Patent Publication Number: US-7593456-B2

Title: Maximum likelihood block decision feedback estimation for CCK demodulation apparatus and method

Description:
The present application is a continuation of application Ser. No. 10/934,275 filed Sep. 3, 2004, now U.S. Pat. No. 7,412,000. 

   FIELD OF THE INVENTION 
   The present invention is related to the demodulation of CCK code symbols, where each symbol is formed from a plurality of CCK chips in a wireless IEEE 802.11b system where these CCK code symbols are presently used for 5.5 Mb and 11 Mb data rates. The invention is extensible to demodulation of code symbols which are Direct Sequence Spread Spectrum (DSSS) phase encoded using Walsh-Hadamard chipping sequences. 
   BACKGROUND OF THE INVENTION 
   Prior art wireless communication systems are defined in the IEEE protocols for 802.11b. IEEE 802.11b is an example of a spread spectrum wireless protocol whereby a transmitter emits an isotropic or multidirectional signal including a sequence of broadband phase modulation of a carrier according to a known sequence, where the sequence is referred to as a chipping code. For 1 Mb and 2 Mb IEEE 802.11b data rates, the chipping code is a Barker code. A receiver may be placed within a reception distance from the source, and it is common for a plurality of signals to arrive at the receiver, which may be a first signal which travels a shortest path, followed by reflected signals which have path lengths which may be longer than the shortest path. Additionally, the shortest path may be through an attenuating medium, such that the shortest path does not correspond to the strongest signal. The receiver must recover this plurality of signals and apply a method of filtering before demodulating the transmitted CCK symbol into decision data. One prior art method for compensating a communications channel uses channel sounding, whereby the communications channel is characterized by sending a known signal such as a packet preamble to determine the transfer function of the channel, and then applying a combining filter function to use the channel reflections in the decision process, thereby improving the decision&#39;s reliability. This filter function is known as a channel matched filter (CMF), which in implementation may be either a separate filter or a filter combined with a feed-forward filter (FFF). Because the topology of a feed forward filter involves complex multiplications and additions, the complexity of the filter grows with the number of taps in the filter, and it is desired to minimize the number of taps in this filter. The channel matched filter and the feed-forward filter become more critical with higher data rates, since the reflections may be fixed in time duration, and the higher data rate causes degradation over a longer interval compared to the symbol rate. For example, using a 11 Mhz clock, 11 bits of a Barker code may be transmitted over 1 us for 1 Mb data rate, and the receiver must correlate a single 11 bit code to single binary value. At 11 Mb, the transmitted data is organized into 8 bit symbols, and there are 64 different CCK symbols to decode rather than two Barker codewords, with phase decoding used to recover the remaining 2 bits of data. At these higher data rates using CCK symbols, the channel is more susceptible to bit errors associated with the same multipath reflections, and greater care must be taken in the receiver signal processing leading up to demodulation of the symbols into decision data. 
     FIGS. 1 and 2  show the phase encoding pattern for encoding data to be transmitted into four phase terms used to form the CCK symbol. Each pair of incoming bit patterns is phase encoded according to  FIG. 1 . For a real-valued signal in a quadrature phase mapping, the phases may be represented as  1  (representing phase  0  phase change), j (representing a phase change of π/2), −1 (representing a phase change of π), and −j (representing a phase change of 3π/2).  FIG. 2  shows 8 bit data  100  applied as an input to encoders  104   a - 104   d , which perform phase encoding by bit pairs into four values of phase information  102 , each having one of the values {1,j,−1,−j} according to the two bit (dibit) mapping shown in  FIG. 1 . The CCK symbol is generated using the expression of  FIG. 3 , whereby each of the 4 phases  102  generates the symbol C, comprising 8 chips of phase information, each chip having one of the values {1, j, −1, −j} where the phases represent the real and imaginary components according to Eulers formula:
   e   jφ =cos(φ)+ j  sin(φ). 
     FIG. 4  shows a typical signal processing system  118  for the demodulation of DSSS CCK symbols such as an IEEE 802.11b system operating at 5.5 Mb or 11 Mb. An RF front end  120  includes an antenna  130  coupled to an RF amplifier  132  and a pair of baseband mixers  134   a  and  134   b  which mix the output of the RF amplifier  132  to baseband using quadrature outputs of local oscillator  138 , which is operating at the carrier frequency of the transmitted data. The baseband quadrature signal from mixers  134   a  and  134   b  is filtered by matched low pass filters (LPF)  140   a  and  140   b . The input to A/D converters  122   a  and  122   b  and the subsequent signal processing operate on analytic signals in quadrature form, where the quadrature signal includes the real and imaginary components in separate signal processing channels. The quadrature signal leaving the anti-aliasing, or low pass, filters  140   a  and  140   b  is sampled by A/D converters  122   a  and  122   b  and converted into a binary representation typically comprising 6 bits of in-phase and 6 bits of quadrature data. The quadrature output of the A/D converters  122   a  and  122   b  is then phase corrected  124  such that during the preamble interval of a received packet, the real (1 and −1) component of the analytic signal is maximized in the I channel, and the imaginary (j and −j) component is therefore in the Q channel. During the preamble part of the packet, the preamble is real valued, rather than complex valued, such that after phase correction  124 , all of the preamble signal is maximized in the real (I) channel  125   a  and minimized in the imaginary (Q) channel  125   b . The Q output  125   b  of the phase corrector  124  has no imaginary component, so the phase corrector operates to maximize the real output  125   a , and later in the packet, when the coding switches to quadrature CCK, the demodulation which follows starts from a known phase rotation established during the preamble interval. The phase correction  124  may also include ongoing phase correction to compensate for phase variations over the duration of the packet, and this is often accomplished by multiplying the quadrature input with a quadrature representation of a sine function which is slowly varied to track variations in phase of the incoming signal, or using look-up tables, or any means known in the prior art. The phase corrector  124  therefore initializes and maintains the phase of the signal presented to the symbol sync  126  and CCK decision  128  over the duration of the packet. Symbol synchronization  126  converts the serial stream of CCK chips  8  into a framed series of CCK symbols of 8-chips each, each complex chip represented as 6 bits of binary data for each real and imaginary component. This framed CCK symbols are sent to the CCK decision block  128 , which converts the framed CCK symbols into decisions as to which is the most likely symbol received. 
   Each CCK symbol, also known as a CCK codeword, to be demodulated into 8-data bits is composed of 8 CCK chips, and each CCK chip represents a particular phase encoded value of part of the 8-data bits that was transmitted, as was described earlier. Two commonly practiced techniques for deciding from the 8 CCK chips of the CCK symbol received which data was transmitted are the use of hard decisions based on individual CCK chip decisions, and soft block decisions using n=8 CCK chips at a time. In a hard decision decoder, each individual CCK chip is examined and a hard binary decision is made on each CCK chip based on a threshold parameter, and after 8 such binary decisions are independently made for an 8 chip symbol, a data decision can then be made based on mapping the 8 independent binary CCK chip decisions to a closest (measured by Hamming distance) data decision. In a soft decision decoder as used in the present invention, a block decision is made using all native I&amp;Q sample values on the 8 CCK chips based on the euclidian distance from 8 received chips to a nearest 8 chip CCK codeword. In poor signal to noise conditions, and with a sufficiently sampled digital signal representing each CCK chip, the soft decision decoder is found to produce better code decisions than the hard decision decoder. 
   An additional degradation of the decision process comes in the form of multipath reflections. Multipath reflections in the communications channel result in bleeding, or spreading, of individual CCK chips, and also generate delayed copies of the CCK chips. While bleeding and delayed copies of CCK chips are all part of the same multipath reflection effect, the prior art corrections for them are handled according to whether they cause the CCK chip degradation inside or outside a symbol boundary. Multipath reflections which cause signal bleeding within a symbol block of 8 chips are known as intra-symbol bleeding, and are also referred to as inter-chip interference ICI, and can be handled within the CCK chip soft decoder, since it is using all 8 CCK chips to make a decision, while signal bleeding which transfers energy from one symbol to another symbol is known as inter-symbol interference ISI and must be corrected using Decision Feedback Equalization, as will be described later. 
     FIG. 5   a  shows the time domain smearing effect of a series of CCK chips  144   a ,  144   b ,  144   c , which are convolved  7  with the Fourier transform of the channel response  145  in the time domain. The time-domain channel response  145  includes multipath reflections which cause smearing of the individual chips  144   a ,  144   b ,  144   c  into the received responses  146   a ,  146   b ,  146   c . After multipath reflection, the separate chips are smeared in duration and for a linear system are additive such that the tail of a previous symbol becomes a post-cursor for the present symbol. The incoming chip stream to the CCK demodulator includes beginning of chip  146   b  adding into the tail of chip  146   a , and the tail of chip  146   b  adding into the beginning of chip  146   c , generating ICI. For clarity in showing ICI,  FIG. 5   a  shows short duration channel smearing  145  which is approximately equal to one CCK chip duration.  FIG. 5   a  shows ICI, however in a a typical WLAN setting, the leading edge of the channel impulse response function  145  has maximum phase change, while the tail is much longer than a single chip, and includes minimum phase components lasting many symbol lengths. 
