Abstract:
A blind carrier frequency offset estimator is based on a single-OFDM-symbol training sequence in multi-user OFDMA uplink. Through multiple access interference modeling and analysis, a virtual user is employed that occupies the all null sub-carriers. By minimizing the energy leakage on the virtual user in term of tentative frequency offsets, the estimator can approach the real frequency offset. The estimator performs only on frequency-domain, simplifies interference calculations, and lowers the rank of the matrix. An iterative computation method is used to approach the real frequency offset.

Description:
RELATED APPLICATION 
     The present application claims priority of Chinese Application No. 200610063945.3 filed Aug. 31, 2006, and is a continuation of, U.S. patent application Ser. No. 11/833,157 filed on Aug. 2, 2007. The disclosure of the foregoing United States patent application is specifically incorporated herein by this reference in its entirety and assigned to STMicroelectronics (Beijing) R&amp;D Company LTD., assignee of the present invention. 
    
    
     BACKGROUND OF THE INVENTION 
     The present invention relates to wireless transmission, and more particularly to a method for estimating a carrier frequency offset for an interleaved OFDMA uplink receiver. 
     As a widely used technique for high data rate wireless transmission, the OFDM (Orthogonal Frequency Division Multiplexing) technique makes use of a set of overlapping but orthogonal sub-carriers to reach high spectrum efficiency. Inheriting from OFDM, OFDMA (orthogonal frequency division multiple access) has been proposed in many broadband wireless systems to provide more flexible wireless access scheme and to take more advantage of diversity gain by allocating a user a set of permutation-driven interleaved sub-carriers that guarantee a large sub-carrier spacing for each user. 
     Working in a mobile wireless environment, OFDMA is subject to synchronization errors, such as the misalignments from the terminals to the base-station, the discordances between the oscillator of the base-station and those of the terminals, and Doppler shifts of the terminals. Like the OFDM technique, OFDMA is so sensitive to the synchronization errors that a small frequency offset would lead to the loss of the orthogonality among the sub-carriers, and that a short time delay would result in the complex exponential twiddle on the frequency-domain. 
     The time-domain received signal in uplink is a multiplex of the multi-user signals that are subject to the different frequency offsets, time delays, and channel distortions. The interleaving topology of OFDMA deteriorates this issue by turning the ICI (inter-channel-interference) among the sub-carriers to the MAI (multiple access interference) among users. Besides, synchronization errors start to fluctuate when a user moves fast. 
     In order to keep the synchronization of terminals and base-station, a ranging process is taken to detect the synchronization errors of a terminal and to control the adjustment of the terminal&#39;s transmission in a close loop between this terminal and the base-station. 
     Functionally, the ranging process is classified into initial ranging and periodic ranging. Initial ranging takes place when a terminal is (re-)registered into the network; while periodic ranging is performed to keep the synchronization between a terminal and the base-station during its constant transmission. Usually, the initial ranging consumes more signaling resources by transmitting multiple OFDM symbols training sequence in uplink by which the base-station receiver is able to estimate the synchronization errors accurately but in relative long time interval; and the periodic ranging needs single OFDM symbol training sequence in uplink by which the base-station receiver can estimate the synchronization errors in a short time interval. 
     The synchronization errors of a low mobile or even fixed terminal may change so slowly that after the initial ranging reduces the synchronization errors of the terminal under an acceptable criterion, the base-station hardly performs any periodic ranging for its maintenance. But, once the terminal speeds up, its synchronization errors may fluctuate dramatically so as to require frequent periodic ranging processes. Among the synchronization errors, frequency offset is the most important one, for it would destroy the orthogonality causing MAI. (The phase rotations resulting from the time delays can be, more or less, compensated by the channel estimator.) 
     In conventional OFDM system uplink, a common pre-defined training sequence (ranging code) is transmitted on the overall sub-carrier in one OFDM symbol. And with a repetitious structure on the time-domain, this training sequence can be taken by ML (maximum likelihood) algorithm to estimate the frequency offset. However, this kind of ranging code doesn&#39;t work in OFDMA uplink, because 1) the ranging code by no means occupies the overall band; and 2) it isn&#39;t a common pre-defined training sequence but a CDMA (code division multiplex access) code generated by PN (pseudo-noise) polynomial in order to distinct terminals. 
     An alternative to estimate the frequency offset in OFDMA uplink is to repeatedly transmit a CDMA ranging code in multiple consecutive OFDM symbols (on the partial band) with phase continuity on the time-domain. Then, the base-station receiver can still apply ML to the repetitious training sequence. This method is taken in the initial ranging. 
     IEEE802.16e OFDMA system adopts initial ranging and periodic ranging.  FIG. 1  shows the mandatory initial ranging and periodic ranging. 
