Abstract:
Gain variations during a packet can lead to significant performance degradation in communications systems that use high order quadrature amplitude modulation (QAM). A method and the associated apparatus track such variations in an OFDM system and completely eliminate any performance degradation. Gain estimation and compensation is employed with the use of pilot subcarriers in the payload of an OFDM data packet. Estimated pilot magnitude ratios are averaged, throughout the processing life of a packet, to yield accurate gain estimations. A gain compensation factor is used to adjust data carriers. An exclusion method is also employed to eliminate pilot carriers which contribute to noise.

Description:
RELATED APPLICATIONS 
       [0001]    This application is a continuation of U.S. application Ser. No. 11/522,793, filed Sep. 18, 2006, which is a continuation-in-part of U.S. application Ser. No. 11/343,736, filed Jan. 30, 2006, which is a continuation of U.S. application Ser. No. 11/202,930, filed Aug. 12, 2005, which claims the benefit of U.S. Provisional Application No. 60/601,099, filed on Aug. 12, 2004. The entire teachings of the above applications are incorporated herein by reference. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    Frequency Division Multiplexing (FDM) is a technology, widely used in communication systems, which allows for the transmission of multiple signals simultaneously over a single transmission path. Each signal travels within its own unique frequency range, or carrier, which is modulated by the data (i.e., text, voice, video, etc.) to represent the information being transmitted. 
         [0003]    Orthogonal Frequency Division Multiplexing (OFDM) is a technique which distributes data over several carriers that are spaced apart at precise frequencies. This spacing provides the orthogonality in this technique which prevents the demodulators from seeing frequencies other than their own. 
         [0004]    From a frequency perspective, a typical OFDM packet structure  100  is divided into subcarriers, as shown in  FIG. 1 . These subcarriers are further classified into pilots (P n ) or data (X n ). Pilots are subcarriers whose value is known prior to the transmission of the packet. Knowing the value of the pilot subcarriers serves as a useful reference to estimate impairments throughout a packet. 
         [0005]    In contrast to the pilot subcarriers, the value of data subcarriers is not known prior to the transmission of the packet. The data subcarriers hold the information which is to be transmitted, and is therefore unknown prior to transmission. Each data carrier has an associated pilot, which is typically the pilot that is closest to it in frequency.  FIG. 1  shows the four pilots (P 0 -P 3 ) and the associated data subcarriers for a 802.11 OFDM. 
         [0006]    From a time perspective, the OFDM packet  200  is divided into a sequence of OFDM symbols, as is shown in  FIG. 2 . At the start of the OFDM packet  200  is a training symbol  201 , which is followed by a number of payload symbols  203 ( 1 )- 203 ( n ). All subcarriers (pilots and data) in the training symbol  201  are known prior to the transmission of the OFDM packet, wherein, only the pilots are known prior to transmission in the payload symbols 203. 
         [0007]    In order to distribute and transmit the various OFDM symbols, OFDM systems may employ a Quadrature Amplitude Modulation (QAM) scheme. A QAM scheme conveys data by changing or modulating the amplitudes of two carrier waves, in response to a data signal. A typical OFDM transmitter and receiver pair  300  employing a QAM scheme is shown in  FIG. 3 . A flow of bits to be transmitted from a digital transmitter baseband  304  is split into two equal parts, therefore generating two independent signals. The two signals are converted to analog signals via digital to analog converters (DAC)  306   a  and  306   b . The analog signals are then passed through low pass filters (LPF)  308   a  and  308   b . The two analog signals are then encoded separately, with one analog signal being multiplied  308   a  by a cosine wave and the other analog signal being multiplied  308   b  by a sine wave. The encoded signals are then added together  311 , and the combined signal is then passed through a power amplifier  312  and transmitted with the use of an antenna  313 . The receiver  302  detects the transmitted signal with the use of an antenna  315  and transmits the signal to a radio frequency receiving device  314 . The received signal is then separated and decoded with the multiplication of a cosine wave  316   a  and a sine wave  316   b . The signals are then passed through a low pass filter  318   a  and  318   b , and thereafter, the signals are digitally transformed with the use of analog to digital converters  320   a  and  320   b . Finally, the signals are then sent to the digital receiver baseband  322 . 
         [0008]    In the radio frequency (RF) portion of the transmitter, electromagnetic waves are generated and transmitted by alternating current fed into antenna  313 . This RF section is also a region that may be susceptible to gain variations. Gain variations may be caused by a number of factors, one of which may be thermal changes in the transmitting device.  FIG. 4  shows a calculated transient electrical response of a self-heating device. The self-heating effect, caused by a temperature rise due to a power dissipation and a temperature dependence of device characteristics, can be regarded as a thermal-electrical feedback inside the device (f). The y-axis of the plot represents the gain change of the device and the x-axis represents normalized time. 
         [0009]    As may be seen in  FIG. 4 , the device is stable, or endures minimal gain variations with respect to time, when f&lt;1. The device becomes unstable, or endures sustainable variations with respect to time, when f≧1. The graphical data illustrates gain changes increasing linearly with time for f=1, and exponentially with time for f&gt;1. 
         [0010]    The effects of gain variations on a signal are also shown in  FIGS. 5A and 5B . In an undistorted QAM constellation, points are usually arranged in a square grid with equal vertical and horizontal spacing and with the number of points on the grid typically being a power of 2, as may be seen in  FIG. 5A .  FIG. 5B  displays a QAM 64 constellation, which may be used to support 48 and 54 Mbps rates in a typical 802.11 transmitter and receiver pair, with a 0.5 dB gain distortion. As is shown in  FIG. 5B , the 0.5 dB gain distortion has caused the constellations of the QAM to become blurred and expanded. Distortions may include impairments such as group delay, discrete echoes (e.g., reflections, ghosts or multipath distortion), micro-reflections and amplitude tilt. Distortions such as the ones previously mentioned are capable of causing serious damage to upstream data. The table below records the signal to noise ratio (SNR) limit that is imposed on a signal due to gain variation or distortion. 
         [0000]    
       