     FIG. 5   b  shows an 8 chip CCK symbol  147 , and when the symbol  147  is passed through a communications channel with the impulse response  148 , a smeared symbol  149  results, which includes ICI as described in  FIG. 5   a , and also smears and broadens the extent of the 8 chip symbol  149  compared to the original symbol  147 . When a series of such smeared CCK symbols  149  are received, information from other symbols bleeds into the current symbol, as shown in  FIG. 5   c . The thick line represents the smeared current symbol  154 , while the dashed line represent the previous symbol  150 . The resulting ISI is the sum of waveforms  150  and  154  of  FIG. 5   c . A feedforward filter may be used to correct the phase shifts and minor delay shifts within each chip, thereby correcting the pre-cursor effects  152  of the current symbol. A technique known as Decision Feedback Equalization (DFE) may be used to smear a previous decision using knowledge of the communication channel impulse response  148 , and then invert and add the previous decision smeared symbol  151  into the current symbol  154  and previous symbol  150 , thereby removing the previous symbol  150  from the symbol stream, leaving only the current symbol  154  shown as the DFE result  155  of  FIG. 5   d , which then contains only the current symbol  155  including ICI. Current symbol post cursor effects are of 2 types: post-cursor ISI representing interference between the current symbol  154  and a post-cursor  150  of a previous symbol, which is removed by the feedback filter of DFE which produced waveform  151 , and post-cursor ICI from the current symbol. Post-cursor ICI can be removed by Maximum Likelihood Estimation (MLE), as will be described. In Decision Feedback Equalization, once the current symbol decision has been made and presented as decision data, it is converted back into an 8 chip symbol, smeared with the minimum phase part of the channel impulse response, and the smeared version of this previous symbol is subtracted from the current symbol, which improves the decision for the current symbol. In Maximum Likelihood Estimation, a soft decision (decision based on euclidian distance based on a block of 8 chips) is made on the current symbol based on subtracting the ICI component for all possible 256 current symbols from the current symbol, which has the effect of increasing the detector response for the correct symbol, and comparatively reducing the detector response for incorrect symbols. By making soft decisions based on subtracting the ICI from all possible current symbols, the accuracy of the decision is improved, shown as cleaned up waveform  157  of  FIG. 5   e.    
     FIG. 6  shows the block diagram for a prior art block decision feedback equalizer such as described in FIGS. 6 and 7 of U.S. Pat. No. 6,233,273 by Webster et al. Incoming quadrature CCK symbols are presented at input  160  to a feedforward filter  162 , which performs phase compensation principally on the leading edge of each of the 8 chips of the symbol where the signal energy and phase errors are highest. This phase compensation improves the accuracy of demodulation of the current symbol where only DFE is used. Block  168  makes a CCK symbol decision  170 , as will be described later. The decision data  170  is converted back to a CCK symbol of 8 chips in block  169  and is smeared to match the original channel distortion using feedback filter  166 , and is presented to the subtractor  164 , which restores the leading edge of each chip of the chip stream, as was described in  FIGS. 5   c  and  5   d . In the implementation of  FIG. 6 , feed forward filter  162  has a small number of taps, since it is used principally to perform phase correction on the comparatively short leading edge of each chip, while the decision feedback filter  166  operates over the comparatively longer temporal range, but minimum phase region of an entire symbol by smearing the previous symbol to match the channel reflections and subtracting this smeared previous symbol from the present symbol. Together, the feedforward filter  162  and feedback filter  166  constitute a transversal filter trained to equalize the channel impulse response of the communications channel. In this manner, the feedforward filter  162  reduces symbol interference due to pre-cursor effects, the feedback filter  166  reduces symbol interference due to post-cursors from previous symbols, and the maximum likelihood estimation removes the ICI effects from the current symbol. 
     FIG. 7  shows a prior art implementation  190  of the CCK symbol decision block  168  of  FIG. 6 . Examining one of the four blocks, a simple Fast Walsh Transform (FWT)  182  operates on the incoming CCK symbol  172 , and produces a complex-valued output based on transforming the incoming CCK chip  172  with a particular butterfly φ 2 . Each of the FWT blocks  182 ,  184 ,  186 ,  188  performs the Fast Walsh Transform using the incoming CCK chip and a unique butterfly value of φ 2 ={1,j,−1,−j}, for  174 ,  176 ,  178 , and  180 , respectively. The output of each Fast Walsh Transform block  182 ,  184 ,  186 ,  188  is 16 complex values, and for the four FWT blocks  182 ,  184 ,  186 ,  188  in aggregate, one of the 64 outputs will have a maximum magnitude. The FWT output which is maximum magnitude will determine the CCK symbol decision, and this one of 64 selection can be decoded to generate 6 bits of data. Examining the real (I), and imaginary (Q) values from the maximum valued output will further indicate which of the four phases {1,j,−1,−j} is present, which will result in 2 additional bits of data, thereby generating an 8 bit decision data value. 
     FIG. 8  shows a Fast Walsh Transform (FWT) block such as one of FWT blocks  182 ,  184 ,  186 ,  188  of  FIG. 7 . CCK symbols are presented to input  222  accompanied by one of the four butterfly phases φ 2 ={1,j, −1, −j} at input  232 . The simple FWT  220  comprises a butterfly configuration of multiplying the phase φ 2   232  with every alternating bit of the incoming symbol, followed by a series of adders which sum the resulting outputs after multiplication by {1,j,−1,−j}, as shown. For example, adder  226   a  adds b 1  to b 0 *e jφ2 , where e jφ2  is from the set {1, j, −1, −j} as was described  9  earlier. The output of each adder  226   a ,  226   b ,  226   c ,  226   d  is multiplied by {1,j,−1,−j} according to the legend shown in  FIG. 8  and these complex results are forwarded to the next stage as shown. The result of the series of additions of these results is seen in the outputs of the 16 adders  230   a  through  230   p , only one of which will have a maximum value for a given input symbol. For a given symbol in the subset associated with a particular phase  232 , one of the adder outputs  230   a  through  230   p  will have a maximum value, corresponding to the decoded symbol. When four of the simple FWT blocks  220  are provided with the same input symbol  222  and each of the four phases  232 , all 64 (16*4) code symbols are available for decoding, and a decision on the set of 64 symbols is made based on the largest value at the outputs of the adders in each FWT block  220 , and the phase of this output provides 2 additional bits of data. 
   As was described in the  FIGS. 5   a  through  5   d  and  FIG. 6 , the decision feedback equalizer can improve the decision made on a current symbol in poor signal to noise conditions. Once the previous symbol has been decided, it can be smeared with a filter which has characteristics similar to the communications channel, and then inverted and added as symbol  151  of  FIG. 5   c  to the current symbol  154 , thereby removing the post-cursor effects of the previous symbol  150  on current symbol  154 . It should also be noted that the decision feedback filter  166  may have as many taps as is required to subtract as many previous symbols from the current symbol as the taps of the filter permit. For the case of a feedback filter  166  with 16 taps, it is possible to subtract the effects of two 8 chip previous symbols which have added into the current symbol. It is also desirable to remove the intra-symbol (inter-chip interference ICI) interference of one chip smearing with another chip within a symbol, as shown in  FIG. 5   a . This can be done by making soft decisions on the entire symbol after subtracting the chip smearing ICI effect.  FIG. 9  shows a CCK symbol demodulator which performs both decision feedback equalization (DFE) (also referred to as pre-equalization), as well as maximum likelihood symbol estimation (MLE), also referred to functionally in  FIG. 9  as a post-equalizer function  312 . There are many ways of realizing the general MLE and DFE functionality shown in  FIG. 9 , but one such realization is described in FIGS. 13 and 14 of prior art U.S. Pat. Nos. 6,690,715 and 6,233,273 by Webster et al. In  FIG. 9 , incoming CCK chips  286  are presented as a digital stream of complex data, sampled with approximately 6 bit data width. This complex-valued digital data may first be combined with a channel matched filter (not shown) performing rake combining, the output of which is filtered by a feed-forward filter (FFF)  288 , which performs maximum phase compensation on the incoming signal, and passes the output as signal  289  to a serial-parallel converter  290 , which converts the serial stream of data into framed  8  complex values, representing a current CCK symbol  291  comprising 8 CCK chips. In addition to the CCK symbol input  291 , previous decision feedback symbol input  304  is provided for pre-equalization (removal of previous symbol post-cursors from the current symbol), and the output of this stage is fed to demodulator  314  which performs an FWT and current symbol subtractor  296  subtracts the 64 post-equalization symbol inputs for maximum likelihood estimates which remove intra-symbol interference ICI (shown in  FIG. 5   a ), where the intra-symbol interference correction is generated by post-equalizer  312 . Examining the decision feedback equalization path from the output  305 , the previous decision data  305  is converted to a CCK symbol of 8 chips in converter  300 , which produces a CCK symbol  301  corresponding to 8 complex chip phase values of I and Q as is done in CCK encoding for transmission of data. These 8 complex phase values  301  I and Q channels are passed through 8 feedback filters  302 - 1  through  302 - 8  which simulate the minimum phase and maximum delay component of the channel impulse response, as was described earlier. The length of the feedback filter  302 - 1  through  302 - 8  governs how many previous symbols of contributions may be removed from the current symbol, and in the case of a length flt_len=16 taps, a channel response function which includes signal energy bleeding from two previous CCK symbols may be subtracted from the current CCK symbol. 