       FIG. 1(   a ) is the time-domain illusion  100  of the initial-ranging transmission. The initial-ranging transmission is performed during two consecutive OFDM symbol periods  102  and  104  with copies of specific duration of last samples as CP  106 . 
       FIG. 1  ( b ) is the time-domain illusion  110  of the periodic ranging transmission. The periodic ranging transmission is performed during one OFDM symbol period  112  with a copy of specific duration of last samples as CP. 
     These repetitions of symbol period, termed CP, provide multipath immunity as well as tolerance for symbol time synchronization errors. 
     Initial ranging serves registering a new terminal into network. The time delays, frequency offsets, and transmission power of an un-registered terminal shall be estimated and adjusted to guarantee its on-going reliable transmission. A base-station grants an initial ranging opportunity by allocating ranging channels in an uplink sub-frame. The grant information is encapsulated into a UL_MAP that is broadcast in the downlink sub-frame. Given a ranging opportunity, terminals collide on these ranging channels by transmitting a CDMA code, denoted as an initial ranging code, which is randomly selected from a CDMA code candidate set specified by the base-station. This ranging code will be detected and transmitted together with the parameter adjustment message in a ranging response during the next downlink opportunity to notify the terminal to be adjusted. 
     Periodic ranging serves re-synchronizing a terminal with the base-station. A base-station grants a periodic ranging opportunity in an uplink sub-frame. The terminals collide on the ranging channel by transmitting a CDMA code that is randomly selected from a candidate set specified by the base-station. 
     A prior art solution has been proposed by Young-Ha Lee et al. This solution is applied to solve synchronization of an uplink between a subscriber station and a base-station by utilizing the ranging system in a multiple access wireless communication system of OFDMA. 
     However, this solution is only restricted to timing synchronization rather than frequency synchronization. Since OFDMA system is very sensitive to frequency synchronization errors in a mobile environment, performing the synchronization process without considering frequency offset becomes inapplicable in practice. 
     Another prior art solution has been proposed by Chang-Wahn Yu et al. This solution is applied to process the ranging channels to measure the propagation delay and the power of each subscriber station. 
     In an OFDMA system, each subscriber station has different carrier frequency offsets if the system is not synchronized. The orthogonality among these subcarriers of the different subscriber stations are thus destroyed due to MAI. Therefore, the insufficiencies of the above solution are that this solution does not take into account of frequency offset either. 
     A challenge to the periodic ranging is how to estimate the CFO based on a single-OFDM-symbol CDMA ranging code, when it comes to a mobile environment. 
     SUMMARY OF THE INVENTION 
     According to the present invention, a method for blind carrier frequency offset estimator is based on a single-OFDM-symbol training sequence in multi-user OFDMA uplink. Through multiple access interference modeling and analysis, a virtual user is employed that occupies all null sub-carriers. By minimizing the energy leakage on the virtual user in term of tentative frequency offsets, the method of the present invention can approach the real frequency offset. Besides, the method of the present invention is performed only in the frequency-domain, simplifies interference calculations, and lowers the rank of the matrix. Finally, an iterative computation method is used to approach the real frequency offset. 
     According to the present invention, a frequency offset estimation method for interleaved OFDMA uplink estimates frequency offset based on a one OFDM symbol (on the partial band) CDMA training sequence, is a blind estimation, i.e. without knowing the transmitted CDMA ranging code, and is a low complexity method. The method of the present invention has the advantage that it is performed only on the frequency-domain. The signals do not need to be transformed to the time-domain as is done with conventional CFO estimators. The method of the present invention has the additional advantage of being a low complexity and memory saving method, because the introduction of an influence factor for MAI modeling greatly reduces the rank of the correction matrix. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The aforementioned and other features and objects of the present invention and the manner of attaining them will become more apparent and the invention itself will be best understood by reference to the following description of a preferred embodiment taken in conjunction with the accompanying drawings, wherein: 
         FIG. 1(   a ) is a diagram of an initial ranging method; 
         FIG. 1(   b ) is a diagram of a periodic ranging method according to the prior art; 
         FIG. 2  is a diagram of a single-user signal model according to the present invention; 
         FIG. 3  is a multi-user signal model according to the present invention; 
         FIG. 4  is a plot of the cost function vs. the tentative CFO according to the present invention; 
         FIG. 5  is a flow chart of an iterative computational method according to the present invention; and 
         FIG. 6  is a diagram of an OFDMA system incorporating the method of the present invention. 
     
