Patent Publication Number: US-11652519-B2

Title: Method and apparatus for equal energy codebooks for coupled antennas with transmission lines

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is related to an application entitled “Method and Apparatus for Optimizing Antenna Precoder Selection with Coupled Antennas,” U.S. application Ser. No. 15/016,140, filed on Feb. 4, 2016, an application entitled “Method and Apparatus for Equal Energy Codebooks for Antenna Arrays with Mutual Coupling,” U.S. application Ser. No. 14/855,693, filed on Sep. 16, 2015, and an application entitled “Method and Apparatus for Equal Energy Codebooks for Antenna Arrays with Mutual Coupling,” U.S. application Ser. No. 15/157,754, filed on May 18, 2016, all commonly assigned to the assignee of the present application, which are hereby incorporated by reference. 
     BACKGROUND 
     1. Field 
     The present disclosure is directed to a method and apparatus for equal energy codebooks for coupled antennas with transmission lines. 
     2. Introduction 
     Presently, wireless communication devices communicate with other communication devices using wireless signals. Many wireless communication devices have multiple antennas that can transmit more focused signals to a receiving device using antenna beamforming. Multiple telecommunication standards define antenna precoder codebooks to support antenna beamforming and or Multiple-Input/Multiple Output (MIMO) transmission with feedback from the receiver. Telecommunication standards that employ codebooks of precoders include the Third Generation Partnership Project High Speed Packet Access (3GPP HSPA) and Long Term Evolution (LTE) standards, the IEEE 802.11 and 802.16 standards. In all of these standards, the precoders that are defined have the property that each precoding vector has equal L2 norm with the assumption that the precoders are applied to the antenna array in such a way that precoders having equal L2 norm yield antenna patterns with equal power in the far field. 
     In each of the above telecommunication standards, precoders are used in combination with reference symbol transmissions from a transmitter so that a receiver can evaluate the channel that would result from application of each of the precoders. The receiver applies each of the precoders to the reference symbols in order to evaluate the channel quality. It then signals the index of the best precoder and the corresponding channel quality back to the transmitter. For some transmission modes, the precoder used for the data transmission is signaled to the receiver, which then applies the precoder to estimate the channel for the data symbols. 
     Implicit in the operation of these types of systems is the assumption that the precoders are applied in a manner such that the antenna pattern corresponding to each precoder has the same transmit power. The reason for this assumption is that it is the objective of the receiver to select the precoder which maximizes its channel quality, and thus the achievable data rate, for a given transmit power. In the case of a single user, this will maximize the transmission range of a fixed data rate, or alternatively, the achievable data rate at a fixed range. Alternatively, for multi-user systems, it is desirable to minimize the transmit power needed to achieve a given data rate for each user, as this transmit power is interference for all users other than the target user. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       In order to describe the manner in which advantages and features of the disclosure can be obtained, a description of the disclosure is rendered by reference to specific embodiments thereof which are illustrated in the appended drawings. These drawings depict only example embodiments of the disclosure and are not therefore to be considered to be limiting of its scope. The drawings may have been simplified for clarity and are not necessarily drawn to scale. 
         FIG.  1    is an example block diagram of a system according to a possible embodiment; 
         FIG.  2    is an example illustration of a Thevenin source and two-port antenna array model with a transmission line according to a possible embodiment; 
         FIG.  3    is an example illustration of a matching network between a Thevenin source and an antenna array with an impedance matrix according to a possible embodiment; 
         FIG.  4    is an example illustration of a matching network combined with an antenna array with a resulting combined impedance matrix according to a possible embodiment; 
         FIG.  5    is an example illustration of variation of transmit power as a function of phase for a two-element dipole array with half-wavelength spacing driven by a 50 ohm Thevenin source and 50 ohm transmission lines according to a possible embodiment; 
         FIG.  6    is an example illustration of a Norton source and two-port antenna model with a transmission line according to a possible embodiment; 
         FIG.  7    is an example illustration of a matching network between a Norton source and an antenna array with an impedance matrix according to a possible embodiment; 
         FIG.  8    is an example illustration of a matching network combined with an antenna array with a resulting combined impedance matrix according to a possible embodiment; 
         FIG.  9    is an example graph of variation of transmit power as a function of phase for a two-element dipole array with half-wavelength spacing driven by a 50 ohm Thevenin source with isolators and 50 ohm transmission lines according to a possible embodiment; 
         FIG.  10    is an example flowchart illustrating the operation of a wireless communication device according to a possible embodiment; 
         FIG.  11    is an example flowchart illustrating the operation of a wireless communication device according to a possible embodiment; and 
         FIG.  12    is an example block diagram of an apparatus according to a possible embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     Embodiments provide a method and apparatus for equal energy codebooks for coupled antennas with transmission lines. According to a possible embodiment, a plurality of precoders can be received from a codebook in a transmitter having an antenna array. Each precoder of the plurality of precoders can be transformed to a transformed precoder such that the transmit power for each transformed precoder is equal to the transmit power for each of the other transformed precoders of the plurality of precoders. The transmit power can be expressed as a quadratic form with respect to the corresponding precoder. The quadratic form can be based on a transmission line impedance of a transmission line between a signal source and the antenna array. A signal can be received from the signal source. A transformed precoder of the plurality of transformed precoders can be applied to the signal to generate a precoded signal for transmission over a physical channel. The precoded signal can be transmitted. 
