Abstract:
The disclosed embodiments provide for methods and systems for transmitting a number of streams of modulation symbols by a multi-antenna transmitter. In one aspect, a method for transmitting a number of streams of modulation symbols by a multi-antenna transmitter includes the acts of transforming a first number of streams of symbols to a second number of streams of symbols, the first number being smaller than or to the second number; and transmitting the second number of streams of symbols by a transmitter having the second number of antennas.

Description:
BACKGROUND 
     1. Field 
     The present invention relates generally to wireless communication and more specifically to techniques for mitigating the effect of correlation in a multi-antenna wireless communication system. 
     2. Background 
     A MIMO system employs multiple (N T ) transmit antennas and multiple (N R ) receive antennas for data transmission. A MIMO channel formed by the N T  transmit and N R  receive antennas may be decomposed into N S  independent channels, which are also referred to as spatial channels. Each of the N S  independent channels corresponds to a dimension. The MIMO system can provide improved performance (e.g., higher throughput and/or greater reliability) if the additional dimensionalities created by the multiple transmit and receive antennas are utilized. 
     A multi-carrier MIMO system employs multiple carriers for data transmission. These multiple carriers may be provided by orthogonal frequency division multiplexing (OFDM) or some other construct. OFDM effectively partitions the overall system bandwidth into a number of (N F ) orthogonal subbands, which are also referred to as tones, frequency bins, and frequency sub-channels. With OFDM, each subband is associated with a respective carrier upon which data may be modulated. For a MIMO system that utilizes OFDM (i.e., a MIMO-OFDM system), the MIMO channel for each of the N F  subbands may be decomposed into N S  independent channels, resulting in a total of N S N F  independent channels. 
     In a wireless communication system, data to be transmitted is initially processed (e.g., coded and modulated) to form a stream of symbols. The symbol stream is then upconverted to radio frequency (RF) to generate an RF modulated signal that is more suitable for transmission over a wireless channel. For a MIMO system, up to N T  RF modulated signals may be generated and transmitted in parallel from the N T  transmit antennas. The N T  transmitted signals may reach the N R  receive antennas via a number of propagation paths and may experience different effective channels due to different effects of fading and multipath. Moreover, for a MIMO-OFDM system, the N F  subbands of each transmitted signal may also experience different effective channels. Consequently, the N T  transmitted signals may be associated with different complex channel gains and received signal-to-noise ratios (SNRs) that can vary across the N F  subbands. 
     Communication systems are widely deployed to provide various communication services such as voice, packet data, and so on. These systems may be time, frequency, and/or code division multiple-access systems capable of supporting communication with multiple users simultaneously by sharing the available system resources. Examples of such multiple-access systems include Code Division Multiple Access (CDMA) systems, Multiple-Carrier CDMA (MC-CDMA), Wideband CDMA (W-CDMA), High-Speed Downlink Packet Access (HSDPA), Time Division Multiple Access (TDMA) systems, Frequency Division Multiple Access (FDMA) systems, and Orthogonal Frequency Division Multiple Access (OFDMA) systems. 
     There is therefore a need in the art for mitigating the effect of correlation in a multi-antenna wireless communication system. 
     SUMMARY 
     The disclosed embodiments provide for methods and systems for transmitting a number of streams of modulation symbols by a multi-antenna transmitter. In one aspect, a method for transmitting a number of streams of modulation symbols by a multi-antenna transmitter includes the acts of transforming a first number of streams of symbols to a second number of streams of symbols, the first number being smaller than or equal to the second number, and transmitting the second number of streams of symbols by a transmitter having the second number of antennas. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The features and nature of the present invention will become more apparent from the detailed description set forth below when taken in conjunction with the drawings in which like reference characters identify correspondingly throughout and wherein: 
         FIGS. 1A and 1B  show two embodiments for transmitting streams of modulation symbols by a multi-antenna transmitter; 
         FIG. 2  shows one embodiment for transforming streams of modulation symbols for transmission by the multi-antenna transmitter of  FIG. 1A  or  FIG. 1B ; 
         FIGS. 3A and 3B  show one embodiment for implementing vector rotation; 
         FIG. 4  shows a block diagram of a transmitter system and a receiver system; and 
         FIG. 5  shows a block diagram of a transmitter unit within the transmitter of  FIG. 4 . 
     
