This study proposes a frequency synchronization scheme that, like PATS, implements an iterative approach during estimation supported by a feedback loop from the compensation stage to the estimation stage. However, not only the estimation and compensation stages are different from the ones in PATS, but also the motivation to include the feedback is different. While PATS uses the feedback to reduce the complexity of the estimation stage, in the proposed scheme the feedback is used to increase the otherwise insufficient accuracy of the estimation stage.
The estimation stage of the proposed scheme is partially based on a technique for CFO tracking for OFDM systems. However, this technique alone cannot estimate medium or high CFO values accurately. This problem is solved by means of an iterative process supported by the feedback from the compensation stage. A second technique, based on using nonconsecutive symbols to obtain the phase rotation, is also applied to increase the accuracy of the estimation and reduce the cost of the fixedpoint hardware implementation.
The compensation scheme used in the proposal is the method in [19]. This method has lower complexity than the one in [18]. It also helps to reduce the ICI and MAI noise during estimation through the proposed iterative scheme, thus improving the accuracy of the estimation stage, as explained in the following sections.
Estimation technique
At the receiver, after applying FFTs on two timedomain blocks (y_{
m
}and y_{m + M}), the frequency domain k th subcarrier from Equation (5) becomes
\begin{array}{ll}{Y}_{m}\left(k\right)=& {X}_{i,m}\left(k\right){H}_{i,m}\left(k\right)\underset{{\Delta}_{i,m}}{\underset{\u23df}{{e}^{j2\Pi \frac{m{N}_{s}+{N}_{g}}{N}{\epsilon}_{i}}}}G(k,k,{\epsilon}_{i})\\ +{W}_{\text{ICI}}+{W}_{\text{MAI}}+W\left(k\right)\end{array}
(6)
\begin{array}{ll}{Y}_{m+M}\left(k\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}=& {X}_{i,m+M}\left(k\right){H}_{i,m+M}\left(k\right)\phantom{\rule{0.3em}{0ex}}\underset{{\Delta}_{i,m+M}}{\underset{\u23df}{{e}^{j2\Pi \phantom{\rule{0.3em}{0ex}}\frac{(m+M){N}_{s}+{N}_{g}}{N}{\epsilon}_{i}}}}G(k,k,{\epsilon}_{i})\\ +{W}_{\text{ICI}}^{\prime}+{W}_{\text{MAI}}^{\prime}+{W}^{\prime}\left(k\right)\end{array}
(7)
As it can be observed, the difference between both expressions is in the factors Δ_{i,m} and Δ_{i,m + M} that multiply the data signal, assuming that the CFO and the channel remain nearly constant over the M OFDM symbols. Hence, it is possible to use a frequencydomain CFO estimation technique as in [3], which uses the pilot subcarriers available in the data frame. The adaptation of this scheme to an OFDMA scenario is done by performing the algorithm D times, once for each user and its specific pilot subcarriers.
In [3], CFO estimates are obtained using M consecutive OFDM symbols. The postFFT correlation of Equations (6) and (7) at the p th pilot subcarrier of the i th user is
{C}_{i}\left(p\right)=\sum _{l=1}^{M1}{Y}_{m+l}^{\ast}\left(p\right){Y}_{m+l1}\left(p\right)
(8)
where p∈Γ_{
pi
} and Γ_{
pi
} are the set of pilot subcarriers assigned to user i(Γ_{
pi
}⊂Γ_{
i
}). This scheme is proposed in [3] for M=2. CFO values can then be estimated as
{\epsilon}_{i}=\frac{N}{4\Pi {N}_{s}}\left[arg\sum _{p}{C}_{i}\left(p\right)\right]
(9)
The idea of using the phase rotations of several consecutive OFDM symbols is taken one step further in the proposed approach. The phase rotation between two consecutive OFDM symbols due to the CFO is
{\Phi}_{2}=2\Pi \frac{{N}_{s}}{N}\epsilon =2\Pi \left(1+\frac{{N}_{g}}{N}\right)\epsilon
(10)
while the phase rotation between the first and third OFDM symbols in a sequence of consecutive OFDM symbols becomes
{\Phi}_{3}=2\Pi \frac{2{N}_{s}}{N}\epsilon =4\Pi \left(1+\frac{{N}_{g}}{N}\right)\epsilon
(11)
Since the rotation in (11) is twice the rotation in (10) for the same noise, more accurate CFO estimations can be performed with (11) than with (10). Even though using the first and third OFDM symbols reduces the CFO estimation range by half, this should not be a problem as long as CFOs are small.
Thus, the CFO can be estimated by comparing the phases of the pilot subcarriers of two OFDM symbols separated by M OFDM symbols. Extending the expression in [3]
