Patent Publication Number: US-10326538-B2

Title: Remote radio head reciprocity calibration

Description:
TECHNICAL FIELD 
     Embodiments presented in this disclosure generally relate to calibrating wireless devices, and more specifically, to calibrating a plurality of radio heads. 
     BACKGROUND 
     In wireless communication systems, a client device communicates with an access point (AP) to utilize the wireless network associated with the AP. An AP may have multiple radio heads for communicating with a client device. Modern wireless communication networks may have multiple APs having multiple radio heads in order to provide multiple client devices simultaneous access to the wireless network. However, as the number of radio heads increases, so does the amount of interference caused by the radio heads. Thus, radio heads may need to be calibrated to improve communication with clients of the wireless network. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       So that the manner in which the above-recited features of the present disclosure can be understood in detail, a more particular description of the disclosure, briefly summarized above, may be had by reference to embodiments, some of which are illustrated in the appended drawings. It is to be noted, however, that the appended drawings illustrate only typical embodiments of this disclosure and are therefore not to be considered limiting of its scope, for the disclosure may admit to other equally effective embodiments. 
         FIG. 1  illustrates a wireless system, according to one embodiment disclosed herein. 
         FIG. 2  illustrates a controller coupled with a plurality of radio heads, according to one embodiment disclosed herein. 
         FIG. 3  is a method of calibrating a plurality of radio heads in a wireless system, according to one embodiment disclosed herein. 
     
    
    