       FIG.  1    is an example block diagram of a system  100  according to a possible embodiment. The system  100  can include a transmitting device  110  and a receiving device  120 . The transmitting device  110  can be a User Equipment (UE), a base station, or any other device that can transmit wireless signals. Similarly, the receiving device  120  can be a UE, a base station, or any other device that can receive wireless signals. A UE can be a wireless terminal, a portable wireless communication device, a smartphone, a cellular telephone, a flip phone, a personal digital assistant, a device having a subscriber identity module, a personal computer, a selective call receiver, a tablet computer, a laptop computer, or any other device that is capable of sending and receiving wireless communication signals. A base station can be a wireless wide area network base station, a wireless local area network base station, an enhanced NodeB (eNB), an access point, or any other base station. 
     The transmitting device  110  can include a precoder transformation controller  112 , a codebook  114 , and an antenna array  116 . The precoder transformation controller  112  can be one element or can be distributed between different elements. For example, the precoder transformation controller  112  can be part of a processor, can be part of a transceiver, can be part of a precoder, can be part of other elements in a transmitting device, and/or can be distributed between combinations of elements in a transmitting device and/or over cloud computing. The receiving device  120  can include at least one antenna  122 . For example, in some embodiments the receiving device  120  can have one antenna and in other embodiments the receiving device  120  can have an array of antennas. The transmitting device  110  can also act as a receiving device and the receiving device  120  can also act as a transmitting device depending on which device is currently transmitting or receiving. 
     If there is no mutual coupling of the antenna array  116 , such as a transmit array, then it will be true that antenna precoding vectors having equal L2 norm will yield antenna patterns with equal power (note that some assumptions may be necessary, such as the antennas having equal self-impedance). However, if the antennas of the antenna array  116  are coupled, then the antenna patterns resulting from two precoders having the same L2 norm can differ in transmit power by several dB. The amount of this difference can depend on multiple factors, including the mutual coupling coefficients, the type of source used to drive the array, the source impedance, and/or other factors. The power delivered to the antenna array  116  can depend on relative phases of the inputs of a vector voltage source or a vector current source, on the source impedances, and/or on other factors. 
     In the case that a transmission line is used between a source of a source signal  118  and the antenna array  116 , the transmitted power can vary with the relative phase of the input voltage vector or input source vector, and this variation can further depend on the source impedance, the transmission line impedance, the length of the transmission line, the antenna impedance (and any matching circuitry), and/or other factors. If a transmission line is used between the source and the antenna array  116  in combination with an isolator at the source, the transmitted power can still vary with the relative phase of the input voltage vector or input source vector, but this variation may now depend only on the transmission line impedance, the antenna impedance (and any matching circuitry), and/or other factors so that the power variation may no longer depend on the source impedance or the length of the transmission line. 
     Embodiments can show that the power variation as a function of the precoder can be expressed as a quadratic form, which is non-negative definite for the cases in which transmission lines are used between a source and an antenna array. Using these quadratic forms for the transmit power, at least two methods can be used for mapping the precoders to antenna patterns with equal transmit power. In a first method, each precoder can be scaled by the inverse square root of the transmit power that results from the unscaled precoder. In the second method, the set of precoders can be transformed by multiplying each precoder by a matrix, so that the resulting set of precoders all map to antenna patterns having the same power. If precoder-based channel estimation is used in combination with common reference symbols, then the same precoder transformation can also be applied to the common reference symbol precoders. 
       FIG.  2    is an example illustration  200  of a Thevenin source and two-port antenna array model with a transmission line according to a possible embodiment.  FIG.  3    is an example illustration  300  of a matching network explicitly shown between a Thevenin source and an antenna array with impedance matrix Z according to a possible embodiment.  FIG.  4    is an example illustration  400  of a matching network combined with an antenna array with a resulting combined impedance matrix Z′ according to a possible embodiment, where the combined impedance matrix Z′ can be used instead of the impedance matrix Z in equations herein to account for a matching network. 
     This embodiment can consider transmit power with a Thevenin Source, transmission lines, and no isolators at the source. For example, the illustration  200  shows an antenna array  230  driven by a Thevenin source  210  with a transmission line  220  between the source  210  and the antenna array  230 . This can be used to determine transmit power with a Thevenin source, transmission lines, and no isolators at the source. The Thevenin source  210  can include an ideal vector voltage source v s  in combination with a series impedance Z S_Thev , where Z S_Thev  is a diagonal matrix with diagonal elements equal to the series impedance Z S_Thev1  and Z S_Thev2  for each voltage source v s1  and v s2 , respectively. The impedance looking into a transmission line can be given by
 
