, the pilot tone overhead does not depend on the percentage of resources assigned to each user, but is a percentage of taken over all resources. This implies a high overhead when many users with small number of bytes per packet are active. 
     If completely orthogonalizing the users in the spatial and polarization domains, then the pilot overhead gets multiplied with the number of beams and polarizations. In other words, reserve a separate delay-Doppler (or time-frequency) resource for the pilot of each beam, which ensures that there is no pilot contamination. The broadcast pilots in the downlink need therefore 16% of the total resources (assuming communication in a sector) or 64% (for a full circular cell). The following examples will mostly concentrate on a single sector. 
     Similarly, for the uplink pilots, orthogonal pilots may be used for each of the users, in each of the beams. This results in a 16% overhead per user; with multiple users, this quickly becomes unsustainable. 
     The overhead for digitized feedback from the users can also be considerable. Since there are 200 channel degrees of freedom, quantization with 24 bit (12 bits each on I and Q branch) results in 4.8 Mbit/s for each user. Equivalently, if assuming on average 16 QAM (4 bit/s/Hz spectral efficiency), 1200 channel degrees of freedom are used up for the feedback of quantized information from a single user. This implies that the feedback of the digitized information is by a factor 6 less spectrally efficient than the transmission of an analog uplink pilot whose information can be used. Furthermore, the feedback is sent for the channel state information (CSI) from each BS antenna element to the customer premises equipment (CPE) or user device. Even though the feedback can be sent in a form that allows joint detection, in other words, the feedback info from users in different beams can be sent simultaneously, the overall effort for such feedback seems prohibitively large. 
     In addition, it is useful to consider the overhead of the embedded pilots for the downlink, where they are transmitted precoded in the same way as the data, and thus are used for the demodulation. By the nature of zero-forcing precoding, pilots can be transmitted on each beam separately. Thus, the overhead for the embedded downlink pilots is about 2% of the resources times the average number of users per beam. 
     For explicit feedback, there is yet another factor to consider, namely the overhead for the uplink pilots that accompany the transmission of the feedback data. This tends to be the dominant factor. Overhead reduction methods are discussed in the next section. 
     Overhead Reduction Methods 
     From the above description, it can be seen that overhead reduction is useful. The main bottlenecks indeed are the downlink broadcast pilots and the uplink pilots, since these pilots have to be sent on different time-frequency (or delay/Doppler) resources in different beams. However, under some circumstances, overhead reduction for the feedback packets is important as well. Before going into details, it is worth repeating why transmitters cannot transmit pilots on all beams all the time. Neither the UL pilots nor the broadcast DL pilots are precoded. To separate the pilots from/to different users, transmitters would have to beamform, but in order to beamform, a transmitter should know the channel, e.g., have decided pilots. Thus, a continuous transmission of pilots leads to “pilot contamination”, e.g., the signals from/to users employing the same pilots interfere with each other and lead to a reduced pilot SINR. Since the pilot quality determines the capability of coherently decoding the received data signal, reduction of the pilot SINR is—to a first approximation—as detrimental as reduction of the data SINR. While countermeasures such as joint equalization and decoding are possible, they greatly increase complexity and still result in a performance loss. 
     One effective method of reducing pilot contamination is minimum mean square error (MMSE) filtering, which achieves separation of users with the same pilot tones by projection of the desired users&#39; pilot onto the null-space of the channel correlation matrix of the interfering user. This reduces interference, though at the price of reduced signal power of the desired user. This method can be combined with any and all of the methods described below, and, in some situations, such a combined method will achieve the best performance. In some embodiments, linearly dependent pilot tones for the different users (instead of sets of users that use the same pilots within such a set, while the pilots in different sets are orthogonal to each other) may be used. Again, such a whitening approach can be used in conjunction with the methods described here. 
     A. Pilot Scheduling 
     The previous derivations assumed that the downlink broadcast and uplink pilots in different beams are on orthogonal resources, in order to reduce the overhead. Such an arrangement may not be needed when the angular spectra of the users are sufficiently separated. The simplest assumption is that each user has only a very small angular spread; then users that are on beams without overlaps (beam 1,3,5, . . . etc.) can be transmitted simultaneously. For a larger angular spread, a larger spacing between the beams is used. Still, if, e.g., every 4th beam can be used, then the overall overhead for the downlink broadcast pilots reduces, e.g., from 32% to 16% in one sector. Equally importantly, the overhead remains at 16% when moving from a sector to a 360 degree cell. 
     However, this consideration still assumes that there is a compact support of the angular power spectrum, and there is no “crosstalk”, e.g., between a beam at 0 degree and one at 60 degree. Often, this is not the case. In the presence of scattering objects, the sets of directions of contributions from/to different user devices can be quite different, and not simply a translation (in angle domain) of each other. If simply basing the beam reuse on the “worst case”, one might end up with complete orthogonalization. Thus, for every deployment, it is useful to assess individually what the best pattern is for a spatial reuse of the pilots. This is henceforth called “pilot scheduling”. 
     Before describing some examples of pilot scheduling embodiments, note that it is based on the knowledge of the power transfer matrix (PTM). The PTM may be a K×M matrix, where M is the number of beams at the BS, and K is the number of UEs. The (I,j)th entry of the PMT is then the amount of power (averaged over small-scale fading or time) arriving at the j-th beam when the i-th UE transmits with unit power (one simplification we assume in this exemplary description is that the PTM is identical for the two polarizations, which is reasonable, e.g., when there is sufficient frequency selectivity such that OTFS averages out small-scale fading over its transmission bandwidth; in any case generalization to having different PMT entries for different polarization ports is straightforward). For example, in the uplink, the receiver (base station) should know when a particular user transmits a pilot tone, in which beams to anticipate appreciable energy. This might again seem like a “chicken and egg” problem, since the aim of the pilot transmission is to learn about the channel between the user and the BS. However, the PTM is based on the knowledge of the average or second order channel state information (CSI). Since this property of a channel changes very slowly (on the order of seconds for mobile systems, on the order of minutes or more for FWA), learning the PTM does not require a significant percentage of time-frequency resources. While provisions should be taken in the protocol for suitable mechanisms, those pose no fundamental difficulty, and the remainder of the report simply assumes that PTM is known. 
     1) Pilot scheduling for the uplink: as mentioned above, the PTM contains information about the amount of power that is transferred from the ith user to the jth beam. Now, given the PTM, the question is: when can two uplink pilots be transmitted on the same time-frequency resources? 
     The answer may depend on the subsequent data transmission, for example, if the criterion is: “is the loss of capacity resulting from the imperfect beamforming vectors is less than the spectral efficiency gain of the reduced pilot overhead”. Conventional techniques do not consider such a criterion. This aspect of inquiry can be used in many advantageous ways: 
     a) It is not necessary to have highly accurate (contamination-free) pilots if the subsequent data transmission uses a low-order QAM anyways. 