         
               
               
               
             
           
               
                   
               
               
                 Power 
                 Amplitude 
                 SNR 
               
               
                 Variation 
                 Variation 
                 Limit 
               
               
                 (dB) 
                 (dB) 
                 (dB) 
               
               
                   
               
             
             
               
                 0.1 
                 0.05 
                 38.7 
               
               
                 0.2 
                 0.10 
                 32.7 
               
               
                 0.3 
                 0.15 
                 29.1 
               
               
                 0.4 
                 0.20 
                 26.5 
               
               
                 0.5 
                 0.25 
                 24.5 
               
               
                 0.6 
                 0.30 
                 22.9 
               
               
                 0.7 
                 0.35 
                 21.5 
               
               
                 0.8 
                 0.40 
                 20.3 
               
               
                 0.9 
                 0.45 
                 19.2 
               
               
                 1.0 
                 0.50 
                 18.3 
               
               
                   
               
             
          
         
       
     
         [0011]    As this table shows, a 0.25 dB gain variation in amplitude and a 0.5 variation in power will impose a 24.5 dB upper bound on the SNR. A QAM 64 or a 54 Mbps system may not function properly with such bounds since this system generally needs 27-28 dB in a typical indoor environment. Also, the 802.11 standard transmitter and receiver pair mandates a minimum error vector magnitude (EVM) of 25 dBc. With gain changes of 0.25 dB or more, the transmitter will not even meet the EVM specification (assuming that the transmitter is the source of gain variation). 
         [0012]    Prior art methods of gain compensation have been developed in an attempt to correct the distortion caused by gain variation. Many prior art methods of gain compensation make use of the training symbols in the OFDM packet to estimate the gain. An estimation may typically be made at a time 0, or once a packet is first received. Thus, many prior art systems will evaluate the pilots of the training symbol once the packet has first arrived, ‘time zero,’ and estimate a gain variation. The data subcarriers of the payload symbols are then compensated based on the initial estimation made at ‘time zero.’ An updated estimation may be made, again at time zero, once a new packet has been received. 
       SUMMARY OF THE INVENTION 
       [0013]    A method and apparatus for gain compensation of a packet in an orthogonal frequency division multiplexing (OFDM) system is discussed. The system establishing an initial magnitude estimate for a packet, determining a variation in magnitude of the packet over time, and applying to the packet a gain that varies over time. The gain may be based on the initial magnitude estimate and the variation in magnitude. The determination of the variation in magnitude may further comprise determining a pilot magnitude ratio estimation over time, and computing a pilot magnitude estimation based on the pilot magnitude ratio estimation. The pilot magnitude ratio may be normalized by a lead pilot. Pilot exclusion may also be comprised in the establishment of the variation in magnitude, wherein a pilot whose magnitude is low in accordance with an exclusion algorithm. The variation in magnitude may be updated for every OFDM symbol or alternatively may be updated after a predetermined number of OFDM symbols have been processed. The system may compute the gain compensation factor by obtaining a ratio of the initial magnitude estimate and the variation in magnitude. The gain compensation factor may comprise an operator that returns an index of a pilot associated with the data subcarrier. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0014]    The foregoing will be apparent from the following more particular description of example embodiments of the invention, as illustrated in the accompanying drawings in which like reference characters refer to the same parts throughout the different views. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating embodiments of the present invention. 
           [0015]      FIG. 1  is an illustrative example of an OFDM packet viewed from a frequency perspective; 
           [0016]      FIG. 2  is an illustrative example of an OFDM packet viewed from a time perspective; 
           [0017]      FIG. 3  is a schematic of a typical OFDM transmitter and receiver pair  300  employing a QAM scheme; 
           [0018]      FIG. 4  is a graphical representation of a calculated transient electrical response of a self-heating device; 
           [0019]      FIGS. 5A and 5B  are schematics of an undistorted and distorted, respectively, QAM 64 constellation grid; 
           [0020]      FIG. 6  is a block diagram of a canonical OFDM receiver; 
           [0021]      FIG. 7  is a block diagram of a OFDM receiver according to an embodiment of the present invention; 
           [0022]      FIG. 8  is a detailed view of the computational gain estimation and compensation steps according to an embodiment of the present invention; and 