    
    
     DETAILED DESCRIPTION 
     The word “exemplary” is used herein to mean “serving as an example, instance, or illustration.” Any embodiment or design described herein is “exemplary” and is not necessarily to be construed as preferred or advantageous over other embodiments or designs. 
     In one embodiment, in a MIMO system, a number of streams of symbols are transmitted by a multi-antenna transmitter and received by a multi-antenna receiver. A model for a MIMO system may be expressed as:
 
 y ( s )= H ( s ) x ( s )+ n ( s ), for sεS,  Eq.(1)
 
where x(s) is an {N T ×1} “transmit” vector with N T  entries for N T  symbols or data streams transmitted from the N T  transmit antennas;
         y(s) is an {N R ×1} “receive” vector with N R  entries for N R  symbols or data streams received on the N R  receive antennas;   H(s) is the {N R }×{N T } channel response matrix;   n(s) is a vector of additive white Gaussian noise (AWGN); and   s may correspond to a time-division (s represents a time instance), frequency-division (s represents a frequency instance), time-frequency-division (s represents an instance in time-frequency space) or code division (s represents a code value) multiplexing algorithm.       

     The vector n(s) is assumed to have zero mean and a covariance matrix of Λ n =σ 2 I, where I is the identity matrix with ones along the diagonal and zeros everywhere else, and σ 2  is the variance of the noise. 
     The channel response matrix H(s) may not have full-rank for all instances of s. Assuming the rank of H(s) to be “r,” which may be determined by the transmitter, receiver, or both, r streams of modulation symbols may be transmitted by r antennas. In this case, letting H(s)=[h 1 (s) h 2 (s) . . . h N     t   (s)] to represent the channel response matrix at a given instance of s, and h i  to represent the {N T ×1} channel response vector corresponding to transmit antenna i and all receive antennas, the received signal at the given instance of s, e.g., time, frequency, time-frequency, or code, would be defined as: 
     
       
         
           
             
               
                 
                   
                     r 
                     ⁡ 
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         r 
                       
                       ⁢ 
                       
                         
                           
                             h 
                             i 
                           
                           ⁡ 
                           
                             ( 
                             s 
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             x 
                             i 
                           
                           ⁡ 
                           
                             ( 
                             s 
                             ) 
                           
                         
                       
                     
                     + 
                     
                       
                         n 
                         ⁡ 
                         
                           ( 
                           s 
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     2 
                     ) 
                   
                 
               
             
           
         
       
     
     Depending on the directions of h i (s), the symbol SNR can vary dramatically. If for instance, the r antennas picked for transmission at a given instance of s have two or more highly correlated h i s, the corresponding SNR will be too low. On the other extreme, if h i s are close to being orthogonal, the SNR will be high. Hence, some packets and/or symbols may enjoy high SNRs while others may suffer low SNRs, depending on the channel characteristics. Moreover, if a packet spans over multiple symbols, different symbols may encounter the same SNR. 
     In one embodiment, instead of choosing r antennas to transmit r modulation symbols, which may result in poor SNR due to correlation effect in the transmit antennas, r modulation symbols are transmitted by N T  antennas. In this case, the r×1 vector x(s) is randomly rotated by an orthogonal vector rotation matrix Θ(s), e.g., of dimension N T ×r. That is, the new transmitted vector of dimension {N T ×1} would be:
 
 {tilde over (x)} ( s )=Θ( s ) x ( s ).  Eq. (3)
 
     The effect of Θ(s) is to randomize the direction by which each symbol x i (s) is received. Accordingly, the received symbols, in vector form, would be: 
     
       
         
           
               
             
               
                 
                   
                     
                       
                         
                           
                             r 
                             ⁡ 
                             
                               ( 
                               s 
                               ) 
                             
                           
                           = 
                           
                             
                               
                                 H 
                                 ⁡ 
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 Θ 
                                 ⁡ 
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 x 
                                 ⁡ 
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                             + 
                             
                               n 
                               ⁡ 
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               
                                 
                                   H 
                                   ~ 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 x 
                                 ⁡ 
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                             
                             + 
                             