     b) The pilot scheduling depends on the receiver type. First, different receivers allow different modulation schemes (even for the same SINR). Second, a receiver with iterative channel estimation and data decoding might be able to deal with more pilot contamination, since it processes the decoded data (protected by forward error correction FEC) to improve the channel estimates and reduce contamination effects. 
     c) The pilot scheduling, and the pilot reuse, may change whenever the transmitting users change. A fixed scheduling, such as beams 1, 5, 9, . . . etc. may be highly suboptimum. 
     d) Given the high overhead for uplink pilots, allowing considerable pilot contamination, and use of associated low SINR and modulation coding scheme (MCS), is reasonable, in particular for small data packets. 
     e) For an FWA system, it may be reasonable to allow uplink transmission without embedded pilots, basing the demodulation solely on the average channel state. However, due to the clock drift, a few pilots for phase/timing synchronizations may still be used, but no pilots may be used for channel re-estimation. For those short packets, a reduced-order MCS may be used. Alternatively, the short packets could be transmitted on a subband of the time-frequency resources, where the subband could even be selected to provide opportunistic scheduling gain. 
     The optimum scheduler may be highly complicated, and may change whenever the combination of considered user devices changes. Due to the huge number of possible user combinations in the different beams, it may not even possible to compute the optimum scheduler for each combination in advance and then reuse. Thus, a simplified (and suboptimum) scheduler may have to be designed. 
     2) Pilot scheduling for the downlink: The scheduler for the downlink broadcast pilots has some similarities to the uplink pilots, in that it is based on the PTM. However, one difference is worth noting: the scheduler has to provide acceptable levels of pilot contamination for all users in the system, since all users are monitoring the broadcast pilots and extrapolate the channel in order to be able to feed back the extrapolated channel when the need arises. Thus, the reuse factor of the broadcast pilots may be large (meaning there is less reuse) than for the uplink pilots. For the computation of the pilot schedule, a few things may be taken into account: 
     a) the schedule may only be changed when the active user devices change, e.g., a modem goes to sleep or wakes up. This happens on a much rarer basis than the schedule change in the uplink, which happens whenever the actually transmitting user devices change. 
     b) In the downlink pilots, it may not be exactly known what pilot quality will be required at what time (e.g., the required SINR), since the transmitting user schedule is not yet known (e.g., when the pilots are transmitted continuously). Thus, it may be assumed that data transmission could occur without interference (e.g., all other beams are silent because there are no data to transmit), so that the data transmission for the user under consideration happens with the MCS that is supported by the SNR. 
     c) It is possible that one (or a few) user devices become a “bottleneck”, in the sense that they require a large reuse factor when all other users might allow dense reuse. It is thus useful to consider the tradeoff of reducing the pilot quality to these bottleneck user devices and reducing the MCS for the data transmission, as this might lead to an increase of sum spectral efficiency, and may be performed by taking minimum (committed) service quality constraints into account. 
     Since broadcast pilots are always transmitted from the BS, and can be only either transmitted or not transmitted (there is no power control for them), the number of possible combinations is manageable (2{circumflex over ( )}8), and it is thus possible to compute the SINR at all users in the cell for all pilot schedules, and check whether they result in acceptable SNR at all users, and pick the best one. As outlined above, there is no need to recompute the schedule, except when the set of active user devices changes. When considering a combination of this scheme with MMSE receivers, scheduling should be based on the SINR that occurs after the MMSE filtering. 
     B. Exploiting the Properties of FWA 
     One way for reducing the overhead is to exploit the special properties of FWA channels, namely that the instantaneous channel is the average channel plus a small perturbation. This can be exploited both for reducing the reuse factor, and for more efficient quantization. 
     1) Reducing the reuse factor: The goal of the pilot tones is to determine the CSI for each user device with a certain accuracy. Let us consider the uplink: for the i-th user in the j-th beam, the CSI can be written as H av   ij +ΔH ij ; the power ratio (ΔH ij /H av   ij ) 2  is the temporal Rice factor for this particular link K ij . Now any pilot contamination based on H av   ij  is known and can be eliminated by interference cancellation. Thus, denoting the kj-th entry of the PTM C kj , then a naïve assessment of the pilot contamination would say that the achievable pilot SIR in the j-th beam is C ij /C kj . However, by first subtracting the known contribution H av   kj  from the overall received signal, K kj C ij /C kj  can be achieved. Having thus improved the SIR for each user, the system can employ a much smaller reuse factor (that is, reduce overhead). In practice this method can probably reduce the reuse factor by about a factor of 2. The same approach can also be applied in the downlink. The improvement that can be achieved will differ from user device to user device, and the overall reuse factor improvement will be determined by the “bottleneck links” (the ones requiring the largest reuse factor). Some embodiments can sacrifice throughput on a few links if that helps to reduce the pilot reuse factor and thus overhead, as described above. When combining this method with MMSE filtering, the procedure may occur in two steps: first, the time-invariant part of the channel is subtracted. The time-variant part is estimated with the help of the MMSE filtering (employing the channel correlation matrix of the time-variant part), and then the total channel is obtained as the sum of the time-invariant and the thus-estimated time-variant channel. 
     2) Improved quantization: Another question is the level of quantization that is to be used for the case that explicit feedback is used. Generally, the rule is that quantization noise is 6 dB for every bit of resolution. The 12 bit resolution assumed above for the feedback of the CSI thus amply covers the desired signal-to-quantization-noise ratio and dynamic range. However, in a fixed wireless system, implementations do not need a large dynamic range margin (the received power level stays constant except for small variations), and any variations around the mean are small. Thus, assume a temporal Rice factor of 10 dB, and an average signal level of −60 dBm. This means that the actual fluctuations of the signal have a signal power of −70 dBm. 4-bit quantization provides −24 dB quantization noise, so that the quantization noise level is at −94 dBm, providing more than enough SIR. Embodiments can thus actually reduce the amount of feedback bits by a factor of 3 (from 12-bit as assumed above to 4 bits) without noticeable performance impact. 
     3) Adaptivity of the methods: The improvements described above use the decomposition of the signal into fixed and time-varying parts, and the improvements are the larger the larger the temporal Rice factor is. Measurements have shown that the temporal Rice factor varies from cell to cell, and even UE to UE, and furthermore might change over time. It is thus difficult to determine in advance the reduction of the reuse factor, or the suitable quantization. For the reduction of the reuse factor, variations of the Rice factor from cell to cell and between user devices such as UEs can be taken care of as a part of the pilot scheduling design, as described above. Changes in the temporal Rice factor (e.g., due to reduced car traffic during nighttime, or reduction of vegetation scatter due to change in wind speed) might trigger a new scheduling of pilots even when the active user set has not changed. For the quantization, the protocol should not contain a fixed number of quantization bits, but rather allow an adaptive design, e.g., by having the feedback packet denote in the preamble how many bits are used for quantization. 
     C. Reduction Methods for Small Packet Size 
     The most problematic situation occurs when a large number of users, each with a small packet, are scheduled within one frame. Such a situation is problematic no matter whether it occurs in the uplink or the downlink, as the pilot overhead in either case is significant. This problem can be combatted in two ways (as alluded to above) 
     1) reduce the bandwidth assigned to each user. This is a deviation from the principle of full-spreading OTFS, but well aligned with other implementations of OTFS that can assign a subband to a particular user, and furthermore to various forms of OFDMA. 
