Patent Publication Number: US-7907593-B2

Title: Staggered pilot transmission for channel estimation and time tracking

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is a divisional application of U.S. patent application Ser. No. 10/926,884, filed Aug. 25, 2004 which claims priority to U.S. Provisional Patent Application Ser. No. 60/568,324, filed May 4, 2004, which is incorporated herein by reference in its entirety. 
    
    
     BACKGROUND 
     I. Field 
     The present invention relates generally to data communication, and more specifically to pilot transmission, channel estimation, and time tracking in a multi-carrier communication system. 
     II. Background 
     Orthogonal frequency division multiplexing (OFDM) is a multi-carrier modulation technique that effectively partitions the overall system bandwidth into multiple orthogonal frequency subbands. These subbands are also referred to as tones, subcarriers, bins, and frequency channels. With OFDM, each subband is associated with a respective subcarrier that may be modulated with data. 
     In an OFDM system, a transmitting entity processes data to obtain modulation symbols and further performs OFDM modulation on the modulation symbols to generate OFDM symbols. The transmitting entity then conditions and transmits the OFDM symbols via a communication channel. A receiving entity typically needs to obtain relatively accurate symbol timing in order to recover the data sent by the transmitting entity. The receiving entity often does not know the time at which each OFDM symbol is sent by the transmitting entity nor the propagation delay introduced by the communication channel. The receiving entity would then need to ascertain the timing of each OFDM symbol received via the communication channel in order to properly perform the complementary OFDM demodulation on the received OFDM symbol. The receiving entity also needs a good estimate of the response of the communication channel in order to perform data detection to obtain good estimates of the modulation symbols sent by the transmitting entity. 
     The transmitting entity expends system resources to support channel estimation and time tracking, and the receiving entity also consumes resources to perform these tasks. The resources used by the transmitting and receiving entities for channel estimation and time tracking represent overhead. Thus, it is desirable to minimize the amount of resources expended by both the transmitting and receiving entities for these tasks. 
     There is therefore a need in the art for techniques to efficiently support channel estimation and time tracking in an OFDM system. 
     SUMMARY 
     Techniques for performing “staggered” pilot transmission, channel estimation, and time tracking in a multi-carrier (e.g., OFDM) communication system are described herein. To allow a receiving entity to derive a longer channel estimate while limiting the amount of resources expended for pilot transmission, a transmitting entity may transmit a pilot on different groups of subbands in different time intervals (e.g., different symbol periods). N subbands in the system may be arranged into M non-overlapping groups. Each group may include P=N/M subbands that are distributed across the N subbands. The transmitting entity may transmit the pilot on a different subband group in each time interval. The transmitting entity may select all M subband groups in M time intervals based on a pilot staggering pattern. Alternatively, the transmitting entity may use many or most of the M subband groups in different time intervals, so that a substantial number of all subbands usable for transmission in the system are used for pilot transmission in different time intervals. The substantial number of subbands may be, for example, all of the usable subbands, three quarter of the usable subbands, at least half of the usable subbands, or some other significant percentage of the usable subbands. The receiving entity may derive an initial impulse response estimate with P channel taps based on the pilot received on one subband group. The receiving entity may derive a longer impulse response estimate (with up to N channel taps) by filtering initial impulse response estimates obtained for a sufficient number of different subband groups, as described below. 
     The receiving entity may derive two longer impulse response estimates of two different lengths L 1  and L 2 , which may be used for data detection/decoding and time tracking respectively, where L 1 =S 1 ·P and L 2 =S 2 ·P. Each longer impulse response estimate may be derived based on a different time-domain filter that filters S or more initial impulse response estimates obtained for S or more different subband groups, where S may be S 1  or S 2 . For each longer impulse response estimate, the first P channel taps are for a “main channel”, and the remaining channel taps are for an “excess channel”. The coefficients for each time-domain filter may be selected based on various criteria. For example, the coefficients for the main channel may be selected to (1) cancel the excess channel, (2) suppress time variation in the main channel, (3) provide an unbiased estimate of the main channel, and so on. Details of the filtering are described below. Various aspects and embodiments of the invention are also described in further detail below. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The features and nature of the present invention will become more apparent from the detailed description set forth below when taken in conjunction with the drawings in which like reference characters identify correspondingly throughout and wherein: 
         FIG. 1  shows a block diagram of a transmitting entity and a receiving entity; 
         FIG. 2  shows an interlace subband structure; 
         FIG. 3  shows an impulse response estimate for one interlace; 
         FIGS. 4A through 4C  show three different pilot staggering patterns; 
         FIG. 5  shows a process for deriving a channel estimate used for data detection; 
         FIGS. 6A and 6B  illustrate ambiguity in a channel impulse response estimate due to timing uncertainty; 
         FIG. 7  shows a process for performing time tracking; 
         FIG. 8  shows a channel estimator and a time tracking unit; and 
         FIG. 9  shows a filter for deriving a longer impulse response estimate. 
     
    
    