                               n 
                               ⁡ 
                               
                                 ( 
                                 s 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 r 
                               
                               ⁢ 
                               
                                 
                                   
                                     
                                       h 
                                       ~ 
                                     
                                     i 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     x 
                                     i 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     s 
                                     ) 
                                   
                                 
                               
                             
                             + 
                             
                               
                                 n 
                                 ⁡ 
                                 
                                   ( 
                                   s 
                                   ) 
                                 
                               
                               . 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       4 
                       ) 
                     
                   
                 
               
             
           
         
       
     
     In this case, irrespective of the correlation between antennas, the modulation symbols are received in random directions. Moreover, for the same instance of s, different modulation symbols encounter different effective channel responses {{tilde over (h)} i (s) }. This randomization in directions prevents severe correlation among multiple transmit antennas. 
       FIG. 1  shows two embodiments for transmitting r streams of modulation symbols by N T  antennas. In  FIG. 1A , a first number, e.g., r, of streams of bits is selected, e.g., by a transmitter, for transmission by N T  antennas. This selection may be based on the rank (r) of the channel response matrix H(k), as discussed above. In one embodiment, the selected streams of modulation symbols may be processed by encoders  102  and mappers  104 , and transformed from r streams of modulation symbols, via the vector rotator  106 , to N T  streams of symbols for transmission by N T  antennas  108 . Alternatively, as in  FIG. 1B , the transmitter may process one or more streams of data bits by encoder  110  and mapper  112 , and transform the stream of modulation symbols, via serial-to-parallel converter  114 , to r streams of modulation symbols. In one embodiment, the r streams of modulation symbols may be transformed, via the vector rotator  116 , to N T  streams of symbols for transmission by N T  antennas  118 . 
     In one embodiment, the r streams of modulation symbols are transformed to N T  streams of symbols by a vector rotator matrix Θ, which may include a discrete Fourier transform (DFT) operation followed by phase shift operation.  FIG. 2  shows one embodiment for transforming r streams of modulation symbols {X 1 , X 2 , . . . , X r } for transmission by the N T -antennas transmitter  108 ,  118 . In one embodiment, the r streams of modulation symbols may be augmented by enough number, e.g., N T -r, of known pilots, e.g., “0” entries, as shown by  202 , to provide N T  inputs to a discrete Fourier transform (DFT) unit  204 . In one embodiment, the discrete Fourier transform  204  is implemented/represented by an N T ×N T  unitary square matrix as shown in  FIG. 3A  and repeated below: 
     
       
         
           
             
                 
             
             ⁢ 
             
               Eq 
               . 
               
                   
               
               ⁢ 
               
                 ( 
                 5 
                 ) 
               
             
           
         
       
       
         
           
             D 
             = 
             
               
                 
                   
                     
                       1 
                       
                         
                           N 
                           T 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                         
                     
                     [ 
                     
                       
                         
                           1 
                         
                         
                           1 
                         
                         
                           1 
                         
                         
                           ⋯ 
                         
                         
                           1 
                         
                       
                       
                         
                           1 
                         
                         
                           
                             
                               e 
                               j 
                             
                             
                               
                                 2 
                                 ⁢ 
                                 π 
                               
                               
                                 N 
                                 T 
                               
                             
                           
                         
                         
                           
                             
                               e 
                               j 
                             
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 
                                   N 
                                   T 
                                 
                               
                               · 
                               2 
                             
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           1 
                         
                         
                           
                             
                               e 
                               j 
                             
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 
                                   N 
                                   T 
                                 
                               
                               · 
                               2 
                             
                           
                         
                         
                           
                             
                               e 
                               j 
                             
                             
                               
                                 
                                   2 
                                   ⁢ 
                                   π 
                                 
                                 
                                   N 
                                   T 
                                 
                               
                               · 
                               2 
                               · 
                               2 
                             
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                             
                                 
                             
                             ⁢ 
                             
                               
                                 d 
                                 
                                   P 
                                   ⁢ 
                                   
                                     , 
                                     q 
                                   
                                 
                               
                               = 
                               
                                 
                                   
                                     
                                       e 
                                       j 
                                     
                                     
                                       