    
    
     DESCRIPTION OF THE INVENTION 
     Prior to the introduction of the CFO estimator, we shall develop a base-band signal model for the interleaving OFDMA uplink. Starting by constructing a single-user signal model, we will deduce a multi-user signal model with CFO and a time-variant frequency-selective channel. 
     An equivalent base-band single-user transmitter/receiver  200  is illustrated in  FIG. 2 .  FIG. 2  illustrates a single-user base-band signal model of an interleaving OFDMA system. The out-modem part of the transmitter, such as randomization, channel coding, etc, is simplified as a modulator  202 , and the receiver part is simplified as a de-modulator  204  respectively. The OFDM Framing/DeFraming block  206 ,  208  is used for constructing/deconstructing the standard compatible OFDMA transmission frame. Also the IFFT transformation  210  converts the frequency-domain signal into time-domain and the CP is appended  212  to each OFDM symbol after IFFT transformation  210 . On the receiver side, FFT transformation  214  and CP removal block  216  are implemented for inverse operation as a transmitter part. 
     Focusing on frequency offset estimation, we can safely simplify the outer-modem part of the transmitter as a (de-)modulator. Define complex-valued vector S M×1 =[s( 0 ), s( 1 ), s( 2 ), . . . , s(M−1)] T  as the signals from the modulator to be transmitted in one OFDM symbol. Then, the M signals are to be mapped onto one frequency-domain OFDM symbol vector (a complex vector of length of N×1) by a set of pre-defined index. The remaining (N−M) entries are set to be zeros. The mapping relationship can be realized by a position index (sc( 0 ), sc( 1 ), sc(M−1)), that is, s(i) is mapped at the sc(i)-th entries of the frequency-domain OFDM symbol vector. Define a position matrix P N×M =[ε sc(0) , ε sc(1) , . . . , ε sc(M−1) ], where ε j  is (N×1) zeros vector but with its j-th entry being 1. Thus, the frequency-domain OFDM symbol signal can be expressed as P·S. IFFT operation can be realized by left-multiplication of a (N×N) IFFT matrix W H . The (k,l) entry of W is defined as 
               N     1   2       ·       ⅇ       -   j     ⁢       2   ⁢     π   ·   k   ·   l       N         .           
The CP (cyclic prefix) operation is also represented by a left multiplication of the matrix CP (Ncp+N)×N =[[O Ncp×(N−Ncp) I Ncp×Ncp ] T I N×N ] T , where N cp  is length of CP. Finally, the transmitted signal vector X (N+Ncp)×1  is:
 
 X   (N+N     CP     )×1   =CP   (N     CP     +N )×N ·W   N×N   H   P   N×M   ·S   M×1   Equation 1
 
The signal vector X is sequentially transmitted through a time-varying frequency selective channel. This continuous channel distortion can be modeled as:
 
                           y   ⁡     (   t   )       =         ⅇ     j2π   ⁢           ⁢     F   0     ⁢   t       ·       h   ⁡     (     τ   ,   t     )       ⊗     x   ⁡     (   t   )           +     n   ⁡     (   t   )                     =         ⅇ     j2π   ⁢           ⁢     F   0     ⁢   t       ⁢       ∫   0           τ     ⁢   max       ⁢       h   ⁡     (     τ   ,   t     )       ⁢     x   ⁡     (     t   -   τ     )       ⁢           ⁢     ⅆ   τ           +     n   ⁡     (   t   )                       Equation   ⁢           ⁢   2               
where h(τ,t) is channel impulse response simplified to h(τ) during one OFDM symbol; F 0  is CFO; n(t) is AWGN noise; τ max  is the maximum excess delay. Discrete-time equivalent of Equation 2 by replacing t with i/F s , (F s  is the sampling frequency), is:
 
                     y   ⁡     (   i   )       =         ⅇ     j2π   ⁢           ⁢     f   0     ⁢   i       ⁢       ∑     l   =   0         N   max     -   1       ⁢       h   ⁡     (   l   )       ⁢     x   ⁡     (     i   -   l     )             +     n   ⁡     (   i   )                 Equation   ⁢           ⁢   3               
where N max =T max ·F s , f 0 =F 0 /F s  is the normalized CFO. If N max &lt;N CP , no ISI (inter-symbol-interference) occurs, so the received time-domain signal vector is:
 
 Y   N×1   =e   j2πf     0     ·N     CP   ·diag( g ( f   0 )) N×N   ·W   N×N   H   ·P   N×M ·diag( h ) M×M   ·S   M×1   Equation 4
 
where g(f)=[e j2π·f·0 , e j2π·f·1 , . . . , e j2π·f·(N−1) ], and h=[H(sc( 0 )), H(sc( 1 )), . . . , H(sc(M−1))] (H(i) is designated for the channel frequency response at the i-th sub-carrier). The received frequency-domain signal can be expressed by left multiplications of a FFT matrix W and position matrix P T :
 