           [0023]      FIG. 9  is a graphical representation of results obtained using methods and systems according to embodiments of the prior art and the present invention. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0024]    A description of preferred embodiments of the invention follows. 
         [0025]    Prior art methods of gain estimation and compensation are generally noisy. It is possible to degrade the performance of a modem when such noisy estimates are used to compensate for gain variations. The noisy gain estimates in prior art systems are primarily due to the fact that all gain estimations are made prior to processing the packet, at ‘time zero.’ Changes in gain may occur over microseconds while packet duration may occur in milliseconds. Thus, gain variations may radically affect a system and may also cause mid-packet gain variation, which may be detrimental to the data packet. Therefore, during the processing life of the packet, significant changes in gain may occur. These mid-packet changes must be accounted for in gain estimation and compensation calculations. 
         [0026]      FIG. 6  displays a canonical OFDM receiver  600 . The receiver  600  typically first applies a timing and frequency offset correction  601  to a received signal. The timing and frequency offset correction  601  is applied to the start of an OFDM packet and synchronizes the packet in time and frequency. A Fast Fourier Transform (FFT) is then performed on the synchronized signal to produce the frequency representation of the OFDM signal comprising pilot and data subcarriers. An equalization  605  is then performed to adjust the signal such that all the pilot subcarriers are of equal value. A phase correction or derotation  607  may also be performed to correct rotational distortions. Finally the signal may be decoded  609  in order to produce bits. 
         [0027]    As is shown in  FIG. 6 , the canonical OFDM receiver  600  does not account for mid-packet gain variations. Thus there is no adjustment for gain over the length, or processing life, of an OFDM packet. Furthermore, pilots may be so severely effected by gain variation that their use in gain estimation and compensation calculations may add significant noise to the estimate. The OFDM receiver  600  does not account for severely degenerated pilots in gain estimation or compensation calculations. 
         [0028]    A system and method of gain estimation and compensation is needed which may adjust gain estimations throughout the length or processing life of an OFDM packet without introducing noise in the system. An embodiment of the present invention is to build a gain estimator that eliminates the problem of gain variation, in the receiver, without causing significant additional performance loss of its own. The receiver estimates and compensates for gain variation throughout the length, or processing time, of the OFDM packet. A method of inclusion/exclusion is also presented, where a pilot may be included or excluded based on the severity of its gain change. Thus, the inclusion/exclusion method prevents the incorporation of substantially degraded pilots in gain compensation and estimation calculations. 
         [0029]    An important feature of first order OFDM systems is that the frequency of the OFDM symbols will remain constant over time. Hence, while pilot magnitudes may change with gain variations, their magnitude ratio will remain constant. Thus, in an embodiment of the present invention, a pilot magnitude ratio estimate is obtained for all pilots in each payload symbol. The pilot magnitude ratio estimation is obtained through a time average of pilot ratios, rather than the pilots themselves, which may be updated for every symbol. Upon obtaining the pilot magnitude ratio estimation, the pilot magnitude estimation may be found using an averaging technique. Finding the pilot magnitude estimation by using the pilot magnitude ratio estimation yields accurate results, as compared to prior art methods, due to the fact that pilot magnitude ratios remain constant during gain variations. Using the pilot magnitude estimation, a gain compensation factor may be computed and applied to each data subcarrier to therefore compensate the data. 
         [0030]    The number of pilots herein will be denoted by the variable N. The pilot indices run from 0 through N-1. The number of payload symbols is denoted by the variable M. The training symbol is assigned index 0. It should be appreciated that if there are several training symbols, they are averaged to yield a single training symbol. The payload symbols are assigned indices 1, 2 , 3 . . . M. The i th  pilot in the m th  OFDM symbol is denoted by P m,i . The k th  data subcarrier in the m th  symbol is similarly denoted by X m,k . It should be noted that complex signals are bold and italicized. Real signals are only bold. Variables that are not signals are only italicized. Estimates of unknown quantities are qualified with a tick (′). Hence, if H denotes a known (complex) signal, then H′ is an estimate of H. Magnitude of complex numbers (i.e., their absolute value) is represented via the |·| operator. 
         [0031]    An illustrative example of an embodiment of the present invention is depicted in  FIG. 7 . The OFDM receiver  700  comprises all of the components featured in OFDM receiver  600  of  FIG. 6 , with the addition of a gain estimation and compensation device  701 .  FIG. 8  displays a block diagram of the computational steps taken by the gain estimation and compensation device  701 . 
         [0032]    Gain estimation is achieved by first performing a ratio estimation  801 . The ratio estimation step  801  is a time averaging technique, which may be updated for every payload symbol. The ratio estimation step  801  may be described using the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     R 
                     