 Z   in ( l )= Z   0 ( Z+jZ   0   I   2  tan(2π l ))( Z   0   I   2   +jZ  tan(2π l )) −1 ,
 
where Z 0  can be the impedance of the transmission line, l can be the length of the transmission line in wavelengths, and Z can be the impedance matrix for the combination of the antenna array and any impedance matching circuitry between the transmission line and the antenna array.
 
     The transmit power can be given by
 
 Re ( v   S   H ( Z   S_Thev   +Z   in ( l )) −H   Z   in ( l )( Z   S_Thev   +Z   in ( l )) −1   v   S )= v   S   H ( Z   S_Thev   +Z   in ( l )) −H   Re ( Z   in ( l ))( Z   S_Thev   +Z   in ( l )) −1   v   S ,
 
where the matrix Z S_Thev  can be a diagonal matrix with elements equal to the source impedances in series with the Thevenin voltage sources. This expression can be further simplified as the quadratic form
 
 v   S   H ( Z   S_Thev   +Z   in ( l )) −H   Re ( Z   in ( l ))( Z   S_Thev   +Z   in ( l )) −   v   S   =v   S   H   Q   Thev ( Z   S_Thev   ,Z   in ( l )) v   S ,
 
where
 
 Q   Thev ( Z   S_Thev   ,Z   in ( l ))=( Z   S_Thev   +Z   in ( l )) −H   Re ( Z   in ( l ))( Z   S_Thev   +Z   in ( l )) −1 .
 
     As an example of a two-element array of half-wavelength dipoles with half-wavelength spacing, the impedance matrix for this array can be given by 
             Z   =       [           73   +     j   ·   42.5               -   13     -     j   ·   25                   -   13     -     j   ·   25             73   +     j   ·   42.5             ]     .           
The additional following parameters for this example can be assumed as: source impedance=50 ohms; transmission line impedance=50 ohms; and transmission line length=one-quarter wavelength.
 
     A voltage source of the form v(θ)=[1 exp(jθ)] T  can be considered for which the L2 norm of the precoder v(θ) can be independent of the phase θ, so that
 
∥ v (θ)∥ 2 =2
 
for all θ.
 
       FIG.  5    is an example illustration  500  of variation of transmit power as a function of phase θ for a two-element dipole array with half-wavelength spacing driven by a 50 ohm Thevenin source and 50 ohm transmission lines according to a possible embodiment. It can be noted that the transmit power varies by 1.3 dB even though the L2 norm of the precoder is held constant. 
       FIG.  6    is an example illustration  600  of a Norton source and two-port antenna model with a transmission line according to a possible embodiment.  FIG.  7    is an example illustration  700  of a matching network explicitly shown between a Norton source and an antenna array with impedance matrix Z according to a possible embodiment.  FIG.  8    is an example illustration  800  of a matching network combined with an antenna array with a resulting combined impedance matrix Z′ according to a possible embodiment, where the combined impedance matrix Z′ can be used instead of the impedance matrix Z in equations herein to account for a matching network. 
     According to this embodiment, the transmit power with a Norton source  610 , transmission lines  620 , an antenna array  630 , and no isolators at the source  610 , the impedance looking into a transmission line can be given by
 
 Z   in ( l )= Z   0 ( Z+jZ   0   I   2  tan(2π l ))( Z   0   I   2   +jZ  tan(2π l )) −1 ,
 
where Z 0  can be the impedance of the transmission line, l can be the length of the transmission line in wavelengths, and Z can be the impedance matrix for the combination of the antenna array and any impedance matching circuitry between the transmission line and an antenna array.
 
     The transmit power can be given by
 
 Re ( i   S   H   Z   S_Nor   H ( Z   S_Nor   +Z   in ( l )) −H   Z   in ( l )( Z   S_Nor   +Z   in ( l )) −1   Z   S_Nor ) i   S   =i   S   H ( Z   S_Nor   H ( Z   S_Nor   +Z   in ( l )) −H   Re ( Z   in ( l ))( Z   S_Nor   +Z   in ( l )) −1   Z   S_Nor ) i   S ,
 
where the matrix Z S_Nor  can be a diagonal matrix with elements equal to the shunt source impedances in parallel with the Norton current sources. This expression can be further simplified as the quadratic form
 
 i   S ( Z   S_Nor   H ( Z   S_Nor   +Z   in ( l )) −H   Re ( Z   in ( l ))( Z   S_Nor   +Z   in ( l )) −1   Z   S_Nor ( i   S   =i   S   H   Q   Nor ( Z   S_Nor   ,Z   in ( l )) i   S ,
 
where
 
 Q   Nor ( Z   S_Nor   ,Z   in ( l ))= Z   S_Nor   H ( Z   S_Nor   +Z   in ( l )) −H   Re ( Z   in ( l ))( Z   S_Nor   +Z   in ( l )) −1   Z   S_Nor .
 
     According to a possible embodiment for transmit power with a Thevenin source, transmission lines, and isolators at the source, when the isolator is used at the source, the transmitter does not see the voltage and current reflected from the antenna array (and any matching circuitry). Instead, the reflected voltage and current are routed away from the transmitter and into a matched load. As a result, the impedance looking into the transmission line can just be the transmission line impedance Z 0  and thus may not depend on the impedance of the antenna array. 
     For a Thevenin source, the forward voltage wave into the transmission line can be given by
 
 V   +   =Z   0   Z   S   −1   v   S .
 
     At the antenna array load, the reflected voltage wave can be given by 
                     V   -     =         (     Z   +       Z   0     ⁢     I   2         )       -   1       ⁢     (     Z   -       Z   0     ⁢     I   2         )     ⁢     V   +                     =     SV   +       ,               
where S can be the scattering matrix given by
 
 S =( Z+Z   0   I   2 ) −1 ( Z−Z   0   I   2 ).
 
     The total voltage at the load can be given by 
     
       
         
           
             
               
                 
                   
                     V 
                     tot 
                   
                   = 
                   
                     
                       
                         V 
                         + 
                       
                       + 
                       
                         V 
                         - 
                       
                     
                     ⁢ 
                       
                     = 
                     
                       
                         ( 
                         
                           
                             I 
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                           + 
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                         ) 
                       
                       ⁢ 
                       
                         V 
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                           Z 
                           0 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               I 
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                           ) 
                         
                       
                       ⁢ 
                       
                         Z 
                         
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                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           v 
                           S 
                         
                         . 
                       