     The two design trade-offs of the approach are that (i) it may use a more sophisticated scheduler, which now considers frequency selectivity as well, and (ii) it is a deviation from the simple transmission structure described above, where different users are designed different timeslots and/or delay/Doppler bins. Both of these issues might be solved by a multi-subband approach (e.g., 4 equally spaced subbands), though this may trade off some performance (compared to full OTFS) and retains some significant pilot overhead, since at least CSI in the 4 chosen subbands has to be transmitted. 
     2) transmit the small packets without any pilots, relying on the average CSI for suppression of inter-beam interference. It is noteworthy that for the downlink, an implementation can sacrifice SIR (due to pilot contamination) on some links without disturbing others. Imagine that precise CSI for UE j is available, while it is not available for UE k. An implementation can thus ensure that the transmission for k lies in the exact null-space of j, since the CSI vector hj=[h 1j ; h 2j ; . . . ] is known accurately, and thus its nullspace can be determined accurately as well. So, if the link to j wants to send a big data packet for which the use of a high-order MCS is essential, then the system can invest more resources (e.g., reduce pilot imprecision) for this link, and reap the benefits. 
     3) For the uplink, the approach 2 may not work: in order to have high SINR for the signal from the j-th user, it is advantageous to suppress the interference from all other users that are transmitting in parallel. Thus, instead one approach may be to provide orthogonalization in time/frequency (or delay/Doppler) between the group of users that needs low pilot contamination (usually large packets, so that the efficiency gain from transmitting pilots outweighs the overhead), and another group of users (the ones with small packets) that do not transmit pilots (or just synchronization pilots) and thus are efficient, yet have to operate with lower-order MCS due to the pilot contamination. It must be noted that methods 2 and 3 only work for FWA systems, where one can make use of the average CSI to get a reasonable channel estimate without instantaneous pilots. When migrating to a mobile system, it is recommended to move to approach 1. 
     Examples for the Achievable Gain 
     This section describes some examples of the gain that can be achieved by the proposed methods versus a baseline system. It should be noted that the gain will be different depending on the number of users, the type of traffic, and particularly the directional channel properties. There are examples where the simple orthogonalization scheme provides optimum efficiency, so that no gain can be achieved, and other examples where the gain can be more than an order of magnitude. The section will use what can be considered “typical” examples. Ray tracing data for typical environments and traffic data from deployed FWA or similar systems, for example, can be used to identify a typical or representative system. 
     A. Gain of Pilot Scheduling 
     One purpose of pilot scheduling is to place pilots on such time-frequency resources that they either do not interfere significantly with each other, or that the capacity loss by using more spectral resources for pilots is less than the loss one would get from pilot contamination. In a strictly orthogonal system, there is no pilot contamination, but 16% of all spectral resources must be dedicated to the downlink pilots, and a fraction 0.16*Nupb of the resources for the uplink pilots, where Nupb is the number of users per beam in the uplink. For a full 360 degree cell, the numbers are 64% and 0.64*Nupb. 
     A possibly simplest form of pilot scheduling is just a reuse of the pilot in every P-th beam, where P is chosen such that the interference between two beams separated by P is “acceptable” (either negligible, or with a sufficiently low penalty on the capacity of the data stream). This scheme achieves a gain of 36/P in a completely homogeneous environment. For a suburban LoS type of environment, P is typically 4, so that the pilot overhead can be reduced by a factor of 9 (almost an order of magnitude) for a 360 degree cell. Equivalently, for the uplink pilots, the number of users in the cell can be increased by a factor of 9 (this assumes that the overhead for the uplink pilots dominates the feedback overhead, as discussed above). 
     Simple scheduling may work only in an environment with homogeneous channel and user conditions. It can be seen that a single (uplink) user with angular spread covering P0 beams would entail a change in the angular reuse factor to P0 (assuming that a regular reuse pattern for all users is used), thus reducing the achievable gain. The more irregular the environment, the more difficult it is to find a reasonable regular reuse factor, and in the extreme case, complete orthogonalization might be necessary for regular reuse patterns, while an irregular scheduling that simply finds the best combination of users for transmitting on the same spectral resources, could provide angular reuse factors on the order of 10. However, in an environment with high angular dispersion (e.g., microcell in a street canyon), where radiation is incident on the BS from all directions, even adaptive scheduling cannot provide significant advantages over orthogonalization. 
     In conclusion, pilot scheduling provides an order-of-magnitude reduction in pilot overhead, or equivalently an order of magnitude larger number of users that can be accommodated for a fixed pilot overhead, compared to full orthogonalization. Compared to simple (regular) pilot reuse, environment-adaptive scheduling retains most of the possible gains, while regular scheduling starts to lose its advantages over complete orthogonalization as the environment becomes more irregular. 
     B. Exploiting FWA Properties for Pilot Scheduling 
     The exploitation of FWA properties can be more easily quantified if we retain the same reuse factor P as we would have with a “regular” scheme, but just make use of the better signal-to-interference ratio of the pilots (e.g., reduced pilot contamination). As outlined in Sec. 3.2, the reduction in the pilot contamination is equal to the temporal Rice factor. Assuming 15 dB as a typical value, and assuming a high-enough SNR that the capacity of the data transmission is dominated by pilot contamination, the SINR per user is thus improved by 15 dB. Since 3 dB SNR improvement provide 1 bit/s/Hz increase in spectral efficiency, this means that for each user, capacity is increased by 5 bit/s/Hz. Assuming 32 QAM as the usual modulation scheme, an implementation can double the capacity through this scheme. 
     A different way to look at the advantages is to see how much the number of users per beam can be increased, when keeping the pilot SIR constant. This can depend on the angular spectrum of the user devices. However, with a 15 dB suppression of the interference, one can conjecture that (with suitable scheduling), a reuse factor of P=2, and possibly even P=1, is feasible. This implies that compared to the case where an implementation does not use this property, a doubling or quadrupling of the number of users is feasible (and even more in highly dispersive environments) 
     In summary, exploiting the FWA properties for pilot scheduling doubles the capacity, or quadruples the number of users 
     C. Exploiting the FWA Properties for Reduction of Feedback Overhead 
     As outlined above, exploiting the FWA properties allow to reduce the feedback from 12 bit to 4 bit, thus reducing overhead by a factor of 3. Further advantages can be gained if the time-variant part occurs only in the parts of the impulse response with small delay, as has been determined experimentally. Then the feedback can be restricted to the delay range over which the time changes occur. If, for example, this range is 500 ns, then the feedback effort is reduced by a further factor of 10 (500 ns/5 microsec). In summary, the reduction of the feedback overhead can be by a factor of 3 to 30. 
     7. Second-Order Statistics for FDD Reciprocity 
     This section covers using second order statistics of a wireless channel to achieve reciprocity in frequency division duplexing (FDD) systems. FDD systems may have the following challenges in implementing such a precoded system:
         The downlink channel response is different from the uplink channel response, due to the different carrier frequencies. On top of that, there is a different response of the transmit and receive RF components in the base-station and user equipment.   For non-static channels, the base-station needs to predict the channel for the time of the transmission.       