     To facilitate understanding, identical reference numerals have been used, where possible, to designate identical elements that are common to the figures. It is contemplated that elements disclosed in one embodiment may be beneficially utilized on other embodiments without specific recitation. 
     DESCRIPTION OF EXAMPLE EMBODIMENTS 
     Overview 
     One embodiment presented in this disclosure is a method for calibrating a first radio head and a second radio head, wherein the first radio head and the second radio head comprise a respective plurality of antennas. The method comprises transmitting, via the plurality of antennas of the first radio head, a first calibration signal to the second radio head, wherein the plurality of antennas of the second radio head receives the first calibration signal. The method also comprises transmitting, via the plurality of antennas of the second radio head, a second calibration signal to the first radio head, wherein the plurality of antennas of the first radio head receives the second calibration signal. The method further comprises calibrating the first radio head and the second radio head based on a calibration vector determined from the calibration signals received at the respective plurality of antennas of the first radio head and the second radio head. 
     One embodiment presented in this disclosure is a system, comprising a processor; and a memory comprising instructions which, when executed on the processor, performs an operation for calibrating a first radio head and a second radio head, wherein the first radio head and the second radio head comprise a respective plurality of antennas. The operation comprises transmitting, via the plurality of antennas of the first radio head, a first calibration signal to the second radio head, wherein the plurality of antennas of the second radio head receives the first calibration signal. The operation also comprises transmitting, via the plurality of antennas of the second radio head, a second calibration signal to the first radio head, wherein the plurality of antennas of the first radio head receives the second calibration signal. The operation further comprises calibrating the first radio head and the second radio head based on a calibration vector determined from the calibration signals received at the respective plurality of antennas of the first radio head and the second radio head. 
     One embodiment presented in this disclosure is a computer program product for calibrating a first radio head and a second radio head, wherein the first radio head and the second radio head comprise a respective plurality of antennas. The computer program product comprises a non-transitory computer-readable medium having program instructions embodied therewith, the program instructions executable by a processor to perform an operation that comprises transmitting, via the plurality of antennas of the first radio head, a first calibration signal to the second radio head, wherein the plurality of antennas of the second radio head receives the first calibration signal. The computer program product also comprises transmitting, via the plurality of antennas of the second radio head, a second calibration signal to the first radio head, wherein the plurality of antennas of the first radio head receives the second calibration signal. The computer program product further comprises calibrating the first radio head and the second radio head based on a calibration vector determined from the calibration signals received at the respective plurality of antennas of the first radio head and the second radio head. 
     Example Embodiments 
     In wireless communications, Multiple-Input and Multiple-Output (MIMO) is a method of communicating between multiple transmit and receive antennas to take advantage of spatial diversity and spatial multiplexing. A common wireless system used today is Wi-Fi, which is a Time Division Duplex (TDD) wireless system that uses the same frequency band to transmit and receive data by using the band over disjoint intervals. MIMO systems that utilize TDD either need explicit channel characteristics to determine spatial processing parameters or attempt to determine the channel characteristics implicitly by using multi-antenna samples to determine the transmit strategy per client. The channel characteristics define the amplitude and phase changes caused when wireless signals are transmitted in a communication channel between two network devices. 
     In wireless networks, radio heads may utilize beamforming to improve communication among radio heads and/or client devices. Beamforming is a method that takes advantage of the channel characteristics of wireless networks. With beamforming, the phase and gain of a plurality of signals are altered in such a way to create constructive interference to improve the combined signals at specific location. Thus, a MIMO system may use beamforming to provide improved signal strength at the location of a user. 
     The explicit approach to determining the channel characteristics requires regular communication between the client and the multiple APs, which is expensive in terms of signaling cost. The APs comprise one or more radio heads (RH), which is a distributed transceiver that mixes down to baseband I/Q and feeds those samples back to the centralized processing system of the AP. The implicit approach relies on channel reciprocity between the communicating RHs. That is, the channel characteristics determined by the transmitter can be used to determine the channel characteristics for the receiver without communication among the radio heads. For example, the effective channel (observed at baseband) from a first RH A to a second RH B is the product of transmitter gain of RH A multiplied by the propagation channel which is multiplied by the receiver gain of RH B. In contrast, the effective channel from RH B to RH A is the product of transmitter gain of RH B multiplied by the propagation channel which is multiplied by the receiver gain of RH A. 
     Although the selected receiver and transmitter gains of RH A and RH B may be known, these are analog quantities that drift from the original specification of the RHs due to external factors such as manufacturing tolerances; aging; infeasibility of measuring due to large number of automatic gain control (AGC) states, power transmit states, bands and/or channels; etc. Thus, the implicit approach discussed above becomes less accurate because the implicit approach relies on the original specification of the RHs and does not take into account interference that is present in real world situations. Therefore, a wireless network can be improved by calibrating the physical characteristics of the RHs. 
