Patent Publication Number: US-8537922-B2

Title: Methods and systems for providing feedback for beamforming and power control

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This application claims the benefit of U.S. Provisional Patent Application No. 60/936,340, filed Jun. 19, 2007, which is hereby incorporated by reference herein in its entirety. 
    
    
     TECHNICAL FIELD 
     The disclosed subject matter relates to methods and systems for providing feedback for beamforming and power control. 
     BACKGROUND 
     Digital wireless communication systems have gained widespread adoption in recent years. Common examples of digital wireless communication systems include wireless routers, which are frequently made according to the IEEE 802.11 standards, and mobile telephones. 
     A common problem with digital communication systems is multi-path fading as multiple copies of a signal propagate from a transmitter to a receiver via different paths. For example, one copy of a signal can propagate via a straight line between a transmitter and a receiver and another copy can propagate via a reflection off a structure between the transmitter and the receiver. Because the two copies of the signals are taking different paths, the copies will be out of phase when they reach the receiver. This can result in constructive or destructive interference. As a receiver moves relative to a fixed transmitter, the receiver will either pick up a stronger or weaker sum of the copies of the signal. This variation is fading in the signal. 
     To address multi-path fading problems, as well as other forms of signal degradation, orthogonal frequency divisional multiplexing (OFDM) has been adopted in many digital wireless systems. OFDM operates by sending digital signals across many different orthogonal subcarriers (or channels). Unlike some other forms of communication which attempt to send a large amount of data over a single carrier at high speed, OFDM spreads the data across multiple subcarriers at lower speeds. This enables OFDM systems to be more robust to interference problems. 
     To further improve the performance of OFDM wireless systems, multiple-input and multiple-output (MIMO) configurations of OFDM systems have been adopted. In a typical configuration, a MIMO system may use two or more transmit antennas and two or more receive antennas. By controlling the signals being output on these transmit antennas, beamforming can be used to control the beam shape of the transmitted signal. By controlling the shape of the beam transmitted by the antennas, various forms of signal degradation, including multi-path fading, can be reduced. 
     SUMMARY 
     Methods and systems for providing feedback for beamforming and power control are provided. In some embodiments, methods for controlling transmit-power between a wireless transmitter and a wireless receiver are provided, the methods comprising: calculating a threshold associated with a subcarrier between the transmitter and the receiver; receiving the subcarrier containing an information symbol at the receiver from the transmitter; determining a channel estimate of the subcarrier and a weighting vector for the information symbol; based at least on the channel estimate and the weighting vector, determining a power level of the subcarrier; comparing the power level to the threshold; generating a feedback signal to the transmitter indicating a first energy level to be used for the subcarrier based on the comparison of the power level to the threshold; subsequent to generating the feedback signal, determining a second power level of the subcarrier based at least on the channel estimate and the weighting vector; comparing the second power level to the threshold; and generating a second feedback signal to the transmitter indicating a second energy level to be used for the subcarrier based on the comparison of the second power level to the threshold. 
     In some embodiments, systems for controlling transmit-power from a wireless transmitter are provided, the systems comprising: a receiver that: calculates a threshold associated with a subcarrier between the transmitter and the receiver; receives the subcarrier containing an information symbol from the transmitter; determines a channel estimate of the subcarrier and a weighting vector for the information symbol; based at least on the channel estimate and the weighting vector, determines a power level of the subcarrier; compares the power level to the threshold; generates a feedback signal to the transmitter indicating a first energy level to be used for the subcarrier based on the comparison of the power level to the threshold; subsequent to generating the feedback signal, determines a second power level of the subcarrier based at least on the channel estimate and the weighting vector; compares the second power level to the threshold; and generates a second feedback signal to the transmitter indicating a second energy level to be used for the subcarrier based on the comparison of the second power level to the threshold. 
     In some embodiments, systems for controlling transmit-power from a wireless transmitter are provided, the systems comprising: means for calculating a threshold associated with a subcarrier from the transmitter; means for receiving the subcarrier containing an information symbol from the transmitter; means for determining a channel estimate of the subcarrier and a weighting vector for the information symbol; means for determining a power level of the subcarrier based at least on the channel estimate and the weighting vector; means for comparing the power level to the threshold; means for generating a feedback signal to the transmitter indicating a first energy level to be used for the subcarrier based on the comparison of the power level to the threshold; means for determining a second power level of the subcarrier based at least on the channel estimate and the weighting vector subsequent to generating the feedback signal; means for comparing the second power level to the threshold; and means for generating a second feedback signal to the transmitter indicating a second energy level to be used for the subcarrier based on the comparison of the second power level to the threshold. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a diagram of a system for providing feedback for beamforming and power control in accordance with some embodiments. 
         FIG. 2  is a diagram of a mechanism for transmitting a signal in accordance with some embodiments. 
         FIG. 3  is a diagram of a system for providing feedback for beamforming in accordance with some embodiments. 
         FIG. 4  is a diagram of processes that can be performing in a transmitter and a receiver for providing and using feedback for beamforming in accordance with some embodiments. 
         FIG. 5  is a diagram of a process for determining a perturbation vector in accordance with some embodiments. 
         FIG. 6  is a graph illustrating a relationship between values of an adoption rate β used in updating weighting vectors and Bit Error Rates in accordance with some embodiments. 
         FIG. 7  is a diagram illustrating odd and even intervals containing information symbols and pilot symbols in accordance with some embodiments. 
         FIG. 8  is a diagram illustrating weighting vectors being applied to different subcarriers (or channels) during different time intervals in accordance with some embodiments. 
         FIG. 9  is a diagram of a process for determining energy levels and thresholds for power control in accordance with some embodiments. 
     