   The output of the feedback filters  302 - 1  through  302 - 8  produce a symbol comprising 8 complex channel-equalized chip values for the previous decision  304 , which are subtracted from the current symbol  291  by the previous decision subtractor  292 , which subtracts the previous decision smeared symbol  304  from the current CCK symbol  291 , as was described for block  164  of  FIG. 6 . Simplifying the operation by ignoring the effect of the current symbol subtractor  296 , Fast Walsh Transform  294  converts the 8 complex values for a decision feedback equalized symbol into 64 complex outputs, from which the largest magnitude output would represent the demodulated symbol, thereby extracting 6 bits of data, and the phase position of the largest magnitude output {1,j,−1,−j} would provide an additional 2 bits of data, generating an 8 bit decision data value. Now including consideration of the operation of the post-equalizer processor  312  and current decision subtractor  296 , it may be understood that the subtraction of post-cursor ICI from chips within the current symbol smearing together may be done prior to the FWT  294  in the time domain, as was done with previous decision subtractor  292 , or it may be done after the FWT  294  in the phase domain, since the FWT is a linear operation. This may be done as long as the value to be subtracted is in the symbol time domain at the input of the FWT, or transformed into the phase domain if subtracted in the post-FWT phase domain. Considering further the implementation of the post-FWT current symbol subtractor  296  as shown in  FIG. 9 , the post-equalizer  312  provides 64 post-FWT phase domain, channel compensated symbol correction values  285  which represent corrections to all of the possible current CCK symbols in the post-FWT phase domain. These possible CCK symbol corrections  285  are presented to current symbol subtractor  296  as 64 complex values. Each of the 64 complex outputs  297  is expanded to four using 1,j,−1 and −j with index variable k  299 . The biggest picker  298  has 256 complex FWT values stored for this input symbol  291 , and one of the 256 values will have maximum real part. This maximum output will include the post-FWT phase domain current symbol, with the post-cursor ISI of the previous decision (shown as  150  of  FIG. 5   c ) removed by the pre-equalizer smeared previous decision  304  which provided symbol  151  of  FIG. 5   c , and the ICI of the chips of the current symbol bleeding into each other removed by post-equalizer  312  outputs  285  iterated over the four values of k. The index of the largest expanded output with largest  9  real-part would give the 8-data bit decision data. 
   Post-equalization value generator  312  produces the error generated by inter-chip interference (ICI) in the symbol, converts it into the post-FWT phase domain by taking the 64 CCK codewords  280 - 1  through  280 - 64 , passing each of the CCK codewords through an identical feedback filter  282 - 1  through  282 - 64  which generates the ICI correction value, and each of these is passed through a correlator  284 - 1  through  284 - 64  for converting the symbol to the post-FWT phase domain. In this manner, the ICI for each possible received symbol is subtracted, cleaning up the FWT output corresponding to the current symbol compared to the other outputs, thereby generating an improved decision for the correct symbol. 
   OBJECTS OF THE INVENTION 
   A first object of the invention is a sequential CCK demodulation processor for demodulating CCK symbols into decision data, the demodulation including a post-equalization value generator for converting CCK symbols into a plurality of post-FWT phase domain ICI correction components, a decision feedback circuit including a first subtractor for subtracting previous demodulated data which has been converted into a channel smeared previous demodulated CCK symbol from a current CCK symbol, the output of the first subtractor coupled to a Fast Walsh Transform, the Fast Walsh Transform generating an output which is coupled to a second subtractor operating on the Fast Walsh Transform output and individually subtracting the plurality of ICI correction components provided by the post-equalization value generator, the second subtractor output coupled to a biggest picker for selecting the largest magnitude said second subtractor output and extracting data bits from which said second subtractor output was selected, and extracting additional data bits from the phase of the largest magnitude of the second subtractor output. 
   A second object of the invention is, after characterizing a channel and generating from this characterization a channel compensation including a feedforward filter part and a DFE feedback part, performing demodulation of a current CCK symbol when a previously demodulated symbol is present by subtracting a channel-compensated previous CCK symbol from the present CCK symbol to form a first subtraction value, performing a Fast Walsh Transform (FWT) on the resulting first subtraction value, and then subtracting each of the possible ICI components for all possible current CCK symbols in the post-FWT phase domain from the resulting FWT value, and choosing a maximum magnitude subtraction result from the available  256  FWT values, where the maximum magnitude subtraction result may then be converted to demodulated data. 
   A third object of the invention is the generation of a channel compensated decision feedback component to compensate for inter-symbol interference due to post-cursors of previous symbols adding to the present symbol, the decision feedback component generated using a shift register and the values of a compensation filter {C 1  . . . C flt     —   len}, and a current chip value {c 1  . . . c nChip } 
   A fourth object of the invention is the generation of an Intra-Symbol Interference, or ICI, correction component, the ICI correction component computed for all possible CCK symbols, the ICI correction component being computed using a sparse set {C 1 ,C 3 ,C 5 ,C 7 } from a complete coefficient set {C 1 ,C 2 ,C 3 ,C 4 ,C 5 ,C 6 ,C 7 ,C 8 }, trained during start of the packet, the generation of the ICI correction component being done using six multipliers and an adder. 
   SUMMARY OF THE INVENTION 
   A CCK symbol demodulator demodulates an incoming current CCK symbol into current data using: 
   1) an iteration variable k which is varied from 0 through 3 where k also generates a φ 2 ={1,j,−1,−j} during a given current CCK symbol, the CCK symbol comprising 8 complex chip values; 
   2) a post-equalization value generator for computing values stored in post-equalize registers during a preamble time. During a preamble interval, a channel characterization is performed, the channel characterization generating a channel compensation function and related channel compensation coefficients {c 1 , c 2 , . . . , c 7 }, for example, for use in a feedback filter. After these coefficients {c 1  . . . c 7 } are known, a reduced set of 16 complex post-equalization register values is iteratively computed by a post-equalization value generator which uses only an iteration variable i with a range from 0 through 15 which selects one of 16 CCK phase combinations {φ 2 ,φ 3 }, and a reduced set of channel compensation coefficients {c 1 ,c 3 ,c 5 ,c 7 } using a reduced number of multipliers and adders. The post-equalize register contains 16 values, 4 of which are associated with a particular φ 2  derived from k for each φ 2 ={1,j,−1,−j} and thereafter expanded into 16 values by multiplication by φ 4 ={1,j,−1,−j}; 
   3) a first subtractor which subtracts the effects of previous CCK symbols from the present symbol to generate a first subtractor output; 
   4) a pre-equalization register containing a value which includes the channel compensated effects of previous decision symbols for removing previous symbol contributions from the current symbol; 
   5) a demodulator having as inputs the first subtractor output, a phase φ 2  derived from the iteration variable k, the four values of the post-equalization register associated with the phase φ 2  derived from k which were expanded into 16 post-equalize values as described earlier, where the output of the first subtractor is applied to a simple FWT, the simple FWT performing an FWT on the first subtraction output using as a butterfly phase the φ 2  which was derived from the iteration variable k, thereby producing 16 FWT outputs. The 16 FWT outputs are presented to a second subtractor, which subtracts from the 16 FWT outputs the 16 post-equalize values that were computed in step  1  for a given iteration variable k. These 16 post-equalize register values contain values the ICI contribution assuming that the corresponding symbol was the one transmitted, and the second subtractor output, is presented to a demodulator biggest picker  374 , which chooses the largest magnitude value from the plurality m of complex values. The maximum complex value MAX(k), which is the largest value for the iteration k, is stored along with which OUTPUT(m) of the plurality m of subtractor outputs contained this MAX(k). Both values MAX(k) and OUTPUT(m) are required to generate the decision data. 
   4) a final biggest picker, which selects the largest magnitude value from the set {MAX( 0 ) . . . MAX( 3 )} and associated {OUTPUT( 0 ) . . . OUTPUT( 3 )}. The current data  384  of 8 bits is formed from the final biggest picker with 2 bits derived from which iteration k had the maximum value MAX(k), 4 bits from the OUTPUT value for the case where m=16, and 2 bits from the phase of the largest amplitude. After the current data is determined using this method, a new current symbol is presented to the demodulator, and the current data becomes previous data for the next cycle. 
   5) a pre-equalize register value generator, which uses the previous data from the biggest picker to convert the previous data into a previous CCK symbol, smear the data using a feedback filter and the channel compensation coefficients to populate the pre-equalization register values. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  shows a dibit phase encoding for CCK. 
       FIG. 2  shows the encoding of 8 bits of data into four phase variables using four encoders as shown in  FIG. 1 . 
       FIG. 3  shows the CCK symbol generation from the four phase variables of  FIG. 2 . 
       FIG. 4  shows the block diagram of a wireless receiver including a front end, A/D converters, a phase corrector, symbol synchronization, and CCK decision demodulator. 
       FIGS. 5   a  through  5   e  show the time domain waveforms for a sequence of CCK symbols passing through a receiver incorporating Decision Feedback Equalization (DFE) and Maximum Likelihood Estimation (MLE). 
       FIG. 6  shows the block diagram for a CCK demodulator using Decision Feedback Equalization (DFE). 
       FIG. 7  shows the block diagram for a CCK symbol decision function. 
       FIG. 8  shows the block diagram for a simple Fast Walsh Transform (FWT) Butterfly which takes the CCK symbol and iteration variable φ 2  as inputs and gives 16 complex correlated outputs. 
       FIG. 9  shows the block diagram for a CCK symbol demodulator including Decision Feedback Equalization and Maximum Likelihood Estimation of a current CCK symbol. 
       FIGS. 10   a  through  10   h  shows the waveforms for a current symbol with a previous symbol and next symbol subtracted, in accordance with the block diagram of  FIG. 11   a.    
       FIG. 11   a  shows a CCK symbol demodulator according to the present invention. 
       FIG. 11   b  shows a flowchart showing the iterative operation of the block diagram of  FIG. 11   a.    
       FIG. 12  shows a block diagram for the post-equalization value generator of  FIG. 11   a.    
       FIG. 12   a  shows a detail block diagram of the adder of  FIG. 12 . 
       FIG. 13  shows a block diagram for the parallel-in serial out feedback filter of  FIG. 11   a.    
     FIG. A- 1  shows a graph diagram of intra-codeword Intersymbol Interference (ISI). 
     FIG. A- 2  shows an amplitude plot for shifted auto-correlation of CCK-codeword. 
     FIG. B- 1  shows a graph diagram of inter-codeword Intersymbol Interference (ISI). 
   