     DETAILED DESCRIPTION 
     The word “exemplary” is used herein to mean “serving as an example, instance, or illustration.” Any embodiment or design described herein as “exemplary” is not necessarily to be construed as preferred or advantageous over other embodiments or designs. 
       FIG. 1  shows a block diagram of a transmitting entity  110  and a receiving entity  150  in an OFDM system  100 . Transmitting entity  110  may be a base station or a wireless device, and receiving entity  150  may also be a base station or a wireless device. A base station is generally a fixed station and may also be referred to as a base transceiver system (BTS), an access point, or some other terminology. A wireless device may be fixed or mobile and may also be referred to as a user terminal, a mobile station, or some other terminology. 
     At transmitting entity  110 , a transmit (TX) data and pilot processor  120  receives different types of data (e.g., traffic/packet data and overhead/control data) and processes (e.g., encodes, interleaves, and symbol maps) the data to generate data symbols. As used herein, a “data symbol” is a modulation symbol for data, a “pilot symbol” is a modulation symbol for pilot (which is data that is known a priori by both the transmitting and receiving entities), and a modulation symbol is a complex value for a point in a signal constellation for a modulation scheme (e.g., M-PSK, M-QAM, and so on). Processor  120  provides data and pilot symbols to an OFDM modulator  130 . 
     OFDM modulator  130  multiplexes the data and pilot symbols onto the proper subbands and further performs OFDM modulation on the multiplexed symbols to generate OFDM symbols. For each symbol period, OFDM modulator  130  performs an N-point inverse fast Fourier transform (IFFT) on N multiplexed symbols for N total subbands and obtains a “transformed” symbol that contains N time-domain samples. Each sample is a complex value to be transmitted in one sample period. OFDM modulator  130  then repeats a portion of each transformed symbol to form an OFDM symbol that contains N+C samples, where C is the number of samples being repeated. The repeated portion is often called a cyclic prefix and is used to combat inter-symbol interference (ISI) caused by frequency selective fading. An OFDM symbol period (or simply, a symbol period) is the duration of one OFDM symbol and is equal to N+C sample periods. OFDM modulator  130  provides a stream of OFDM symbols to a transmitter unit (TMTR)  132 . Transmitter unit  132  processes (e.g., converts to analog, amplifies, filters, and frequency upconverts) the OFDM symbol stream to generate a modulated signal, which is then transmitted from an antenna  134 . 
     At receiving entity  150 , the transmitted signal from transmitting entity  110  is received by an antenna  152  and provided to a receiver unit (RCVR)  154 . Receiver unit  154  processes (e.g., filters, amplifies, frequency downconverts, and digitizes) the received signal and provides a stream of input samples. An OFDM demodulator (Demod)  160  performs OFDM demodulation on the input samples and provides received data and pilot symbols. A detector  170  performs data detection (e.g., equalization or matched filtering) on the received data symbols with a channel estimate from a channel estimator  172  and provides detected data symbols, which are estimates of the data symbols sent by transmitting entity  110 . A receive (RX) data processor  180  processes (e.g., symbol demaps, deinterleaves, and decodes) the detected data symbols and provides decoded data. In general, the processing by OFDM demodulator  160  and RX data processor  180  is complementary to the processing by OFDM modulator  130  and TX data and pilot processor  120 , respectively, at transmitting entity  110 . 
     Channel estimator  172  derives impulse response estimates based on the received pilot symbols from OFDM demodulator  160  and further derives frequency response estimates used by detector  170 . A synchronization unit  162  performs time tracking and determines symbol timing based on the impulse response estimates from channel estimator  172 . OFDM demodulator  160  performs OFDM demodulation based on the symbol timing from unit  162 . 
     Controllers  140  and  190  direct operation at transmitting entity  110  and receiving entity  150 , respectively. Memory units  142  and  192  provide storage for program codes and data used by controllers  140  and  190 , respectively. 
     Data and pilot may be transmitted in various manners in system  100 . For example, data and pilot may be transmitted (1) simultaneously in the same symbol period using frequency division multiplexing (FDM), (2) sequentially in different symbol periods using time division multiplexing (TDM), or (3) using a combination of FDM and TDM. The N total subbands may also be used for data and pilot transmission in various manners. An exemplary data/pilot transmission scheme is described below. 
       FIG. 2  shows an interlace subband structure  200  that may be used for data and pilot transmission in system  100 . System  100  has an overall system bandwidth of BW MHz, which is partitioned into N orthogonal frequency subbands using OFDM. Each subband has a bandwidth of BW/N MHz. Of the N total subbands, only U subbands may be used for data and pilot transmission, where U≦N, and the remaining G=N−U subbands may be unused and serve as guard subbands. As a specific example, system  100  may utilize an OFDM structure with N=4096 total subbands, U=4000 usable subbands, and G=96 guard subbands. For simplicity, the following description assumes that all N subbands may be used for data and pilot transmission. These N subbands are assigned indices of k=1 . . . N. 
     The N total subbands may be arranged into M “interlaces” or disjoint subband groups. The M interlaces are disjoint or non-overlapping in that each of the N total subbands belongs in only one interlace. Each interlace contains P subbands, where P·M=N. The M interlaces are given indices of m=1 . . . M, and the P subbands in each interlace are given indices of p=1 . . . P. 
     The P subbands for each interlace may be uniformly distributed across the N total subbands such that consecutive subbands in the interlace are spaced apart by M subbands. Each interlace m, for m=1 . . . M, may include P subbands with the following k indices:
 
(p−1)·M+m, for p=1 . . . P.  Eq (1)
 
As shown in  FIG. 2 , interlace  1  contains subbands with indices k=1, M+1, 2M+1, and so on, interlace  2  contains subbands with indices k=2, M+2, 2M+2, and so on, and interlace M contains subbands with indices k=M, 2M, 3M, and so on. The P subbands in each interlace are thus interlaced with the P subbands in each of the other M−1 interlaces. Each interlace is further associated with a staggering phase m, which is equal to the index k of the first subband in the interlace.
 
     In general, system  100  may utilize any OFDM structure with any number of total, usable, and guard subbands. Any number of interlaces may also be formed. Each interlace may contain any number of subbands and any one of the N total subbands. The interlaces may contain the same or different numbers of subbands. For clarity, the following description is for the interlace subband structure shown in  FIG. 2  with M interlaces and each interlace containing P uniformly distributed subbands. This interlace subband structure provides several advantages. First, frequency diversity is achieved since each interlace contains subbands taken from across the entire system bandwidth. Second, a receiving entity may recover data/pilot symbols sent on a given interlace by performing a partial P-point FFT instead of a full N-point FFT, which can simplify the processing at the receiving entity. 
     A communication channel between transmitting entity  110  and receiving entity  150  in OFDM system  100  may be characterized by either a time-domain channel impulse response or a corresponding frequency-domain channel frequency response. As used herein, and which is consistent with conventional terminology, a “channel impulse response” or “impulse response” is a time-domain response of the channel, and a “channel frequency response” or “frequency response” is a frequency-domain response of the channel. In a sampled-data system, the channel frequency response is the discrete Fourier transform (DFT) of the channel impulse response. This relationship may be expressed in matrix form, as follows:
 
   H     N×1   = W     N×N   · h     N×1  and    h     N×1   = W     N×N   H   · H     N×1 ,  Eq (2)
         where  h   N×1  is an N×1 vector for the impulse response of the communication channel;     H   N×1  is an N×1 vector for the frequency response of the communication channel;     W   N×N  is an N×N Fourier matrix; and   “H” denotes a conjugate transpose.
 
The Fourier matrix  W   N×N  is defined such that the (l,n)-th entry, W N   l,n , is given as:
       

                       W   N     l   ,   n       =     ⅇ       -   j     ⁢           ⁢   2   ⁢           ⁢   π   ⁢         (     l   -   1     )     ⁢     (     n   -   1     )       N           ,       for   ⁢           ⁢   l     =       1   ⁢           ⁢   …   ⁢           ⁢   N   ⁢           ⁢   and   ⁢           ⁢   n     =     1   ⁢           ⁢   …   ⁢           ⁢   N                 Eq   ⁢           ⁢     (   3   )                 
where l is a row index and n is a column index.
 
     The channel impulse response  h   N×1  is composed of N channel taps, with each channel tap h l  being defined by a zero or non-zero complex gain value at a specific tap delay l. The channel frequency response  H   N×1  is composed of N channel gains for the N total subbands, with each channel gain H k  being a complex gain value for a specific subband k. 
     If pilot symbols are transmitted on the P subbands in interlace m, then the received pilot symbols for this interlace may be expressed as:
 
   Y     m   = H     m   ∘ X     m   + N     m ,  Eq (4)
 
where
           X   m  is a P×1 vector with P pilot symbols sent on the P subbands in interlace m;     Y   m  is a P×1 vector with P received pilot symbols obtained by the receiving entity for the P subbands in interlace m;     H   m  is a P×1 vector for the actual channel frequency response for interlace m;   N m  is a P×1 noise vector for the P subbands in interlace m; and   “∘” denotes the Hadamard product, which is an element-wise product, where the i-th element of  Y   m  is the product of the i-th elements of  X   m  and  H   m .
 