                     
                   
                 
               
             
           
         
       
     
     The total current at the load can be given by 
     
       
         
           
             
               
                 
                   
                     I 
                     tot 
                   
                   = 
                   
                     
                       
                         I 
                         + 
                       
                       + 
                       
                         I 
                         - 
                       
                     
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                         Z 
                         0 
                         
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                         ⁡ 
                         
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                         ⁡ 
                         
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                       ⁢ 
                       
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                       ⁢ 
                       
                         
                           v 
                           S 
                         
                         . 
                       
                     
                   
                 
               
             
           
         
       
     
     The power delivered to the load can then be given by the quadratic form 
     
       
         
           
             
               
                 
                   
                     Re 
                     ⁡ 
                     
                       ( 
                       
                         
                           V 
                           tot 
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                         ⁢ 
                         
                           I 
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                       ) 
                     
                   
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                       Z 
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                         ( 
                         
                           
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                                 Z 
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                       . 
                     
                   
                 
               
             
           
         
       
     
     In the event that the source impedances are equal, this expression for the transmitted power can be expressed as 
                 Re   ⁡     (       V     t   ⁢   o   ⁢   t     H     ⁢     I     t   ⁢   o   ⁢   t         )       =         Z   0              Z   S_Thev          2       ⁢       v   S   H     ⁡     (       I   2     -     Re   ⁡     (       S   H     ⁢   S     )         )       ⁢   v       ,         
or more simply as
 
 Re ( V   tot   H   I   tot )= v   S   H   Q   Thev_iso ( Z   0   ,Z   S_Thev   ,S ) v,  
 
where
 
     
       
         
           
             
               
                 Q 
                 Thev_iso 
               
               ⁡ 
               
                 ( 
                 
                   
                     Z 
                     0 
                   
                   , 
                   
                     Z 
                     S_Thev 
                   
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                   S 
                 
                 ) 
               
             
             = 
             
               
                 
                   Z 
                   0 
                 
                 
                   
                      
                     
                       Z 
                       S_Thev 
                     
                      
                   
                   2 
                 
               
               ⁢ 
               
                 
                   ( 
                   
                     
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                       2 
                     
                     - 
                     
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                       ⁡ 
                       
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                   ) 
                 
                 . 
               
             
           
         
       
     
       FIG.  9    is an example graph  900  of variation of transmit power as a function of phase for a two-element dipole array with half-wavelength spacing driven by a 50 ohm Thevenin source with isolators and 50 ohm transmission lines according to a possible embodiment. The same example as previously given of a two-element array of half-wavelength dipoles with half-wavelength spacing can be considered. The impedance matrix for this array can be given by 
     
       
         
           
             Z 
             = 
             
               
                 [ 
                 
                   
                     
                       
                         
                           7 
                           ⁢ 
                           3 
                         
                         + 
                         
                           j 
                           · 
                           42.5 
                         
                       
                     
                     
                       
                         
                           - 
                           13 
                         
                         - 
                         
                           j 
                           · 
                           25 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           13 
                         
                         - 
                         
                           j 
                           · 
                           25 
                         
                       
                     
                     
                       
                         
                           7 
                           ⁢ 
                           3 
                         
                         + 
                         
                           j 
                           · 
                           42.5 
                         
                       
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
     
     As before, the transmission line impedance can be assumed as 50 ohms and a voltage source of the form v(θ)=[1 exp (jθ)] T  for which the L2 norm of the precoder v(θ) is independent of the phase θ can be considered so that
 
∥ v (θ)∥ 2 =2
 
for all θ.
 
     For this example, the variation of the transmit power as a function of the phase θ is shown in the graph  900 . It can be noted that the transmit power varies by 1.3 dB even though the L2 norm of the precoder is held constant. 
     According to a possible embodiment for transmit power with a Norton source, transmission lines, and isolators at the source, when an isolator is used at the source, the transmitter does not see the voltage and current reflected from the antenna array (and any matching circuitry). Instead, the reflected voltage and current are routed away from the transmitter and into a matched load. As a result, the impedance looking into the transmission line can just be the transmission line impedance Z 0  and thus may not depend on the impedance of the antenna array. 
     For a Norton source, the forward voltage wave into the transmission line can be given by
 
 V   + =( Z   0   I   2   +Z   S_Nor ) −1   Z   0   Z   S_Nor   i   S .
 
     As in the previous case, the antenna array load, the reflected voltage wave can be given by 
                     V   -     =       ⁢         (     Z   +       Z   0     ⁢     I   2         )       -   1       ⁢     (     Z   -       Z   0     ⁢     I   2         )     ⁢     V   +                     =       ⁢     S   ⁢     V   +         ,               
where S can be the scattering matrix given by
 
 S =( Z+Z   0   I   2 ) −1 ( Z−Z   0   I   2 ).
 
     The total voltage at the load can be given by 
     
       
         
           
             
               
                 
                   
                     V 
                     tot 
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
                         V 
                         + 
                       
                       + 
                       
                         V 
                         - 
                       
                     
                     = 
                     
                       
                         ( 
                         
                           
                             I 
                             2 
                           
                           + 
                           S 
                         
                         ) 
                       
                       ⁢ 
                       
                         V 
                         + 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         Z 
                         0 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             I 
                             2 
                           
                           + 
                           S 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             
                               Z 
                               0 
                             
                             ⁢ 
                             
                               I 
                               2 
                             
                           
                           + 
                           
                             Z 
                             S_Nor 
                           
                         
                         ) 
                       
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       Z 
                       S_Nor 
                     
                     ⁢ 
                     
                       
                         i 
                         S 
                       
                       . 
                     