     For example, when the RHs are distributed to cover a larger area and are coordinated to form a coordinated remote radio head cluster implementing MIMO technology, all transmit and receive chains of a remote radio head cluster can be jointly calibrated for reciprocity. However, because remote radio head are separated from each other to cover a large area, channel estimates for calibrating tend to be noisy compared to calibrating the antennas of a single AP with a small distance between antennas. 
       FIG. 1  illustrates a wireless network having multiple remote radio heads (RRHs) and a controller, according to one embodiment herein. As shown in  FIG. 1 , the wireless network  100  includes four RRHs  105 . Further, the RRHs  105  are connected with a controller  110 . In one embodiment, the controller  110  is implemented as a single computing device having any suitable form. In other embodiments, the controller  110  is a distributed computing device and includes a plurality of discrete computing devices that are connected through wired or wireless networking. While four RRHs  105  have been illustrated for simplicity, the system may contain any number of RRHs and should not be limited to four. 
     As shown in  FIG. 1 , each of the RRHs  105  has four wireless antennas for communicating with client devices (not shown). In one exemplary embodiment, the RRHs communicate using the 802.11 protocol, and the RRHs  105  form a single network for communicating with client devices, such as a MIMO network. Further, the RRHs  105  may communicate with a client device using beamforming, as is known in the art. While the RRHs  105  have been illustrated with four wireless antennas, it is to be understood that the RRHs may have any number of antennas and should not be limited only to four. 
     In one exemplary embodiment, the four RRHs  105  and the wireless antennas of the RRHs  105  are synchronized via the controller  110 . That is, each of the RRHs  105  operates in unison with one another based upon commands from the controller  110 . In this manner, all the RRHs  105  are precisely synchronized during communication to client devices. In one embodiment, the controller  110  synchronizes the RRHs  105  by transmitting a high precision clock signal to each of the RRHs  105 . 
     Turning to  FIG. 2 , a detailed version of controller  110  and RRHs  105  of  FIG. 1  are illustrated according to an exemplary embodiment. The controller  110  comprises a processor  205 , input/output (I/O) interface  210 , and memory  215 . Processor  205  controls the operation of the controller  110 . The I/O  210  allows the controller  110  to communicate with other electronic devices (e.g., the RRHs, client devices, network devices, etc.). For example, the I/O  210  may have physical data ports to establish a wired connection with the RRHs  105 . 
     The processor  205  may include any processing element suitable for performing functions described herein, and may include single or multiple core processors, as well as combinations thereof. While the controller  110  is shown as having a single processor  205 , in alternative embodiments the controller  110  has a plurality of processors  205 . The processor  205  may be included in a single computing device, or may represent an aggregation of processing elements included across a number of networked computing devices. 
     In one exemplary embodiment, the controller  110  communicates with the RRHs  105  via the I/O  210 . For example, the RRHs  105  may be physically connected with the controller  110  via a wired connection, such as Ethernet cables, and the controller  110  communicates with the RRHs via the Ethernet cables. Further, the controller  110  may be connected to an external network (not shown), such as the Internet, or to additional RRHs  105 . 
     Memory  215  may include a variety of computer-readable media selected for their size, relative performance, or other capabilities: volatile and/or non-volatile media, removable and/or non-removable media, etc. Memory  215  may include cache, random access memory (RAM), Flash memory, a hard disk drive, a solid state drive, an optical storage device, and/or a magnetic storage device. In one embodiment, the memory  215  includes software applications for controlling the digital part of the PHY layer, as well as the MAC layer function of the RRHs  105 . In one embodiment, the digital PHY layer includes the functions for processing digital signals received at the RRHs  105 . Further, the MAC layer includes function for channel access control. Thus, in one exemplary embodiment, the central controller  110  performs the digital PHY layer functions and the MAC layer functions for all of the RRHs  105 . While the digital PHY layer and the MAC layer have been described as being within memory  215 , a person skilled in the art would appreciate that the PHY layer and the MAC layer may be hardware, software, or a combination of hardware and software. 
     As shown, memory  215  comprises calibration module  220  and calibration estimates  225 . In one exemplary embodiment, calibration module  220  is a method for calibrating a plurality of RRHs  105 , as will be discussed in further detail below with regards to  FIG. 3 . Those skilled in the art should appreciate that memory  215  may contain additional items such as firmware for operating the controller  110 , which have not been illustrated for simplicity. 
     Calibration estimates  225  are estimates compiled from the calibration module  220 . In one exemplary embodiment, the calibration estimates  225  comprise a matrix containing calibration estimates for all of the antennas of each of the respective RRHs  105  being calibrated. In another exemplary embodiment, the calibration estimates  225  comprise channel estimates obtained by each RRH  105  at each of the respective RRH&#39;s antennas for each of the antennas of the remaining RRHs  105 . For example, if there are two RRHs  105  having four antennas each, each antenna will have four calibration estimates, which is one calibration estimate for each of the other RRH&#39;s  105  antennas. Thus, with there will be thirty-two calibration estimates in total. While calibration estimates  225  have been described as a being stored in a matrix for simplicity, it should be understood by a person skilled in the art that any method of storing the calibration estimates  225  is possible. 