    
    
     DETAILED DESCRIPTION 
     In accordance with various embodiments, as described in more detail below, mechanisms for providing feedback for updating beamforming weighting vectors and energy levels are provided. These mechanisms can be used in any suitable digital-wireless-communication application. For example, beamforming weighting vectors and energy levels can be applied to symbols being transmitted in wireless MIMO-OFDM systems to improve the performance of the transmissions, or in any other suitable systems (such as other types of MIMO systems). By updating these vectors and energy levels using feedback, the signal power seen by the receivers can be made to respond to dynamic environmental conditions. 
     As a more particular example, in some embodiments, after initializing a base weighting vector, odd and even weighting vectors can be farmed. These odd and even weighting vectors can then be multiplexed with information symbols to be transmitted over various OFDM subcarriers during alternating odd and even intervals, and transmitted by a transmitter. At a receiver, channels estimates for each of the subcarriers can be formed (as known in the art) and used, along with the odd and even weighting vectors (which were locally generated), to form a feedback bit indicating which of the weighting vectors results in greater power at the receiver. This feedback bit can then be transmitted to the transmitter, where it is can be used to update the base weighting vector. After updating the base weighting vector, the actions above can be repeated. 
     As another example, in some embodiments, additionally or alternatively to beamforming, the energy level(s) used to transmit symbols on one or more subcarriers from a transmitter can be controlled for a subcarrier or a group of subcarriers. These energy levels can be updated using feedback from a receiver. The feedback can be generated by comparing the power seen by the receiver for a subcarrier channel to one or more thresholds. Based on how the power seen compares to the threshold(s), the receiver can instruct the transmitter to modify the power output for the corresponding subcarrier (or a group of subcarriers). When two energy levels and one threshold are used, the feedback signal can be implemented with a single bit. 
     For the purpose of illustration, various embodiments that implement a MIMO-OFDM system are described below, although the mechanisms described herein can be used in systems other than MIMO-OFDM systems. 
     For example, as illustrated in  FIG. 1 , a MIMO-OFDM system  100  can employ N T  transmit antennas  106  at a base station  102  and N R  receive antennas  108  at a mobile terminal  104  for down link transmissions. Separate feedback antenna  110  and  112  can be provided at the transmitter and receiver, respectively. 
     A block diagram of a transmitter  200  that can be used in some embodiments of such a system is illustrated in  FIG. 2 . The processing within transmitter  200  can be performed as follows. First, the input data stream  202  can be mapped into a sequence of M Phase-Shift Keying (PSK) or M Quadrature Amplitude Modulation (QAM) symbols by S/P &amp; Mapper  204 . Then, this sequence of symbols can be serial-to-parallel converted (also by S/P &amp; Mapper  204 ), producing a series of non-overlapping blocks; where each block consists of N u  symbols and where N u  represents the number of useful subcarriers. Then, N vc =N−N u  virtual carriers can be inserted in each block by Virtual carrier insertion  206 , resulting in an N-dimensional vector:
 
a [a 0 ,a 1 , . . . , a N−1 ] T ,
 
where a n  represents the channel symbol associated with the n-th subcarrier. These data blocks can then be fed to an adaptive beamforming mechanism  208  operating on a subcarrier-by-subcarrier basis. For the symbol transmitted over the n-th subcarrier, mechanism  208  can generate an N T -dimensional complex weight vector:
 
w n   [W n,1 ,W n,2 , . . . , w n ,N T ] T ,
 
where w n,1  is the weight for the i-th transmit antenna  106 . The signal transmitted from the N T  antennas  106  over the n-th subcarrier frequency is then w n a n . This technique, employed for each subcarrier, can thus produce a parallel stream of N T  vectors (each consisting of N elements), which can then be fed to a bank of N T  OFDM modulators  210  where they can undergo an N-th order inverse discrete Fourier transform (IDFT) by IDFT  212  followed by cyclic prefix insertion by CP  214 , serial to parallel conversion and transmit filtering by P/S &amp; p(t)  216 . Finally, the resultant signal can be transmitted by antennas  106 .
 
     In the following, for the purpose of illustration, it is assumed that N cp , is the cyclic prefix length, that the impulse response p(t) of the transmit filter in P/S &amp; p(t)  216  is time-limited to the interval (N p T s ,N p T s ) (where T s  is the channel symbol interval), and that the Fourier transform P(f) of p(t) is the root of a raised cosine with roll-off α, so that N u =int[N(1−α)]. 
     The OFDM signal can be transmitted over a wide sense stationary uncorrelated scattering (WSS-US) multipath fading channel. The tapped delay line model: 
                       h     i   ,   j       ⁡     (   t   )       ⁢     =   Δ     ⁢       ∑     l   =   0       L   -   1       ⁢         h     i   ,   j       ⁡     [   l   ]       ⁢     δ   ⁡     (     t   -     l   ⁢           ⁢     T   s         )                   (   1   )               
can be adopted for the channel impulse response between the i-th transmit antenna  106  (i=1, . . . , N T ) and the j-th receive antenna  108  (j=1, . . . , N R ). Here, δ(t) denotes the Dirac delta function, L is the number of channel distinct taps and h ij [l] is the complex gain of the l-th tap. For the purpose of illustration, it is assumed that: (a) the channel is static over each OFDM symbol interval (quasi static channel) and (b) N cp &gt;=2N p +L−1, so that inter-block interference can be avoided in the detection of each OFDM symbol. At mobile terminal  104 , after matched filtering and sampling (wherein the first N cp , samples (i.e., the samples associated with the cyclic prefix) can be discarded from each OFDM time interval and ideal timing is assumed), the N samples collected at the j-th receive antenna undergo an N-th order DFT producing the N-dimensional vector:
 
r j   [r j [0],r j [1], . . . , r j [N−1]] T .
 
It is not difficult to show that this vector can be expressed as:
 
                       r   j     =           ME   b       N   T       ⁢       ∑     i   =   1       N   T       ⁢       W   i     ⁢     AF   L     ⁢     h     i   ,   j             +     n   j         ,           (   2   )               
where:
         A  diag{a n , n=0, 1, . . . , N−1} is an N×N diagonal matrix containing all the elements of a along its main diagonal;   h i,j   [h i,j [0], h i,j [1], . . . , h i,j [L−1]] T  collects the channel gains of h i,j (t) (see equation (1));   F L  is an N×L DFT matrix with [F L ] p,q =exp[−j2πpq/N];   p=0, 1, . . . , N−1;   q=0, 1, . . . , L−1;   w i     diag g {w 0,i w 1,i , . . . w N−1,i } is an N×N diagonal matrix collecting the complex weights applied to the OFDM symbol sent by the i-th transmit antenna;   E b  is the total average transmitted energy per information bit over time; and:   z j   [z j [0], z j [1], . . . z j [N−1]] T ˜N c (0, σ z   2 I N ) is a noise vector.       