The vector  H   m  contains only P entries of the vector  H   N×1  for the P subbands in interlace m. For simplicity, the noise  N   m  is assumed to be additive white Gaussian noise (AWGN) with zero mean and a variance of σ 2 .
       

     An initial frequency response estimate may be obtained for interlace m, as follows:
 
   Ĥ     m   = Y     m   / X     m   = H     m   + N     m   / X     m ,  Eq (5)
         where
             Y   m / X   m =[y m,1 /p m,1  . . . y m,P /p m,P ], and y m,i  and p m,i  are respectively the received and transmitted pilot symbols for the i-th subband in interlace m; and     Ĥ   m  is a P×1 vector for the initial frequency response estimate for interlace m.
 
 Ĥ   m  contains P channel gain estimates for the P subbands in interlace m, which may be obtained based on P element-wise ratios of the received pilot symbols to the transmitted pilot symbols, as shown in equation (5). If interlace m contains unused subbands with no received pilot symbols, then extrapolation, interpolation, and/or some other technique may be used to estimate the channel gains for these unused subbands.
   
               

     A P-tap impulse response estimate using interlace m may be obtained by performing a P-point IFFT on the initial frequency response estimate  Ĥ   m , as follows:
 
   ĥ     m   = W     m   · W     P×P   H   · Ĥ     m ,  Eq (6)
 
where  ĥ   m  is a P×1 vector for the impulse response estimate for interlace m;
 
 W   P×P  is a P×P Fourier matrix with elements defined as shown in equation (3); and
 
 W   P×P  is a P×P diagonal matrix containing W N   −m,p  for the p-th diagonal element, for p=1 . . . P, and zeros elsewhere, where
 
               W   N       -   m     ,   p       =       ⅇ     j   ⁢           ⁢   2   ⁢           ⁢   π   ⁢         (     m   -   1     )     ⁢     (     p   -   1     )       N         .           
The channel component in the P elements of vector  W   P×P   H · Ĥ   m  contains a phase ramp which may be expressed as: h m,p =h p ·W N   m,p , for p=1 . . . P. The slope of the phase ramp is determined by the staggering phase m of interlace m. The phase ramp may be removed by multiplying each element of  W   P×P   H · Ĥ   m  with W N   −m,p  to obtain a corresponding element of  ĥ   m . The P elements of  ĥ   m  may be expressed as: h p =h m,p ·W N   −m,p , for p=1 . . . P.
 
       ĥ   m  contains P channel taps and is obtained based on  Ĥ   m , which contains P channel gain estimates for the P subbands in interlace m. Since the actual channel impulse response  h   N×1  is composed of N channel taps, the initial impulse response estimate  ĥ   m  is undersampled in the frequency domain by the P subbands in interlace m. This undersampling in the frequency domain causes aliasing of the channel impulse response  h   N×1 , in the time domain. The initial impulse response estimate  ĥ   m  may be expressed as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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             where  h   N×1 =[ h   1   T   h   2   T  . . .  h   M   T ]T is the full-length actual channel impulse response; 
               h   s , for s=1 . . . M, is a P×1 vector containing P channel taps in  h   N×1  with tap indices of (s−1)·P+1 through s·P; 
               n  is a P×1 vector of noise for the initial impulse response estimate  ĥ   m ; 
           
         
       
    
                 W   M     s   ,   m       =     ⅇ       -   j     ⁢           ⁢   2   ⁢           ⁢   π   ⁢         (     s   -   1     )     ⁢     (     m   -   1     )       M           ;   and         
“T” denotes a transpose.
 
An “aliasing pattern” corresponding to staggering phase m may be defined as {W M   s,m }, for s=1 . . . M, and include the coefficients used for equation (7). The full-length actual channel impulse response  h   N×1  is composed of M segments. Each segment s contains P consecutive channel taps in  h   N×1  and is represented by a vector  h   s . Equation (7) indicates that the M segments alias and combine when undersampled in the frequency domain, and the combining coefficients are given by the aliasing pattern.
 
       FIG. 3  shows an impulse response estimate  300  obtained based on pilot symbols received on P subbands in one interlace. The full-length channel impulse response  h   N×1  is composed of N channel taps with indices of 1 through N. The first P channel taps in  h   N×1  are contained in h 1  and may be referred to as the main channel. The remaining N−P channel taps in  h   N×1  are contained in  h   2  through  h   M  and may be referred to as the excess channel. The excess channel taps alias when undersampled in the frequency domain. The aliasing results in the excess channel taps at indices of P+l, 2P+l, . . . , and (M−1)·P+l all appearing at tap index l, for l=1 . . . P. The P channel taps in  ĥ   m  thus contain P main channel taps as well as N−P excess channel taps. Each aliased excess channel tap causes error in the estimation of the corresponding main channel tap. 
     A longer impulse response estimate with more than P channel taps may be obtained by transmitting pilot symbols on multiple interlaces. One interlace may be used for pilot transmission in each symbol period, and different interlaces may be used for pilot transmission in different symbol periods. The use of multiple interlaces for pilot transmission allows the receiving entity to obtain a longer channel estimate, which may improve performance. By using all M interlaces for pilot transmission, it is possible to estimate the entire full-length channel impulse response with N channel taps. 
     The specific interlace to use for pilot transmission in each OFDM symbol period may be determined by a pilot staggering pattern. Various staggering patterns may be used for pilot transmission. In an embodiment, a staggering pattern may select one interlace for pilot transmission in each symbol period based on the following:
 
 m   t =[( m   t−1 −1+Δ m )mod  M]+ 1, with (Δ m,M )=1,  Eq (8)
 
where
         t is an index for symbol period;   Δm is the difference between interlace indices for two consecutive symbol periods;   m t  is the interlace to use for pilot transmission in symbol period t; and   (x,y)=1 means that x and y are relatively prime (i.e., the greatest common divisor for both x and y is one).
 
The −1 and +1 in equation (8) account for an interlace index numbering scheme that starts with ‘1’ instead of ‘0’. The interlace used for the first symbol period is m 1 , where m 1 ε{1 . . . M}. Different “complete” staggering patterns may be formed with different values of Δm. A complete staggering pattern is one that selects all M interlaces for pilot transmission, e.g., in M symbol periods. As an example, with Δm=1, the M interlaces are selected in sequential order, and the staggering pattern may be expressed as {1, 2, 3, . . . , M}. For the case with M=8, values of 1, 3, 5, and 7 may be used for Δm to obtain different complete staggering patterns. Of these four values, 7 is equivalent to 1 (in terms of performance) since Δm=1 is an increment of one and Δm=7 is a decrement of one, and 5 is equivalent to 3 for the same reason.
       