\phantom{\rule{14.0pt}{0ex}}\begin{array}{ll}{C}_{i}\left(p\right)& ={Y}_{m}^{\ast}\left(p\right){Y}_{m+M}\left(p\right)\\ =\underset{{\Delta}_{i,M}}{\underset{\u23df}{{e}^{j2\Pi \phantom{\rule{0.3em}{0ex}}\frac{\phantom{\rule{0.3em}{0ex}}M\phantom{\rule{0.3em}{0ex}}{N}_{s}\phantom{\rule{0.3em}{0ex}}}{N}{\epsilon}_{i}}}}{\alpha}^{2}\left(\phantom{\rule{0.3em}{0ex}}{\epsilon}_{i}\phantom{\rule{0.3em}{0ex}}\right)\leftH\right(\phantom{\rule{0.3em}{0ex}}p\phantom{\rule{0.3em}{0ex}}){}^{2}{\beta}^{2}{\sigma}_{a}^{2}+\phantom{\rule{0.3em}{0ex}}w\end{array}
(12)
where {\sigma}_{a}^{2} is the power of the transmitted symbols, β^{2} is the boosted power factor of the pilot subcarriers, and α is the attenuation caused by the CFO as described in (4). Notice that the phase factor Δ_{i,M} allows that the CFO value can be obtained as
{\epsilon}_{i}=\frac{N}{2\mathrm{\Pi M}{N}_{s}}\left[arg\sum _{p}{C}_{i}\left(p\right)\right]
(13)
This proposed method, that has also been applied in [6], is called dataaided phase incremental technique (DAPIT), and it assumes that ε_{
i
} and the channel are nearly static over M consecutive OFDM symbols.
One important issue of the UL PUSC frame is that there are no continual pilot subcarriers, since not every OFDM symbol has pilot subcarriers. As it can be observed in Figure 1, there are two simple ways of using the pilot subcarriers in the UL PUSC mode to perform the necessary correlation prior to CFO estimation. It can either be calculated using the pilot subcarriers of two consecutive OFDM symbols, or the pilot subcarriers of one OFDM symbol and the third consecutive OFDM symbol. Therefore, Equations (12) and (13) can be applied directly to both cases, whereas (8) and (9) are not valid for the second one, since they only perform the correlation over consecutive OFDM symbols.
Another important advantage of using DAPIT is that it can also be used with the other permutation schemes of the uplink 802.16e, as UL OPUSC or UL AMC, which have different pilot and tile structures.
Reduced complexity compensation method
As it was previously mentioned, the compensation stage is performed with the DCSC method [19]. In this scheme, an interference matrix is built for each group of K subcarriers that form a tile or cluster. For example, the interference matrix for UL PUSC has a 4×4 size, e.g., K=4. By using this small matrix, the total correction of the ICI and MAI noise cannot be performed as it is proposed in [18]. Instead, ICI interference is removed in each tile by multiplying the subcarriers of the tile by the inverse of this small ICI matrix. This stage is called decorrelation. A second procedure, called successive cancellation, is used to remove the MAI of the current tile on the other tiles. The K×K matrix π_{
i
} is the interference matrix, whose entries are π_{
i
}(u k)=G(u k ε_{
i
}), where G is given in (4). This matrix is invertible for −1<ε_{
i
}<1, as it is shown in [19].
In the DCSC scheme, the clusters are first sorted in descending order according to their average power. Then, starting from the cluster c with the largest power which belongs to i th user, decorrelation is applied to every subcarrier in the cluster. This is represented as
{\widehat{\mathit{y}}}_{c}={\mathit{\pi}}_{i}^{1}{\mathit{y}}_{c}
(14)
where y_{
c
}=[Y_{
i
}(k)…Y_{
i
}(k + K)], c\in {\Gamma}_{i}^{c}, and {\Gamma}_{i}^{c} span the subcarriers from k to k + K(i.e., {\Gamma}_{i}^{c}=k,\dots ,k+K). This decorrelation corrects the ICI of the tile and, after that, the channel is cancelled on the {\widehat{\mathit{y}}}_{c} signal and the demapping is performed. Therefore, in ideal conditions, the transmitted data subcarriers of the tile are obtained. These data subcarriers are used in the next step that is called successive cancellation.
After obtaining the partially corrected subcarriers in the current cluster, the MAI from the neighboring clusters is reconstructed and canceled using the knowledge of the frequency offset value. ICI from another cluster of the same i th user is treated as MAI in this procedure. The data subcarriers and the estimation of the channel (assuming that it was obtained from previous OFDM symbols) are used to accomplish that. Therefore, the MAI can be reconstructed without noise or interference if the data decisions are correct. After removal of the MAI from the c th cluster, the u th (u\notin {\Gamma}_{i}^{c}) subcarrier value becomes [19]
\u0176\left(u\right)=Y\left(u\right){\left(\widehat{\mathit{X}}\widehat{\mathit{H}}\right)}_{c}^{T}{\mathit{g}}_{c,u}\phantom{\rule{8.5359pt}{0ex}}