 R   M×1   =e   j2πf     0     N     CP     P   T   ·W ·diag( g ( f   0 ))· W   H   ·P ·diag( h )· S   M×1   Equation 5
 
Equation 5 is the base-band signal transmission model of the inner-modem part. If f 0 =0, i.e. no CFO presents, because the matrix diag(g(f 0 ))=I N×N . And with W·W H =I N×N  and P T ·P=I M×M , we can rewrite R:
 
 R   M×1   =e   j2πf     0     N     CP   ·diag( h )· S   M×1   Equation 6
 
Apparently, f 0 =0 gives rise to no interference between sub-carriers.
 
     An equivalent base-band multi-user transmitter/receiver  300  is illustrated in  FIG. 3 .  FIG. 3  illustrates a multi-user based band signal model for an interleaving OFDMA system. The out-modem part of transmitter, such as randomization, channel coding, etc, is simplified as a modulator  302 , and the receiver part is simplified as a de-modulator  304 , respectively. The OFDM Framing/DeFraming block  306 ,  308  is used for constructing/deconstructing the standard compatible OFDMA transmission frame. Also, the IFFT transformation  310  converts the frequency-domain signal into time-domain and the CP is appended  312  to each OFDM symbol after IFFT transformation  310 . On the receiver side, FFT transformation  314  and CP removal block  316  are implemented for inverse operation as a transmitter part. 
     Comparing with the single user model, the user De-Mux block  318  is implemented on the receiver side to extract each single user data from its location within the OFDMA transmission frame. 
     In order to give out a multi-user signal model, we denote the superscript (•) (k)  as the assignment to the k-th user. Since one OFDM symbol is shared by several users without collision (overlapping), we have: 
                         (     P     (   k   )       )     T     ·     (     P     (   l   )       )       =     {           I       M     (   k   )       ×     M     (   l   )                 k   =   l               O       M     (   k   )       ×     M     (   l   )                 k   ≠   l                     Equation   ⁢           ⁢   7               
And
 
                 ∑     k   =   0         N   user     -   1       ⁢     M     (   k   )         ≤     N   -     N   left_guard     -       N     right_guard   .       .             
Thus, the received time-domain signal can be a sum of those of individual users:
 
                           Y     N   ×   1       =       ⁢       ∑     k   =   0         N   user     -   1       ⁢     Y     N   ×   1       (   k   )                     =       ⁢       ∑     k   =   0         N   user     -   1       ⁢       ⅇ     j2π   ⁢           ⁢     f   0     (   k   )       ⁢     N   CP         ·     diag   ⁡     (     g   ⁡     (     f   0     (   k   )       )       )       ·                       ⁢       W   H     ·     P     (   k   )       ·     diag   ⁡     (     h     (   k   )       )       ·     S     (   k   )                       Equation   ⁢           ⁢   8               
Similarly, the received frequency-domain of the k-th user is:
 
                     R       M     (   k   )       ×   1       (   k   )       =             ⅇ     j2π   ⁢           ⁢     f   0     (   k   )       ⁢     N   CP         ⁡     (     P     (   k   )       )       T     ·   W   ·     diag   ⁡     (     g   ⁡     (     f   0     (   k   )       )       )       ·     W   H     ·     P     (   k   )       ·     diag   ⁡     (     h     (   k   )       )       ·     S       M     (   k   )       ×   1       (   k   )         +       ∑         l   =   0     ,       l   ≠   k           N   user     -   1       ⁢           ⅇ     j2π   ⁢           ⁢     f   0     (   l   )       ⁢     N   CP         ⁡     (     P     (   k   )       )       T     ·   W   ·     diag   ⁡     (     g   ⁡     (     f   0     (   l   )       )       )       ·     W   H     ·     P     (   l   )       ·     diag   ⁡     (     h     (   l   )       )       ·     S       M     (   l   )       ×   1       (   l   )                     Equation   ⁢           ⁢   9               
The first term on the right-hand side of Equation 9 is the received signal from the k-th user. If f 0   (k) ≠0, the sub-carrier interference presents, denoted as self-interference. The second term on the right-hand side of Equation 9 includes the signals from the other users. If f 0   (l) =0, ∀l, l≠k, the sum of the second term is zero because of Equation 7; otherwise, it introduces the interference from other users, denoted as MAI.
 