     The RRHs  105  are illustrated with an analog PHY layer  230  that receives and transmits wireless signals (i.e., an analog interface), while the central controller  110  includes a digital portion of a PHY layer and a MAC layer for processing received packets and transmitting packets to the RRHs  105 . Thus, the PHY layer may be split between the RRHs  105  and the central controller  110  where the higher level processing layers are in the central controller  110 . However, in one embodiment, the RRHs  105  comprise a processor (not shown) and memory (not shown) for processing digital signals received via the PHY layer. The RRHs  105  may also include high-level processing layers (e.g., MAC or physical coding sublayer (PCS)) to evaluate the received packets. 
     Although not shown, the RRHs  105  may include one or more antennas for wireless communication. In one exemplary embodiment, the RRHs  105  are capable of communicating using the 802.11 wireless communication standard via the one or more antennas. In another exemplary embodiment, the RRHs  105  are capable of communicating with a cellular network. Further, the RRHs  105  may comprise a plurality of antennas for communicating using different standards. For example, one antenna may be capable of communicating using the 802.11 wireless standard, while another antenna is capable of communicating with a cellular network. Thus, the RRHs  105  are capable of communicating with a plurality of wireless networks and a plurality of clients using the antennas. 
       FIG. 3  illustrates an exemplary method for calibrating the RRHs  105  of  FIG. 1 . While four RRHs  105  each having four antennas are used for ease of explanation of the method, any number of RRHs having any number of antennas can be calibrated using the method described below such as 2, 6, 16, 100, etc. For ease of explanation, the four RRHs  105  of  FIG. 1  will be referred to as W, X, Y, and Z. Each antenna of the respective RRH is designated with a number. For example, the first antenna of RRH W is referred to as W 1 , the second as W 2 , and so forth. Further, while the method has been described as applying to remote radio heads, a person skilled in the art will appreciate the method may also be applied to non-remote radio heads, and the terms RRH and radio head (RH) will be used interchangeably in the following paragraphs. 
     The method begins at block  305 , where a first RH transmits a calibration signal. In one exemplary embodiment, the calibration signal is a calibration packet, which is a Null Data Packet (NDP) consisting of a PHY preamble containing one or more 802.11 wireless standard training fields to fully sound the channel. In one exemplary embodiment, the controller  110  derives the channel estimate from the High Throughput (HT), Very High Throughput (VHT), or High Efficiency-Long Training Field (HT/VHT/HE-LTF field) of the NDP. The NDP may include multiple copies of the HT/VHT/HE-LTF field for increased immunity against interference and noise. 
     For example, W transmits a calibration signal via the four antennas of W that is received at X, Y, and Z. Thus, each antenna of X, Y, and Z receives a calibration signal from each of the four antennas of W. Therefore, X 1  receives a separate signal from W 1 , W 2 , W 3 , and W 4 ; X 2  receives a separate calibration signal from W 1 , W 2 , W 3 , and W 4  and so forth for each of the antennas of X, Y, and Z. Based upon each of the received signals, a respective channel estimate representing the channel characteristics is determined between the antennas that sent the calibration signals and the antennas that received the calibration signals. In one exemplary embodiment, the channel estimates are derived from the preamble of the calibration signal and stored in a matrix for later calibration. In one exemplary embodiment, the RHs W, X, Y, and Z each take turns transmitting the calibration signal to the antennas of the other RHs. In one exemplary embodiment, the periods in which the RHs transmit are non-overlapping such that only a single RH is transmitting at a time. 
     After block  310 , if all the RHs of the system have transmitted a calibration signal, the method continues to block  315 . However, if all the RHs of the system have not transmitted a calibration signal, the method returns to block  310  to allow the remaining RHs to transmit a calibration signal. Stated differently, at block  310  the RHs transmit a calibration signal one at a time in a round robin fashion until all RHs have transmitted a calibration signal. 
     To better explain how the channel estimate is determined and stored, an example is provided for channel estimates between two radio heads A and B. As mentioned above, the effective channel (observed at baseband) from a first RH A to a second RH B is the product of transmitter gain of RH A multiplied by the propagation channel multiplied by the receiver gain of RH B. Thus, rewritten in equation form, B=R B H AB T A  where B is the channel estimate at RH B when radio head A is transmitting. In one exemplary embodiment, B is an M×M matrix that is not sparse. R B  is a matrix comprising receive path responses for radio head B. T A  is a matrix holding transmit path response for radio head A. H AB  is the propagation channel matrix between the antennas of RH A and the antennas of RH B. The channel estimate at the j-th antenna of RH B when the i-th antenna of RH A is transmitting is defined as b j,i . 
     Similarly, the effective channel from RH B to RH A is the product of transmitter gain of RH B multiplied by the propagation channel multiplied by the receiver gain of RH A. Rewritten in equation form, A=R A H BA T B . Due to reciprocity, A=R A H BA T B =R A (H AB ) T T B . In one exemplary embodiment, A is a M×M matrix that is not sparse, R A  is a matrix comprising receive path responses for radio head A, and T B  is a matrix holding transmit path response for radio head B. H BA  is the propagation channel matrix between the antennas of RH B and the antennas of RH A. The channel estimate at the j-th antenna of RH A when the i-th antenna of RH B is transmitting is defined as a j,i . 
     Given the above definitions, the channel estimate at the i-th antenna of A when the j-th antenna of B is transmitting is a i,j =r a,i *h i,j *t b,j , where t b,j  is the transmit gain at the j-th antenna of B, r a,i  is the receive gain at the i-th antenna of A, and h i,j  is the propagation channel between the two radio heads. Of note, the propagation channel is reciprocal, which means that h i,j =h j,i . 
     Similarly, the channel estimate at the j-th antenna of B when the i-th antenna of A is transmitting is b j,i =r b,j *h j,i *t a,i , where t a,i  is the transmit gain at the i-th antenna of A, r b,j  is the receive gain at the j-th antenna of B, and h i,j  is the propagation channel between the two radio heads. As stated above, the propagation channel is reciprocal, which means that h j,i =h i,j . 
     Using the above formulas, the equivalent of the matrix relationship for calibration coefficients is obtained which can be proved by substitution as follows: 
     