   DETAILED DESCRIPTION OF THE INVENTION 
   The approach as shown in  FIG. 9  describes the advantages of using decision feedback equalization for removing inter-codeword ISI by estimating current symbol ICI and subtracting it to improve the decision on the current symbol. An implementation of the circuitry in  FIG. 9  would entail a very extensive set of multipliers and adders. Each of the 64 feedback filters  282 - 1  through  282 - 64  would require approximately many adders and multipliers, and each of the correlators  284 - 1  through  284 - 64  would require many adders and multipliers. The 16 feedback filters  302 - 1  through  302 - 8  each require many adders and multipliers, and the current symbol subtractor  296  requires 64 simultaneous complex subtractions. 
     FIG. 11   a  shows the block diagram for a reduced complexity CCK demodulator including decision feedback equalization for pre-equalization as well as for post-equalization, and may be understood in combination with the operating waveforms of  FIG. 10   a - 10   f . It is understood that the waveforms of  FIG. 10   a - 10   f  are for illustrative purposes, as the signals of the actual system are typically digitized samples with real and imaginary components.  FIG. 10   a  shows an 8 chip symbol  600 , etc convolved with a channel impulse response  602  to produce the baseband chip stream shown in  FIG. 10   b , which is applied to the input  382  of  FIG. 11   a . It can be seen that that chip stream of  FIG. 10   b  includes a current symbol  606  with previous symbol  604  signal adding in to generate ISI, and there is also ICI within the current chip from adjacent intra-symbol smearing. Incoming baseband CCK chip stream  382 , also shown in  FIG. 10   b , is provided as a quadrature stream of unframed complex signals (I,Q) to Feed Forward Filter  364 , which performs short term pre-cursor ISI correction to compensate for the channel response, using feedforward coefficients derived from the preamble interval of the packet. The feedforward filter  364  performs maximum phase corrections, and reduces pre-cursors, as shown in the waveform of  FIG. 10   c . For clarity,  FIG. 10   c  through  FIG. 10   f  are shown as a sequential series of samples, although the processing is performed on blocks of 8 chips as symbols. During the preamble period, and before CCK symbols begin to arrive in the stream  382 , the channel characterization is done which generates channel compensation coefficients {c 1  . . . c 7 }, where these coefficients are suitable for use in a feedback filter. Once these coefficients are known, it is possible to populate the Post-Equalization register  360  and initialize the coefficients of the parallel-in Serial-Out feedback filter  378  of the pre-equalization value generator, as will be described in detail later. The training of pre-equalization value coefficients {c 1 -c 7 } used for DFE occurs from the interval starting with the short (56 us) or long (128 us) preamble, the 16 us start of frame delimiter, through the header of length 22 us or 46 us depending on packet type. The calculation of post-equalization register values occurs thereafter in roughly 2 us. When the preamble period has ended, a stream of CCK chips arrives from the phase corrector as was described for  FIG. 4 , and is applied to the feedforward filter  364 , which performs short term pre-cursor ISI corrections shown in the current symbol  610  and previous symbol  608  waveform of  FIG. 10   c . The output of the FFF  364  is applied to the serial to parallel converter  366 , which groups the incoming serial data stream into a current symbol  386  comprising 8 complex chip values to be demodulated, and the current  7  symbol  386  is stored in latch  396 . The symbol is applied to a complex first subtractor  368 , which subtracts a channel-equalized previous symbol  388  stored in pre-equalization register  380  from the current symbol  386 . The contents of the pre-equalization register  380  are shown in  FIG. 10   d  as  614 , which is an inverted form of previous symbol  612 . The output  390  of the first subtractor  368  is shown as symbol  616  of  FIG. 10   d , which is now free of previous symbols, and only contains ICI interference from the adjacent chips in the symbol. For each first subtractor  368  output  390 , the demodulator  362  performs the same operation four times, using the first subtractor output  390 , an FWT phase (φ 2  generated by the iteration variable k  399 , a previous CCK symbol stored in pre-equalization register  380 , and  16  possible post-cursor ICI values  394  generated by multiplying the contents of post equalization register  360  associated with that φ 2  by fours phase value φ 4 ={1,j,−1,−j}. As was described earlier, the multiplication of a value a+jb by {1,j,−1,−j} results in the trivial remapping real and imaginary values into {a+jb,−b+ja,−a−jb,b−ja}. The output  390  of the first subtractor  368  represents the current CCK symbol with the previous CCK symbol subtracted out, as shown in symbol  616  of  FIG. 10   d . The output  390  of the first subtractor  368  is presented to a simple FWT  370  as described in  FIG. 8 , accompanied by the FWT phase φ 2  generated by the current iteration k  399 . The output of the FWT  370  is 16 complex values. When the iteration value k  399  generates all four values, this produces all 64 complex FWT values, as was done at block  294  of  FIG. 9 . FWT output  400  will have one of m outputs with a maximum magnitude for one of the four iterations k, and for the case of the simple FWT  370 , there are 16 complex outputs. Since the FWT function  370  is a linear operation, it is possible to remove the effect of the ICI due to smearing of chips within the current symbol by using the communication channel response and computing ICI corrections for all 256 possible CCK codewords (or symbols), transforming them to the post-FWT phase domain, and subtracting this ICI effect from the FWT  370  output. Although the subtraction is done in the post-FWT domain, reference  FIGS. 10   e  and  10   f  show the effect of performing these ICI subtractions as if there were done in the pre-FWT domain, in order to show continuity of relationship to pre-FWT domain  FIGS. 10   a  through  10   d . As described earlier, the FWT is a linear operation, allowing ICI to be subtracted in the pre-FWT domain as shown in the waveforms of  FIGS. 10   e  and  10   f . For the block diagram of  FIG. 11   a , the ICI correction is performed in the post-FWT domain as shown in  FIG. 10   g  with solid lines representing a particular FWT  370  output response to a particular CCK symbol as one input to the second subtractor  372  and the dashed lines representing the second subtractor ICI compensation input from the post-equalizer for a particular CCK symbol. In  FIG. 10   h , the output FWT- 1  corresponds to the current CCK codeword, and subtracting ICI generates the improvement in decision for the associated CCK codeword as shown by comparison to output FWT- 1  of  FIG. 10   g . Second subtractor  372  performs a subtraction of post-cursor ISI effect by multiplying  393  each of the 4 complex values of the post-equalization register  360  corresponding to the phase φ 2  (FWT butterfly number) with 4 possible φ 4   371  values, and subtracting the 16 complex values from this multiplication  393  from the output of FWT  370 , which also comprises 16 complex values. The second subtractor  372  output  402  of 16 complex values is presented to demodulator biggest picker  374 , which selects one of the 16 outputs having the largest magnitude, and passes along as an output the biggest magnitude complex value as MAX, accompanied by the particular output of the 16 which carried this value as the value OUTPUT.  FIG. 10   f  shows the effect of subtracting the ICI from the current codeword in the time domain (shown for clarity as if the second subtractor  372  were performed pre-FWT), as the cleaned-up symbol in the 8 chip block would correspond to a single maximum value in the post-FWT phase domain. In the post-FWT domain of second subtractor  372 ,  FIG. 10   g  shows the FWT output as thick lines, the post equalization register outputs  394  as dashed lines, and  FIG. 10   h  shows the resulting second subtractor outputs  402  for the case where output FWT- 1  is the maximum response output. As stated earlier, the linearity of the FWT allows either subtraction of phase-domain post-equalization values post-FWT, or subtraction of time domain post-equalization values pre-FWT to compensate for ISI. In this manner, and for all values of k=0-3, the final biggest picker  376  selects the largest magnitude value which includes the effects of pre-cursor ISI and post-cursor ICI cancellation, and generating decision data  384  as described earlier. At the end of each demodulation cycle, address generator  398  asserts signal NEW_CCK, which resets k=0, latches the new current symbol  386 , latches the decision data  384  as previous data, which is used to reload pre-equalization register  380  before the next data decision iteration for k=0 begins. 
     FIG. 11   b  describes the sequence of operation of the block diagram of  FIG. 11   a . During the preamble interval of the incoming packet, sequence  442  is performed once, and when each current CCK symbol  386  is presented in  FIG. 11   a , sequence  444  is performed for each CCK symbol through the last CCK symbol of the packet. During the preamble sequence  442 , the feedback filter coefficients {c 1 ,c 3 ,c 5 ,c 7 } are characterized and computed using decision feedback equalization. These filter coefficients describe the coefficients which perform the same smearing as the communications channel, so that the symbol corrections fed back for pre-equalization and for post-equalization include the effect of the channel. After the coefficients {c 1  . . . c 7 } are known, they are loaded into parallel-in serial-out feedback filter  378 , as will be described later. The coefficient subset {c 1 ,c 3 ,c 5 ,c 7 } are loaded into block  356  of  FIG. 11   a . The post equalization value generator  352  initializes iteration variable i=0 in step  422 , and computes each of the 16 post-equalization register values in steps  424 ,  426 , and  428 . When all 16 registers are initialized, the post-equalization register  360  of  FIG. 11   a  is fully populated, and the demodulator is ready to start decoding CCK symbols using the contents of post-equalization register  360 . Sequence  444  shows the process of demodulation for each current CCK symbol presented in step  430  and iteration variable k=0. In step  432 , each value of k is associated with four corresponding φ 2  values read from the Post equalize register, which are expanded by multiplication by φ 4  to produce the set of 16 complex values that are subtracted from 16 complex FWT outputs at the second subtractor outputs  402 . As described earlier, the biggest picker  374  of  FIG. 11   a  selects the largest value MAX, and also passes along the value OUTPUT which identifies which largest output was selected. This process repeats with incrementing values of iteration variable k, as shown in steps  434  and  436 . When all four iterations are complete, final biggest picker  376  chooses the biggest of MAX( 0 ) through MAX( 3 ), and its associated OUTPUT( 0 ) through OUTPUT( 3 ), as was described earlier. In step  440 , the process exits for end of packet, or continues for the balance of the packet with the next current CCK symbol. 
   The appendix shows the derivation of equations used for the post-equalization generator, and make use of the symmetries found in CCK chip codes. The 16 post equalization register values Reg i  are found to be:
 