     For the purpose of illustration, equation (2) can be converted into a subcarrier-based equation. Specifically, the N R -dimensional vector:
 
r[n] [r 1 [n], . . . , r N     R   [n]] T ,
 
containing the samples from all the receive antennas at the n-th subcarrier, can be expressed as:
 
                       r   ⁡     [   n   ]       =           ME   b       N   T       ⁢     H   n     ⁢     w   n     ⁢     a   n       +     z   n         ,           (   3   )               
where:
         z n ˜N c (0, σ z   2 I N     R   ) is an N R -dimensional Gaussian noise vector; and   H n =[H ij [n]] (with i=1, . . . , N T , j=1, . . . , N R ) represents an N R ×N T  matrix collecting the responses of the MIMO channel at the n-th subcarrier frequency.
 
Note that the N-dimensional vector:
 
H i,j   [H i,j [0],H i,j [1], . . . , H i,j [N−1]] T ,
 
collecting the values of the channel frequency response between the i-th transmit and the j-th receive antennas  108 , is
 
H i,j =F L h i:j .  (4)
       

     The received signal vectors r[n], n=0, 1, . . . , N−1, in equation (3), at a mobile terminal  104  can be used to generate feedback signals that can be used to (1) update the beamforming vector w n  and (2) set a proper energy level E n  at the base station. 
     Turning to  FIGS. 3 and 4 , two diagrams illustrating the beamforming of one subcarrier in a MIMO-OFDM system in accordance with some embodiments are shown. Although  FIGS. 3 and 4  illustrate the operation of one subcarrier in such a system, any suitable number of subcarriers can be used in systems in accordance with some embodiments, and any one or more of such subcarriers can operate as illustrated in  FIGS. 3 and 4 . 
     As illustrated in  FIG. 3 , data  306  that is input to a transmitter  302  can be transmitted as a signal  316  and then received and output as data  308  by a receiver  304 . Data  306  can be provided from any suitable source, such as a digital processing device that is part of, or coupled to, transmitter  302 , and can be for any purpose. Similarly, data  308  can be provided to any suitable destination, such as a digital processing device that is part of, or coupled to, receiver  304 , and can be for any purpose. Data  306  and data  308  can be in a modulated form using any suitable technique. For example, this data can be Quadrature Phase-Shift Keying (QPSK) modulated, Quadrature Amplitude Modulation (QAM) modulated, etc. In modulated form, data  306  and data  308  can be referred to as information symbols. 
     To provide beamforming in signal  316  during transmission, a weighting vector  342  can be multiplexed with data  306  by a multiplexer  310 . This weighted data can then be provided to an OFDM modulator  312 , which can modulate the weighted data. The modulated and weighted data can then be transmitted by a transmitter array  314  to receiver  304 . The transmission energy level can be under the control of power control mechanism  370  (as described in more detail below). Any suitable multiplexer, OFDM modulator, power control mechanism, and transmitter array can be used to multiplex, modulate and transmit the data signal. Once the transmitted data signal arrives at receiver  304 , the data signal can be received by receiver array  318  and then be provided to OFDM demodulator  320 , which can then demodulate signal  316  into data  308 . Any suitable receiver array and OFDM demodulator can be used to receive and demodulate the data. Because the data received by the OFDM demodulator is weighted, a weighting vector  360  can be provided to OFDM demodulator  320  to facilitate receiving the signal. 
     The weighting vectors that are multiplexed with data  306  can be generated and updated in transmitter  302  to take advantage of feedback from receiver  304 . More particularly, as illustrated, a feedback signal  322  can be provided by receiver  304  to transmitter  302  using feedback transmitter  323  and feedback receiver  324 . Any suitable transmitter and receiver can be used for feedback transmitter  323  and feedback receiver  324 . The feedback signal can then be used to select one of two weighting vectors used during a previous interval—an even weighting vector (w even )  326  or an odd weighting vector (w odd )  328 —at multiplexer  330 . The selected weighting vector  332  can then be provided to a weighting adaptation mechanism  334 , which can produce a new even weighting vector  326  and a new odd weighting vector  328  using a perturbation  336  created by a perturbation generator  338  (which is described below in connection with  FIG. 5 ). A multiplexer  340  can then select one of the new even weighting vector  326  and new odd weighting vector  328  as the new base weighting vector  342  based on a current even or odd interval slot so that the selected weighting vector can be multiplexed with the data to be transmitted. The application of weighting vectors to data is described further below in connection with  FIGS. 7 and 8 . Any suitable multiplexers, weighting adaptation mechanisms, and perturbation generators can be used. 
     To generate feedback signal  322 , receiver  304  can include a feedback generation mechanism  344 . In generating the feedback signal, feedback generation mechanism  344  can use one or more channel estimates (H)  346 , even weighting vector  348 , and odd weighting vector  350 . The channel estimate(s) can be generated by channel estimate generator  347  using any suitable technique known in the art, such as by using pilot symbols as described in T. Cui, C. Tellambura, “Robust Joint Frequency Offset and Channel Estimation for OFDM Systems,” IEEE Veh. Tech. Conf. 2004, vol. 1, pp. 603-607, 26-29 Sep. 2004, which is hereby incorporated by reference herein in its entirety. The odd and even weighting vectors can be produced by weighting adaptation mechanism  352  using perturbation  354  created by perturbation generator  356  (which can operate in the same manner as perturbation generator  338 , and which is further described below in connection with  FIG. 5 ). In addition to being used to generate the feedback signal, one of these weighting vectors can be selected by multiplexer  353  to producing weighting vector  360  used by OFDM demodulator  320  to receive signal  316 . Any suitable mechanisms can be used to form feedback generation mechanism  344 , weight adaptation mechanism  352 , perturbation generator  356 , and multiplexer  353 . 
     In some embodiments, a digital processing device (e.g., such as a microprocessor) can be used to control weighting vector formation, feedback, and power control. For example, in transmitter  302 , perturbation generator  338 , weighting adaptation mechanism  334 , multiplexers  310 ,  330 , and  340 , and power control mechanism  370  can be implemented in a digital processing device running suitable software. As another example, in receiver  304 , perturbation generator  356 , weighting adaptation mechanism  352 , multiplexer  353 , and feedback generation mechanism  344  can be implemented in a digital processing device running suitable software. 
     Turning to  FIG. 4 , mechanisms for weighting, transmitting, and receiving data, generating feedback, and updating weighting vectors in connection with some embodiments are further illustrated. As mentioned above,  FIG. 4  illustrates actions that can be taken in a transmitter  402  and a receiver  404  for one subcarrier of a MIMO-OFDM system, although the same actions can be taken in some embodiments in multiple subcarriers. 
     As shown at  406 , the weighting vector (w) for the subcarrier can be initialized. Any suitable process for initializing the weighting vector can be used. For example, in some embodiments, the initial value for the weighting vector can be calculated randomly. As another example, in some embodiments, a code book can be used to calculate the initial value of w. More particularly, using the values from the codebook of Table 1, γ k   LMMSE  can be calculated for each entry in the table using the following equation (5): 
     