     In some embodiments, the base-station may send, before every precoded downlink transmission, reference signals (pilots) to the user equipment. The users will receive them and send them back to the base-station as uplink data. Then, the base-station will estimate the downlink channel and use it for precoding. However, this solution is very inefficient because it takes a large portion of the uplink capacity for sending back the received reference signals. When the number of users and/or base-station antennas grow, the system might not even be implementable. Also, the round-trip latency, in non-static channels, may degrade the quality of the channel prediction. 
     Second-Order Statistics Training 
     For simplicity, the case with a single user antenna and the L base-station antennas is considered, but can be easily extended to any number of users. The setup of the system is shown in  FIG. 84 . The base-station predicts from the uplink channel response, the downlink channel response in a different frequency band and N latency  subframes later. 
     To achieve this, the system preforms a preliminary training phase, consisting of multiple sessions, where in each session i=1, 2, . . . , N training , the following steps are taken:
         At subframe n, the user equipment transmits reference signals (RS) in the uplink. The base-station receives them and estimate the uplink channel H UL   (i)  over the L base-station antennas.   At subframe n+N latency , the base-station transmits reference signals in the downlink from all its antennas. The user equipment receives it and sends it back as uplink data in a later subframe. The base-station computes the downlink channel estimation for it, H DL   (i) . In a different implementation, it is possible that the UE will compute the channel estimation and send it to the base-station as uplink data.   The base-station computes the second-order statistics       