Reg i =(− C   1   e   −j(φ2+φ3) +2 C   3   e   (−jφ2)   +C   3   e   j(φ2−φ3)+ 2 C   5   e   j(φ2)   +C   5   e   j(φ3−φ2)   +C   7   e   j(φ2+φ3) )  (Equation A of Appendix)
 
   Second Subtractor Input=e jφ4 (Reg i ) 
   Where Reg i  is the value stored in the ith register  360 , and i is the iteration variable generated by  355  of  FIG. 11   a;    
   Second Subtractor Input is  394  of  FIG. 11   a;    
   {C 1 ,C 3 ,C 5 ,C 7 } are the coefficients derived from the preamble training sequence. 
   and the overall phase term e jφ4  is realized by multiplication  371  by the four phases derived from φ 4   399 . 
     FIG. 12  shows the detail of the implementation of the above equation 1 in the post equalization value generator  352  of  FIG. 11   a . The coefficients {C 1 ,C 3 ,C 5 ,C 7 }  450  are initialized, after which the 16 combinations of the phases  452 , corresponding to block  354  of  FIG. 11   a , are generated and each term is multiplied by some combination of the phases {φ 2 ,φ 3 }  452 . As described earlier, a multiplication by a phase term as shown in  454 ,  456 ,  458 ,  460 ,  462 ,  464  is actually a remapping and/or negation of real and imaginary components, as is known in the art, and the multiplication by the factor  2  for terms C 3  and C 5  is accomplished by left shifting the bits of the coefficients C 3  and C 5  by one position. Post equalization registers  360 , register values  392  prior to multiplication  393 , and after multiplication  394  are shown to match the reference numerals of  FIG. 11   a  for clarity.  FIG. 12   a  shows an implementation of adder  466 , which is formed from a plurality of 2 input adders. Subtraction is realized by inverting the term prior to presentation to the adder, as is known to one skilled in the art. 
   Referring to the appendix for pre-equalization register generator equation derivations, it can be seen that equation B shows the equation for calculation of the ISI effect due to post-cursors of one codeword on the codewords that follow using a feedback filter  378  in  FIG. 11   a , and is reproduced below: 
   
     
       
         
           
             
               
                 
                   
                     P 
                     
                       i 
                       , 
                       j 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         u 
                         = 
                         1 
                       
                       
                         j 
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         k 
                         
                           i 
                           , 
                           u 
                         
                         c 
                       
                       ⁢ 
                       
                         c 
                         
                           ( 
                           
                             j 
                             - 
                             u 
                           
                           ) 
                         
                       
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     for 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     j 
                   
                   = 
                   
                     1 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     to 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     m 
                   
                 
               
             
             
               
                 ( 
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   B 
                 
                 ) 
               
             
           
         
       
     
   
   where Qj is the post-cursor ISI effect to be stored in pre-equalize register  380  of  FIG. 11   a;    
   C u  are the channel compensation coefficients as described earlier; 
   r (vm+j)−u  are the previous CCK chip values. 
     FIG. 13  shows one implementation of the parallel-in serial-out feedback filter  378  of  FIG. 11   a . Previous data  384  of  FIG. 11   a  is converted to a CCK symbol in converter  377  of  FIG. 11   a , and loaded as a parallel value in input  480  of  FIG. 13  with the signal Load  481  asserted, which copies these values into registers  488 ,  490  through  492 . The coefficients C 1  through C flt     —     len  were loaded into a coefficient register (not shown) during the preamble interval, and the four phases of those coefficients is selected by the phase of the outputs of each register  488 ,  490  . . .  492  . . .  494 , and a serial output stream at adder  504  produces a serial version of the data to be loaded into pre-equalize register  380  of  FIG. 11   a . The previous symbol  480  is updated at the end of each decision. Many variations  7  of multiplications by phase terms can be accomplished using 8 multiplexers as shown in  FIG. 13 , or as a multiplier as shown in  FIG. 11   a.    
   While specific examples for the elements of the CCK processor are shown for clarity, these examples are not intended to limit the invention, but to assist in understanding the operation. 
   APPENDIX 
   Derivations of Equations for Post Equalization and Pre Equalization 
   Consider the data stream s i  (s 1  being transmitted first in time), which gets corrupted by a multipath channel, which can be modeled as a linear FIR filter. Now the corrupted stream r i  is received and has to be equalized. In DFE architecture the Feedback Filter (FBF) removes the post-cursors. Let us consider that we have a trained FBF with coefficients c i  (i=1 to FILTER_LEN), where c i  represents the effect of s (j−i)  on s j . The FBF removes the effect of s (j−FILTER     —     LEN)  to s (j−1)  on s j . In this approach we use r i   d , the decision on r i  as an estimate of s i . This decision if taken at the output of the correlators would give a certain improvement in performance under noise. Below we derive the equations for post-cursor removal in the case of Block Decision using a Correlator for the general case, and extend it to the CCK modulation case. 
   While using block demodulation cancelling the post-cursor effect of chips in the same codeword becomes a problem because the decisions are not yet available. Hence we have to use the Maximum Likelihood approach. That is if we have n-codewords in a multi-codeword modulation system (like we have 256 in case of CCK) and if we have n-correlators, then at the input of each correlator we cancel the effect of post-cursors (due to chips in the same codeword symbol), assuming that the particular symbol was the one that was transmitted. This condition is: 
   If k 1  to k n  are the codewords in the system. And if there are in-chips to a codeword, that is, the i th  codeword is k i,1  to k i,m . Then at the input of the i th  correlator the effect of post-cursors due to chips in the same codeword is given by: 
   
     
       
         
           
             
               
                 
                   
                     P 
                     
                       i 
                       , 
                       j 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           u 
                           = 
                           1 
                         
                         
                           j 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           k 
                           
                             i 
                             , 
                             u 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           c 
                           
                             ( 
                             
                               j 
                               - 
                               u 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         for 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         i 
                       
                     
                     = 
                     
                       1 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       to 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       n 
                     
                   
                 
                 ; 
                 
                   j 
                   = 
                   
                     1 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     to 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     m 
                   
                 
                 ; 
               
             
             
               
                 ( 
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
                 ) 
               
             
           
         
       
     
   