       FIG. 4A  shows a complete staggering pattern  400  that may be used for pilot transmission. The vertical axis represents interlace indices, and the horizontal axis represents time. For this example, M=8 and one interlace is used for pilot transmission in each symbol period. Staggering pattern  400  is generated with Δm=1 in equation (8), and the complete staggering pattern may be expressed as {1, 2, 3, 4, 5, 6, 7, 8}. The pilot is thus transmitted on interlace  1  in symbol period  1 , then interlace  2  in symbol period  2 , and so on, then interlace  8  in symbol period  8 , then back to interlace  1  in symbol period  9 , and so on. All eight interlaces are used for pilot transmission in each 8-symbol period duration. 
       FIG. 4B  shows a complete staggering pattern  410  that may also be used for pilot transmission. Again, M=8 and one interlace is used for pilot transmission in each symbol period. Staggering pattern  410  is generated with Δm=3 in equation (8), and the complete staggering pattern may be expressed as {1, 4, 7, 2, 5, 8, 3, 6}. The pilot is thus sent on interlace  1  in symbol period  1 , then interlace  4  in symbol period  2 , then interlace  7  in symbol period  3 , and so on. Again, the pilot is transmitted on all eight interlaces in each 8-symbol period duration. Over three symbol periods, staggering pattern  410  selects interlaces with relative offsets of {1, 4, 7} while staggering pattern  400  selects interlaces with relative offsets of {1, 2, 3}. Staggering pattern  410  is thus more “spread out” than staggering pattern  400  and may provide better performance. 
       FIG. 4C  shows a complete staggering pattern  420  that does not satisfy equation (8) but may also be used for pilot transmission. This complete staggering pattern may be expressed as {1, 5, 2, 6, 3, 7, 4, 8}. The pilot is transmitted on all eight interlaces in each 8-symbol period duration. 
     In general, the pilot may be transmitted on any number of interlaces and on any one of the M interlaces in each symbol period. The particular interlace to use for pilot transmission in each symbol period may be selected based on any staggering pattern, three of which are shown in  FIGS. 4A through 4C . The pilot may be transmitted on all M interlaces using a complete staggering pattern or on a subset of the M interlaces using a “partial” staggering pattern. 
     A longer impulse response estimate  {tilde over (h)}   L×1 (t) with L channel taps, where P&lt;L≦N, may be obtained by filtering multiple P-tap initial impulse response estimates obtained for multiple interlaces. This time-domain filtering may be performed, e.g., with a finite impulse response (FIR) filter, as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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             where  ĥ (t)=[ĥ 1 (t)ĥ 2 (t) . . . ĥ P (t)] T  is an initial impulse response estimate obtained for symbol period t based on a pilot received on interlace m t ; 
               {tilde over (h)}   s (t)=[{tilde over (h)} s,1 (t){tilde over (h)} s,2 (t) . . . {tilde over (h)} s,P (t)] is a P×1 vector that is an estimate of the channel impulse response  h   s (t) for segment s in symbol period t; 
             α s,l (i) is a coefficient for the i-th filter tap used to derive the l-th channel tap in segment s; 
             N f  is the number of non-causal taps for the time-domain filter; and 
             N b  is the number of causal taps for the time-domain filter.
 
The L-tap impulse response estimate  {tilde over (h)}   L×1 (t) is composed of S segments and may be given as:  {tilde over (h)}   L×1 (t)=[ {tilde over (h)}   1   T (t) {tilde over (h)}   2   T (t) . . .  ĥ   S   T (t)] T , where S&gt;1 and L=S·P. Each segment s, for s=1 . . . S, contains P channel taps that are included in the vector  {tilde over (h)}   s (t).  {tilde over (h)}   s (t) is an estimate of  h   s (t), which is the actual channel impulse response for segment s.
 
           
         
       
    
     Equation (9) indicates that the P channel taps for each segment s may be obtained by filtering N f +N b  initial impulse response estimates  ĥ (t+N f ) through  ĥ (t−N b +1), which may be obtained over N f +N b  symbol periods for N f +N b  different interlaces. The initial impulse response estimate  ĥ (t) for the current symbol period t is aligned at filter tap i=0. Equation (9) also indicates that each channel tap {tilde over (h)} s,l (t) in  {tilde over (h)}   L×1 (t) may be obtained by multiplying N f +N b  channel taps ĥ l (t−N b +1) through ĥ l (t+N f ) with N f +N b  coefficients α s,l (N b −1) through α s,l (−N f ), respectively, and combining the N f +N b  resultant products. 
     In general, the coefficients for each channel tap {tilde over (h)} s,l (t) of each segment s may be selected separately. Furthermore, N f  and N b  may be selected for each channel tap of each segment s. For simplicity, one set of N f +N b  coefficients may be used for all P channel taps in each segment, and S sets of coefficients may be defined for the S segments of  {tilde over (h)}   L×1 (t) In this case, the coefficients {α s (i)} for each segment s are not a function of channel tap index. 
     The time-domain filtering may also be performed using other types of filter, such as an infinite impulse response (IIR) filter. The time-domain filtering may also be performed using a causal filter (with N f =0 and N b ≧1), a non-causal filter (with N f ≧1), or a filter with both causal and non-causal taps. For clarity, the following description is for the time-domain filter shown in equation (9). 
     1. Channel Impulse Response Estimate of Length 2P 
     To obtain a longer impulse response estimate  {tilde over (h)}   2P×1 (t) with L=2P channel taps, the initial impulse response estimate  ĥ (t) obtained in symbol period t for one interlace may be expressed as:
 
   ĥ   ( t )=   h     1 ( t )+   h     2 ( t )· W   M   m     l     + n   ( t ),  Eq (10)
 
where
 
               W   M     m   t       =       ⅇ       -   j     ⁢           ⁢   2   ⁢           ⁢   π   ⁢       (       m   t     -   1     )     M         .           
Equation (10) is derived based on equation (7) and assumes that segments 3 through M contain channel taps with zero magnitude. The vector  h   1 (t) contains the first P channel taps in  h   N×1 (t) for the main channel. The vector  h   2  (t) contains the next P channel taps in  h   N×1 (t) for the excess channel.
 
     The coefficients for the time-domain filter for the main channel estimate  {tilde over (h)}   1 (t) may be selected based on various constraints such as: 
     Cancel excess channel: 
                         ∑     i   =     -     N   f             N   b     -   1       ⁢         α   1     ⁡     (   i   )       ·     W   M     m     t   -   i             =   0     ,           Eq   ⁢           ⁢     (     11   ⁢   a     )                 
Suppress time variation:
 
                         ∑     i   =     -     N   f             N   b     -   1       ⁢         α   1     ⁡     (   i   )       ·   i       =   0     ,           Eq   ⁢           ⁢     (     11   ⁢   b     )                 
Provide unbiased estimate:
 
                         ∑     i   =     -     N   f             N   b     -   1       ⁢       α   1     ⁡     (   i   )         =   1     ,           Eq   ⁢           ⁢     (     11   ⁢   c     )                 
and
 
Minimize noise variance:
 
                     min   (       ∑     i   =     -     N   f             N   b     -   1       ⁢       α   1   2     ⁡     (   i   )         )     ,           Eq   ⁢           ⁢     (     11   ⁢   d     )                 
where m t−i  is the interlace used for pilot transmission in symbol period t−i, which corresponds to the i-th filter tap. An unbiased estimate is one for which the mean of the estimate (over noise) is equal to the perfect channel value.
 