       
         
           
             
               
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     This relationship translates into the following matrix equation for the calibration coefficients for both radio heads. In the below matrix, Zx=0 and x≠0. In this manner, a matrix is populated with the channel estimates for each antenna of RH A and RH B. In one exemplary embodiment, the channel estimates are based upon the received calibration signals. This matrix can then be used to solve the calibration coefficients for each of the antennas of RH A and RH B in order to calibrate the RHs. 
     
       
         
           
             
               
                 
                   
                       
                     
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     Returning to  FIG. 3 , at block  315 , the controller calibrates the RHs. In one exemplary embodiment, the phase and amplitude imbalances between the transmit and receive paths of the RHs are calibrated as a function of frequency. In one exemplary embodiment, all of the transmit and receive paths of all the RHs are calibrated. In another exemplary embodiment, some (e.g., 99%, 95%, 75%, 50%, etc.) of the transmit and receive paths of the RHs are calibrated. The calibration may be per band on the current channel, per band on a different channel, per block of subcarrier, and/or per individual subcarrier. Further, the bulk timing differences between the RHs and the controller are calibrated. In one exemplary embodiment, calibration coefficients are used to calibrate the RHs, which are determined from the transmitted and received calibration signals. In one exemplary embodiment, the calibration coefficients are complex numbers that represent the phase and amplitude adjustment required for the receive path and transmit path so that the multi-RH channel becomes reciprocal. By calibrating the RHs, complete knowledge of the wireless channel is determined to allow for improved communication to client devices. That is, once the RHs are calibrated, the RHs are capable of using beamforming to improve communications with client devices with minimal impact from external factors. Further, the methods of calibration described below provide significant cost savings in terms of processing compared to explicit calibration, while at the same time providing similar benefits to explicit calibration. 
     In one exemplary embodiment, the gains of RHs X, W, Y, and Z are determined from the sent and received calibration signals. In another exemplary embodiment, each antenna of the RHs is calibrated with a respective gain based on the sent and received calibration signals. In one exemplary embodiment, the calibration of the RHs is a calibration vector determined as the right singular vector of a matrix (Z) comprising the estimated channels for each RH. In another exemplary embodiment, the calibration of the RHs is a calibration vector determined as the eigenvector with the smallest eigenvalue of R, which is a Z′ *Z matrix. 
     Returning to the example above with RHs A and B, an introduction to solving the calibration coefficients will be provided. The calibration vector c A  for RH A is an M×1 vector which is the diagonal elements of R A (T A ) −1 =(T A ) −1 R A . Likewise, the calibration vector c B  for RH B is an M×1 vector which is the diagonal elements of R B (T B ) −1 =(T B ) −1 R B . The solution for the calibration coefficients of the radio heads is the matrix x above, which is the right singular vector of the matrix Z, defined above, with the smallest singular value. Ideally, the smallest singular value is zero. Further, the solution to Zx=0, as the right singular vector of Z with the lowest singular value is the least squares solution. 
     Since Z is a M 2 ×2M matrix, its rank can be at most 2M with estimation errors when M&gt;1. In one exemplary embodiment, the rank of Z is 2M−1 for a unique solution to the calibration coefficients. However, under adverse conditions, the rank of Z may be less than that. For example, if the first antenna of RH B does not receive any energy from RH A transmissions, and due to reciprocity the RH A receivers do not receive any energy from the first antenna of RH B, the rank of Z drops by two since the first column and the (M+1)th columns of Z are zero. However, because the channel estimates are determined when forming Z, the channel estimates that are close to zero may be removed from Z, and Z can be solved for the valid set of identifiable calibration factors. In one exemplary embodiment, the unidentifiable calibration factors may be estimated via calibration with a different RH. In another exemplary embodiment, the unidentifiable calibration factors may determine from self-calibration among the antennas of the same RH. 
     Thus, an example of a calibration vector when two RHs (A and B) are signaling each other in block form is produced below: 
     
       
         
           
             
               
                 
                   F 
                   A 
                 
                 ⁡ 
                 
                   ( 
                   B 
                   ) 
                 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           b 
                           
                             1 
                             , 
                             1 
                           
                         
                       
                       
                         0 
                       
                       
                         … 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           b 
                           
                             2 
                             , 
                             1 
                           
                         
                       
                       
                         0 
                       
                       
                         ⋱ 
                       
                       
                         ⋮ 
                       
                     
                     