     In order to design a blind CFO estimator, we shall introduce a concept of a “virtual user”. As its name suggests, UL PUSC of IEEE802.16e system, one of the mandatory transmission structure, uses some of the sub-channels, that is, some sub-channels are deliberately set to zeros. These null sub-channels are uniformly distributed on the overall band in a given permutation way to separate different users. In a practical system, about 60%˜75% sub-channels are used, while the remainders are null sub-channels against the MAI. Thus, despite the presence of the interferences due to the existing CFOs, their influence from one user on another user would greatly diminish along with the increasing of the sub-carrier distances between the two users. 
     Another issue is sectorization. Like other cellular systems, IEEE802.16e system sectorizes its cell. All of the available sub-channels, excluding the null sub-channels, are grouped into three segments. Each segment is assigned to a sector&#39;s usage. Concurrently, three sets of directional transmitter/receiver antenna arrays are installed at the BS for the sectors. Therefore, in a given sector, the signal energies (or interferences) from the neighboring sectors can be low enough to be considered as white noise. Accordingly, the sub-channels of the other segments can be regarded as null sub-channels too. 
     Taking into account the two points above, there are a number of null sub-channels in OFDMA uplink. These null sub-channels can be regarded as a special user that transmits only zero signals without CFO. We denote this null sub-channel set as “virtual user.” Logically, it contributes no interferences on the other users; whereas the other users present interferences on it. Equation 10 expresses this relationship: 
                     R       M     (   null   )       ×   1       (   null   )       =       N       M     (   null   )       ×   1       +       ∑       l   =   0     ,         N   user     -   1       ⁢         (     P     (   null   )       )     T     ·   W   ·     diag   ⁡     (     g   ⁡     (     f   0     (   l   )       )       )       ·     W   H     ·     P     (   l   )       ·     diag   ⁡     (     h     (   l   )       )       ·     S       M     (   l   )       ×   1       (   l   )                     Equation   ⁢           ⁢   10               
where the superscript (•) (null)  is designated for the assignment to the virtual user. The mapping relationship of the virtual user can be realized by a position index (null( 0 ), null( 1 ), . . . , null(M (null) −1)). And a null position matrix P N×M =[ε null(0) , ε null(1) , . . . , ε null(M−1) ].
 
     In an ideal synchronous system, i.e., ω 0   (l) =0, ∀l, the virtual user only transmits the white noise; otherwise, the MAI from other users leaks on the virtual user&#39;s band. We name this MAI as “signal energy leakage.” 
     Due to the fact that CFO gives rise to the signal energy leakage that augments the signal energy on the virtual user&#39;s band, we design a CFO estimator that minimizes the energy. 
     Ranging can be regarded as a specific user. Denote the superscript (•) (ranging)  as the assignment to the ranging user. Define a joint position index of the virtual user and the ranging user as ((null+ranging)( 0 ), (null+ranging)( 1 ), . . . , (null+ranging)(M (null) +M (ranging) −1))=(null( 0 ), null( 1 ), . . . , null(M (null) −1))∪(sc (ranging) ( 0 ), sc (ranging) ( 1 ), . . . , sc (ranging) (M (ranging) −1)). The joint position matrix of the virtual user and the ranging user is:
 
 P   N×(M     (null)     +M     (ranging)     )   (null+ranging) =└ε (null+ranging)(0) ,ε (null+ranging)(1) , . . . ,ε (null+ranging)(M     (null)     +M     (ranging)     −1) ┘  Equation 11
 
The observed frequency-domain joint signal of the virtual user and the ranging user is:
 
                     R       (       M     (   nll   )       +     M     (   ranging   )         )     ×   1       (     null   +   ranging     )       =             ⅇ     j2π   ⁢           ⁢     f   0     (   ranging   )       ⁢     N   CP         ⁡     (     P     (     null   +   ranging     )       )       T     ·   W   ·     diag   ⁡     (     g   ⁡     (     f   0     (   ranging   )       )       )       ·     W   H     ·     P     (   ranging   )       ·     diag   ⁡     (     h     (   ranging   )       )       ·     S       M     (   ranging   )       ×   1       (   ranging   )         +       ∑         l   =   0     ,       l   ≠   ranging           N   user     -   1       ⁢           ⅇ     j2π   ⁢           ⁢     f   0     (   l   )       ⁢     N   CP         ⁡     (     P     (     null   +   k     )       )       T     ·   W   ·     diag   ⁡     (     g   ⁡     (     f   0     (   l   )       )       )       ·     W   H     ·     P     (   I   )       ·     diag   ⁡     (     h     (   l   )       )       ·     S       M     (   l   )       ×   1       (   l   )                     Equation   ⁢           ⁢   12               
Assuming that the other users have already been synchronized with the BS, that is, f 0   (l) =0, ∀l, l≠k, the second term of the right hand side of Equation 12 turns to zeros:
 