     Equation (11b) cancels the linear component of the channel variation over the N f +N b  symbol periods, which would be the dominant component at low speeds and/or small N f +N b . The first constraint in equation (11a) cancels the contribution from the excess channel  h   2  (t), so that  {tilde over (h)}   1 (t) contains mostly components from the main channel {tilde over (h)} 1,l (t). The second constraint in equation (11b) suppresses time variation in the main channel  h   1 (t) across the N f +N b  symbol periods. The third constraint in equation (11c) provides an unbiased estimate of  h   1 (t), so that the expected magnitude of {tilde over (h)} 1,l (t) is equal to h 1,l (t). The fourth constraint in equation (11d) minimizes the noise variance in the main channel estimate  {tilde over (h)}   1 (t). The number of taps (N f +N b ) for the time-domain filter determines (1) the number of degrees of freedom for selecting the coefficients and (2) the number of constraints that may be applied in selecting the coefficients. 
     The coefficients for the time-domain filter for the excess channel  {tilde over (h)}   2 (t) may be selected based on the various constraints such as: 
     Cancel main channel: 
                         ∑     i   =     -     N   f             N   b     -   1       ⁢       α   2     ⁡     (   i   )         =   0     ,           Eq   ⁢           ⁢     (     12   ⁢   a     )                 
Suppress time variation of main channel:
 
                         ∑     i   =     -     N   f             N   b     -   1       ⁢         α   2     ⁡     (   i   )       ·   i       =   0     ,           Eq   ⁢           ⁢     (     12   ⁢   b     )                 
Suppress time variation of excess channel:
 
                         ∑     i   =     -     N   f             N   b     -   1       ⁢           α   2     ⁡     (   i   )       ·   ⅈ     ⁢           ⁢     W   M     m     t   -   i             =   0     ,           Eq   ⁢           ⁢     (     12   ⁢   c     )                 
and
 
Provide unbiased estimate:
 
                       ∑     i   =     -     N   f             N   b     -   1       ⁢         α   2     ⁡     (   i   )       ·     W   M     m     t   -   i             =   1.           Eq   ⁢           ⁢     (     12   ⁢   d     )                 
The first constraint in equation (12a) cancels the contribution from the main channel  h   1 (t), so that  {tilde over (h)}   2  (t) contains mostly components from the excess channel  h   2  (t). The second constraint in equation (12b) suppresses time variation in the main channel  h   1 (t). The third constraint in equation (12c) provides an unbiased estimate of  h   2 (t).
 
     As a specific example, a 3-tap time-domain filter may be used to derive the 2P channel taps in  {tilde over (h)}   2P×1 (t) based on  ĥ (t−1),  ĥ (t), and  ĥ (t+1) for three symbol periods. The 3-tap time-filter may be designed as follows. Using equation (10), the l-th channel tap in symbol periods t−1, t, and t+1, prior to the time-domain filtering, may be expressed as:
 
 ĥ   l ( t− 1)= h   1,l ( t− 1)+ h   2,l ( t− 1)· W   M   m     t−1     +n   l ( t− 1),
 
 ĥ   l ( t )= h   1,l ( t )+ h   2,l ( t )· W   M   m     t−1     +n   l ( t ), for l=1 . . . P,
 
 ĥ   l ( t− 1)= h   1,l ( t− 1)+ h   2,l ( t− 1)· W   M   m     t−1     +n   l ( t− 1),  Eq (13)
         where ĥ l (t), h 1,l (t), h 2,l (t), and n l (t) are the l-th element of  ĥ (t),  h   1 (t),  h   2 (t), and  n (t), respectively; and   m t−1 , m t , and m t+1  are the interlaces used for pilot transmission in symbol periods t−1, t, and t+1, respectively.       

     For the 3-tap time-domain filter for staggering pattern  410  shown in  FIG. 4B , with M=8, m t−1 =m t −3, and m t+1 =m t +3, the constraints used to select the coefficients for the main channel estimate  {tilde over (h)}   1 (t) may be expressed as: 
                                    Cancel excess channel:   α 1 (−1) · e −j3π/4  + α 1 (0) + α 1 (1) · e j3π/4  = 0,       Suppress time variation:   α 1 (−1) − α 1 (1) = 0, and       Provide unbiased estimate:   α 1 (−1) + α 1 (0) + α 1 (1) = 1.                    
The first equation above (to cancel the excess channel) is from equation (11a) and has the form: α 1 (−1)·W 8   m     t     +3 +α 1 (0)·W 8   m     t   +α 1 (1)·W 8   m     t     −3 =0, which may be simplified as: α 1 (−1)·W 8   3 +α 1 (0)+α 1 (1)·W 8   −3 =0, where W 8   3 =e −j3π/4  and W 8   −3 =e +j3π/4 .
 
     The solution to the above set of equations for the main channel is given as: 
                     [             α   1     ⁡     (     -   1     )                   α   1     ⁡     (   0   )                   α   1     ⁡     (   1   )             ]     =       [           1   -     1   /     2                     2     -   1               1   -     1   /     2               ]     .             Eq   ⁢           ⁢     (   14   )                 
Equation (14) indicates that the coefficients for the main channel estimate  {tilde over (h)}   1 (t) are independent of symbol period t. This set of coefficients suppresses time variation in the main channel  h   1 (t) but does not suppress time-variation in the excess channel  h   2  (t). Time-variation error is proportional to the energy of the channel taps, which is typically small for the excess channel and significant only when the transmitting and/or receiving entity is moving at high speeds. Thus, not suppressing time variation in the excess channel h 2 (t) may only marginally degrade performance, if at all.
 
     For the 3-tap time-domain filter for staggering pattern  410  shown in  FIG. 4B , the constraints used to select the coefficients for the excess channel estimate  {tilde over (h)}   2  (t) may be expressed as: 
                                    Cancel main channel:   α 2 (−1) + α 2 (0) + α 2 (1) = 0,       Suppress time variation:   α 2 (−1) − α 2 (1) = 0, and       Provide unbiased estimate:   α 2 (−1) · e −j3π/4  + α 2 (0) + α 2 (1) · e j3π/4  =           e j2π·(m     t     −1)/8 .                    
The third equation above (to provide an unbiased estimate) is from equation (12c) and has the form: α 2 (−1)·W 8   m     t     +3 +α 2 (0)·W 8   m     t   +α 2 (1)·W 8   m     t     −3 =1, which may be simplified as: α 1 (−1)·W 8   3 +α 1 (0)+α 1 (1)·W 8   −3 =W 8   m     t   , where m t ε{1 . . . M}.
 
     The solution to the above set of equations for the excess channel is given by: 
                     [             α   2     ⁡     (     -   1     )                   α   2     ⁡     (   0   )                   α   2     ⁡     (   1   )             ]     =       [             -   1     +     1   /     2                   2   -     2                   -   1     +     1   /     2               ]     ·       ⅇ     j   ⁢           ⁢   2   ⁢           ⁢     π   ·       (       m   t     -   1     )     /   8           .               Eq   ⁢           ⁢     (   15   )                 
Equation (15) indicates that the coefficients for the excess channel are dependent on the staggering phase m t  of interlace m t  used for pilot transmission in symbol period t.
 