                       
                         ⋮ 
                       
                       
                         ⋮ 
                       
                       
                         ⋱ 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           b 
                           
                             M 
                             , 
                             1 
                           
                         
                       
                       
                         0 
                       
                       
                         … 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           b 
                           
                             1 
                             , 
                             2 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           b 
                           
                             2 
                             , 
                             2 
                           
                         
                       
                       
                         ⋮ 
                       
                       
                         ⋮ 
                       
                     
                     
                       
                         ⋮ 
                       
                       
                         ⋮ 
                       
                       
                         ⋮ 
                       
                       
                         ⋮ 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           b 
                           
                             M 
                             , 
                             2 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                             
                         
                       
                       
                         
                           ⋮ 
                           ⁢ 
                           
                               
                           
                         
                       
                       
                         
                             
                         
                       
                       
                         
                             
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           b 
                           
                             1 
                             , 
                             M 
                           
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           b 
                           
                             2 
                             , 
                             M 
                           
                         
                       
                     
                     
                       
                         ⋮ 
                       
                       
                         ⋮ 
                       
                       
                         ⋮ 
                       
                       
                         ⋮ 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           b 
                           
                             M 
                             , 
                             M 
                           
                         
                       
                     
                   
                   ] 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 and 
               
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 G 
                 B 
               
               ⁡ 
               
                 ( 
                 A 
                 ) 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       - 
                       
                         a 
                         
                           1 
                           , 
                           1 
                         
                       
                     
                   
                   
                     0 
                   
                   
                     … 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       - 
                       
                         a 
                         
                           1 
                           , 
                           2 
                         
                       
                     
                   
                   
                     ⋱ 
                   
                   
                     ⋮ 
                   
                 
                 
                   
                     ⋮ 
                   
                   
                     ⋱ 
                   
                   
                     ⋱ 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     … 
                   
                   
                     0 
                   
                   
                     
                       - 
                       
                         a 
                         
                           1 
                           , 
                           M 
                         
                       
                     
                   
                 
                 
                   
                     
                       - 
                       
                         a 
                         
                           2 
                           , 
                           1 
                         
                       
                     
                   
                   
                     0 
                   
                   
                     … 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       - 
                       
                         a 
                         
                           2 
                           , 
                           2 
                         
                       
                     
                   
                   
                     ⋱ 
                   
                   
                     ⋮ 
                   
                 
                 
                   
                     ⋮ 
                   
                   
                     ⋱ 
                   
                   
                     ⋱ 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     … 
                   
                   
                     0 
                   
                   
                     
                       - 
                       
                         a 
                         
                           2 
                           , 
                           M 
                         
                       
                     
                   
                 
                 
                   
                     
                         
                     
                   
                   
                     ⋮ 
                   
                   
                     
                         
                     
                   
                   
                     
                         
                     
                   
                 
                 
                   
                     
                       - 
                       
                         a 
                         
                           M 
                           , 
                           1 
                         
                       
                     
                   
                   
                     0 
                   
                   
                     … 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     
                       - 
                       
                         a 
                         
                           M 
                           , 
                           2 
                         
                       
                     
                   
                   
                     ⋱ 
                   
                   
                     ⋮ 
                   
                 
                 
                   
                     ⋮ 
                   
                   
                     ⋱ 
                   
                   
                     ⋱ 
                   
                   
                     0 
                   
                 
                 
                   
                     0 
                   
                   
                     … 
                   
                   
                     0 
                   
                   
                     
                       - 
                       
                         a 
                         
                           M 
                           , 
                           M 
                         
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
     F A (B) consists of the channel estimates obtained by RH B when RH A is transmitting. That is, b i,j  is the channel estimate at RH B&#39;s i-th antenna when RH A&#39;s j-th antenna is transmitting. G B (A) consists of channel estimates obtained by RH A when RH B is transmitting. That is, a i,j  is the channel estimate of RH A&#39;s i-th antenna when RH B&#39;s j-th antenna is transmitting. 
     Applying the definitions above: 
     
       
         
           
             
                 
             
             ⁢ 
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               F 
                               A 
                             
                             ⁡ 
                             
                               ( 
                               B 
                               ) 
                             
                           
                         
                         
                           
                             
                               G 
                               B 
                             
                             ⁡ 
                             
                               ( 
                               A 
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   
                     ︸ 
                     Z 
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         
                           c 
                           A 
                         
                       
                     
                     
                       
                         
                           c 
                           B 
                         
                       
                     
                   
                   ] 
                 
               
               = 
               
                 0 
                 
                   2 
                   ⁢ 
                   M 
                   × 
                   1 
                 
               
             
           
         
       
     
     Using the aforementioned matrix, and applying it to the system of  FIG. 1  (i.e., four RHs (W, X, Y, and Z) each having four antennas) that have provided the calibration signal in a round robin fashion results in the following matrix, which will be referred to as Z system . The calibration vectors for W, X, Y, and Z are also shown. 
     