         R   UL   (i)   =H   UL   (i) ·( H   UL   (i) ) H  
 
         R   DL,UL   (i)   =H   DL   (i) ·( H   UL   (i) ) H  
 
         R   DL   (i)   =H   DL   (i) ·( H   DL   (i) ) H  
 
     Herein, (·) H  is the Hermitian operator. For the case that the channel has non-zero-mean, both the mean and the covariance matrix should be determined. When the training sessions are completed, the base-station averages out the second order statistics: 
     
       
         
           
             
               
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     Then, it computes the prediction filter and the covariance of the estimation error: 
         C   prediction   =R   DL,UL ·( R   UL ) −1  
 
         R   E   =R   DL   −C   prediction ·( R   DL,UL ) H  
 
     The inversion of R UL  may be approximated using principal component analysis techniques. We compute {λ}, the K most dominant eigenvalues of R UL , arranged in a diagonal matrix D=diag(λ 1 , λ 2 , . . . , λ K ) and their corresponding eigenvectors matrix V. Typically, K will be in the order of the number of reflectors along the wireless path. The covariance matrix can then be approximated by R UL ≈V·D·(V) H  and the inverse as R UL   −1 ≈V·D −1 ·(V) H . 
     Note, that there is a limited number of training sessions and that they may be done at a very low rate (such as one every second) and therefore will not overload the system too much. 
     To accommodate for possible future changes in the channel response, the second-order statistics may be updated later, after the training phase is completed. It may be recomputed from scratch by initializing again new N training  sessions, or by gradually updating the existing statistics. 
     The interval at which the training step is to be repeated depends on the stationarity time of the channel, e.g., the time during which the second-order statistics stay approximately constant. This time can be chosen either to be a system-determined constant, or can be adapted to the environment. Either the base-station or the users can detect changes in the second-order statistics of the channel and initiate a new training phase. In another embodiment, the base-station may use the frequency of retransmission requests from the users to detect changes in the channel, and restart the process of computing the second-order statistics of the channel. 
     Scheduling a Downlink Precoded Transmission 
     For each subframe with a precoded downlink transmission, the base-station should schedule all the users of that transmission to send uplink reference signals N latency  subframes before. The base-station will estimate the uplink channel responses and use it to predict the desired downlink channel responses 
     
       
      
       H 
       DL 
       =C 
       prediction 
       ·H 
       UL  
      
     
     Then, the downlink channel response H DL  and the prediction error covariance R E  will be used for the computation of the precoder. 
     8. Second-Order Statistics for Channel Estimation 
     This section covers using second order statistics of a wireless channel to achieve efficient channel estimation. Channel knowledge is a critical component in wireless communication, whether it is for a receiver to equalize and decode the received signal, or for a multi-antenna transmitter to generate a more efficient precoded transmission. 
     Channel knowledge is typically acquired by transmitting known reference signals (pilots) and interpolating them at the receiver over the entire bandwidth and time of interest. Typically, the density of the pilots depends on characteristics of the channel. Higher delay spreads require more dense pilots along frequency and higher Doppler spreads require more dense pilots along time. However, the pilots are typically required to cover the entire bandwidth of interest and, in some cases, also the entire time interval of interest. 
     Embodiments of the disclosed technology include a method based on the computation of the second-order statistics of the channel, where after a training phase, the channel can be estimated over a large bandwidth from reference signals in a much smaller bandwidth. Even more, the channel can also be predicted over a future time interval. 
     Second-Order Statistics Training for Channel Estimation 
       FIG. 85  shows a typical setup of a transmitter and a receiver. Each one may have multiple antennas, but for simplicity we will only describe the method for a single antenna to a single antenna link. This could be easily extended to any number of antennas in both receiver and transmitter. 
     The system preforms a preliminary training phase, consisting of multiple sessions, where in each session i=1, 2, . . . , N training , the following steps are taken:
         The transmitter sends reference signals to the receiver. We partition the entire bandwidth of interest into two parts BW 1  and BW 2 , as shown in  FIGS. 86A-86C , where typically the size of BW 1  will be smaller or equal to BW 2 . Note, that these two parts do not have to from a continuous bandwidth. The transmitter may send reference signals at both parts at the same time interval (e.g.,  FIG. 87 ) or at different time intervals (e.g.,  FIG. 88 ).       

     The receiver receives the reference signals and estimates the channel over their associated bandwidth, resulting in channel responses H 1   (i)  and H 2   (i) . 
     The receiver computes the second-order statistics of these two parts: 
         R   1   (i)   =H   1   (i) ·( H   1   (i) ) H  
 