                                         2 
                                         ⁢ 
                                         π 
                                       
                                       
                                         N 
                                         T 
                                       
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       P 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     q 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                             ⋱ 
                             ⁢ 
                             
                                 
                             
                           
                         
                         
                           
                               
                           
                         
                       
                       
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                               
                           
                         
                         
                           
                             
                                 
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     e 
                                     j 
                                   
                                   
                                     
                                       2 
                                       ⁢ 
                                       π 
                                     
                                     
                                       N 
                                       T 
                                     
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       N 
                                       T 
                                     
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                               2 
                             
                           
                         
                       
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
                 
                   
                     N 
                     T 
                   
                   × 
                   
                     N 
                     T 
                   
                 
               
               . 
             
           
         
       
     
     In one embodiment, the N T  outputs of the DFT unit  204  are phase shifted by phase rotators  206 . In one embodiment, phase rotations are implemented/represented by an N T ×N T  unitary diagonal square matrix, as shown in  FIG. 3B , and repeated below: 
     
       
         
           
             
               
                 
                   Δ 
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   
                                       
                                   
                                   
                                     
                                         
                                     
                                     
                                         
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   j 
                                   e 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                 
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                 
                                   
                                       
                                   
                                   
                                     
                                         
                                     
                                     
                                         
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   j 
                                   e 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   2 
                                 
                               
                             
                             
                               
                                   
                               
                             
                             
                               ◯ 
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               ⋱ 
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                           
                           
                             
                               
                                   
                               
                             
                             
                               ◯ 
                             
                             
                               
                                   
                               
                             
                             
                               
                                   
                               
                             
                             
                               
                                 
                                   
                                       
                                   
                                   
                                     
                                         
                                     
                                     
                                         
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   j 
                                   e 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   
                                     N 
                                     T 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       
                         
                           N 
                           T 
                         
                         × 
                         
                           N 
                           T 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   
                     ( 
                     6 
                     ) 
                   
                 
               
             
           
         
       
     
     where θ i sε[−π π] may be uniformly distributed random variables. The random variables θ i , may be generated from a “seed,” which may be communicated to the receiver side, in real-time or at a predetermined time, for generating similar random variables to reconstruct the vector rotator matrix Θ(s) 
     In one embodiment, the vector rotator matrix Θ(s) is implemented by:
 