   Where P i,j  is the effect of post-cursors on the j th  chip of the i th  codeword, in other words it is the intra-codeword post-cursor contribution to be subtracted from the j th  input to the i th  correlator. 
   Hence, the m inputs to the i th  correlator will now be (r i,j -P i,j ), and its output will be: 
   
     
       
         
           
             
               
                 
                   
                     M 
                     i 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         m 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               r 
                               
                                 i 
                                 , 
                                 j 
                               
                             
                             - 
                             
                               P 
                               
                                 i 
                                 , 
                                 j 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           k 
                           
                             i 
                             , 
                             j 
                           
                           * 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         for 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         i 
                       
                     
                     = 
                     
                       1 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       to 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       n 
                     
                   
                 
                 ; 
               
             
             
               
                 ( 
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2 
                 
                 ) 
               
             
           
         
       
     
   
   The correlator with the maximum output will determine the decision taken on the received codeword. When we expand the above equation we get the following equation for post-equalization: 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           M 
                           i 
                         
                         = 
                         
                           
                             
                               ∑ 
                               
                                 j 
                                 = 
                                 1 
                               
                               m 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 r 
                                 
                                   i 
                                   , 
                                   j 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 k 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 * 
                               
                             
                           
                           - 
                           
                             
                               ∑ 
                               
                                 j 
                                 = 
                                 1 
                               
                               m 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 P 
                                 
                                   i 
                                   , 
                                   j 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 k 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 * 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             Rcorr 
                             i 
                           
                           - 
                           
                             Pcorr 
                             i 
                           
                         
                       
                     
                   
                 
                 ⁢ 
                 
                   
 
                 
                 ⁢ 
                 
                   
                     
                       for 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       i 
                     
                     = 
                     
                       1 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       to 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       n 
                     
                   
                   ; 
                 
               
             
             
               
                 ( 
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   3 
                 
                 ) 
               
             
           
         
       
     
   
   This equation shows that we can first correlate the incoming stream with the 256 codewords to get Rcorr i  and then from the output stream of each correlator we can subtract the corresponding Pcorr i  (i.e., correlated P i,j ). But this method is extremely hardware intensive. Let us look at the inherent symmetry of CCK codewords to reduce the complexity. 
   Calculation of P i,j  and Pcorr i  for the case of CCK Modulation 
   The 256 CCK codewords are given by the formula:
 
 k   i   c   =[e   j(φ     1     +φ     2     +φ     3     +φ     4     )   ,e   j(φ     1     +φ     3     +φ     4     )   ,e   j(φ     1     +φ     2     +φ4   )   ,−e   j(φ     1     +φ     4     )   ,e   j(φ     1     +φ     2     +φ     3     )   ,e   j(φ     1     +φ     3     )   ,−e   j(φ     1     +φ     2     )   ,e   j(φ     1     ) ]
 
 k   i   nc   =[e   9(φ     1     +φ     2     +φ     3     +φ     4     )   ,e   j(φ     1     +φ     3     +φ     4     )   ,e   9(φ     1     +φ     2     +φ     4     )   ,e   j(φ     1     +φ     4     )   ,e   9(φ     1     +φ     2     +φ     3     )   ,e   9(φ     1     +φ     3     )   ,e   9φ     1     +φ     2     )   ,e   j(φ     1     ) ]
 
for i=1 to 256; i.e., ∀φ 1 ,φ 2 ,φ 3 ,φ 4 ε{1,j,−1,−j}  (Equation 4)
 
   We can see that here n=256 and m=8 as defined previously. For k i   c  the fourth and seventh symbols are negated (cover code inserted), whereas k i   nc  are codewords without cover 
   
     
       
         
           
             
               
                 
                   
                     P 
                     
                       i 
                       , 
                       j 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           u 
                           = 
                           1 
                         
                         
                           j 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           k 
                           
                             i 
                             , 
                             u 
                           
                           c 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           c 
                           
                             ( 
                             
                               j 
                               - 
                               u 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         for 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         i 
                       
                     
                     = 
                     
                       1 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       to 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       256 
                     
                   
                 
                 ; 
                 
                   j 
                   = 
                   
                     1 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     to 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     8 
                   
                 
               
             
             
               
                 ( 
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   5 
                 
                 ) 
               
             
           
         
       
     
   
   
     
       
         
           
             
               
                 
                   
                     P 
                     
                       corr 
                       i 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         8 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           P 
                           
                             i 
                             , 
                             j 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           k 
                           
                             i 
                             , 
                             j 
                           
                           
                             nc 
                             * 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         for 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         i 
                       
                     
                     = 
                     
                       1 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       to 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       256 
                     
                   
                 
                 ; 
               
             
             
               
                 ( 
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   6 
                 
                 ) 
               
             
           
         
       
     
   
   First we substitute equation-4 into equation-5, and then we substitute equation-4 and 5 into equation-6. Then simplifying we find that many terms get cancelled due to the symmetry of CCK code set and we finally get the reduced equation as:
 
 P   corr     i     =e   jφ     4   (− c   1   e   −j(φ     2     +φ     3     ) +2 c   3   e   −jφ     2     +c   3   e   j(φ     2     −φ     3     ) +2 c   5   e   j(φ     2     )   −c   5   e   j(φ     3     −φ     2     )   +c   7   e   j(φ     2     +φ     3     ) )
 
for i=1 to 256; i.e. ∀φ 1 ,φ 2 ,φ 3 ,φ 4 ε{1,j,−1,−j}  (Equation A)
 
   Derivation of the P Corri  Equation A is as shown below 
   Alternate Notation:
 
 e   j(φ     1     +φ     2     +φ     3     +φ     4     ) ={1,2,3,4}
 
 e   j(φ     1     +φ     3     +φ     4     ) ={2,3,4}
 
 e   j(φ     1     +φ     2     +φ     4     ) ={1,2,4}
 
 e   j(φ     1     +φ     4     ) ={1,4}
 
 e   j(φ     1     +φ     2     +φ     3     ) ={1,2,3}
 
 e   j(φ     1     +φ     3     ) ={1,3}
 
 e   j(φ     1     +φ     2     ) ={1,2}
 
 e   j(φ     1     ) ={1}
 
{c 1 , c 2 , c 3 , c 4 , c 5 , c 6 , c 7 } Channel equalization coefficients
 
   
     
       
         
           
             P 
             0 
           
           = 
           0 
         
       
     
     
       
         
           
             P 
             1 
           
           = 
           
             
               C 
               1 
             
             · 
             
               { 
               2 
               } 
             
           
         
       
     
     
       
         
           
             
               
                 
                   P 
                   2 
                 
                 = 
                 
                   
                     { 
                     
                       
                         
                           C 
                           1 
                         
                         · 
                         
                           { 
                           
                             1 
                             , 
                             3 
                             , 
                             4 
                           
                           } 
                         
                       
                       + 
                       
                         
                           C 
                           2 
                         
                         ⁢ 
                         
                           { 
                           
                             1 
                             , 
                             2 
                             , 
                             3 
                             , 
                             4 
                           
                           } 
                         
                       
                     
                     } 
                   
                   · 
                   
                     
                       { 
                       
                         1 
                         , 
                         2 
                         , 
                         4 
                       
                       } 
                     
                     * 
                   
                 
               
             
           
           
             
               
                 = 
                 
                   
                     
                       C 
                       1 
                     
                     ⁢ 
                     
                       { 
                       
                         1 
                         , 
                         3 
                         , 
                         4 
                       
                       } 
                     
                     ⁢ 
                     
                       
                         { 
                         
                           1 
                           , 
                           2 
                           , 
                           4 
                         
                         } 
                       
                       * 
                     
                   
                   + 
                   
                     
                       C 
                       2 
                     
                     ⁢ 
                     
                       { 
                       
                         1 
                         , 
                         2 
                         , 
                         3 
                         , 
                         4 
                       
                       } 
                     
                     ⁢ 
                     
                       
                         { 
                         
                           1 
                           , 
                           2 
                           , 
                           4 
                         
                         } 
                       
                       * 
                     
                   
                 
               
             
           
           
             
               
                 = 
                 
                   
                     
                       C 
                       1 
                     
                     ⁢ 
                     
                       { 
                       
                         3 
                         , 
                         
                           - 
                           2 
                         
                       
                       } 
                     
                   
                   + 
                   
                     
                       C 
                       2 
                     
                     ⁢ 
                     
                       { 
                       3 
                       } 
                     
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   P 
                   3 
                 
                 = 
                 
                   
                     - 
                     
                       { 
                       
                         
                           
                             C 
                             1 
                           
                           ⁢ 
                           
                             { 
                             
                               1 
                               , 
                               2 
                               , 
                               4 
                             
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             2 
                           
                           ⁢ 
                           
                             { 
                             
                               1 
                               , 
                               3 
                               , 
                               4 
                             
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             3 
                           
                           ⁢ 
                           
                             { 
                             
                               1 
                               , 
                               2 
                               , 
                               3 
                               , 
                               4 
                             
                             } 
                           
                         
                       
                       } 
                     
                   
                   · 
                   
                     
                       { 
                       
                         1 
                         , 
                         4 
                       
                       } 
                     
                     * 
                   
                 
               