 R   (M     (null)     +M     (ranging)     )×1   (null+ranging)   =e   j2πf     0       (ranging)     N     CP   ( P   (null+ranging) ) T   ·W ·diag( g ( f   0   (ranging) ))· W   H   ·P   (ranging) ·diag( h   (ranging) )· S   M     (ranging)     ×1   (ranging)   Equation 13
 
Multiple R (null+ranging)  by correction matrixes in term of a tentative CFO f (ranging)  and extract the signal of the virtual user:
 
 R′   M     (nll)     ×1   (null) =( P   (null) ) T   ·W ·diag( g ( f   (ranging) ))· W   H   ·P   (ranging+null)   ·R   (nll+ranging)   Equation 14
 
Replace Equation 13 into Equation 14:
 
 R′   M     (nll)     ×1   (null)   =e   j2πf     0       (ranging)     N     CP   ·( P   (null) ) T   ·W ·diag( g ( f   (ranging)   +f   0   (ranging) ))· W   H   ·P   (ranging) ·diag( h   (ranging) )· S   M     (ranging)     ×1   (ranging)   Equation 15
 
Equation 15 indicates that R′ (null)  is the signal leakage from the ranging user that is corrected by a tentative CFO f (ranging) . Define a correct matrix C (ranging) (f)=(P (null) ) T ·W·diag(g(f (ranging) ))·W H ·P (null+ranging) . We re-write (14):
 
 R′   M     (nll)     ×1   (null)   =C   (ranging) ( f )· R   (nll+ranging)   Equation 16
 
We can introduce a cost function in terms of the signal energy of R′ (null) :
 
 J   (ranging) ( f )=( R′   M     (nll)     ×1   (null) ) H   ·R′   M     (nll)     ×1   (null) =( R   (null+ranging) ) H ·( C   (ranging) ( f )) H   ·C   (ranging) ( f )· R   (null+ranging)   Equation 17
 
To explain why the estimated CFO minimizes the cost function J (ranging) (f), we note that in Equation 15 f (ranging) +f 0   (ranging) =0 leads to diag(g(f (ranging) +f 0   (ranging) ))=I N×N , i.e. R′ (null) =0. Thus, relying on the cost function, CFO estimator is given by:
 
                   f   ⋒     0     (   ranging   )       =         arg   ⁢           ⁢   min     f     ⁢       J     (   ranging   )       ⁡     (   f   )           ⁢                   FIG. 4  shows the cost function in terms of tentative CFO.
 
     As noted above, the CFO estimator is based on signal energy detection on the virtual user. However, the generation of the correction matrix C (ranging) (f) is high complex operation to be performed once f (ranging)  is updated. To address the simplification of the CFO estimator, we start by analyzing MAI property in OFDMA uplink. 
     From Equation 9, the interference from the l-th user on the k-th user due to the CFO of the l-th user can be modeled as:
 
 MAI ( k,l,f   o   (l) )= e   j2πf     0       (l)     N     CP   ( P   (k) ) T   ·W ·diag( g ( f   0   (l) ))· W   H   ·P   (l)   ·R   M     (l)     ×1   (l)   Equation 18
 
Define a MAI function m(k,l,f)=(P (k) ) T ·W·diag(g(f))·W H ·P (l) , so the interference is re-written as:
 
 MAI ( k,l,f   0   (l) )= e   j2πf     0       (l)     N     CP     ·m ( k,l,f   0   (l) )· R   M     (l)     ×1   (l)   Equation 19
 
To investigate the MAI property, we discard the term e j2πf     0       (l)     N     CP   , for it only causes the phase rotation. Without loss of generality, we can investigate MAI property by studying m(k,l,f). The function m(k,l,f) returns a M (k) ×M (l)  matrix, the (u,v) entry of which is
 
               I   ⁡     (     u   ,   v     )       =       (     1   /   N     )     ·       ∑     n   =   0       N   -   1       ⁢       ⅇ       j2π   ⁡     (     f   +       sc     (   l   )       ⁡     (   u   )       -       sc     (   k   )       ⁡     (   v   )         )       ⁢     n   /   N         .               
It equals:
 