Θ( s )=Λ D   Eq.7
 
     where D is the unitary N T -point DFT matrix, as defined above by Eq. (5), and Λ is the N T -point unitary diagonal square matrix, as defined by Eq. (6). This choice of Θ(s) may facilitate its implementation at the receiver side, e.g., when the number of transmit antennas (N T ) is a power of two, or N T  may be decomposed into at least two prime numbers, efficient FFT techniques can be used to implement Θ(s). 
     The disclosed embodiments may be applied to any one or combinations of the following technologies: Code Division Multiple Access (CDMA) systems, Multiple-Carrier CDMA (MC-CDMA), Wideband CDMA (W-CDMA), High-Speed Downlink Packet Access (HSDPA), Time Division Multiple Access (TDMA) systems, Frequency Division Multiple Access (FDMA) systems, and Orthogonal Frequency Division Multiple Access (OFDMA) systems. 
       FIG. 4  shows a block diagram of an embodiment of a transmitter system  410  and a receiver system  450  in a MIMO system  400 , e.g., in an OFDMA environment. At transmitter system  410 , data for one or multiple streams is provided by a data source  412 , coded by a transmit (TX) data processor  414 , and modulated by a modulator  420  to provide modulation symbols. The data rate, coding, and modulation for each stream may be determined by controls provided by a controller  430 . The modulation symbols for all streams and pilot symbols are then multiplexed and further processed to provide N T  symbol streams, which are further processed by N T  transmitters (TMTR)  422   a  through  422 T to provide N T  RF modulated signals that are then transmitted from N T  antennas  424   a  through  424 T. 
     At receiver system  450 , the N T  transmitted signals are received by N R  antennas  452   a  through  452 R. Each receiver (RCVR)  454  processes a received signal from an associated antenna  452  to provide a corresponding received symbol stream. A receive (RX) spatial/data processor  460  then processes the N R  received symbol streams from N R  receivers  454  to provide N T  detected symbol streams, and further processes each detected symbol stream to obtain decoded data for the stream. 
     RX spatial/data processor  460  may also derive an estimate of the channel response between the N T  transmit and N R  receive antennas (e.g., based on the pilot symbols) for each subband used for data transmission. The channel response estimate may be used to perform equalization at the receiver. RX spatial/data processor  460  may further estimate the SNRs of the detected symbol streams. Controller  470  may provide channel state information (CSI) regarding the MIMO channel and/or the received symbol streams (e.g., the received SNRs or rates for the symbol streams). The CSI is then processed by a TX data processor  478 , modulated by a modulator  480 , conditioned by transmitters  454   a  through  454 R, and sent back to transmitter system  410 . 
     At transmitter system  410 , the modulated signals from receiver system  450  are received by antennas  424 , conditioned by receivers  422 , demodulated by a demodulator  440 , and processed by an RX data processor  442  to recover the CSI sent by the receiver system. The CSI is then provided to controller  430  and may be used to (1) determine the number of symbol streams to transmit, (2) determine the rate and coding and modulation scheme to use for each symbol stream, (3) generate various controls for TX data processor  414  and modulator  420 , and (4) DFT and phase rotate the streams of symbols, as discussed above. 
     Controllers  430  and  470  direct the operation at the transmitter and receiver systems, respectively. Memory units  432  and  472  provide storage for program codes and data used by controllers  430  and  470 , respectively. 
       FIG. 5  shows a block diagram of a transmitter unit  500 , which is an embodiment of the transmitter portion of transmitter system  410  in  FIG. 4 , corresponding to  FIG. 1A . In this embodiment, TX data processor  414   a  includes a demultiplexer  510 , N D  encoders  512   a  through  512 D, and N D  channel interleavers  514   a  through  514 D (i.e., one set of encoder and channel interleaver for each stream). Demultiplexer  510  demultiplexes the data into N D  data streams, where N D  may be any integer from one to N T , e.g., rank “r.” Each data stream is coded and interleaved by a respective set of encoder  512  and channel interleaver  514 . The N D  coded data streams are then provided to modulator  420   a.    
     In this embodiment, modulator  420   a  includes N D  symbol mapping elements  522   a  through  522 D, a Vector rotator  524 , and N T  (OFDM) modulators. Each OFDM modulator includes an inverse fast Fourier transform (IFFT) unit  526  and a cyclic prefix generator  528 . Each of the N D  coded data streams is symbol mapped by a respective symbol mapping element  522  to provide a respective stream of modulation symbols, which is referred to as a transmitted symbol stream. Vector rotator  524  then performs the DFT and phase shifting and provides N T  symbol streams to the N T  OFDM modulators. 
     Within each OFDM modulator, for each symbol period, N F  symbols for the N F  sub-carriers are transformed by IFFT unit  526  to obtain a corresponding time-domain “transformed” symbol that includes N F  samples. To combat frequency selective fading, cyclic prefix generator  528  repeats a portion of each transformed symbol to obtain a corresponding OFDM symbol. A stream of OFDM symbols is formed for each transmit antenna and further processed by an associated transmitter  422  to obtain an RF modulated signal. N T  RF modulated signals are generated and transmitted in parallel from the N T  transmit antennas. 
     The signaling transmission techniques described herein may be implemented by various means. For example, these techniques may be implemented in hardware, software, or a combination thereof. For a hardware implementation, the processing units used to process (e.g., compress and encode) signaling may be implemented within one or more application specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field programmable gate arrays (FPGAs), processors, controllers, micro-controllers, microprocessors, other electronic units designed to perform the functions described herein, or a combination thereof. The processing units used to decode and decompress the signaling may also be implemented with one or more ASICs, DSPs, and so on. 
     For a software implementation, the signaling transmission techniques may be implemented with modules (e.g., procedures, functions, and so on) that perform the functions described herein. The software codes may be stored in a memory unit (e.g., memory unit  432  or  472  in  FIG. 4 ) and executed by a processor (e.g., controller  430  or  470 ). The memory unit may be implemented within the processor or external to the processor. 
     The previous description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the invention. Thus, the present invention is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.