       
         
           
             
               
                 
                   
                     γ 
                     k 
                     LMMSE 
                   
                   = 
                   
                     
                       1 
                       
                         
                           
                             σ 
                             2 
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   w 
                                   H 
                                 
                                 ⁢ 
                                 
                                   H 
                                   H 
                                 
                                 ⁢ 
                                 Hw 
                               
                               + 
                               
                                 
                                   σ 
                                   2 
                                 
                                 ⁢ 
                                 I 
                               
                             
                             ] 
                           
                         
                         
                           k 
                           , 
                           k 
                         
                         
                           - 
                           1 
                         
                       
                     
                     - 
                     1 
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     This equation assumes that receiver  404  is a least-minimum-mean-square-error (LMMSE) receiver. When w possesses a Householder structure, further simplification in equation (5) can be attained. Particularly for the case of a 4×N R ×2 system (i.e., four transmitter antennas and two transmission ranks, and any number of receiver antennas), the product term inside the inverse can be written as follows in equation (6): 
                       W   H     ⁢     H   H     ⁢   HW     =       [                  h   1          2             h   1   H     ⁢     h   2                   h   2   H     ⁢     h   1                    h   2          2           ]     -     2   ⁡     [             u   1     ⁢     u   H     ⁢     H   H     ⁢     h   1               u   1     ⁢     u   H     ⁢     H   H     ⁢     h   2                   u   2     ⁢     u   H     ⁢     H   H     ⁢     h   1               u   2     ⁢     u   H     ⁢     H   H     ⁢     h   2             ]       -     2   ⁡     [             u   1   *     ⁢     h   1   H     ⁢   Hu             u   2   *     ⁢     h   1   H     ⁢   Hu                 u   1   *     ⁢     h   2   H     ⁢   Hu             u   2   *     ⁢     h   2   H     ⁢   Hu           ]       +     4   ⁡     [                    u   1          2     ⁢     u   H     ⁢     H   H     ⁢   Hu             u   1     ⁢     u   2   *     ⁢     u   H     ⁢     H   H     ⁢   Hu                 u   2     ⁢     u   1   *     ⁢     u   H     ⁢     H   H     ⁢   Hu                    u   2          2     ⁢     u   H     ⁢     H   H     ⁢   Hu           ]                 (   6   )               
Here, h i  denotes the i-th column vector of the channel matrix (or channel estimate) H and u j  is the j-th element of the vector u. Notice that the above expression only involves matrix-vector operations of three different terms: u H H H h 1 , u H H H h 2 , and u H H H Hu. The rest are simply scalar complex multiplications and additions. This allows significant complexity reduction. A similar technique can be applied to 4×N R ×3 scenario or any other configuration.
 
     The value of u that produces the largest value of γ k   LMMSE  can then be used as the initial value for w. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 4 × N R  × 2 Householder codebook: 4 bits 
               
            
           
           
               
               
               
               
               
               
            
               
                 # 
                 u 
                 Rows 
                 # 
                 u 
                 Rows 
               
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 1 
                 0.2927 
                 1, 2 
                 9 
                 0.4326 
                 1, 2 
               
               
                   
                   0.1298 + 0.6512i 
                   
                   
                 −0.4304 + 0.0481i   
               
               
                   
                 −0.0335 − 0.5622i 
                   
                   
                 0.6963 − 0.1860i 
               
               
                   
                   0.1662 + 0.3586i 
                   
                   
                 0.1500 + 0.2888i 
               
               
                 2 
                 0.5754 
                 1, 2 
                 10 
                 0.4225 
                 1, 2 
               
               
                   
                   0.0530 + 0.2187i 
                   
                   
                 −0.0582 − 0.3111i   
               
               
                   
                 −0.7569 + 0.1852i 
                   
                   
                 −0.4713 − 0.1316i   
               
               
                   
                   0.0209 + 0.1035i 
                   
                   
                 −0.5823 − 0.3779i   
               
               
                 3 
                 0.4379 
                 1, 2 
                 11 
                 0.6569 
                 1, 2 
               
               
                   
                   0.4527 − 0.0588i 
                   
                   
                 0.1485 − 0.0776i 
               
               
                   
                 −0.3630 − 0.3639i 
                   
                   
                 −0.0998 − 0.4801i   
               
               
                   
                 −0.3692 + 0.4465i 
                   
                   
                 0.5359 + 0.1125i 
               
               
                 4 
                 0.5062 
                 1, 2 
                 12 
                 0.7509 
                 1, 2 
               
               
                   
                 −0.1806 − 0.1003i 
                   
                   
                 −0.4061 − 0.1814i   
               
               
                   
                   0.6056 + 0.0988i 
                   
                   
                 0.0274 + 0.2670i 
               
               
                   
                   0.2603 − 0.5068i 
                   
                   
                 0.0993 − 0.3954i 
               
               
                 5 
                 0.0774 
                 1, 2 
                 13 
                 0.3816 
                 1, 2 
               
               
                   
                 −0.3019 − 0.1046i 
                   
                   
                 0.5020 + 0.1775i 
               
               
                   