(15)
where g_{c,u}=G(k u ε_{
i
}),G(k + 1,u ε_{
i
})…G(k + K u ε_{
i
})] and {\left(\widehat{\mathit{X}}\widehat{\mathit{H}}\right)}_{c}^{T} is the transmitted signal for this tile, obtained after the demapping is performed with the channel values. The successive cancellation stage needs correct data decisions on the demapper to perform the cancelation accurately.
In the proposed scheme (see below), the DCSC method uses (14) and (15) in Y_{
m
} and Y_{m + M} to cancel the ICI and MAI noise in order to improve the accuracy of the estimation stage.
Proposed integrated scheme
Among the previously proposed schemes, the combined application of the SAGE estimation [15] and the DCSC compensation [19] probably constitutes the most balanced solution when considering cost as well as performance (see 2 Figure 2a). When SAGE performs two iterations during estimation, this scheme achieves good BER results with less computational resources than other existing schemes. However, the amount of computations is still very large, leading to very costly hardware implementations, and SAGE also requires the use of midambles in mobile WiMAX, resulting in reduced system throughput.
A pilotbased scheme that requires less resources than SAGE + DCSC can be obtained by combining the DAPIT estimation technique with DCSC (see Figure 2b). However, as shown in the next section, if medium or high CFO values must be estimated in high SNR conditions, DAPIT alone is not sufficient to achieve acceptable BER results in OFDMA systems due to the high ICI and MAI noise.
An integrated CFO estimation and compensation (IEC) scheme is proposed here to solve this problem (see Figure 2c). It implements an iterative scheme through a feedback from the DCSC compensation stage that increases the accuracy of the DAPIT estimation stage, as it allows the iterative cancellation of the ICI and MAI noise [Equations (6) and (7)] during the estimation process. The estimation accuracy is further improved by considering nonconsecutive symbols when obtaining the phase shift used by DAPIT to estimate the CFO.
The proposed approach avoids the internal iterations for estimating the different CFOs that occur in SAGE. In addition, the phase changes produced by the CFOs are not corrected. In particular, the ICI and MAI are removed from (6) and (7), but the phase factors Δ_{
m
}and Δ_{m + M}produced by the CFO still remain. These factors are not corrected as part of the proposed IEC scheme because of two main reasons.
First, the amount of operations is reduced as a consequence of using the DAPIT estimator which is based on these phase shift factors [Equations (12) and (13)]. And second, each new CFO estimation is not iteratively added to the previous one, as it occurs in PATS where a similar iterative scheme is also proposed [17]. By not correcting this phase shift, the new CFO estimation substitutes the previous one, thus maintaining the range of its numeric value, instead of decreasing it as it happens in [17]. In consequence, the digital representation of the CFO estimates in a hardware implementation (obtained after a quantization process) can be maintained across iterations, thus requiring fewer bits than in other schemes to obtain the same accuracy. This favors the use of reduced size fixedpoint operators and, therefore, improves the area (cost), power consumption, and performance of the hardware implementation.
When the iterative estimation process converges and the last ICI and MAI compensation is performed, the phase shift Δ is finally corrected.
In summary, the iterative algorithm can be described in terms of the following steps.

1.
Iterative part:

(a)
Estimation of residual CFOs according to (12) and (13) for D different users →ε _{
i
}

(b)
Updating of π _{
i
}(u,k) using (4) with ε _{
i
}

(c)
Compensation of ICI and MAI according to (14) and (15) using π _{
i
}(u,k)

(d)
Return to (a) using the OFDM symbols corrected in (c) until convergence is reached

2.
Phase shift cancellation
It is important to realize that the compensation in step (c) is always applied to the original OFDM symbols, not to the symbols corrected in previous iterations.
The simplest criterion to stop the iterative process is to consider a predefined fixed number of iterations. As in other previous iterative schemes, simulations show that after two iterations improvements are negligible, so IEC always performs only two iterations.