         R   2,1   (i)   =H   2   (i) ·( H   1   (i) ) H  
 
         R   2   (i)   =H   2   (i) ·( H   2   (i) ) H  
 
     Herein, (·) H  is the Hermitian operator. For the case that the channel has non-zero-mean, both the mean and the covariance matrix should be determined. When the training sessions are completed, the base-station averages out the second-order statistics in a manner similar to that described in Section 6. 
     Efficient Channel Estimation 
     After the training phase is completed, the transmitter may only send reference signals corresponding to BW 1 . The receiver, estimated the channel response H 1  and use it to compute (and predict) and channel response H 2  over BW 2  using the prediction filter: 
         H   2   =C   prediction   ·H   1 . 
       FIGS. 89 and 90  show examples of prediction scenarios (same time interval and future time interval, respectively). 
     9. Multi-User Support Using Spatial Separation 
     Embodiments of the disclosed technology include systems that comprise a central base station with multiple spatial antennas and multiple users, which can be configured to transmit or receive simultaneously to or from multiple users, over the same time and frequency resources, using spatial separation. In an example, this may be achieved by creating multiple beams, where each beam is focused towards a specific user. To avoid cross-interference between the beams, each beam should be nulled at the directions of the other users, thus maximizing the SINR for each user.  FIG. 91A  shows an example of overlaid beam patterns for the case with four users (UEs or mobile devices). 
     In some embodiments, the configuration of these beams may depend on the detection of the main radiation pattern coming from the users, or in other words, their angles-of-arrival, which is shown in the example in  FIG. 91B  for the users shown in  FIG. 91A . 
     In some embodiments, a method for detecting the angle-of-arrival of users in a wireless system comprises processing the received uplink transmissions from users to create multiple beams, wherein these beams have minimal cross-interference between them, and subsequently, transmitting or receiving to or from the multiple users. 
     In some embodiments, each beam has its energy focused towards the angle-of-arrival of a specific user and has minimal energy towards the angle-of-arrival of the other users. 
     In some embodiments, the angle-or-arrival is derived from uplink reference signals. 
     In some embodiments, the beams are created for a selected subset of the users under some angular separation criterion. 
     Detecting the Aliased Angle-of-Arrival 
     In some embodiments, and when the spacing between the spatial antennas is larger than half of the wavelength of the uplink transmission, there is an aliasing phenomenon, where the angle-of-arrival folds into a smaller range than 180 degrees. If the downlink has a different frequency (FDD), then the true angle is needed for correct beam configuration. 
     Embodiments of the disclosed technology include an algorithm for detecting whether the angle-of-arrival is aliased or not. If it is known to be aliased, the angle can be unfolded. In an example, the algorithm measures the angle-of-arrival at two different frequencies within the band. For OFDM, the different frequencies may be different subcarriers, preferably at the band edges. Then, a function of these two detected angles is compared to the ratio of these two frequencies, which enables the detection of aliasing because the function results in different outcomes for aliased and non-aliased angles. In an example, the function may be the ratio of these angles. 
     In some embodiments, a method for detecting aliased angle-of-arrival in a system with antenna spacing larger than half of the wavelength comprises measuring the angle-of-arrival at two different frequencies and comparing a function of these two measurements to the ratio of the frequencies. 
     In some embodiments, the true angle-of-arrival is derived from a detected aliased angle-of-arrival. 
     In some embodiments, the beam creation for the downlink is based on the true angle-of-arrival. 
     10. Embodiments of Fractional COMP 
     One practical problem faced by COMP operation is the availability of backhaul bandwidth to wireless devices, such as base station towers, for communicating both user data and control messages that need to be exchanged for COMP operation. For example, in embodiments in which millimeter wave wireless connections are used by network devices for backhaul communication, 2 Gbps is a typical bandwidth available under present-day technologies. A significant amount of this bandwidth may be utilized for COMP coordination messages. 
       FIG. 92  shows an example of a wireless system in which multiple neighboring cells operate to provide wireless connectivity services to wireless devices. The calculation shown in  FIG. 92  shows that approximately 350 Mbps per channel per cell may be needed for backhaul coordination messaging for COMP operation for 20 MHz channel bandwidth that is being used at 80% utilization. 
     As further described in the present document, a fractional COMP method may be used in some embodiments to alleviate the bandwidth pressure on backhaul. Wireless devices services by a network device may be logically divided into a first fraction of devices that are serviced using a joint COMP mode in which backhaul message exchange occurs for coordination among multiple transmissions towers or network nodes or network devices and a remaining fraction in which user devices or the corresponding transmission resources such as frequency bands, are controlled by the corresponding network device. In the latter mode, no backhaul COMP messaging bandwidth may be needed for scheduling or channel quality related messaging, thereby substantially alleviating bandwidth pressure on the backhaul link. For example, in some cases, network devices may be able to operate with zero backhaul bandwidth utilization for the locally managed mode for the remaining fraction of devices. 
       FIG. 93  shows an example antenna configuration for use in backhaul communication. This figure explains why 2 Gbps link is a practically realistic bandwidth available for backhaul connections using two dual polarization antennas configured with requisite azimuth/elevation and operating using 100 MHz bandwidth. 
       FIG. 94  shows a three-dimensional graph in which performance of various wireless system operations is depicted for comparison. The vertical dimension represents number of layers used for communication, and as expected, a greater number of layers provides a greater amount of throughput or performance. The horizontal axis depicts amount of spectrum used for communication, such as for example channel bonding used for simultaneously using multiple conventional channels. Greater bandwidth (or bonding) results in greater amount of performance. The Z (depth) axis represents quality of channel in terms of SNR, and how many bits per second per Hertz per layer can be fit into the communication. As depicted, the smallest volume (left bottom) is the conventional wireless communication, the medium volume represents current state of the art which has greatly expanded throughput. Finally, the largest volume cube depicts performance achieved by embodiments that use some techniques described herein, including channel quality prediction and fractional COMP utilization. 
       FIG. 95  is an example depiction of backhaul bandwidth utilization. As shown in the figure, the total backhaul bandwidth usage by a network device may be due to multiple COMP interconnects with neighboring network device (cell towers, base stations, gNBs, etc.). Furthermore, the backhaul may also carry wide area connectivity traffic for user devices that are services by each network device in its radio access network (RAN) using, for example, MU-MIMO communication, as described herein. 
       FIG. 96  is a block diagram of an example system that controls MAC-PHY interfaces across a scheduler interface, and illustrates how existing MU-MIMO schedulers can be enhanced to implement the techniques described in the present document. 
     In some embodiments, and with reference to the scheduler interface shown in  FIG. 96 , the eNodeB scheduler makes all decisions about bandwidth grants. Thus, when adding MU-MIMO functionality to the eNodeB, the scheduler should be configured to be directly involved in all decisions and bandwidth grants. This includes, but is not limited to, deciding when to precode or not, deciding which UEs to pair on the same Physical Resource Block&#39;s (PRB&#39;s) different layers, and determining which MCS to use. 
     In some embodiments, these functionality could be offloaded to the channel estimation (ChEst) and pairing estimation (PairingEst) embodiments described in the present document. This would typically be implemented in a manner that was cognizant of the scheduler timing constraints. 
       FIG. 97  is a block diagram of another example system that shows the multi-user MIMO system Application Programming Interfaces (APIs). As shown therein, the required data can be directly shared with the scheduler, which can then make decisions with very minimal overhead. The APIs shown in  FIG. 97  can advantageously provide real-time updates of this information as the channel conditions change. 
     As shown in  FIGS. 96 and 97 , the enhancements, which include the channel prediction block, the pairing estimate and MCS calculation block and the coefficient calculation, are configured to support fractional COMP. That is, they can support both local and backhaul communications for a first subset of users, and only local communications for a second subset of users thereby relieving the pressure on the backhaul. 
     In some embodiments, the scheduler interface for precoding can be configured to support a per-UE state for precoding with the following characteristics and rules:
         As UEs enter and leave eNodeB, they must be announced to ChEst
           RRC state: RRC-Connected, RRC-Idle   
           Before channel estimation is complete for a UE, it must be scheduled SISO
           The scheduler can do those grants as in existing implementations   The ChEst block will inform the eNodeB of the new UE   ChEst must inform eNodeB of precoding state/readiness   A new state: PrecodingEnable, can be configured   
               