                 −0.2880 + 0.8634i 
                   
                   
                 0.3039 − 0.5125i 
               
               
                   
                   0.2237 + 0.1160i 
                   
                   
                 0.4323 − 0.1702i 
               
               
                 6 
                 0.1056 
                 1, 2 
                 14 
                 0.6961 
                 1, 2 
               
               
                   
                 −0.5156 + 0.2733i 
                   
                   
                 0.1961 + 0.5260i 
               
               
                   
                 −0.5343 − 0.2538i 
                   
                   
                 −0.0002 − 0.0892i   
               
               
                   
                 −0.5021 + 0.2152i 
                   
                   
                 −0.3517 + 0.2622i   
               
               
                 7 
                 0.3474 
                 1, 2 
                 15 
                 0.1991 
                 1, 2 
               
               
                   
                   0.1465 − 0.4442i 
                   
                   
                 0.0523 − 0.4184i 
               
               
                   
                   0.6611 − 0.4650i 
                   
                   
                 0.0165 + 0.4263i 
               
               
                   
                 −0.0535 + 0.0662i 
                   
                   
                 0.1029 − 0.7681i 
               
               
                 8 
                 0.3585 
                 1, 2 
                 16 
                 0.1471 
                 1, 2 
               
               
                   
                   0.0138 + 0.3114i 
                   
                   
                 −0.2071 − 0.4459i   
               
               
                   
                 −0.0018 + 0.1268i 
                   
                   
                 0.2385 − 0.1701i 
               
               
                   
                 −0.7014 + 0.5160i 
                   
                   
                 0.6400 + 0.4912i 
               
               
                   
               
            
           
         
       
     
     In order to facilitate demodulation of data and updating of weighting vectors, the weighting vectors used in the transmitter and the receiver can be synchronized. To do so, for example, as illustrated at  414 , receiver  404  can also initialize a weighting vector in the receiver in the same manner as performed at  406 . In some embodiments, the weighting vector can be determined as described only at receiver  404  and then provided to transmitter  402  through a feedback channel. 
     Next, at  408 , the odd and even weighting vectors can be calculated using the following equations (7) and (8): 
                     w   n   e     =         w   n   b     +     β   ⁢           ⁢     p   n                  w   n   b     +     β   ⁢           ⁢     p   n                        (   7   )                 w   n   o     =         w   n   b     -     β   ⁢           ⁢     p   n                  w   n   b     -     β   ⁢           ⁢     p   n                        (   8   )               
where:
         n indicates the subcarrier corresponding to this weighting vector.   p n  is a perturbation for the subcarrier.   β is the adaptation rate, which determines how fast the weighting matrices change.   w n   b  is the preferred weight vector (or base weighting vector) previously selected based on feedback, or is the initial value of w, for the subcarrier.   w n   e  is the even weighting vector for the subcarrier.   w n   o  is the odd weighting vector for the subcarrier.   The perturbation vector (p n ) can be calculated using any suitable technique. For example, the perturbation vector can be calculated using a Quasi Monte Carlo technique  500  illustrated in  FIG. 5 . As shown at  502 , a first set of B prime numbers can be determined. B can be any suitable number and is based on the number of transmission antennas reflected in the dimensions of the weighting vectors. For example, for B equal to 4 for four transmission antennas, the set of prime numbers can be {3, 5, 7, 11}. Next, at  504 , a random number n 0  can be generated and the first N natural numbers beginning at n 0  identified. Any suitable value for N can be selected, and can be based on the accuracy of tracking the channel modes. For example, for n 0  equal to 15 and N equal to 1000, the N identified numbers can be {15, 16, 17, . . . , 1014}. Then, at  506  through  514 , for each of the N identified numbers, a set of B elements can be created using respective ones, b, of the B prime numbers. Particularly, for each one, n, of the N identified numbers (looping between  506  and  514 ), each one of the B elements (looping between  508  and  512 ) can then be determined by taking the sequence of digits of n when expressed in base b (where b is a respective one of the B prime numbers) and using those digits to form a base 10 number. For example, with n=−15 and b=2, the digits of n when expressed in base b are “1111.” These digits can then be used to form the number 1,111 (in base 10). This formation of numbers can be performed for each combination of n and b to form a set of N×B numbers (u(N,B)). As illustrated in the figure, this can be represented in pseudo-code as u(n,b)=val(itoa(n(base b))), where n(base b) is the number n in base b, itoa( ) converts an integer number to a string, and val( ) converts a string to a base 10 number. This set of N×B numbers can be referred to as a Halton Sequence. Next at  516 , for each one, n, of the values N in the Halton Sequence, p n  can be calculated using the following equation (9):       

                     p   n     =       1   B     ⁢       ∑     b   =   1     B     ⁢       ϕ     -   1       ⁡     (     u   ⁡     (     n   ,   b     )       )                   (   9   )               
where:
         φ −1  is an inverse mapping of the Gaussian distribution; and   u(n,b) are the B numbers corresponding to each of the N values in the Halton Sequence.   p n  is a gaussian random vector with zero mean and covariance Matrix I (identity matrix).
 
This process is further described in D. Guo and X. Wang, “Quasi-Monte Carlo filtering in nonlinear dynamic systems”, IEEE Trans. Sign. Proc., vol. 54, no. 6, pp. 2087-2098, June 2006, which is hereby incorporated by reference herein in its entirety.
       