             
           
           
             
               
                 = 
                 
                   
                     
                       - 
                       
                         C 
                         1 
                       
                     
                     ⁢ 
                     
                       { 
                       2 
                       } 
                     
                   
                   - 
                   
                     
                       C 
                       2 
                     
                     ⁢ 
                     
                       { 
                       3 
                       } 
                     
                   
                   - 
                   
                     
                       C 
                       3 
                     
                     ⁢ 
                     
                       { 
                       
                         2 
                         , 
                         3 
                       
                       } 
                     
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   P 
                   4 
                 
                 = 
                 
                   
                     { 
                     
                       
                         
                           - 
                           
                             C 
                             1 
                           
                         
                         ⁢ 
                         
                           { 
                           
                             1 
                             , 
                             4 
                           
                           } 
                         
                       
                       + 
                       
                         
                           C 
                           2 
                         
                         ⁢ 
                         
                           { 
                           
                             1 
                             , 
                             2 
                             , 
                             4 
                           
                           } 
                         
                       
                       + 
                       
                         
                           C 
                           3 
                         
                         ⁢ 
                         
                           { 
                           
                             1 
                             , 
                             3 
                             , 
                             4 
                           
                           } 
                         
                       
                       + 
                       
                         
                           C 
                           4 
                         
                         ⁢ 
                         
                           { 
                           
                             1 
                             , 
                             2 
                             , 
                             3 
                             , 
                             4 
                           
                           } 
                         
                       
                     
                     } 
                   
                   · 
                   
                     
                       { 
                       
                         1 
                         , 
                         2 
                         , 
                         3 
                       
                       } 
                     
                     * 
                   
                 
               
             
           
           
             
               
                 = 
                 
                   
                     
                       - 
                       
                         C 
                         1 
                       
                     
                     ⁢ 
                     
                       { 
                       
                         
                           - 
                           2 
                         
                         , 
                         
                           - 
                           3 
                         
                         , 
                         4 
                       
                       } 
                     
                   
                   + 
                   
                     
                       C 
                       2 
                     
                     ⁢ 
                     
                       { 
                       
                         
                           - 
                           3 
                         
                         , 
                         4 
                       
                       } 
                     
                   
                   + 
                   
                     
                       C 
                       3 
                     
                     ⁢ 
                     
                       { 
                       
                         
                           - 
                           2 
                         
                         , 
                         4 
                       
                       } 
                     
                   
                   + 
                   
                     
                       C 
                       4 
                     
                     ⁢ 
                     
                       { 
                       4 
                       } 
                     
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   P 
                   5 
                 
                 = 
                 
                   
                     { 
                     
                       
                         
                           C 
                           1 
                         
                         ⁢ 
                         
                           { 
                           
                             1 
                             , 
                             2 
                             , 
                             3 
                           
                           } 
                         
                       
                       - 
                       
                         
                           C 
                           2 
                         
                         ⁢ 
                         
                           { 
                           
                             1 
                             , 
                             4 
                           
                           } 
                         
                       
                       + 
                       
                         
                           C 
                           3 
                         
                         ⁢ 
                         
                           { 
                           
                             1 
                             , 
                             2 
                             , 
                             4 
                           
                           } 
                         
                       
                       + 
                       
                         
                           C 
                           5 
                         
                         ⁢ 
                         
                           { 
                           
                             1 
                             , 
                             2 
                             , 
                             3 
                             , 
                             4 
                           
                           } 
                         
                       
                     
                     } 
                   
                   · 
                   
                     
                       { 
                       
                         1 
                         , 
                         3 
                       
                       } 
                     
                     * 
                   
                 
               
             
           
           
             
               
                 = 
                 
                   
                     
                       C 
                       1 
                     
                     ⁢ 
                     
                       { 
                       2 
                       } 
                     
                   
                   - 
                   
                     
                       C 
                       2 
                     
                     ⁢ 
                     
                       { 
                       
                         
                           - 
                           3 
                         
                         , 
                         
                           - 
                           4 
                         
                       
                       } 
                     
                   
                   + 
                   
                     
                       C 
                       3 
                     
                     ⁢ 
                     
                       { 
                       
                         2 
                         , 
                         
                           - 
                           3 
                         
                         , 
                         4 
                       
                       } 
                     
                   
                   + 
                   
                     
                       C 
                       4 
                     
                     ⁢ 
                     
                       { 
                       4 
                       } 
                     
                   
                   + 
                   
                     
                       C 
                       5 
                     
                     ⁢ 
                     
                       { 
                       
                         2 
                         , 
                         4 
                       
                       } 
                     
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   P 
                   6 
                 
                 = 
                   
                 ⁢ 
                 
                   - 
                   
                     { 
                     
                       
                         
                           C 
                           1 
                         
                         ⁢ 
                         
                           { 
                           
                             1 
                             , 
                             3 
                           
                           } 
                         
                       
                       + 
                       
                         
                           C 
                           2 
                         
                         ⁢ 
                         
                           { 
                           
                             1 
                             , 
                             2 
                             , 
                             3 
                           
                           } 
                         
                       
                       - 
                       
                         
                           C 
                           3 
                         
                         ⁢ 
                         
                           { 
                           
                             1 
                             , 
                             4 
                           
                           } 
                         
                       
                       + 
                       
                         
                           C 
                           4 
                         
                         ⁢ 
                         
                           { 
                           
                             1 
                             , 
                             2 
                             , 
                             4 
                           
                           } 
                         
                       
                       + 
                       
                         
                           C 
                           5 
                         
                         ⁢ 
                         
                           { 
                           
                             1 
                             , 
                             3 
                             , 
                             4 
                           
                           } 
                         
                       
                       + 
                     
                   
                 
               
             
           
           
             
               
                 
                     
                   ⁢ 
                   
                     
                       C 
                       6 
                     
                     ⁢ 
                     
                       { 
                       
                         1 
                         , 
                         2 
                         , 
                         3 
                         , 
                         4 
                       
                       } 
                     
                   
                   } 
                 
                 · 
                 
                   
                     { 
                     
                       1 
                       , 
                       2 
                     
                     } 
                   
                   * 
                 
               
             
           
           
             
               
                 = 
                   
                 ⁢ 
                 
                   
                     
                       - 
                       
                         C 
                         1 
                       
                     
                     ⁢ 
                     
                       { 
                       
                         
                           - 
                           2 
                         
                         , 
                         3 
                       
                       } 
                     
                   
                   - 
                   
                     
                       C 
                       2 
                     
                     ⁢ 
                     
                       { 
                       3 
                       } 
                     
                   
                   + 
                   
                     
                       C 
                       3 
                     
                     ⁢ 
                     
                       { 
                       
                         
                           - 
                           2 
                         
                         , 
                         4 
                       
                       } 
                     
                   
                   + 
                   
                     
                       C 
                       4 
                     
                     ⁢ 
                     
                       { 
                       4 
                       } 
                     
                   
                   - 
                   
                     
                       C 
                       5 
                     
                     ⁢ 
                     
                       { 
                       
                         
                           - 
                           2 
                         
                         , 
                         3 
                         , 
                         4 
                       
                       } 
                     
                   
                   - 
                   
                     
                       C 
                       6 
                     
                     ⁢ 
                     
                       { 
                       
                         3 
                         , 
                         4 
                       
                       } 
                     
                   
                 
               
             
           
         
       
     
     
       
         
           
             
               
                 
                   
                     
                       
                         P 
                         7 
                       
                       = 
                         
                       ⁢ 
                       
                         { 
                         
                           
                             
                               - 
                               
                                 C 
                                 1 
                               
                             
                             ⁢ 
                             
                               { 
                               
                                 1 
                                 , 
                                 2 
                               
                               } 
                             
                           
                           + 
                           
                             
                               C 
                               2 
                             
                             ⁢ 
                             
                               { 
                               
                                 1 
                                 , 
                                 3 
                               
                               } 
                             
                           
                           + 
                           
                             
                               C 
                               3 
                             
                             ⁢ 
                             
                               { 
                               
                                 1 
                                 , 
                                 2 
                                 , 
                                 3 
                               
                               } 
                             
                           
                           - 
                           
                             
                               C 
                               4 
                             
                             ⁢ 
                             
                               { 
                               
                                 1 
                                 , 
                                 4 
                               
                               } 
                             
                           
                           + 
                           
                             
                               C 
                               5 
                             
                             ⁢ 
                             
                               { 
                               
                                 1 
                                 , 
                                 2 
                                 , 
                                 4 
                               
                               } 
                             
                           
                           + 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               C 
                               6 
                             
                             ⁢ 
                             
                               { 
                               
                                 1 
                                 , 
                                 3 
                                 , 
                                 4 
                               
                               } 
                             
                           
                           + 
                           
                             
                               C 
                               7 
                             
                             ⁢ 
                             
                               { 
                               
                                 1 
                                 , 
                                 2 
                                 , 
                                 3 
                                 , 
                                 4 
                               
                               } 
                             
                           
                         