     In some embodiments, channel estimation can be performed using the sounding reference signal (SRS), with the following (non-limiting) characteristics:
         Channel estimation is slow periodic, for all connected UEs, on order of 100 ms   Mobile UEs will require higher periodicity   ChEst block will inform scheduler of new SRS request   ChEst block will update scheduler if/when SRS periodicity for a UE changes       

     In some embodiments, the coefficient computation block shown in  FIGS. 96 and 97  can be configured to ensure that there is tight time constraint from DL grant info from Scheduler, moving to the compute platform, and to getting the DL coefficients back to the PHY layer precoding block. However, the UL timing is more relaxed. Furthermore, the scheduler must compensate for any latency incurred by DL coefficient computation. 
     In some embodiments, the DL coefficient calculations could be moved into the PHY layer to advantageously reduce this latency. 
     In some embodiments, the API PHY interfaces (e.g., as shown in  FIG. 97 ) can be configured to as to enable a GigE or 10GigE interface to connect the eNodeB and the channel estimation (ChEst) platform. 
     Some embodiments of the disclosed technology can be configured to account for uplink antenna usage and the impact on downlink precoding. In an example, to meet the objective of MU-MIMO with 1 layer for each multiplexed UE, the uplink SRS can use a single antenna; e.g., 2 UEs each with one precoded layer in the DL for 2T2R (2×2 MIMO), or 4 UEs each with one precoded layer in the DL for 4T4R (4×4 MIMO). 
     Some embodiments of the disclosed technology can be configured to alleviate potential problems of using Transmission Mode 4 (TM4) for MU-MIMO precoding. Existing systems currently face the following potential problems:
         Implementations of precoding algorithms require non-codebook precoding   Non-codebook precoding requires UE-specific reference signals (e.g., DMRS)   TM4 does not support UE-specific reference signals (introduced for dual-layer with Transmission Mode 8 (TM8))       

     The embodiments described herein can provide the following potential solutions:
         eNB precodes cell-specific reference signals in conjunction with TM4   Introduces conflicts related to how the UE equalizes control and data, but resolves these conflicts without requiring changes to the UE       

       FIGS. 98A-98D  show example hardware platforms used for implementing some of the techniques described in the embodiments.  FIG. 98A  shows a computing platform that:
         supports 16T16R configuration   utilizes commercial off-the-shelf GPUs   provides scalable spectral efficiency up to 72× reuse   provides multi-protocol support (OTFS, 4G and 5G)   provides a cloud-based platform, including software-defined radio (SDR)       

       FIGS. 98B-98D  show examples of 2T2R and 4T4R platforms that:
         utilize off-the-shelf X86 (Xeon) based platforms   utilize x86/low cost GPU Hybrid Platform   utilize low-cost GPU based platforms   utilize plug-in GPU cards or Xeon cards       