2. Channel Impulse Response Estimate of Length 3P
 
     To obtain a longer impulse response estimate  {tilde over (h)}   3P×1 (t) with L=3P channel taps, the initial impulse response estimate  ĥ (t) obtained in symbol period t for one interlace may be expressed as:
 
   ĥ   ( t )=   h     1 ( t )+   h     2 ( t )· W   M   m     t     + h     3 ( t )· W   M   2m     t     + n   ( t ),  Eq (16)
 
Equation (16) is derived based on equation (7) and assumes that segments 4 through M contain channel taps with zero magnitude. The vectors  h   1 (t),  h   2 (t), and  h   3 (t) contain P channel taps for the first, second, and third segments, respectively, of  h   N×1 (t).
 
     A 3-tap time-domain filter may also be used to derive the 3P elements of  {tilde over (h)}   3P×1 (t) based on  ĥ (t−1),  ĥ (t), and  ĥ (t+1) obtained in three symbol periods. Using equation (16), the l-th channel tap in symbol periods t−1, t, and t+1, prior to the time-domain filtering, may be expressed in matrix form, as follows: 
                     [           ⁢               h   ^     l     ⁡     (     t   -   1     )                     h   ^     l     ⁡     (   t   )                     h   ^     l     ⁡     (     t   +   1     )             ]     =         [           ⁢         1         W   M       -   Δ     ⁢           ⁢   m             W   M       -   2     ⁢           ⁢   Δ   ⁢           ⁢   m               1       1       1           1         W   M     Δ   ⁢           ⁢   m             W   M     2   ⁢           ⁢   Δ   ⁢           ⁢   m             ]     ·     [           ⁢             h     1   ,   l       ⁡     (   t   )                     h     2   ,   l       ⁡     (   t   )       ·     W   M     m   t                       h     3   ,   l       ⁡     (   t   )       ·     W   M     2   ⁢     m   t                 ]       +             [           ⁢             n   l     ⁡     (     t   -   1     )                   n   l     ⁡     (   t   )                   n   l     ⁡     (     t   +   1     )             ]     ,     
     ⁢           ⁢       for   ⁢           ⁢   l     =     1   ⁢           ⁢   …   ⁢           ⁢   P       ,                 Eq   ⁢           ⁢     (   17   )                 
where
 
               W   M     Δ   ⁢           ⁢   m       =       ⅇ       -   j     ⁢           ⁢   2   ⁢   π   ⁢       Δ   ⁢           ⁢   m     M         .           
Equation (17) assumes that m t−1 =m t −Δm and m t+1 =m t +Δm. The 3-tap time-domain filter does not have enough degrees of freedom to suppress time variation in  h   1 (t),  h   2  (t), or  h   3  (t). Thus, equation (17) further assumes that  h   1 (t),  h   2  (t), and  h   3 (t) are constant over the three symbol periods t−1, t, and t+1.
 
     A least-squares estimate of  h   1 (t),  h   2  (t), and  h   3  (t) may be obtained as follows: 
     
       
         
           
             
               
                 
                   
                     
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                   ⁢ 
                   
                       
                   
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                     ( 
                     18 
                     ) 
                   
                 
               
             
           
         
       
     
     The 3-tap time-domain filter for  h   1 (t),  h   2 (t), and  h   3  (t) may be expressed in matrix form, as follows: 
     
       
         
           
             
               
                 
                   
                     
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                   Eq 
                   ⁢ 
                   
                       
                   
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                     ( 
                     19 
                     ) 
                   
                 
               
             
           
         
       
     
     For staggering pattern  410  shown in  FIG. 4B , with M=8 and Δm=3, the coefficients for the 3-tap time-domain filter may be derived based on equation (18) and given as: 
     
       
         
           
             
               
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                         0.3536 
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     The main channel estimate  {tilde over (h)}   1 (t) may be obtained by applying the coefficients α 1 (1), α 1 (0), and α 1 (−1) to  ĥ (t−1), ĥ(t), and  ĥ (t+1), respectively. The excess channel estimate  {tilde over (h)}   2 (t) may be obtained by applying the coefficients α 2  (1), α 2  (0), and α 2 (−1) to  ĥ (t−1),  ĥ (t), and  ĥ (t+1), respectively. The excess channel estimate  {tilde over (h)}   3 (t) may be obtained by applying the coefficients α 3 (1), α 3 (0), and α 3 (−1) to  ĥ (t−1),  ĥ (t), and  ĥ (t+1), respectively. 
     The 3-tap time-domain filter does not have sufficient degrees of freedom to apply many of the constraints shown in equation sets (11) and (12). The coefficients for this time-domain filter do not suppress time variation in the main channel  h   1 (t) or the excess channel  h h 2 (t) and  h   3 (t). The various constraints described above may be applied by using a time-domain filter with more than three taps. 
     In general, a different set of coefficients {α s (i)} may be derived for the time-domain filter for the impulse response estimate  {tilde over (h)} h s (t) for each segment s. The coefficients for each segment s may be selected based on various constraints such as: canceling the other segments, suppressing estimation error due to time variation in the channel, providing an unbiased estimate of  h   s (t), minimizing the noise variance in  {tilde over (h)}   s (t), and so on. The number of taps for the time-domain filter determines the number of constraints that may be applied to the coefficients. Several exemplary 3-tap time-domain filter designs have been described above. Other time-domain filters may also be designed based on the description above and are within the scope of the invention. 
     In general, a longer impulse response estimate with L channel taps may be obtained based on pilot symbols received on L different subbands in one or more symbol periods. The pilot may be transmitted on one interlace in each symbol period to limit the amount of overhead for the pilot. The pilot may be transmitted on different interlaces with staggered subbands in different symbol periods. This allows the receiving entity to obtain a longer impulse response estimate with more than P channel taps. A full-length impulse response estimate with all N channel taps may be obtained if the pilot is transmitted on all M interlaces using a complete staggering pattern. 
     The receiving entity may derive a longer impulse response estimate  {tilde over (h)}   L×1 (t) of length L by filtering initial impulse response estimates  ĥ  of length P for a sufficient number of (S or more) different interlaces. If the pilot is transmitted on a different interlace in each symbol period, then the time-domain filtering may be performed over a sufficient number of (S or more) symbol periods to obtain  {tilde over (h)}   L×1 (t). A progressively longer impulse response estimate may be obtained by filtering over more symbol periods. Time-domain filtering over fewer symbol periods provides better tracking of changes in the channel, is thus more robust to Doppler effects, and can provide an impulse response estimate with a shorter length. Time-domain filtering over more symbol periods increases error in  {tilde over (h)}   L×1 (t) due to changes in the channel over time, is less robust to Doppler effects, but can provide an impulse response estimate with a longer length. 
     A longer impulse response estimate contains excess channel taps. Since each channel tap contains the complex channel gain at that tap position as well as noise, a progressively longer impulse response estimate contains more information regarding the channel but also contains more noise. The noise from the excess channel taps may be viewed as noise enhancement resulting from extending the length of the channel estimate beyond P. If the excess channel energy is relatively small, or if the excess channel taps are not needed, then better performance may be achieved with a shorter impulse response estimate (e.g.,  {tilde over (h)}   2P×1 (t)). If the excess channel energy is relatively large, or if the excess channel taps are pertinent, then a longer impulse response estimate (e.g.,  {tilde over (h)}   3P×1 (t)) may provide better performance even with the noise enhancement. Channel estimates with different lengths may be derived and used for different purposes at the receiving entity. 
     3. Data Detection 
     For data detection, a longer impulse response estimate  {tilde over (h)}   2P×1 (t) with 2P channel taps may provide a good trade-off between a longer channel estimate and additional noise from the excess channel. The longer channel estimate mitigates the deleterious aliasing effect shown in equation (7) due to undersampling the frequency domain, provides a more accurate estimate of the main channel  h   1 (t), and allows for estimation of the excess channel  h   2 (t). The longer impulse response estimate  {tilde over (h)}   2P×1 (t) may be derived as described above. 
       FIG. 5  shows a flow diagram of a process  500  for deriving a channel estimate used for data detection and decoding. Received pilot symbols are obtained for the subbands in interlace m t  used for pilot transmission in the current symbol period t (block  512 ). An initial frequency response estimate  Ĥ (t) is derived based on the received pilot symbols, as shown in equation (5) (block  514 ). An initial impulse response estimate  ĥ (t) is derived based on the initial frequency response estimate  Ĥ (t), as shown in equation (6) (block  516 ). Initial impulse response estimates for at least S 1  symbol periods are filtered with a time-domain filter having at least S 1  taps to obtain a longer impulse response estimate  {tilde over (h)}   L     1     ×1 (t) with L 1  channel taps, where L 1 =S 1 ·P (block  518 ). 
     Post-processing may be performed on the L 1  channel taps in  {tilde over (h)}   L     1     ×1 (t) to further improve channel estimation performance (block  520 ). The post-processing may include truncation, e.g., setting channel taps P+1 through L 1  for the excess channel estimate to zeros. The post-processing may alternatively or additionally include thresholding, e.g., setting channel taps in the main and/or excess channel estimates having energy below a given threshold to zeros. The unprocessed or post-processed longer impulse response estimate  {tilde over (h)}   L     1     ×1 (t) may then be extended to length N by zero-padding to obtain a vector  {tilde over (h)}   N×1 (t) of length N (also block  520 ). An N-point FFT may then be performed on  {tilde over (h)}   N×1 (t) to obtain a frequency response estimate  {tilde over (H)}   N×1 (t) for all N subbands (block  522 ), as follows:
 