                   
                 
               
             
           
         
       
     
     The total current at the load can be given by 
     
       
         
           
             
               
                 
                   
                     I 
                     
                       t 
                       ⁢ 
                       o 
                       ⁢ 
                       t 
                     
                   
                   = 
                     
                   ⁢ 
                   
                     
                       
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     The power delivered to the load can then be given by the quadratic form 
     
       
         
           
             
               
                 
                   
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     In the event that the source impedances are equal, this expression for the transmitted power can be expressed as 
               Re   ⁡     (       V     t   ⁢   o   ⁢   t     H     ⁢     I     t   ⁢   o   ⁢   t         )       =           Z   0     ⁢            Z   S_Nor          2                  Z   0     +     Z   S_Nor            2       ⁢       i   S   H     ⁡     (       I   2     -     Re   ⁡     (       S   H     ⁢   S     )         )       ⁢   i           
or more simply as
 
 Re ( V   tot   H   I   tot )= i   S   H   Q   Nor_iso ( Z   0   ,Z   S_Nor   ,S ) i,  
 
where
 
     
       
         
           
             
               
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     According to a possible embodiment for power variation of precoders with equal Euclidean norm, in the four cases above, the transmit power can be expressed as a quadratic form with respect to the currents or voltages used to drive the antenna arrays. It is apparent that even if the Euclidean norm of the precoder is held constant, the transmit power can vary as a function of the relative phases of the input voltages or input currents. As a result, the implicit assumption used in the design of the 3GPP LTE and IEEE 802.16 codebooks—that precoders with equal Euclidean norm map to antenna patterns with equal transmit power—is incorrect when the antennas are coupled and transmission lines are used between the transmitter and the antenna array. 
     Two methods can be used for mapping precoders with equal Euclidean norm to antenna patterns with equal power. In a first method, a separate real-valued scaling can be applied to each precoder so that the transmit powers are equalized. For example, for the case in which a Thevenin source is used to drive the array and an isolator is used at the transmitter, the transmit power can be given by
 
 Re ( V   tot   H   I   tot )= v   S   H   Q   Thev_iso ( Z   0   ,Z   S_Thev   ,S ) v.  
 
     In order to equalize the transmit power for precoders having equal Euclidean norm, a precoder v k  can be normalized into a normalized precoder v k,norm  by defining v k,norm  as 
                 v     k   ,   norm       =       α   ⁢     v   k           (       v   k   H     ⁢       Q   Thev_iso     ⁡     (       Z   0     ,     Z   S_Thev     ,   S     )       ⁢     v   k       )       1   /   2           ,         
where α is a constant that is the same for all precoders. It should be noted that some equivalent variables in equivalent equations throughout this disclosure may be different for ease of description in the corresponding sections. In this method, the receiver should know the scaling factor applied to each precoder. In particular, the scaling factor for each precoder should be signaled to the receiver. So, for the k-th precoder, the scaling factor
 
             α       (       v   k   H     ⁢       Q   Thev_iso     ⁡     (       Z   0     ,     Z   S_Thev     ,   S     )       ⁢     v   k       )       1   /   2             
can be sent to the receiver. There are other ways to signal the scaling information to the receiver. For example, the parameter α and the matrix
 
 Q   Thev_iso ( Z   0   ,Z   S_Thev   ,S )
 
can be signaled to the receiver. Since the matrix Q Thev_iso  can be Hermitian, the matrix coefficients can be signaled to the receiver in the form of M×(M−1)/2 complex values and M real values, where M can be the number of transmit antennas.
 
     For a second method, a transformation can be performed on the set of equal Euclidean norm precoders so that the transformed precoders can all map to antenna patterns with equal power. For this reason, the matrix Q Thev_iso  can be factored as
 
 Q   Thev_iso ( Z   0   ,Z   S_Thev   ,S )= PP   H ,
 
where this factorization can be non-unique. One factorization of this type is the Cholesky factorization where the matrix P can be lower triangular (and thus P H  can be upper triangular). Other factorizations having this form can be generated by noting that because Q Thev_iso  can be Hermitian, the eigendecomposition of Q Thev_iso  can have the form
 
 Q   Thev_iso   =UΛU   H ,
 
where the columns of U can be the eigenvectors of Q Thev_iso  and the matrix Λ can be diagonal. The diagonal elements of Λ can be the eigenvalues corresponding to the eigenvectors of Q Thev_iso , where the eigenvalues in Λ can be in the same order as the corresponding eigenvectors in U. Using this eigendecomposition, the following can be defined:
 
 P=UΛ   1/2 ,
 
where Λ 1/2  is the square root of the matrix Λ. It can be noted that the eigendecomposition of the matrix Q Thev_iso  may not be unique since the eigenvectors forming the columns of U can be placed in any order. If the dimension of Q Thev  is M×M, then Q Thev_iso  has M eigenvectors and there are M factorial (M!=M*(M−1)*(M−2)* . . . *1) possible orderings of these eigenvectors. Also, given the matrix U and the diagonal matrix of the corresponding eigenvalues Λ, the square root matrix Λ 1/2  can be non-unique since each eigenvalue has both a positive and a negative square root (all of the eigenvalues of Q Thev_iso  are non-negative). Thus, given the matrix Λ, there can be 2 M  possible matrices Λ 1/2 . However, given the matrix Λ there is only one matrix Λ 1/2  for which all of the values are non-negative and this can be referred to as the positive square root.
 