       
         
           
               
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               F 
                               A 
                             
                             ⁡ 
                             
                               ( 
                               B 
                               ) 
                             
                           
                         
                         
                           
                             
                               G 
                               B 
                             
                             ⁡ 
                             
                               ( 
                               A 
                               ) 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           
                             
                               F 
                               A 
                             
                             ⁡ 
                             
                               ( 
                               C 
                               ) 
                             
                           
                         
                         
                           0 
                         
                         
                           
                             
                               G 
                               C 
                             
                             ⁡ 
                             
                               ( 
                               A 
                               ) 
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           
                             
                               F 
                               A 
                             
                             ⁡ 
                             
                               ( 
                               D 
                               ) 
                             
                           
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             
                               G 
                               D 
                             
                             ⁡ 
                             
                               ( 
                               A 
                               ) 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             
                               F 
                               B 
                             
                             ⁡ 
                             
                               ( 
                               C 
                               ) 
                             
                           
                         
                         
                           
                             
                               G 
                               C 
                             
                             ⁡ 
                             
                               ( 
                               B 
                               ) 
                             
                           
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             
                               F 
                               B 
                             
                             ⁡ 
                             
                               ( 
                               D 
                               ) 
                             
                           
                         
                         
                           0 
                         
                         
                           
                             
                               G 
                               D 
                             
                             ⁡ 
                             
                               ( 
                               B 
                               ) 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           
                             
                               F 
                               C 
                             
                             ⁡ 
                             
                               ( 
                               D 
                               ) 
                             
                           
                         
                         
                           
                             
                               G 
                               D 
                             
                             ⁡ 
                             
                               ( 
                               C 
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   
                     ︸ 
                     
                       Z 
                       system 
                     
                   
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       
                         
                           c 
                           A 
                         
                       
                     
                     
                       
                         
                           c 
                           B 
                         
                       
                     
                     
                       
                         
                           c 
                           C 
                         
                       
                     
                     
                       
                         
                           c 
                           D 
                         
                       
                     
                   
                   ] 
                 
               
               = 
               0 
             
           
         
       
     