                       m 
                       , 
                       i 
                     
                     ′ 
                   
                   = 
                   
                     
                       1 
                       m 
                     
                      
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         m 
                       
                        
                       
                           
                       
                        
                       
                         ( 
                         
                           
                              
                             
                               P 
                               
                                 j 
                                 , 
                                 i 
                               
                             
                              
                           
                           / 
                           
                              
                             
                               P 
                               
                                 j 
                                 , 
                                 lead 
                               
                             
                              
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0033]    where R′ m,i  represents the estimated ratio value of the pilots in the OFDM system; m represents the number of payload symbols included in the estimation; |P j,i | represents the absolute value of the i th  pilot in the j th  payload symbol; and |P j,lead | the absolute value of the lead pilot, or the pilot with the largest absolute value, in the j th  payload system. The training pilot with the highest power (i.e., with the largest absolute value) is termed the lead pilot. The lead pilot is used as a reference when calculating relative pilot magnitude ratios. Its index is denoted by the subscript “lead.” 
         [0034]    An example output from pilot magnitude ratio estimation  801  is provided below: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       [ 
                       i 
                       ] 
                     
                     b 
                   
                    
                   
                     
                       [ 
                       
                         
                           
                             
                               R 
                               
                                 1 
                                 , 
                                 4 
                               
                               ′ 
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 2 
                               
                                
                               
                                 R 
                                 
                                   2 
                                   , 
                                   4 
                                 
                                 ′ 
                               
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 3 
                               
                                
                               
                                 R 
                                 
                                   3 
                                   , 
                                   4 
                                 
                                 ′ 
                               
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 4 
                               
                                
                               
                                 R 
                                 
                                   4 
                                   , 
                                   4 
                                 
                                 ′ 
                               
                             
                           
                         
                         
                           
                             
                               R 
                               
                                 1 
                                 , 
                                 3 
                               
                               ′ 
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 2 
                               
                                
                               
                                 R 
                                 
                                   2 
                                   , 
                                   3 
                                 
                                 ′ 
                               
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 3 
                               
                                
                               
                                 R 
                                 
                                   3 
                                   , 
                                   3 
                                 
                                 ′ 
                               
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 4 
                               
                                
                               
                                 R 
                                 
                                   4 
                                   , 
                                   3 
                                 
                                 ′ 
                               
                             
                           
                         
                         
                           
                             
                               R 
                               
                                 1 
                                 , 
                                 2 
                               
                               ′ 
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 2 
                               
                                
                               
                                 R 
                                 
                                   2 
                                   , 
                                   2 
                                 
                                 ′ 
                               
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 3 
                               
                                
                               
                                 R 
                                 
                                   3 
                                   , 
                                   2 
                                 
                                 ′ 
                               
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 4 
                               
                                
                               
                                 R 
                                 
                                   4 
                                   , 
                                   2 
                                 
                                 ′ 
                               
                             
                           
                         
                         
                           
                             
                               R 
                               
                                 1 
                                 , 
                                 1 
                               
                               ′ 
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 2 
                               
                                
                               
                                 R 
                                 
                                   2 
                                   , 
                                   1 
                                 
                                 ′ 
                               
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 3 
                               
                                
                               
                                 R 
                                 
                                   3 
                                   , 
                                   1 
                                 
                                 ′ 
                               
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 4 
                               
                                
                               