   {tilde over (H)}     N×1 ( t )=   W     N×N   · {tilde over (h)}     N×1 ( t ).  Eq (20)
 
Process  500  may be performed for each symbol period with pilot transmission.
 
       {tilde over (H)}   N×1 (t) contains N channel gains for the N total subbands and may be expressed as:  {tilde over (H)}   N×1 (t)=[ {tilde over (H)}   1   T (t) {tilde over (H)}   2   T (t) . . .  {tilde over (H)}   M   T (t)] T , where  {tilde over (H)}   m (t) contains P channel gain estimates for P subbands in interlace m. The M frequency response estimates  {tilde over (H)}   m (t) for the M interlaces may have different noise variances depending on the particular staggering pattern used for pilot transmission. In general, a staggering pattern that is more spread out (e.g., staggering pattern  410 ) may result in less noise variation across  {tilde over (H)}   m (t) for the M interlaces than a staggering pattern that is more closely spaced (e.g., staggering pattern  400 ). 
     4. Time Tracking 
     The receiving entity performs time tracking to estimate and track symbol timing across different OFDM symbols. The symbol timing is used to capture a window of N input samples (often called an FFT window) from among the N+C input samples for each received OFDM symbol. Accurate symbol timing is pertinent since performance of both channel estimation and data detection is affected by the placement of the FFT window. The timing of the received OFDM symbol for each symbol period may be estimated by deriving a longer impulse response estimate for that symbol period and detecting for the timing based on an appropriate criterion, e.g. maximizing the energy that falls within the cyclic prefix. 
     If pilot symbols are available on L different subbands and a timing reference is not available, then a longer impulse response estimate with L channel taps may be derived but only L/2 channel taps may be resolved without any ambiguity. This is because a negative timing error results in earlier channel taps aliasing and appearing at the end of the impulse response estimate. Thus, it is not possible to determine whether the channel taps at the end of the impulse response estimate are later channel taps (if the symbol timing is correct) or earlier channel taps that have aliased (if there is a negative timing error). A longer channel impulse response estimate with up to N channel taps may be obtained by filtering the initial impulse response estimates for M different interlaces. The resolvable length of the communication channel is increased by the use of the longer impulse response estimate. 
       FIGS. 6A and 6B  illustrate ambiguity in a channel impulse response estimate due to timing uncertainty.  FIG. 6A  shows a channel impulse response estimate  610  of length L for an actual channel with an impulse response of length greater than L/2. In  FIG. 6A , the symbol timing is correct and channel impulse response estimate  610  properly includes responses  612  and  614  of the actual channel at the proper locations. 
       FIG. 6B  shows an impulse response  620  of length greater than L/2 for another actual channel. If there is no timing error, then a channel impulse response estimate for this channel would include responses  622  and  624  at the locations as shown in  FIG. 6B . However, if there is a timing error of x, then response  622  would alias and appear as response  632 . Thus, a channel impulse response estimate for this channel, with timing error of x, would be similar to channel impulse response estimate  610  in  FIG. 6A   
       FIGS. 6A and 6B  illustrate that channel impulse response estimate  610  may be obtained for (1) a channel having the impulse response shown in  FIG. 6A , with no timing error, or (2) a channel having the impulse response shown in  FIG. 6B , with a timing error of x, and these two cases cannot be distinguished. However, this ambiguity problem would not occur if the channel response lengths are always assumed to be less than L/2. Since the actual channel in  FIG. 6B  would then have to be longer than L/2 to be mistaken with the channel in  FIG. 6A , it can be concluded that the channel response estimate in  FIG. 6A  does correspond to the true channel. Thus, an initial estimate of length L can resolve a channel of length L/2 with timing uncertainty. A longer channel impulse response estimate is thus desirable for time tracking. 
     The longer channel impulse response estimate has additional noise due to the excess channel taps, and greater error due to channel time-variations. However, time tracking is likely to be less sensitive to the additional noise since the goal of time tracking is to determine less detailed information such as the general location of the channel energy rather than the complex channel gains of each tap. Thus, the tradeoff between channel quality and length is consistent with the requirements for data detection and time tracking. Specifically, for time tracking, a longer impulse response estimate  {tilde over (h)}   3P×1 (t) with 3P channel taps may provide a good trade-off between resolvable channel length and noise enhancement. For example, if P=512, then  {tilde over (h)}   3P×1 (t) contains 1536 channel taps, and up to 768 channel taps may be resolved without ambiguity. Once the symbol timing is known, the communication channel may be assumed to be 3P/2 taps long for data detection purpose. A 3P/2-tap channel may be estimated by obtaining a longer impulse response estimate with 2P channel taps and truncating the last 256 channel taps. 
       FIG. 7  shows a flow diagram of a process  700  for performing time tracking. Blocks  712 ,  714 ,  716 , and  718  in  FIG. 7  are as described above for block  512 ,  514 ,  516 , and  518 , respectively, in  FIG. 5 . However, a longer impulse response estimate  {tilde over (h)}   L     2     ×1 (t) with a different length L 2  may be used for time tracking, and a different time-domain filter with at least S 2  channel taps may be used to derive  {tilde over (h)}   L     2     ×1 (t), where L 2 =S 2 ·P. The channel estimate  {tilde over (h)}   L     2     ×1 (t) is then processed to determine the timing of the received OFDM symbol for the current symbol period t (block  720 ). One method to determine the timing is as follows. A window of length L 2 /2 is placed such that the left edge of the window is initially at tap index 1. The energy of all channel taps falling within the window is computed. The window is then moved to the right, one tap position at a time until tap index L 2 /2 is reached. The channel tap energy is computed for each tap position. The peak energy among all of the L 2 /2 window starting positions is then determined. If multiple window starting positions have the same peak energy, then the leftmost window starting position with the peak energy is identified. The leftmost window starting position with the peak energy uniquely determines the FFT window for the received OFDM symbol. Timing detection may also be performed using other techniques. In any case, symbol timing estimate is updated with the timing information obtained for the current received OFDM symbol (block  722 ). 