     Every block submatrix of the coefficient matrix Z system  is (M 2 ×M), resulting in a (M 2 *H*(H−1)/2×M*H) coefficient matrix. Although the size of Z system  is large, we can determine the calibration coefficients by determining the eigenvector with the smallest eigenvalue. In one exemplary embodiment, singular value decomposition (SVD) is used to determine the eigenvector of the matrix. 
     Once the calibration coefficients are determined from the matrix, the gains can be determined for each of the RHs. That is, the effective channel (observed at baseband) from a first RH A to a second RH B is the product of transmitter gain of RH A multiplied by the propagation channel multiplied by the receiver gain of RH B. Thus, rewritten in equation form, B=R B H AB T A  where B is the channel estimate at RH B when radio head A is transmitting. Since each of the RHs has received a signal from the remaining RHs which allows the RH to determine the receiver gain, as well as the channel estimates, the controller can use the reciprocity described above to solve for the transmitter gain. Thus, the determined calibration coefficients are used to determine the gains of the antennas. In one exemplary embodiment, the calibration coefficients are complex numbers that represent the phase and amplitude adjustment required for each receive and transmit path so that the channel becomes reciprocal among the plurality of RHs. In one exemplary embodiment, the channel estimates may be used to determine the implicit channel (i.e., the uplink channel) measurements among the RHs and calibrate the implicit channel measurements to determine the downlink channel. 
     Returning to  FIG. 3 , at block  320  the calibrated RHs communicate with one or more client devices. In one exemplary embodiment, the RHs send data packets to the client devices, as well as receive data packets. In one exemplary embodiment, the calibrated RHs communicate with the client devices using beamforming. The method concludes after block  320 . 
     In the preceding, reference is made to embodiments presented in this disclosure. However, the scope of the present disclosure is not limited to specific described embodiments. Instead, any combination of the described features and elements, whether related to different embodiments or not, is contemplated to implement and practice contemplated embodiments. Furthermore, although embodiments disclosed herein may achieve advantages over other possible solutions or over the prior art, whether or not a particular advantage is achieved by a given embodiment is not limiting of the scope of the present disclosure. Thus, the preceding aspects, features, embodiments and advantages are merely illustrative and are not considered elements or limitations of the appended claims except where explicitly recited in a claim(s). 
     As will be appreciated by one skilled in the art, the embodiments disclosed herein may be embodied as a system, method or computer program product. Accordingly, aspects may take the form of an entirely hardware embodiment, an entirely software embodiment (including firmware, resident software, micro-code, etc.) or an embodiment combining software and hardware aspects that may all generally be referred to herein as a “circuit,” “module” or “system.” Furthermore, aspects may take the form of a computer program product embodied in one or more computer readable medium(s) having computer readable program code embodied thereon. 
     Any combination of one or more computer readable medium(s) may be utilized. The computer readable medium may be a computer readable signal medium or a computer readable storage medium. A computer readable storage medium may be, for example, but not limited to, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any suitable combination of the foregoing. More specific examples (a non-exhaustive list) of the computer readable storage medium would include the following: an electrical connection having one or more wires, a portable computer diskette, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), an optical fiber, a portable compact disc read-only memory (CD-ROM), an optical storage device, a magnetic storage device, or any suitable combination of the foregoing. In the context of this document, a computer readable storage medium is any tangible medium that can contain, or store a program for use by or in connection with an instruction execution system, apparatus or device. 
     A computer readable signal medium may include a propagated data signal with computer readable program code embodied therein, for example, in baseband or as part of a carrier wave. Such a propagated signal may take any of a variety of forms, including, but not limited to, electro-magnetic, optical, or any suitable combination thereof. A computer readable signal medium may be any computer readable medium that is not a computer readable storage medium and that can communicate, propagate, or transport a program for use by or in connection with an instruction execution system, apparatus, or device. 
     Program code embodied on a computer readable medium may be transmitted using any appropriate medium, including but not limited to wireless, wireline, optical fiber cable, RF, etc., or any suitable combination of the foregoing. 
     Computer program code for carrying out operations for aspects of the present disclosure may be written in any combination of one or more programming languages, including an object oriented programming language such as Java, Smalltalk, C++ or the like and conventional procedural programming languages, such as the “C” programming language or similar programming languages. The program code may execute entirely on the user&#39;s computer, partly on the user&#39;s computer, as a stand-alone software package, partly on the user&#39;s computer and partly on a remote computer or entirely on the remote computer or server. In the latter scenario, the remote computer may be connected to the user&#39;s computer through any type of network, including a local area network (LAN) or a wide area network (WAN), or the connection may be made to an external computer (for example, through the Internet using an Internet Service Provider). 
     Aspects of the present disclosure are described below with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems) and computer program products according to embodiments presented in this disclosure. It will be understood that each block of the flowchart illustrations and/or block diagrams, and combinations of blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions/acts specified in the flowchart and/or block diagram block or blocks. 
     These computer program instructions may also be stored in a computer readable medium that can direct a computer, other programmable data processing apparatus, or other devices to function in a particular manner, such that the instructions stored in the computer readable medium produce an article of manufacture including instructions which implement the function/act specified in the flowchart and/or block diagram block or blocks. 
     The computer program instructions may also be loaded onto a computer, other programmable data processing apparatus, or other devices to cause a series of operational steps to be performed on the computer, other programmable apparatus or other devices to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide processes for implementing the functions/acts specified in the flowchart and/or block diagram block or blocks. 
     The flowchart and block diagrams in the Figures illustrate the architecture, functionality and operation of possible implementations of systems, methods and computer program products according to various embodiments. In this regard, each block in the flowchart or block diagrams may represent a module, segment or portion of code, which comprises one or more executable instructions for implementing the specified logical function(s). It should also be noted that, in some alternative implementations, the functions noted in the block may occur out of the order noted in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each block of the block diagrams and/or flowchart illustration, and combinations of blocks in the block diagrams and/or flowchart illustration, can be implemented by special purpose hardware-based systems that perform the specified functions or acts, or combinations of special purpose hardware and computer instructions. 
     In view of the foregoing, the scope of the present disclosure is determined by the claims that follow.