                                 R 
                                 
                                   4 
                                   , 
                                   1 
                                 
                                 ′ 
                               
                             
                           
                         
                       
                       ] 
                     
                     
                       [ 
                       m 
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     1 
                      
                     A 
                   
                   ) 
                 
               
             
           
         
       
     
         [0035]    where the ratio estimates are arranged horizontally by increasing payload symbol number [m] and vertically by increasing pilot number [i]. The pilot magnitude ratio is the absolute value of the pilot, for which the magnitude ratio is being estimated for, normalized by the absolute value of the lead pilot. Calculated averages of previously estimated ratios may also be taken into account and averaged over time for later processed pilots. In other words, consider row b, the calculation of R′ 4,3  yields the following: 
         [0000]        R′   4,3 =(¼)[ R′   1,3   +R′   2,3   +R′   3,3   +|P   4,3   |/P   4,lead |]  (1B)
 
         [0036]    In the example provided above, the estimated magnitude ratio for the pilot (i=3) in the symbol (m=4) is the average of all the previous estimates (for symbols m=1-3) for the same pilot (i=3), as well as the normalized absolute value of the pilot for which the magnitude is being estimated for (|P 4,3 |/|P 4,lead |). Therefore, determining the pilot magnitude estimation is achieved by obtaining the normalized value of the pilot in question and averaging that value over time, similarly indexed pilots in previous symbols. 
         [0037]    Next, the pilot magnitude, or variation in pilot magnitude,  803  is computed using the data obtained by the magnitude ratio estimation step  801 . The computation of the pilot magnitude  803  may be represented by the following averaging formula: 
         [0000]    
       
         
           
             
               
                 
                   
                     A 
                     
                       m 
                       , 
                       i 
                     
                     ′ 
                   
                   = 
                   
                     
                       R 
                       
                         m 
                         , 
                         i 
                       
                       ′ 
                     
                     · 
                     
                       
                         ( 
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               1 
                             
                             N 
                           
                            
                           
                               
                           
                            
                           
                              
                             
                               P 
                               
                                 m 
                                 , 
                                 j 
                               
                             
                              
                           
                         
                         ) 
                       
                       / 
                       
                         ( 
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               1 
                             
                             N 
                           
                            
                           
                               
                           
                            
                           
                             R 
                             
                               m 
                               , 
                               j 
                             
                             ′ 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0038]    where A′ m,i  represents the estimated pilot magnitude, R′ m,i  represents the pilot magnitude ratio estimation obtained in equation (1), and N is the total number of pilots in the m th  symbol. The pilot magnitude estimate is obtained by summing the absolute value of all the pilots in the symbol and dividing that sum by the sum of all the pilot magnitude ratio estimates obtained for the pilots in the symbol. The result is then multiplied by the pilot ratio estimation of the pilot whose magnitude is being solved for. As an example, consider the following example: 
         [0000]    
       
         
           
             
               
                 
                   
                     A 
                     
                       2 
                       , 
                       2 
                     
                     ′ 
                   
                   = 
                   
                     
                       R 
                       
                         2 
                         , 
                         2 
                       
                       ′ 
                     
                     [ 
                     
                       
                          
                         
                           
                             P 
                             
                               2 
                               , 
                               1 
                             
                           
                           + 
                           
                             P 
                             
                               2 
                               , 
                               2 
                             
                           
                           + 
                           
                             P 
                             
                               2 
                               , 
                               3 
                             
                           
                           + 
                           
                             P 
                             
                               2 
                               , 
                               4 
                             
                           
                         
                          
                       
                       
                         
                           R 
                           
                             2 
                             , 
                             1 
                           
                           ′ 
                         
                         + 
                         
                           R 
                           
                             2 
                             , 
                             2 
                           
                           ′ 
                         
                         + 
                         
                           R 
                           
                             2 
                             , 
                             3 
                           
                           ′ 
                         
                         + 
                         
                           R 
                           
                             2 
                             , 
                             4 
                           
                           ′ 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   
                     2 
                      
                     A 
                   
                   ) 
                 
               
             
           
         
       
     
         [0039]    where the estimated pilot magnitude for the pilot indexed as two (j=2) in the symbol indexed as two (m=2), is the estimated pilot magnitude ratio of the pilot in question multiplied by the sum of the absolute value of all the pilots comprised in the symbol m=2, divided by the sum of all the estimated pilot magnitude ratio estimates obtained the symbol m=2. It should be appreciated that although the example illustrates only four pilots per symbol, a symbol may comprise any number of pilots. Expanding and approximating the equation, the following results: 
         [0000]    
       