                         } 
                       
                       · 
                       
                         
                           { 
                           1 
                           } 
                         
                         * 
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             - 
                             
                               C 
                               1 
                             
                           
                           ⁢ 
                           
                             { 
                             2 
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             2 
                           
                           ⁢ 
                           
                             { 
                             3 
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             3 
                           
                           ⁢ 
                           
                             { 
                             
                               2 
                               , 
                               3 
                             
                             } 
                           
                         
                         - 
                         
                           
                             C 
                             4 
                           
                           ⁢ 
                           
                             { 
                             4 
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             5 
                           
                           ⁢ 
                           
                             { 
                             
                               2 
                               , 
                               4 
                             
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             6 
                           
                           ⁢ 
                           
                             { 
                             
                               3 
                               , 
                               4 
                             
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             7 
                           
                           ⁢ 
                           
                             { 
                             
                               2 
                               , 
                               3 
                               , 
                               4 
                             
                             } 
                           
                         
                       
                     
                   
                 
               
             
           
           
             
               
                 
                   
                     
                       
                         Pcorr 
                         i 
                       
                       = 
                         
                       ⁢ 
                       
                         
                           P 
                           0 
                         
                         + 
                         
                           P 
                           1 
                         
                         + 
                         
                           P 
                           2 
                         
                         + 
                         
                           P 
                           3 
                         
                         + 
                         
                           P 
                           4 
                         
                         + 
                         
                           P 
                           5 
                         
                         + 
                         
                           P 
                           6 
                         
                         + 
                         
                           P 
                           7 
                         
                         + 
                         
                           P 
                           8 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         0 
                         + 
                         
                           
                             C 
                             1 
                           
                           · 
                           
                             { 
                             2 
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             1 
                           
                           [ 
                           
                             3 
                             , 
                             
                               - 
                               2 
                             
                           
                           } 
                         
                         + 
                         
                           
                             C 
                             2 
                           
                           ⁢ 
                           
                             { 
                             3 
                             } 
                           
                         
                         - 
                         
                           
                             C 
                             1 
                           
                           ⁢ 
                           
                             { 
                             2 
                             } 
                           
                         
                         - 
                         
                           
                             C 
                             2 
                           
                           ⁢ 
                           
                             { 
                             3 
                             } 
                           
                         
                         - 
                         
                           
                             C 
                             3 
                           
                           ⁢ 
                           
                             { 
                             
                               2 
                               , 
                               3 
                             
                             } 
                           
                         
                         - 
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         
                           
                             C 
                             1 
                           
                           ⁢ 
                           
                             { 
                             
                               
                                 - 
                                 2 
                               
                               , 
                               
                                 - 
                                 3 
                               
                               , 
                               4 
                             
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             2 
                           
                           ⁢ 
                           
                             { 
                             
                               
                                 - 
                                 3 
                               
                               , 
                               4 
                             
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             3 
                           
                           ⁢ 
                           
                             { 
                             
                               
                                 - 
                                 2 
                               
                               , 
                               4 
                             
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             4 
                           
                           ⁢ 
                           
                             { 
                             4 
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             1 
                           
                           ⁢ 
                           
                             { 
                             2 
                             } 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         
                           
                             - 
                             
                               C 
                               2 
                             
                           
                           ⁢ 
                           
                             { 
                             
                               
                                 - 
                                 3 
                               
                               , 
                               
                                 - 
                                 4 
                               
                             
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             3 
                           
                           ⁢ 
                           
                             { 
                             
                               2 
                               , 
                               
                                 - 
                                 3 
                               
                               , 
                               4 
                             
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             4 
                           
                           ⁢ 
                           
                             { 
                             4 
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             5 
                           
                           ⁢ 
                           
                             { 
                             
                               2 
                               , 
                               4 
                             
                             } 
                           
                         
                         + 
                         
                           
                             - 
                             
                               C 
                               1 
                             
                           
                           ⁢ 
                           
                             { 
                             
                               
                                 - 
                                 2 
                               
                               , 
                               3 
                             
                             } 
                           
                         
                         - 
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         
                           
                             C 
                             2 
                           
                           ⁢ 
                           
                             { 
                             3 
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             3 
                           
                           ⁢ 
                           
                             { 
                             
                               
                                 - 
                                 2 
                               
                               , 
                               4 
                             
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             4 
                           
                           ⁢ 
                           
                             { 
                             4 
                             } 
                           
                         
                         - 
                         
                           
                             C 
                             5 
                           
                           ⁢ 
                           
                             { 
                             
                               
                                 - 
                                 2 
                               
                               , 
                               3 
                               , 
                               4 
                             
                             } 
                           
                         
                         - 
                         
                           
                             C 
                             6 
                           
                           ⁢ 
                           
                             { 
                             
                               3 
                               , 
                               4 
                             
                             } 
                           
                         
                         + 
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         
                           
                             - 
                             
                               C 
                               1 
                             
                           
                           ⁢ 
                           
                             { 
                             2 
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             2 
                           
                           ⁢ 
                           
                             { 
                             3 
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             3 
                           
                           ⁢ 
                           
                             { 
                             
                               2 
                               , 
                               3 
                             
                             } 
                           
                         
                         - 
                         
                           
                             C 
                             4 
                           
                           ⁢ 
                           
                             { 
                             4 
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             5 
                           
                           ⁢ 
                           
                             { 
                             
                               2 
                               , 
                               4 
                             
                             } 
                           
                         
                         + 
                         
                           
                             C 
                             6 
                           
                           ⁢ 
                           
                             { 
                             
                               3 
                               , 
                               4 
                             
                             } 
                           
                         
                         + 
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         
                           C 
                           7 
                         
                         ⁢ 
                         
                           { 
                           
                             2 
                             , 
                             3 
                             , 
                             4 
                           
                           } 
                         
                       
                     
                   
                 
                 
                   
                     
                       = 
                         
                       ⁢ 
                       
                         
                           
                             - 
                             
                               C 
                               1 
                             
                           
                           ⁢ 
                           
                             { 
                             
                               
                                 - 
                                 2 
                               
                               , 
                               
                                 - 
                                 3 
                               
                               , 
                               4 
                             
                             } 
                           
                         
                         + 
                         
                           2 
                           ⁢ 
                           C 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           3 
                           ⁢ 
                           
                             { 
                             
                               
                                 - 
                                 2 
                               
                               , 
                               4 
                             
                             } 
                           
                         
                         + 
                         
                           C 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           3 
                           ⁢ 
                           
                             { 
                             
                               2 
                               , 
                               
                                 - 
                                 3 
                               
                               , 
                               4 
                             
                             } 
                           
                         
                         + 
                         
                           2 
                           ⁢ 
                           C 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           5 
                           ⁢ 
                           
                             { 
                             
                               2 
                               , 
                               4 
                             
                             } 
                           
                         
                         - 
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         
                           C 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           5 
                           ⁢ 
                           
                             { 
                             
                               
                                 - 
                                 2 
                               
                               , 
                               3 
                               , 
                               4 
                             
                             } 
                           
                         
                         + 
                         
                           C 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           7 
                           ⁢ 
                           
                             { 
                             
                               2 
                               , 
                               3 
                               , 
                               4 
                             
                             } 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
   
   Several observations that can be made from this reduction:
         1. The ISI due to post-cursors of a codeword on the chips of the same codeword does not depend on c 2 , c 4 , c 6 . This is because of the zero “shifted auto-correlation” of CCK codewords at these shifts as can be seen from the shifted auto-correlation profile of the CCK-codewords drawn below:   2. It also does not depend upon the phase φ 1  of the CCK codeword.   3. Moreover the φ 4  phase of the CCK codewords seems to act as the phase of the error contribution. Coupled with the above result, this means that to apply a constant equalization correction at the output of the ML-CCK correlator we need to store Pcorr i  for all possible combinations of φ 2  and φ 3  only. That means we store only 16 corrections, and to get the remaining we just multiply these by all possible φ 4  (i.e., 1, j, −1, j).       

   Derivation of Equation B generating pre-equalization coefficients for the “chip smearing” pre-equalization value generator of FIG.  13   
   Let the inputs r 1  to r m , r m+1  to r m+m , . . . , r vm+1  to r vm+m , and so on, correspond to the codewords R 1 , R 2 , . . . , R v , and so on. On each chip there will be the effect of post-cursors due to the chips of q previous codewords. Where q=[(FILTER_LEN−1)/m]+1 ([.] denotes the smallest integer greater than or equal to). That is, on r vm+i  there will be effect of post-cursors of the chips of R v−q  to R v−1 . 
   Hence the previous codeword post-cursor effect on r vm+j  is given by: 
   
     
       
         
           
             
               
                 
                   
                     Q 
                     j 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           u 
                           = 
                           j 
                         
                         FILTER_LEN 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           c 
                           u 
                         
                         ⁢ 
                         
                           r 
                           
                             
                               ( 
                               
                                 vm 
                                 + 
                                 j 
                               
                               ) 
                             
                             - 
                             u 
                           
                           d 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         for 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         j 
                       
                     
                     = 
                     
                       1 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       to 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       m 
                     
                   
                 
                 ; 
               
             
             
               
                 ( 
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   B 
                 
                 )