                     I   ⁡     (     u   ,   v     )       =         sin   ⁢           ⁢     π   ⁡     (     f   +       sc     (   l   )       ⁡     (   u   )       -       sc     (   k   )       ⁡     (   v   )         )           N   ⁢           ⁢   sin   ⁢     π   N     ⁢     (     f   +       sc     (   l   )       ⁡     (   u   )       -       sc     (   k   )       ⁡     (   v   )         )         ·     ⅇ       -     jπ   ⁡     (     1   -     1   N       )         ⁢     (     f   +       sc     (   l   )       ⁡     (   u   )       -       sc     (   k   )       ⁡     (   v   )         )                   Equation   ⁢           ⁢   20               
Noting that for a large N:
 
                       lim     N   →   ∞       ⁢       sin   ⁢           ⁢     π   ⁡     (     f   +       sc     (   l   )       ⁢     (   u   )       -       sc     (   k   )       ⁡     (   v   )         )           N   ⁢           ⁢   sin   ⁢     π   N     ⁢     (     f   +       sc     (   l   )       ⁡     (   u   )       -       sc     (   k   )       ⁡     (   v   )         )           =     sin   ⁢           ⁢     c   ⁡     (     f   +       sc     (   l   )       ⁡     (   u   )       -       sc     (   k   )       ⁡     (   v   )         )                 Equation   ⁢           ⁢   21               
the normalized power of l(u,v) decreases dynamically with the increase of the distance |sc (l) (u)−sc (k) (v)| (it is an integer) for different f (f≠0). It can be concluded that the interference from the u-th sub-carrier of the l-th user has the influence only on its limited neighboring sub-carriers. In the case of f=0, sinc(sc (l) (u)−sc (k) (v))≡0 if and only if |sc (l) (u)−sc (k) (v)|≠0.
 
We introduce an interference influence factor d to express the “effective” interference limitation:
 
                     I   ⁡     (     u   ,   v     )       =     {           sin   ⁢           ⁢       c   ⁡     (     f   +       sc     (   l   )       ⁡     (   u   )       -       sc     (   k   )       ⁡     (   v   )         )       ·     ⅇ       -     jπ   ⁡     (     1   -     1   N       )         ⁢     (     f   +       sc     (   l   )       ⁡     (   u   )       -       sc     (   k   )       ⁡     (   v   )         )                              sc     (   l   )       ⁡     (   u   )       -       sc     (   k   )       ⁡     (   v   )              ≤   d             0       else                   Equation   ⁢           ⁢   22               
With Equation 22, we can lower the rank of the correction matrix C (ranging) (f) by discarding those null sub-carriers from which the distances to the nearest ranging sub-carriers are greater than a pre-defined distance d. Re-define null sub-carrier position index:
 
{ sc   (null′) ( l )εnull∥ sc   (null) ( l )− sc   (ranging) ( m )|≦ d,m= 0,1 , . . . M   (ranging) }
 
And the (u,v) entry of C (ranging) (f) is l(u,v) in Equation 22 where sc (l) (u)=sc (null′) (u), sc (k) (v)=sc (ranging) , and f=f (ranging) .
 
The cost function of J (ranging) (f) is illustrated in  FIG. 5 . Knowing that f 0   (ranging) ε(f 1 ,f 2 ), we can use iterative computation to reach the estimated f (ranging) :
         1. f (ranging) =f 1 , J 0 =+∞; (step  502 ))   2. Calculate C (ranging) (f (ranging) ) with Equation 22; (step  504 )   3. Calculate R′ M     (nll′)     ×1   (null′)  based on Equation 16; (step  506 )   4. From R′ M     (nll′)     ×1   (null′) , calculate J (ranging) (f (ranging) ) with Equation 17, and J 1 =J (ranging) (f (ranging) ); (step  518 )   5. if J 1 &gt;J 0 , go to end and return f (ranging) ; (decision  508 )   6. f (ranging) =f (ranging) +Δf step , if f (ranging) &gt;f 2 , go to end and estimate failure, otherwise J 0 =J 1  and go to step 2; (steps  510 ,  512 , and  516 )   7. End.       