     Additionally or alternatively, the perturbation can be calculated using normal Monte-Carlo techniques or randomly. As stated above, any other suitable technique for determining the perturbation vector can be used. 
     As mentioned above, β is the adaptation rate of the perturbation p n  into the even and odd weighting vectors. In fast changing environments, it can be desirable for the even and odd weighting vectors to change rapidly, and, thus, to use a large value of β. Example values for β versus resultant Bit Error Ratios (BER) are illustrated in  FIG. 6  for different normalized Doppler bandwidths (B D NT s ) for motion of the receiver and/or transmitter. Any suitable values for β can be used. For example, β equal to 0.2 can be used. 
     Returning to  FIG. 4 , once again, to maintain synchronization between the weighting vectors in the transmitter and receiver, at  416 , the odd and even weighting vectors can also be calculated in the same manner as performed at  408   
     Next, at  410 , the data can be multiplexed with the odd and even weighting vectors in any suitable manner. For example,  FIGS. 7 and 8  illustrate ways in which odd and even weighting vectors can be multiplexed with data. As shown in  FIG. 7 , for different subcarriers of a MIMO-OFDM system (which are represented at different points on f-axis  706 ), pilot symbols  702  and information symbols  704  can be transmitted at different time intervals  710  and  712  (which are represented at different points on t-axis  708 ). As indicated in legend  714 , pilot symbols  702  can be not-weighted and used for channel estimation (as known in the art), information symbols  704  can be weighted using the even weighting vector during even intervals  710 , and information symbols  704  can be weighted using the odd weighting vector during odd intervals  712 . 
     Although pilot symbols are illustrated as being transmitted in every third interval in  FIG. 7 , and at an even spacing across subcarriers, any suitable approach to transmitting pilot symbols can be used. For example, in some embodiments, pilot symbols can be transmitted more or less frequently and/or on more or less subcarriers. In some embodiments, pilot symbols can be omitted. 
     As also shown, multiple intervals  710  and  712  can form perturbation probing periods (PPP)  716  and  718 . During a PPP, a feedback signal can be generated. This feedback signal can then be used to determine the odd and even weight vectors to be used during the next PPP. 
     Turning to  FIG. 8 , the weighting vectors  804 ,  806 ,  808 , and  810  that can be used over even intervals  710  and odd intervals  712  are illustrated. As shown, a weighting vector w 1   e    804  can be used to weight subcarrier  1  during even interval  710 . A weighting vector w 2   e    806  can be used to weight subcarrier  2  during even interval  710 . A weighting vector w 1   o    808  can be used to weight subcarrier  1  during interval  712 . And, a weighting vector w 2   o    810  can be used to weight subcarrier  2  during interval  712 . Although two subcarriers are specifically identified in connection with  FIG. 8 , any suitable number of the subcarriers can have weighting applied. 
     Although odd and even intervals are illustrated in  FIGS. 7 and 8  as occurring every other interval, any suitable approach for designating intervals as odd or even can be used. For example, a first pair (or any other suitable number) of intervals can be even, and the next pair (or any other suitable number) of intervals can be odd. 
     Referring back to  FIG. 4 , after the data is weighted at  410 , the weighted data can be transmitted to receiver  404  at  412 . This weighted data can then be received at  418 . Any suitable technique can be used to transmit and receive the weighted data/information symbols. For example, the data/information symbols can be transmitted using MIMO-OFDM transmitters and receivers. 
     Next, at  420 , the receiver can generate the feedback signal. For example, the feedback signal can be generated as a feedback bit using the following equation (10):
 
 b   n =sign(∥ H   n   e   w   n   e ∥ F   2   −∥H   n   o   w   n   o ∥ F   2 )  (10)
 
where:
         H n   e  is the channel gain matrix (or channel estimate) for the even ODFM intervals of a given PPP; and   H n   o  is the channel gain matrix (or channel estimate) for the odd ODFM intervals of a given PPP.
 
Equation (10) results in b n  being equal to +1 if the even weighting vector results in greater power for a given subcarrier n, and being equal to −1 if the odd weighting vector results in greater power for a given subcarrier n. In some embodiments, rather than using different channel gain matrices for the odd and even intervals, a single channel gain matrix can be used.
       

     This feedback bit can then be transmitted from receiver  404  to transmitter  402  at  422  and  424 . Any suitable technique for transmitting the feedback bit from the receiver to the transmitter can be used. 
     Next, at  426  and  428 , the weighting vector in each of the receiver and the transmitter can be updated. For example, using the feedback bit, the weighting vector can be updated using the following equation: 
                     w   n   b     =     {             w   n   e     ,       if   ⁢           ⁢     b   n       =     +   1                     w   n   o     ,       if   ⁢           ⁢     b   n       =     -   1                         (   11   )               
That is, if the feedback bit is +1, then the weighting vector is set to the even weighting vector. Otherwise, if the feedback bit is −1, then the weighting vector is set to the odd weighting vector.
 
     After updating the weighting vectors in the receiver and the transmitter, the processes shown loop back to  416  and  408 , respectively. 
     As mentioned above, feedback can also be used to control transmit power on subcarriers of MIMO systems (such as MIMO-OFDM systems having beamformed channels as described above) in some embodiments. Controlling of the energy level E n  for each subcarrier n can be performed according to the following rule: 
                     E   n     =     {               E   1     ,       if   ⁢           ⁢     λ   n       &lt;   τ                   E   2     ,       if   ⁢           ⁢     λ   n       &gt;   τ             ,     n   =   0     ,   …   ⁢           ,     N   -   1                 (   12   )               
where:
         E 1  is a first energy level;   E 2  is a second energy level;   τ represents a threshold level; and   λ n  represents the main power of a beamformed channel seen by a mobile terminal at an n-th subcarrier.
 
More particularly, λ n  can be used to represent the main power of a beamformed channel by the following equation (see also equation (6):
 
λ n   w n   H H n   H H n w n   (13)
 
where:
   w n   H  represents the Hermetian operation on w n ;   H n   H  represents the Hermetian operation on H n ;   H n  represents the channel estimate of subcarrier n; and   w n  represents the beamforming weight applied to subcarrier n.       