     For the remainder of this section, the following definition can be used:
 
 P=UΛ   1/2 ,
 
where Λ 1/2  can be the positive square root of the matrix Λ. The ordering of the eigenvectors of Q Thev_iso  within the columns of U may not matter, though the ordering of the eigenvalues in Λ should correspond to the ordering of the eigenvectors in U. It can be noted that because the eigenvectors are orthonormal, it follows that
 
 P   −H   =UΛ   1/2 .
 
     Now define 
                     v   S     =       ⁢       P     -   H       ⁢   w                   =       ⁢     U   ⁢           ⁢     Λ       -   1     /   2       ⁢   w       ,               
so that v S  is the sum of the projections of w onto the eigenvectors of Q Thev_iso  scaled by the inverse square root of the corresponding eigenvalues. Note that
 
     
       
         
           
             
               
                 
                   
                     
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     So long as the reference symbol precoders use the same transformation as the data symbol precoders, the receiver can use the existing precoders to estimate the channel, and the receiver does not need to know the precoder transformation that was used at the transmitter. Thus, if each precoder w is transformed into a voltage vector v S  using the transformation v S =P Thev   −H  w, all of the precoders can map to equal energy patterns so long as all of the precoders have the same L 2  norm. 
     Methods described for mapping equal Euclidean norm precoders to antenna patterns with equal power can be used for all four of the cases given above in which the transmit power is quadratic with respect to the vector voltage (for a Thevenin source) or vector current (for a Norton source). Therefore, embodiments can be applied at least for the following four cases considered in this description: transmit power with a Thevenin source, transmission lines, and no isolators at the source; transmit power with a Norton source, transmission lines, and no isolators at the source; transmit power with a Thevenin source, transmission lines, and isolators at the source; and transmit power with a Norton source, transmission lines, and isolators at the source. 
       FIG.  10    is an example flowchart  1000  illustrating the operation of a wireless communication device, such as the device  110  and/or the device  120 , according to a possible embodiment. For example, the method of the flowchart  1000  can be used in cases where transmission lines are used without isolators, in cases where isolators are used at a source of the signal for transmission along with transmission lines, and other cases. At  1010 , a plurality of precoders can be received from a codebook in a transmitter having an antenna array. 
     At  1020 , each precoder of the plurality of precoders can be transformed to a transformed precoder such that the transmit power for each transformed precoder is equal to the transmit power for each of other transformed precoders of the plurality of precoders. The transmit power can be expressed as a quadratic form with respect to the corresponding precoder. The quadratic form can be non-negative definite. The quadratic form can be based on a transmission line impedance of a transmission line between a signal source and the antenna array. The quadratic form can also be based on an impedance matrix of the antenna array. This can be used both when isolators are used at a source of the signal for transmission and also when isolators are not used. The quadratic form can also be based on a matching network between the signal source and the antenna array. A matching network can transform the impedance matrix of the antenna array to improve the power transfer between the source and the antenna array. The quadratic form can further be based on an impedance of the antenna array, a signal source impedance, and length of the transmission line. For example, the quadratic form may be based on the length of the transmission line when isolators are not used at the signal source. Additionally, the quadratic form can be a function of the transmission line impedance of a transmission line between the signal source and the antenna array, an impedance of a source of the signal for transmission, an impedance matching network, a scattering matrix, and other information, where the quadratic form can be a function of some or all of the information. 
     The transformation can equalize the transmit power for all of the plurality of precoders. Transforming can also transform the precoders to equal energy precoders when transmission lines are used between a source of the signal and the antenna array. Furthermore, both data symbol precoders and reference symbol precoders can be transformed by the same transformation. 
     Each precoder can be transformed by scaling each precoder by an inverse square root of a transmit power that results from the corresponding precoder before scaling is applied. Scaling can include normalizing a precoder into a normalized precoder based on the quadratic form. For example, scaling can include normalizing a precoder w k , into a normalized precoder v S,norm  based on 
                 v     S   ,   norm       =       α   ⁢     w   k           (       w   k   H     ⁢       Q   Thev_iso     ⁡     (       Z   0     ,     Z   S_Thev     ,   S     )       ⁢     w   k       )       1   /   2           ,         
where α can be a constant that is the same for all precoders and Q Thev_iso  can be a matrix that can be a function of a scattering matrix S, an impedance Z S_Thev  of a source of the signal for transmission, and a transmission line impedance Z 0  of the transmission line. The scattering matrix S can be a function of the impedance matrix of the antenna array. This equation can cover the case in which a Thevenin source model is used with isolators. Other similar equations can cover other cases, such when a Norton source model is used, when an isolator is not used, and other cases.
 