     In general, the same or different impulse response estimates may be used for data detection/decoding and time tracking. The use of the same impulse response estimate can reduce the amount of computation at the receiving entity. In this case, the channel length L and the time-domain filter for this impulse response estimate may be selected to provide good performance for both data detection and time tracking. Different impulse response estimates may also be used for data detection/decoding and time tracking in order to achieve better performance for both, and may be derived with two time-domain filters. The channel length and the time-domain filter coefficients for each impulse response estimate may be selected to provide good performance for data detection or time tracking. 
       FIG. 8  shows an embodiment of OFDM demodulator  160 , channel estimator  172 , and time tracking unit  162  at receiving entity  150 . Within OFDM demodulator  160 , a cyclic prefix removal unit  812  captures N input samples for each received OFDM symbol based on the symbol timing provided by time tracking unit  162 . An FFT unit  814  performs an N-point FFT on each window of N input samples and obtains N received symbols for the N subbands. FFT unit  814  provides received data symbols to detector  170  and received pilot symbols to channel estimator  172 . Detector  170  also receives the frequency response estimate  {tilde over (H)}   N×1 (t) from channel estimator  172 , performs data detection on the received data symbols, and provides detected data symbols. 
     Within channel estimator  172 , a pilot detector  822  removes the modulation on the received pilot symbols and may perform extrapolation and/or interpolation to obtain the initial frequency response estimate  Ĥ (t) composed of P channel gains for the P subbands in the interlace used for pilot transmission in the current symbol period t. An IFFT unit  824  performs a P-point IFFT on  Ĥ (t) to obtain the modulated impulse response estimate  ĥ   m (t) with P channel taps. A rotator  826  removes the phase ramp in the P elements of  ĥ   m (t) and provides the initial impulse response estimate  ĥ (t). A time-domain filter  830  filters the initial impulse response estimates  ĥ (t) obtained for S 1  or more interlaces obtained in S 1  or more symbol periods and provides the longer impulse response estimate  {tilde over (h)}   L×1 (t) with L 1  channel taps. A post-processor  832  performs post-processing (e.g., truncation, thresholding, and so on) and zero-padding on  {tilde over (h)}   L     1     ×1 (t) and provides a vector  {tilde over (h)}   N×1 (t) with N channel taps. An FFT unit  834  performs an N-point FFT on  {tilde over (h)}   N×1  (t) to obtain the frequency response estimate  {tilde over (H)}   N×1 (t) for the N total subbands. Channel estimator  172  may also derive a frequency response estimate  {tilde over (H)}   m (t) for just one or more selected interlaces. 
     Within time tracking unit  162 , a time-domain filter  840  filters the initial impulse response estimates  ĥ (t) for S 2  or more interlaces obtained in S 2  or more symbol periods and provides the longer impulse response estimate  {tilde over (h)}   L     2     ×1 (t) with L 2  channel taps. A timing detector  842  determines the timing for the current received OFDM symbol, e.g., based on the energy of the channel taps in  {tilde over (h)}   L     2     ×1 (t). A time tracking loop  844  (which may be a loop filter) adjusts the symbol timing from the timing used for the current received OFDM symbol. 
       FIG. 9  shows a block diagram of a time-domain filter  830   x , which may be used for filters  830  and  840  in  FIG. 8 . Within filter  830   x , the l-th channel tap in  ĥ (t) is provided to N f +N b −1 series-coupled delay elements  912 . Each delay element  912  delays its input channel tap by one symbol period. N f +N b −1 multipliers  914  couple to the input of the N f +N b −1 delay elements, and one multiplier  914  couples to the output of the last delay element. The N f +N b  multipliers receive and multiply their channel taps ĥ l (t+N f ) through ĥ l (t−N b +1) with coefficients α s,l (−N f ) through α s,l (N b −1), respectively. The same coefficients may be used for all P channel taps in each segment, in which case the coefficients may be denoted as α 8 (−N f ) through α s (N b −1), without subscript l for tap index. A summer  916  receives and sums the outputs of all N f +N b  multipliers and provides the l-th channel tap in segment s of  {tilde over (h)}   L×1 (t). L may be equal to L 1  for data detection and to L 2  for time tracking. The filtering for only one channel tap in  {tilde over (h)}   L×1 (t) is shown in  FIG. 9 . The filtering for each of the remaining channel taps in  {tilde over (h)}   L×1 (t) may be performed in similar manner. 
     The pilot transmission, channel estimation, and time tracking techniques described herein may be implemented by various means. For example, these techniques may be implemented in hardware, software, or a combination thereof. For a hardware implementation, the processing units used for pilot transmission at the transmitting entity may be implemented within one or more application specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DSPDs), programmable logic devices (PLDs), field programmable gate arrays (FPGAs), processors, controllers, micro-controllers, microprocessors, other electronic units designed to perform the functions described herein, or a combination thereof. The processing units used for channel estimation and time tracking at the receiving entity may also be implemented within one or more ASICs, DSPs, and so on. 
     For a software implementation, these techniques may be implemented with modules (e.g., procedures, functions, and so on) that perform the functions described herein. The software codes may be stored in a memory unit (e.g., memory unit  142  or  192  in  FIG. 1 ) and executed by a processor (e.g., controller  140  or  190 ). The memory unit may be implemented within the processor or external to the processor. 
     Headings are included herein for reference and to aid in locating certain sections. These headings are not intended to limit the scope of the concepts described therein under, and these concepts may have applicability in other sections throughout the entire specification. 
     The previous description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the invention. Thus, the present invention is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.