     In some embodiments, where beamforming is not used, the power determined in equation (13) can be represented as:
 
λ n   H n   H H n  
 
     Referring to  FIG. 3 , feedback generation mechanism  344  in receiver  304  can be used to determine whether the energy level E n  for a subcarrier n should be set to E 1  or E 2  based on current values E 1 , E 2 , λ n , and τ using equations (12) and (13), and return any suitable feedback (such as a bit set to 0 for E 1  or to 1 for E 2 ) to transmitter  302 . Within transmitter  302 , a power control mechanism  370  can be used to receive this feedback and control the energy level on the subcarrier generated by transmit array  314 . This determination, feedback, and control can be performed for every symbol transmitted on a subcarrier, or periodically (e.g., every m symbols or t time increments), and can be performed at the same and/or different times from the updating of the beamforming weights. 
     In order to provide this feedback and set E n , the receiver needs to have values for E 1 , E 2 , and τ, and the transmitter needs to have values for E 1  and E 2 . In some embodiments, E 1 , E 2 , and τ can be calculated in receiver  304  (as described below, for example) and E 1  and E 2  provided to transmitter  302  through the feedback channel. In some embodiments, the transmitter can perform separate and similar calculations as being performed in the receiver to determine E 1  and E 2 . 
     The energy levels E 1  and E 2 , and the threshold τ, can be selected to minimize the bit error rate (BER) on each subcarrier n. BER expressions depend on the constellation (e.g., QAM or QPSK) of symbols that is adopted in a transmitter. Formulas for some common constellations can be found in J. G. Proakis, “Digital Communications,” 4 th  ed., McGraw-Hill, New York, N.Y., 2001, which is hereby incorporated by reference herein in its entirety. 
     Using QPSK as an example (although any other suitable constellation can also be used), the signal-to-noise ratio per bit for the n-th subcarrier in a MIMO-OFDM system can be given by: 
                         [   SNR   ]     n     =         2   ⁢     E   b         σ   z   2       ⁢     λ   n         ,           (   14   )               
and the corresponding average bit error probability can be given by:
 
                       P   n     =       ∫   0   ∞     ⁢       Q   ⁡     (         SNR   n     ⁡     (     λ   n     )         )       ⁢     p   ⁡     (     λ   n     )             ,           (   15   )               
where:
         σ z   2  represents the noise variance of the subcarrier signal;   Q represents the probabilistic function related to the subcarrier signal, and is described in J. G. Proakis, “Digital Communications,” 4 th  ed., McGraw-Hill, New York, N.Y., 2001; and   p(λ n ) represents the probability density function of λ n .
 
Substituting equation (14) into equation (15) and taking into account rule (12) yields the bit error probability:
       

                     P   n     =         ∫   0   τ     ⁢       Q   ⁡     (       2   ⁢     E   1     ⁢       λ   n     /     σ   z   2           )       ⁢     p   ⁡     (     λ   n     )       ⁢     ⅆ     λ   n           +       ∫   τ   ∞     ⁢       Q   ⁡     (       2   ⁢     E   2     ⁢       λ   n     /     σ   z   2           )       ⁢     p   ⁡     (     λ   n     )       ⁢     ⅆ     λ   n                     (   16   )               
Energy levels E 1  and E 2 , and threshold τ can then be selected while attempting to minimize this bit error probability P n  under the following time-based energy constraint:
 
 q   1   E   1   +q   2   E   2   =E   b ,  (17)
 
where:
 
                 q   1     ⁢     =   Δ     ⁢       ∫   0   τ     ⁢       p   ⁡     (     λ   n     )       ⁢     ⅆ     λ   n             ;                   q   2     ⁢     =   Δ     ⁢       ∫   τ   ∞     ⁢       p   ⁡     (     λ   n     )       ⁢     ⅆ     λ   n             ;         
and
         E b  is the total average transmitted energy per information bit over time.
 
The time-based energy constraint attempts to control the energy level over time so that the average energy level is E b .
       

     This selection can be performed in some embodiments using Lagrange multipliers. For example, to do so, the following Lagrangian function can be defined: 
                     L   =         ∫   0   τ     ⁢       Q   ⁡     (       2   ⁢     E   1     ⁢       λ   n     /     σ   z   2           )       ⁢     p   ⁡     (     λ   n     )       ⁢     ⅆ     λ   n           +       ∫   τ   ∞     ⁢       Q   ⁡     (       2   ⁢     E   2     ⁢       λ   n     /     σ   z   2           )       ⁢     p   ⁡     (     λ   n     )       ⁢     ⅆ     λ   n           +     μ   ⁢           ⁢     q   1     ⁢     E   1       +     μ   ⁢           ⁢     q   2     ⁢     E   2       -     μ   ⁢           ⁢     E   b           ,           (   18   )               
and the following system solved:
 
                   {               ⅆ   L       ⅆ     E   1         =           ∫   0   τ     ⁢       E   1       -   1     /   2       ⁢     exp   ⁡     (       -     E   1       ⁢     λ   n       )       ⁢     λ   n       -   1     /   2       ⁢     p   ⁡     (     λ   n     )       ⁢     ⅆ     λ   n           +     μ   ⁢           ⁢     q   1         =   0                     ⅆ   L       ⅆ     E   2         =           ∫   τ   ∞     ⁢       E   2       -   1     /   2       ⁢     exp   ⁡     (       -     E   2       ⁢     λ   n       )       ⁢     λ   n       -   1     /   2       ⁢     p   ⁡     (     λ   n     )       ⁢     ⅆ     λ   n           +     μ   ⁢           ⁢     q   2         =   0                     ⅆ   L       ⅆ   τ       =         Q   ⁡     (       2   ⁢     E   1     ⁢     τ   /     σ   z   2           )       +     μ   ⁢           ⁢     E   1       -     Q   ⁡     (       2   ⁢     E   2     ⁢     τ   /     σ   z   2           )       -     μ   ⁢           ⁢     E   2         =   0                     ⅆ   L       ⅆ   μ       =           q   1     ⁢     E   1       +       q   2     ⁢     E   2       -     E   b       =   0.                     (   19   )               
From the third equation of system (19) it can be inferred that:
 
                   μ   =         Q   ⁡     (       2   ⁢     E   2     ⁢     τ   /     σ   z   2           )       -     Q   ⁡     (       2   ⁢     E   1     ⁢     τ   /     σ   z   2           )             E   1     -     E   2                 (   20   )               
and from the fourth equation of system (19) that:
 
                     E   2     =         E   b     -       q   1     ⁢     E   1           q   2               (   21   )               
Substituting equations (20) and (21) into the second equation of system (19) yields the following system based on E 1  and τ:
 
             {               f   ⁡     (       E   1     ,   τ     )       =   0                 g   ⁡     (       E   1     ,   τ     )       =   0                       
This system can be solved iteratively using the Newton-Ralphson method. Given an initial point (E 1   (i) , T (i) ), to simplify the problem f (E 1 , τ) and g(E 1 , τ) can be replaced with the first order Taylor approximation around (E 1   (i) , τ (i) ):
 