     Scattering parameters may or may not be used for the case in which no isolators are used at the source. Scattering parameters can depend on the transmission line impedance, the antenna impedance, and any matching circuitry. If isolators are used at the source, the quadratic form for the transmit power can be based on only the scattering parameters without individual knowledge of the transmission line impedance and the impedance matrix of the antenna array. In the case with no isolators at the source, the transmission line impedance and the antenna array impedance can be used separately. In the case that isolators are used at the source, a power variation may only depend on a scattering matrix without the need for the additional knowledge of the transmission line impedance and the antenna array impedance separately. 
     Also or alternately, each precoder can be transformed by multiplying each precoder by a transformation matrix such that the resulting set of precoders each map to antenna patterns having the same power. The transformation matrix can be based on an impedance matrix of the antenna array seen at the signal source as a function of the transmission line length, the transmission line impedance, and an impedance of the antenna array. The transmission line length can be measured in wavelengths. The transformation matrix can also be based on a diagonal matrix of transmitter source impedances. As an example for a Thevenin source model, the transmitter can include a transmitter source of the signal for transmission and the transformation can be based on
 
 v   S   =P   Thev   −H   w,  
 
where w can be a precoder from a set of precoders, v s  can be the transformed precoder in terms of voltage, and P Thev  can be based on
 
 Q   Thev   =P   Thev   P   Thev   H ,
 
where
 
 Q   Thev ( Z   S_Thev   ,Z   in ( l ))=( Z   S_Thev   +Z   in ( l )) −H   Re ( Z   in ( l ))( Z   S_Thev   +Z   in ( l )) −1 ,
 
where l can be a transmission line length measured in wavelengths, Z S_Thev  can be a diagonal matrix of transmitter source impedances, and Z in  can be the impedance matrix of the antenna array seen at the source as a function of the transmission line length, the transmission line impedance, and an impedance of the antenna array. v s  can be a two element vector including two source voltages v s1  and v s2 . This equation can be used for a Thevenin source model where another similar equation can be used for a Norton source model.
 
     As an example for a Norton source model, the transmitter can include a transmitter source of the signal for transmission and the transformation can be based on
 
 i   S   =P   Nor   −H   w,  
 
where w can be a precoder from a set of precoders, i can be the transformed precoder, and P Nor  can be based on
 
 Q   Nor   =P   Nor   P   Nor   H ,
 
where
 
 Q   Nor ( Z   S_Nor   ,Z   in ( l ))= Z   S_Nor   H ( Z   S_Nor   +Z   in ( l )) −H   Re ( Z   in ( l ))( Z   S_Nor   +Z   in ( l )) −1   Z   S_Nor ,
 
where l can be a transmission line length measured in wavelengths, Z S_Nor  can be a diagonal matrix of transmitter source impedances, and Z in  can be the impedance matrix of the antenna array seen at the source as a function of the transmission line length, the transmission line impedance, and an impedance of the antenna array.
 