         
           
             
               
                 
                   
                     A 
                     
                       2 
                       , 
                       2 
                     
                     ′ 
                   
                   = 
                   
                     
                       R 
                       
                         2 
                         , 
                         2 
                       
                       ′ 
                     
                     [ 
                     
                       
                          
                         
                           
                             P 
                             
                               2 
                               , 
                               1 
                             
                           
                           + 
                           
                             P 
                             
                               2 
                               , 
                               2 
                             
                           
                           + 
                           
                             P 
                             
                               2 
                               , 
                               3 
                             
                           
                           + 
                           
                             P 
                             
                               2 
                               , 
                               4 
                             
                           
                         
                          
                       
                       
                         
                           1 
                           2 
                         
                         [ 
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                          
                                         
                                           P 
                                           
                                             1 
                                             , 
                                             1 
                                           
                                         
                                          
                                       
                                       
                                          
                                         
                                           P 
                                           
                                             1 
                                             , 
                                             lead 
                                           
                                         
                                          
                                       
                                     
                                     + 
                                     
                                       
                                          
                                         
                                           P 
                                           
                                             2 
                                             , 
                                             1 
                                           
                                         
                                          
                                       
                                       
                                          
                                         
                                           P 
                                           
                                             2 
                                             , 
                                             lead 
                                           
                                         
                                          
                                       
                                     
                                   
                                   ) 
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                          
                                         
                                           P 
                                           
                                             1 
                                             , 
                                             2 
                                           
                                         
                                          
                                       
                                       
                                          
                                         
                                           P 
                                           
                                             1 
                                             , 
                                             lead 
                                           
                                         
                                          
                                       
                                     
                                     + 
                                     
                                       
                                          
                                         
                                           P 
                                           
                                             2 
                                             , 
                                             2 
                                           
                                         
                                          
                                       
                                       
                                          
                                         
                                           P 
                                           
                                             2 
                                             , 
                                             lead 
                                           
                                         
                                          
                                       
                                     
                                   
                                   ) 
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     1 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                            
                                           
                                             P 
                                             
                                               1 
                                               , 
                                               3 
                                             
                                           
                                            
                                         
                                         
                                            
                                           
                                             P 
                                             
                                               1 
                                               , 
                                               lead 
                                             
                                           
                                            
                                         
                                       
                                       + 
                                       
                                         
                                            
                                           
                                             P 
                                             
                                               2 
                                               , 
                                               3 
                                             
                                           
                                            
                                         
                                         
                                            
                                           
                                             P 
                                             
                                               2 
                                               , 
                                               lead 
                                             
                                           
                                            
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                        
                                       
                                         P 
                                         
                                           1 
                                           , 
                                           4 
                                         
                                       
                                        
                                     
                                     
                                        
                                       
                                         P 
                                         
                                           1 
                                           , 
                                           lead 
                                         
                                       
                                        
                                     
                                   
                                   + 
                                   
                                     
                                        
                                       
                                         P 
                                         
                                           2 
                                           , 
                                           4 
                                         
                                       
                                        
                                     
                                     
                                        
                                       
                                         P 
                                         
                                           2 
                                           , 
                                           lead 
                                         
                                       
                                        
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   
                     2 
                      
                     B 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           A 
                           
                             2 
                             , 
                             2 
                           
                           ′ 
                         
                         ≈ 
                           
                          
                         
                           
                             R 
                             
                               2 
                               , 
                               2 
                             
                             ′ 
                           
                            
                           
                             [ 
                             
                               
                                  
                                 P 
                                  
                               
                               
                                 ( 
                                 
                                   
                                      
                                     P 
                                      
                                   
                                   
                                      
                                     
                                       P 
                                       lead 
                                     
                                      
                                   
                                 
                                 ) 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         ≈ 
                           
                          
                         
                           
                             1 
                             2 
                           
                            
                           
                             
                               ( 
                               
                                 
                                   
                                      
                                     
                                       P 
                                       
                                         1 
                                         , 
                                         2 
                                       
                                     
                                      
                                   
                                   
                                      
                                     
                                       P 
                                       
                                         1 
                                         , 
                                         lead 
                                       
                                     
                                      
                                   
                                 
                                 + 
                                 
                                   
                                      
                                     
                                       P 
                                       
                                         2 
                                         , 
                                         2 
                                       
                                     
                                      
                                   
                                   
                                      
                                     
                                       P 
                                       
                                         2 
                                         , 
                                         lead 
                                       
                                     
                                      
                                   
                                 
                               
                               ) 
                             