     11. Methods and Implementations of the Disclosed Technology 
       FIG. 99  shows an example of a wireless transceiver apparatus  9900 . The apparatus  9900  may be used to implement the node or a UE or a network-side resource that implements channel estimation/prediction tasks. The apparatus  9900  includes a processor  9902 , an optional memory ( 9604 ) and transceiver circuitry  9906 . The processor  9902  may be configured to implement techniques described in the present document. For example, the processor  9902  may use the memory  9904  for storing code, data or intermediate results. The transceiver circuitry  9906  may perform tasks of transmitting or receiving signals. This may include, for example, data transmission/reception over a wireless link such as Wi-Fi, millimeter wavelength (mmwave) or another link, or a wired link such as a fiber optic link. Further details of some embodiments of the apparatus  9900  are disclosed with reference to  FIGS. 22-31, 33 and 98A-98D . 
       FIG. 100  is a flowchart for an example method  10000  of wireless communication. The method  10000  includes, at operation  10002 , determining, by a network device, a cooperative multipoint (COMP) management status of wireless devices served by the network device. The network device may be implemented using a software and hardware platform as described with respect to  FIG. 99 . 
     The method  10000  includes, at operation  10004 , providing, by the network device, wireless connectivity to the one or more wireless devices. For example, the wireless connectivity may allow the wireless devices to send and receive wireless signals that include user and control data. The wireless connectivity may provide access to the wireless devices to wireless services and application servers such as for voice, video and other user level communication with other devices via the wireless network. 
     In some embodiments, the network device jointly manages transmission resources for a first wireless device due to the COMP management status being a joint COMP status and the network device locally manages transmission resources for a second wireless device due to the COMP management status being a local COMP status. Some examples of calculations performed for COMP management of wireless devices are described with reference to  FIGS. 14 to 19 . 
     In some embodiments, wireless devices at cell edge are given the joint COMP status and wireless device towards cell center are given the local COMP status. 
     Alternatively, or in addition, in some embodiments, wireless devices that exhibit high mobility may be managed locally by giving them local COMP status while wireless device that are less mobile may be managed in a distributed manner by giving them joint COMP status. To help with this, wireless devices may be classified into two or more mobility classes (e.g., high mobility, low mobility) based on an order of mobility. A first number of classes may be assigned local COMP status and a second number of classes lower in the order (having lower mobility) will be given a joint COMP status. 
     For example, the amount of backhaul traffic generated for scheduling and other management of a high mobility device may be more than similar traffic for a low mobility device having a mobility less than the high mobility device. Therefore, backhaul bandwidth pressure may be effectively addressable by moving a high mobility device under full local control. 
     In some embodiments, in addition to the above discussed statuses of relative location in a cell and mobility, other operational features may be taken into account when classifying wireless devices into local or joint COMP status. For example, a number of antennas of a wireless device has an impact on the amount of channel quality information generated for or by the wireless device and this could be used as another status for deciding whether to manage locally or in COMP manner. 
     In some embodiments, a weighted average of various such criteria may be used in deciding which wireless devices to manage locally and which ones to manage using COMP framework. This decision may also depend on how much backhaul bandwidth is currently being used or available and availability of computational resources locally to a base station. 
     In some embodiments, the following technical solutions can be implemented: 
     1. A wireless communication method, comprising determining, by a network device, a cooperative multipoint (COMP) management status of wireless devices served by the network device; and providing, by the network device, wireless connectivity to the one or more wireless devices, wherein the network device jointly manages transmission resources for a first wireless device due to the COMP management status being a joint COMP status and the network device locally manages transmission resources for a second wireless device due to the COMP management status being a local COMP status. 
     2. The method of solution 1, wherein the COMP management status is determined using channel quality measurements for the one or more wireless devices. 
     3. The method of any of solution 1 or 2, wherein the transmission resources for the first wireless device include resource blocks used for transmissions in both directions between the network device and the first wireless device. 
     4. The method of any of solutions 1 to 3, further including grouping the wireless devices into groups of wireless devices; mapping the groups of wireless devices to layers of transmission; and blending the layers of transmission together into multi-user multi-input multi-output (MU-MIMO) signals for transmission or reception using multiple antenna. 
     5. The method of solution 4, wherein the mapping is performed by predicting channel for user devices at a future time and/or a different frequency band and by calculating modulation coding scheme for the user devices. 
     6. The method of solution 1, wherein the COMP status of wireless devices is determined based on a signal to noise ratio (SNR) estimate or a signal to interference plus noise (SINR) estimate. 
     7. The method of solution 1, wherein wireless devices at cell edge are given the joint COMP status and wireless device towards cell center are given the local COMP status. 
     8. The method of any of solutions 1 to 7, wherein the providing wireless connectivity includes exchanging COMP messages for the joint COMP status with neighboring network devices on a backhaul connection on which a fractional bandwidth is used for the COMP messages compared to that used for providing wireless connectivity for entirety of wireless device serviced by the network node. 
     9. The method of solution 8, wherein the backhaul connection is a microwave wireless connection. 
     10. A wireless communication system, comprising an arrangement of a plurality of network nodes in which each network node is configured to provide wireless connectivity to wireless devices using a mode that includes a joint cooperative multipoint (COMP) mode and a local COMP mode, wherein, in the joint COMP mode, transmission resources for wireless devices are managed cooperatively with other network nodes, and wherein, in the local COMP mode, transmission resources for wireless devices are managed locally, without explicit coordination with other network nodes. Some embodiments of such a system are described with reference to  FIGS. 14, 17, 19 and 20  of the present document. The system may be spread over a wide geographic range that covers more than 100 miles of radius, with individual network nodes (e.g., base stations) in communication with each other with shorter range (e.g., 1 to 10 kilometer) distance. 
     11. The system of solution 10, wherein the mode further includes a deferred COMP mode in which transmission resources for wireless device are managed based on control commands received from a remote controller. 
     12. The system of solution 10, wherein a different mode is used for wireless devices at cellular edge and wireless devices not at cellular edge. Other possible classifications of the wireless devices into joint or local management status are described with reference to  FIG. 99 . 
     13. The system of solution 12, wherein wireless devices are determined to be at cellular edge based on signal to noise ratio (SNR) calculations. 
     14. A wireless communication apparatus comprising one or more processors configured to implement a method recited in any one or more of solutions 1 to 9. 
     15. A non-transitory computer readable medium storing instructions, which when executed by at least one computing device, perform any of the methods in solutions 1 to 9. 
     The disclosed and other embodiments, modules and the functional operations described in this document can be implemented in digital electronic circuitry, or in computer software, firmware, or hardware, including the structures disclosed in this document and their structural equivalents, or in combinations of one or more of them. The disclosed and other embodiments can be implemented as one or more computer program products, i.e., one or more modules of computer program instructions encoded on a computer readable medium for execution by, or to control the operation of, data processing apparatus. The computer readable medium can be a machine-readable storage device, a machine-readable storage substrate, a memory device, a composition of matter effecting a machine-readable propagated signal, or a combination of one or more them. The term “data processing apparatus” encompasses all apparatus, devices, and machines for processing data, including by way of example a programmable processor, a computer, or multiple processors or computers. The apparatus can include, in addition to hardware, code that creates an execution environment for the computer program in question, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them. A propagated signal is an artificially generated signal, e.g., a machine-generated electrical, optical, or electromagnetic signal, that is generated to encode information for transmission to suitable receiver apparatus. 
     A computer program (also known as a program, software, software application, script, or code) can be written in any form of programming language, including compiled or interpreted languages, and it can be deployed in any form, including as a standalone program or as a module, component, subroutine, or other unit suitable for use in a computing environment. A computer program does not necessarily correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program in question, or in multiple coordinated files (e.g., files that store one or more modules, sub programs, or portions of code). A computer program can be deployed to be executed on one computer or on multiple computers that are located at one site or distributed across multiple sites and interconnected by a communication network. 
     The processes and logic flows described in this document can be performed by one or more programmable processors executing one or more computer programs to perform functions by operating on input data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit). 
     Processors suitable for the execution of a computer program include, by way of example, both general and special purpose microprocessors, and any one or more processors of any kind of digital computer. Generally, a processor will receive instructions and data from a read-only memory or a random access memory or both. The essential elements of a computer are a processor for performing instructions and one or more memory devices for storing instructions and data. Generally, a computer will also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto optical disks, or optical disks. However, a computer need not have such devices. Computer readable media suitable for storing computer program instructions and data include all forms of non-volatile memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto optical disks; and CD ROM and DVD-ROM disks. The processor and the memory can be supplemented by, or incorporated in, special purpose logic circuitry. 
     While this patent document contains many specifics, these should not be construed as limitations on the scope of an invention that is claimed or of what may be claimed, but rather as descriptions of features specific to particular embodiments. Certain features that are described in this document in the context of separate embodiments can also be implemented in combination in a single embodiment. Conversely, various features that are described in the context of a single embodiment can also be implemented in multiple embodiments separately or in any suitable sub-combination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a sub-combination or a variation of a sub-combination. Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. 
     Only a few examples and implementations are disclosed. Variations, modifications, and enhancements to the described examples and implementations and other implementations can be made based on what is disclosed.