     At  1030 , a signal can be received from the signal source. At  1040 , a transformed precoder of the plurality of transformed precoders can be applied to the signal to generate a precoded signal for transmission over a physical channel. At  1050 , the precoded signal can be transmitted. At  1060 , a scaling factor used for the scaling can be transmitted. 
       FIG.  11    is an example flowchart  1100  illustrating the operation of a wireless communication device, such as the device  110  and/or the device  120 , according to a possible embodiment. At  1110 , a precoded signal including reference symbols can be received. At  1120 , channels for the reference symbols can be estimated. At  1130 , a channel for the data symbols can be estimated by taking an inner product of a conjugate of a data symbol precoder and the reference symbol channel estimates. The channel estimate for the reference symbols and the data symbols can be performed when the antenna array is used to receive precoded signals. For example, when a device including the antenna array is transmitting, it can transmit precoded signals and when the device is receiving, it can receive precoded signals. At  1140 , received data symbols can be demodulated based on the estimated channel. 
     It should be understood that, notwithstanding the particular steps as shown in the figures, a variety of additional or different steps can be performed depending upon the embodiment, and one or more of the particular steps can be rearranged, repeated or eliminated entirely depending upon the embodiment. Also, some of the steps performed can be repeated on an ongoing or continuous basis simultaneously while other steps are performed. Furthermore, different steps can be performed by different elements or in a single element of the disclosed embodiments. 
       FIG.  12    is an example block diagram of an apparatus  1200 , such as the transmitting device  110 , according to a possible embodiment. The apparatus  1200  can be a base station, a UE, or any other transmitting and/or receiving apparatus. The apparatus  1200  can include a housing  1210 , a controller  1220  coupled to the housing  1210 , audio input and output circuitry  1230  coupled to the controller  1220 , a display  1240  coupled to the controller  1220 , a transceiver  1250  coupled to the controller  1220 , an antenna array including plurality of antennas, such as antennas  1255  and  1257 , coupled to the transceiver  1250 , a user interface  1260  coupled to the controller  1220 , a memory  1270  coupled to the controller  1220 , and a network interface  1280  coupled to the controller  1220 . The apparatus  1200  can perform the methods described in all the embodiments. 
     The display  1240  can be a viewfinder, a liquid crystal display (LCD), a light emitting diode (LED) display, a plasma display, a projection display, a touch screen, or any other device that displays information. The transceiver  1250  can include a transmitter and/or a receiver. The transceiver  1250  can also include a signal source or the signal source can be located elsewhere on the apparatus  1200 . The plurality of antennas  1255  and  1257  can be an antenna array. A transmission line (not shown) can be coupled between the signal source  1252  and the antenna array. The transmission line can have a transmission line impedance. The plurality of antennas  1255  and  1257  can include two or more antennas. The antennas  1255  and  1257  can be mutually coupled in that voltage or current applied to one antenna element induces a voltage or current on another antenna element in the antenna array. The audio input and output circuitry  1230  can include a microphone, a speaker, a transducer, or any other audio input and output circuitry. The user interface  1260  can include a keypad, a keyboard, buttons, a touch pad, a joystick, a touch screen display, another additional display, or any other device useful for providing an interface between a user and an electronic device. The network interface  1280  can be a Universal Serial Bus (USB) port, an Ethernet port, an infrared transmitter/receiver, an IEEE 1398 port, a WLAN transceiver, or any other interface that can connect an apparatus to a network, device, or computer and that can transmit and receive data communication signals. The memory  1270  can include a random access memory, a read only memory, an optical memory, a flash memory, a removable memory, a hard drive, a cache, or any other memory that can be coupled to a wireless communication device. 
     The apparatus  1200  or the controller  1220  may implement any operating system, such as Microsoft Windows®, UNIX®, or LINUX®, Android™, or any other operating system. Apparatus operation software may be written in any programming language, such as C, C++, Java or Visual Basic, for example. Apparatus software may also run on an application framework, such as, for example, a Java® framework, a .NET@framework, or any other application framework. The software and/or the operating system may be stored in the memory  1270  or elsewhere on the apparatus  1200 . The apparatus  1200  or the controller  1220  may also use hardware to implement disclosed operations. For example, the controller  1220  may be any programmable processor. Disclosed embodiments may also be implemented on a general-purpose or a special purpose computer, a programmed microprocessor or microprocessor, peripheral integrated circuit elements, an application-specific integrated circuit or other integrated circuits, hardware/electronic logic circuits, such as a discrete element circuit, a programmable logic device, such as a programmable logic array, field programmable gate-array, or the like. In general, the controller  1220  may be any controller or processor device or devices capable of operating a wireless communication device and implementing the disclosed embodiments. While the controller  1220  is illustrated as one block and operations of the controller  1220  can be performed in one element, the controller  1220  can alternately be distributed between different elements of the apparatus  1200  as well as distributed through cloud computing. For example, different controllers can exist on the apparatus  1200  to perform different operations for different elements on the apparatus  1200 , a master controller can perform all of the operations of the apparatus  1200 , and/or a master controller can perform some overall operations and distributed controllers can perform other operations for other elements on the apparatus  1200 . 
     In operation, the memory  1270  can store a codebook including a plurality of precoders. The controller  1220  can receive a plurality of precoders from the codebook in the memory  1270 . 
     The controller  1220  can transform each precoder of the plurality of precoders to a transformed precoder such that the transmit power for each transformed precoder is equal to the transmit power for each of the other transformed precoders of the plurality of precoders. The transmit power can be expressed as a quadratic form with respect to the corresponding precoder. The quadratic form can be based on the transmission line impedance. The quadratic form can also be based on an impedance matrix of the antenna array, a matching network between the signal source and the antenna array, an impedance of a transmission line, a signal source impedance, and/or a length of the transmission line. 
     Each precoder can be transformed by scaling each precoder by an inverse square root of a transmit power that results from the corresponding precoder before scaling is applied. Scaling can include normalizing a precoder into a normalized precoder based on the quadratic form. Also or alternately, each precoder can be transformed by multiplying each precoder by a transformation matrix such that the resulting set of precoders each map to antenna patterns having the same power. The transformation matrix can be based on an impedance matrix of the antenna array seen at the signal source as a function of the transmission line length, the transmission line impedance, and an impedance of the antenna array. 
     The controller  1220  can receive a signal from the signal source  1252 . The controller  1220  can apply a transformed precoder of the plurality of transformed precoders to the signal to generate a precoded signal for transmission over a physical channel. The transceiver  1250  can transmit the precoded signal. 
     The method of this disclosure can be implemented on a programmed processor. However, the controllers, flowcharts, and modules may also be implemented on a general purpose or special purpose computer, a programmed microprocessor or microcontroller and peripheral integrated circuit elements, an integrated circuit, a hardware electronic or logic circuit such as a discrete element circuit, a programmable logic device, or the like. In general, any device on which resides a finite state machine capable of implementing the flowcharts shown in the figures may be used to implement the processor functions of this disclosure. 
     While this disclosure has been described with specific embodiments thereof, it is evident that many alternatives, modifications, and variations will be apparent to those skilled in the art. For example, various components of the embodiments may be interchanged, added, or substituted in the other embodiments. Also, all of the elements of each figure are not necessary for operation of the disclosed embodiments. For example, one of ordinary skill in the art of the disclosed embodiments would be enabled to make and use the teachings of the disclosure by simply employing the elements of the independent claims. Accordingly, embodiments of the disclosure as set forth herein are intended to be illustrative, not limiting. Various changes may be made without departing from the spirit and scope of the disclosure. 
     In this document, relational terms such as “first,” “second,” and the like may be used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions. The phrase “at least one of,” “at least one selected from the group of,” or “at least one selected from” followed by a list is defined to mean one, some, or all, but not necessarily all of, the elements in the list. The terms “comprises,” “comprising,” “including,” or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. An element proceeded by “a,” “an,” or the like does not, without more constraints, preclude the existence of additional identical elements in the process, method, article, or apparatus that comprises the element. Also, the term “another” is defined as at least a second or more. The terms “including,” “having,” and the like, as used herein, are defined as “comprising.” Furthermore, the background section is written as the inventor&#39;s own understanding of the context of some embodiments at the time of filing and includes the inventor&#39;s own recognition of any problems with existing technologies and/or problems experienced in the inventor&#39;s own work.