                             [ 
                             
                                
                               
                                 P 
                                 lead 
                               
                                
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         ≈ 
                           
                          
                         
                           
                             ( 
                             
                               
                                  
                                 P 
                                  
                               
                               
                                  
                                 
                                   P 
                                   lead 
                                 
                                  
                               
                             
                             ) 
                           
                           [ 
                           
                              
                             
                               P 
                               lead 
                             
                              
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         ≈ 
                           
                          
                         
                            
                           P 
                            
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     2 
                      
                     C 
                   
                   ) 
                 
               
             
           
         
       
     
         [0040]    Therefore, through the use of averaging, equation (2) will yield the strength or magnitude of the i th  pilot averaging over all the pilots and pilot magnitude ratio estimates in the m th  payload symbol. Thus, equation (2) makes use of averaging over time with respect to payload symbols and averaging over frequency with respect to pilots. It should be appreciated that the following simplified equation may also yield a good approximation of pilot magnitude estimation: 
         [0000]    
       
         
           
             
               
                 
                   
                     A 
                     
                       m 
                       , 
                       i 
                     
                     ′ 
                   
                   = 
                   
                     
                       1 
                       m 
                     
                      
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         m 
                       
                        
                       
                           
                       
                        
                       
                          
                         
                           P 
                           
                             j 
                             , 
                             i 
                           
                         
                          
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     2 
                      
                     D 
                   
                   ) 
                 
               
             
           
         
       
     
         [0041]    Equation 2D is the mathematically the same as equation (1) but for the normalization with P lead . A few of the pilots in the payload symbols may be greatly affected by the gain variation such that their magnitude is approximately equal to noise. Normalizing each pilot to the lead pilot will help to reduce the variations among pilots and reduce the amount of noise in the final estimation. Thus, as may be seen in equations (1)-(2D), a pilot magnitude may be obtained by averaging pilot signals over successive symbols over time. 
         [0042]    Finally, a gain compensation factor is computed  805 . The computation of the gain compensation factor may be represented by the following formula: 
         [0000]        C′   m,i   =|P   0,i   |/A′   m,i    (3)
 
         [0043]    where |P 0,i | is the absolute magnitude of the pilot indexed i at ‘time zero,’ A′ m,i  is the estimated pilot magnitude obtained during the processing life of the OFDM symbol and from equation (2), and C′ m,i  is the gain compensation factor, or an estimated value of how much the pilot has changed in magnitude. Thus, the gain compensation factor for an i th  pilot in an m th  symbol is obtained by obtaining the ratio of the absolute value of the pilot at a ‘time zero,’ in the symbol to be compensated, by the estimated pilot magnitude obtained for the pilot in equation (2). Thus, in contrast to prior art systems, the gain compensation factor  805  adjusts the gain throughout the length of the OFDM packet and does not simply estimate the gain using ‘time zero’ analyses. Therefore, mid-packet gain variations may be accounted for and compensated. 
         [0044]    The gain compensation factor is applied to all the data subcarriers in the payload to compensate for changes in the front-end gain  807 . The compensation of the data carriers may be represented by the following formula: 
         [0000]        G   m,k   =X   m,k    C′   m,assoc(k)    (4)
 
         [0045]    where X m,k  represents the k th  data carrier in the m th  payload symbol, C′ m, assoc(k)  represents the gain compensation which must be applied to the data carrier, with the assoc(k) operator returning the index of the pilot associated with data subcarrier k, and G m,k  represents the compensated data carrier. Thus, by applying the gain compensation factor to the data subcarriers, variations due to gain may be compensated for. 
         [0046]    In the examples provided, equations (1)-(4) have been described as being updated for every incoming payload symbol. It should be appreciated that equations (1)-(4) may be updated after a set number of payload symbols have been processed. As was previously shown in  FIG. 4 , gain changes may vary rapidly with time (i.e., f&lt;0), or the gain changes may be relatively stable with time (i.e., f≧0). Therefore, an optimal number of payload symbols for updating may be determined based on the rate of change of gain variations. For example, estimation and compensation steps may be updated for every payload symbol in the case of a rapidly changing gain, or may be updated for every twenty payload symbols for a relatively stable gain. It may be advantageous to include multiple payload symbols before updating estimation and compensation calculations, since doing so will decrease the amount of noise introduced in the system. 
         [0047]    The scheme discussed above is best suited for the ideal case in which all pilots comprise an identical SNR. Frequently it may be the case that one or more pilots comprise a SNR that is significantly lower than that of the lead pilot. In fact, some pilots may be so severely faded that the faded pilots are comprised mostly of noise. Therefore, a significant amount of noise may be incorporated into the gain estimation and compensation. 
         [0048]    As was previously mentioned, normalization of pilots may help alleviate the problem of severely faded pilot magnitudes but a more sophisticated method further reduces the incorporation of noise in estimation and compensation calculations. A pilot inclusion and exclusion method and algorithm has been developed to eliminate pilot noise from gain estimation and compensation. The elimination of pilot noise is accomplished by excluding pilots whose absolute magnitude has been greatly reduced by gain variations. To determine which pilots are to be included or excluded, all pilots are ordered according to their absolute magnitude values in the training symbol. Therefore, the lead pilot will always have an index of zero and so on. The lead pilot will also always be included in the estimation calculations. Inclusion decisions of all the other pilots are made in descending order of their strength. A pilot, indexed i, will be included if and only if the previous pilot, or the pilot indexed i-1, has been included and if the following equation, or algorithm, is satisfied for the pilot indexed i: 
         [0000]    
       