                   {             f   ⁡     (       E   1     ,   τ     )       ≈       f   ⁡     (       E   1     (   i   )       ,     τ     (   i   )         )       +         f     E   1       ⁡     (       E   1     (   i   )       ,     τ     (   i   )         )       ⁢     (       E   1     -     E   1     (   i   )         )       +         f   τ     ⁡     (       E   1     (   i   )       ,     τ     (   i   )         )       ⁢     (     τ   -     τ     (   i   )         )                       g   ⁡     (       E   1     ,   τ     )       ≈       g   ⁡     (       E   1     (   i   )       ,     τ     (   i   )         )       +         g     E   1       ⁡     (       E   1     (   i   )       ,     τ     (   i   )         )       ⁢     (       E   1     -     E   1     (   i   )         )       +         g   τ     ⁡     (       E   1     (   i   )       ,     τ     (   i   )         )       ⁢     (     τ   -     τ     (   i   )         )                         (   22   )               
An iterative algorithm for the joint estimation of E 1  and τ is given by:
 
 E   1   (i+1)   =E   1   (i) +δ E     1   τ (i+1) =τ (i) +δ τ   (23)
 
where the vector δ=[δ E     1   , δ τ ] T  is the solution of the following system:
 
                       J   ⁡     (       E   1     (   i   )       ,     τ     (   i   )         )       ⁢   δ     =     -     F   ⁡     (       E   1     (   i   )       ,     τ     (   i   )         )                 (   24   )             where                           J   ⁡     (       E   1     (   i   )       ,     τ     (   i   )         )       =     (             f     E   1       ⁡     (       E   1     (   i   )       ,     τ     (   i   )         )               f   τ     ⁡     (       E   1     (   i   )       ,     τ     (   i   )         )                   g     E   1       ⁡     (       E   1     (   i   )       ,     τ     (   i   )         )               g   τ     ⁡     (       E   1     (   i   )       ,     τ     (   i   )         )             )             (   25   )             and                           F   ⁡     (       E   1     (   i   )       ,     τ     (   i   )         )       =     (           f   ⁡     (       E   1     (   i   )       ,     τ     (   i   )         )                   g     E   1       ⁡     (       E   1     (   i   )       ,     τ     (   i   )         )             )             (   26   )               
The partial derivatives in equations (25) and (26) are given by equations (27), (28), (29), and (30):
 
     
       
         
           
             
               
                 
                   
                     
                       
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                       . 
                     
                   
                 
               
               
                 
                   ( 
                   30 
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     The probability density function p(λ n ) in equations (28) and (30) can be estimated using standard numerical techniques and the integrals in equations (27) and (29) can be solved using Monte Carlo techniques, as known in the art. 
     Then, if N it  represents the maximum number of the iterations in the Newton-Ralphson method and E 1   (0) =E b , τ (0) =median(λ n ) are selected at a first iteration, E 1  and τ can be iteratively estimated, and then E 2  computed as illustrated in process  900  of  FIG. 9 . As shown, this can be performed by setting i=1 at  902 , and then computing J(E 1   (i) , τ (i) ) from equation (25) at  904 , F(E 1   (i) , τ (i) ) from equation (26) at  906 , and a new estimate E 1   (i+1)  for E 1  and τ (i+1)  for τ from equations (23) and (24) at  908 . Next, at  910 , it can be determined whether i=N it . If not, i can be incremented at  912 . Otherwise, E 2  can be computed from equation (21) at  914  using the final estimate of E 1  and τ. 
     In some embodiments, rather than initializing the weighting vectors in the transmitter and the receiver only once, these vectors are initialized periodically (e.g., every 100 intervals, every 30 PPP, etc.). Similarly, the energy levels E 1  and E 2  and threshold τ can be initialized periodically. This can be advantageous in fast changing environments where conditions can rapidly move back to preferring the initial weighting, or something close to it. 
     As mentioned above, techniques for updating beamforming and power control are illustrated for a single subcarrier. These techniques can be used across multiple subcarriers in some embodiments. For example, rather than performing these techniques for each subcarrier, in some embodiments, multiple subcarriers can be grouped and these techniques can be performed on a subset of the subcartiers—e.g., the center subcarrier. The weighting vector, energy levels, and/or threshold determined for the center subcarrier can then be used for all subcarriers in the group. 
     While single-bit feedback approaches to updating beamforming and power control have been described above, any suitable number of bits for feedback can be used in accordance with some embodiments. For example, in a two bit approach, rather that generating odd and even weighting vectors, four weighting vectors, labeled  1  through  4 , can be generated. These weighting vectors can then each be applied to information symbols across every four intervals. As another example, rather than using two energy levels and one threshold, three energy levels and two thresholds, four energy levels and three thresholds, etc., can be used. Obviously, any other suitable approach can be used. 
     As described above, beamforming and power control can be used separately or together to improve the performance of wireless digital communication systems, such as MIMO (or more particularly, MIMO-OFDM) systems. Because these techniques can increase processing requirements at a transmitter and receiver, and thus increase power usage and decrease battery life of portable devices, in some embodiments the usage of the techniques can be controlled automatically or manually. In an automatic approach, a mobile terminal can automatically determine if it is in a dynamic environment by monitoring whether the channel estimate for a subcarrier changes or by detecting whether the mobile terminal is moving (e.g., based on an attached GPS device). In a manual approach, a mobile terminal can receive a setting from a user selecting whether the device is moving or not, or specifically selecting whether beamforming and/or power control are to be used, and if so at what frequency the beamforming and power control are to be updated. When selections are based on movement (e.g., whether detected automatically or in response to user input), any suitable selection of beamforming and/or power control can automatically be configured. For example, if a mobile terminal is determined to be moving, beamforming can be selected to be active, while power control can be selected to be disabled in order to decrease power demands and increase battery life of the mobile terminal. 
     Although the invention has been described and illustrated in the foregoing illustrative embodiments, it is understood that the present disclosure has been made only by way of example, and that numerous changes in the details of implementation of the invention can be made without departing from the spirit and scope of the invention, which is only limited by the claims which follow. Features of the disclosed embodiments can be combined and rearranged in various ways.