     The flow chart of the iterative method of the present invention is shown in  FIG. 5 . The flow chart of  FIG. 5  also includes step  518 . Step  518  denotes J as J1 for the next step comparison and iteration. The flow chart also includes decision diamond  514  that ends the method of the present invention if the upper end of the carrier frequency offset is reached. The iterative method of the present invention can be enhanced if desired. For instance, Δf step  can be adjusted according the ΔJ=J 1 −J 0 ; a LMS (least Mean Square) algorithm can also be proposed to improve better tracking performance. 
     The minimum value of the cost function is determined during the iterative method of the present invention, as is shown in  FIG. 4 . For the first time iteration, it is impossible for J1 to be greater than J0 (positive infinity), and in step  512  J1 is denoted as J0, so for the next step, the new frequency resulted cost function value is compared with J0, which has the same value of J1 (not positive infinity). By successive iterations, the method of the present invention is continued until the minimum value of cost function is found. 
     Referring now to  FIG. 6 , an entire OFDMA system  600  is shown, including all of the blocks previously described. In addition, system  600  includes a CFO estimator block  620  incorporating the method of the present invention as was previously described. In addition, system  600  includes a frequency offset corrector block  622 , which compensates for the frequency offset estimated by the CFO estimator block  620 . 
     As is known in the art, the entire system  600 , and blocks  620  and  622  in particular, can be implemented in an integrated circuit using DSP technology to realize all of the various mathematical steps and transformations in the method of the present invention. 
     While there have been described above the principles of the present invention in conjunction with specific components, circuitry and bias techniques, it is to be clearly understood that the foregoing description is made only by way of example and not as a limitation to the scope of the invention. Particularly, it is recognized that the teachings of the foregoing disclosure will suggest other modifications to those persons skilled in the relevant art. Such modifications may involve other features which are already known per se and which may be used instead of or in addition to features already described herein. Although claims have been formulated in this application to particular combinations of features, it should be understood that the scope of the disclosure herein also includes any novel feature or any novel combination of features disclosed either explicitly or implicitly or any generalization or modification thereof which would be apparent to persons skilled in the relevant art, whether or not such relates to the same invention as presently claimed in any claim and whether or not it mitigates any or all of the same technical problems as confronted by the present invention. The applicants hereby reserve the right to formulate new claims to such features and/or combinations of such features during the prosecution of the present application or of any further application derived therefrom. 
     
       
         
               
             
               
               
               
               
             
           
               
                   
               
               
                 The following Abbreviations used herein are listed in Table I: 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                   
                 AWGN 
                 Additive White Gaussian Noise 
               
               
                   
                   
                 BS 
                 Base Station 
               
               
                   
                   
                 CDMA 
                 Code Division Multiple Access 
               
               
                   
                   
                 CFO 
                 Carrier Frequency Offset 
               
               
                   
                   
                 CP 
                 Cyclic Prefix 
               
               
                   
                   
                 FFT 
                 Fast Fourier Transform 
               
               
                   
                   
                 ICI 
                 Inter-Channel-Interference 
               
               
                   
                   
                 IFFT 
                 Inverse Fast Fourier Transform 
               
               
                   
                   
                 LS 
                 Least Square 
               
               
                   
                   
                 ISI 
                 Inter Symbol Interference 
               
               
                   
                   
                 LMS 
                 Least Mean Square 
               
               
                   
                   
                 MAI 
                 Multiple Access Interference 
               
               
                   
                   
                 ML 
                 Maximum Likelihood 
               
               
                   
                   
                 OFDM 
                 Orthogonal Frequency Division Multiplexing 
               
               
                   
                   
                 OFDMA 
                 Orthogonal Frequency Division Multiple Access 
               
               
                   
                   
                 PN 
                 Pseudo Noise 
               
               
                   
                   
                 PUSC 
                 Partial Used Sub Channel 
               
               
                   
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
             
           
               
                   
               
               
                 The following Parameters are used herein in Table II: 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 M 
                 number of the available sub-carriers 
               
               
                   
                 N 
                 OFDM modulation size 
               
               
                   
                 S (k)   
                 complex data vector of user k 
               
               
                   
                 S c   (k)   
                 position index of user k 
               
               
                   
                 P 
                 position matrix 
               
               
                   
                 ε j   
                 N × 1 zero vector but with its j-th entry being 1 
               
               
                   
                 CP 
                 cyclic prefix matrix 
               
               
                   
                 N cp   
                 length of CP 
               
               
                   
                 W H   
                 IFFT matrix 
               
               
                   
                 W 
                 FFT matrix 
               
               
                   
                 X 
                 transmitted signal vector 
               
               
                   
                 h(τ, t) 
                 channel impulse response 
               
               
                   
                 n(t) 
                 AWGN noise 
               
               
                   
                 τ max   
                 maximum excess delay 
               
               
                   
                 F s   
                 sampling frequency 
               
               
                   
                 F 0   
                 CFO 
               
               
                   
                 f 0   
                 normalized CFO, F 0 /F s   
               
               
                   
                 Y 
                 received time-domain signal vector 
               
               
                   
                 I 
                 identity matrix 
               
               
                   
                 N left-guard   
                 left virtual guard size 
               
               
                   
                 N right-guard   
                 right virtual guard size 
               
               
                   
                 R 
                 received frequency-domain signal vector 
               
               
                   
                 R′ 
                 corrected received frequency-domain signal vector 
               
               
                   
                 C 
                 correction matrix 
               
               
                   
                 J 
                 cost function in term of the signal energy 
               
               
                   
                 m(k, I.f) 
                 MAI function