         
           
             
               
                 
                   
                      
                     
                         
                     
                      
                     
                       P 
                       
                         0 
                         , 
                         i 
                       
                     
                      
                   
                   &gt; 
                   
                     
                       ( 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             0 
                           
                           
                             i 
                             - 
                             1 
                           
                         
                          
                         
                             
                         
                          
                         
                            
                           
                               
                           
                            
                           
                             P 
                             
                               0 
                               , 
                               j 
                             
                           
                            
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             ( 
                             
                               i 
                               
                                 
                                   ∑ 
                                   
                                     j 
                                     = 
                                     0 
                                   
                                   
                                     i 
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                      
                                     
                                         
                                     
                                      
                                     
                                       P 
                                       
                                         0 
                                         , 
                                         j 
                                       
                                     
                                      
                                   
                                   2 
                                 
                               
                             
                             ) 
                           
                         
                         - 
                         1 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0049]    where |P 0,i | drepresents the absolute value of the i th  pilot, 
         [0000]    
       
         
           
             
               
                 ∑ 
                 
                   j 
                   = 
                   0 
                 
                 
                   i 
                   - 
                   1 
                 
               
                
               
                   
               
                
               
                  
                 
                     
                 
                  
                 
                   P 
                   
                     0 
                     , 
                     j 
                   
                 
                  
               
             
             , 
             
               
 
             
              
             and 
           
         
       
       
         
           
             
               ∑ 
               
                 j 
                 = 
                 0 
               
               
                 i 
                 - 
                 1 
               
             
              
             
                 
             
              
             
               
                  
                 
                     
                 
                  
                 
                   P 
                   
                     0 
                     , 
                     j 
                   
                 
                  
               
               2 
             
           
         
       
     
         [0000]    represent the sum and squared sum of the absolute value, respectively, of all pilots indexed 0-‘i-1.’ 
         [0050]    Alternatively, a simpler inclusion/exclusion scheme may be devised. For example, setting a threshold t, wherein t&lt;1. All the pilots may be multiplied by the threshold t, where only the pilots whose absolute magnitude is greater than the lead pilot, upon multiplication by t, will be included in the estimation. While more sophisticated inclusion/exclusion strategies may allow for better performance, the improved performance may come at the cost of additional hardware. Thus a suitable balance may be determined for individual signal processing needs. 
         [0051]    All the gain estimation and compensation steps, represented by equations (1)-(4) will remain the same for the included pilots. If the associated pilot of a data carrier has been excluded, the closest included pilot to that data carrier will be used for gain compensation. 
         [0052]    A graphical representation of results, obtained in part by using the above mentioned embodiments of the invention, is shown in  FIG. 9 . The benchmark represents a 54 Mbps packet error rate (PER) vs. SNR curve  900  in a 50 ns multipath environment with no gain distortion. A curve with a −2 dB uncompensated gain variation  901  is also plotted. It is desirable to estimate and compensate for gain in order to close the gap  903  between the benchmark curve  900  and the uncompensated −2 dB gain variation curve  901 , so that curve  903  may resemble the benchmark  900  as much as possible. Curves  905 - 908  show plotted data with 2 dB, 1 dB, −1 dB, and −2 dB gain variations, respectively; whose gain has been estimated and compensated according to an embodiment of the present invention. As may be seen from  FIG. 9 , curves  905 - 908  all show a significant decrease in PER while maintaining a high SNR. Curves  905 - 908  also over lap the benchmark curve  900  as well as decrease gap  903 . 
         [0053]    While this invention has been particularly shown and described with references to example embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the scope of the invention encompassed by the appended claims.