The preamble suggested in this paper has a duration of four multicarrier symbols, i.e., 4T. The n th preamble symbol on the k th subcarrier will be denoted by p_{
k
}[ n M/2] in the transmitter and the corresponding received samples by y_{
k
}[ m] in the receiver (similarly to d_{
k
}[ m] and x_{
k
}[ m] for data symbols). The preamble can now be defined as
{p}_{k}\left[n\frac{M}{2}\right]=\left\{\begin{array}{cc}\pm \phantom{\rule{2.77626pt}{0ex}}\sqrt{G}& \phantom{\rule{1em}{0ex}}\text{if}n\in \{0,4\}\text{and}\phantom{\rule{0.5em}{0ex}}\mathit{k}\mathrm{is\; even}\\ 0& \phantom{\rule{1em}{0ex}}\text{otherwise}\end{array}\right.
(7)
The power of one nonzero symbol is G. The sign of a nonzero symbol can be chosen arbitrarily to improve the peaktoaverage power ratio (PAPR) of the preamble but should be the same for symbols on the same subcarrier. On odd subcarriers, the preamble only has zeros. This is to avoid the interference on even subcarriers which can not easily be mitigated before estimation of the channel, the STO and the CFO. On even subcarriers, the preamble has exactly two nonzero symbols spaced 2T from each other. This relatively large spacing, while still reasonable, allows highprecision CFO estimation and also alleviates the OQAM interference issues, making it possible to estimate the STO via the earlylate tracking technique presented below. The tail of the preamble only consists of zeros to avoid the interference coming from subsequent data symbols.
The received preamble is processed right after the IFFT in the receiver, i.e., the subchannel processing blocks in Figure 1. According to (4), and assuming that the channel is approximately flat inside each subcarrier, the received symbols on subcarrier k (denoted by y_{
k
}[ m] instead of x_{
k
}[ m] when they correspond to the preamble) can be written as
\begin{array}{rcl}{y}_{k}\left[\phantom{\rule{0.3em}{0ex}}m\right]& =& \sum _{n=\infty}^{\infty}{p}_{k}\left[n\frac{M}{2}\right]{C}_{k}\phantom{\rule{2.77626pt}{0ex}}{\xe2}_{k,k}\left[mn\frac{M}{2}\right]+{\nu}_{k}\left[\phantom{\rule{0.3em}{0ex}}m\right]\\ =& \pm \phantom{\rule{2.77626pt}{0ex}}\sqrt{G}\phantom{\rule{2.77626pt}{0ex}}{C}_{k}\left({\xe2}_{k,k}\right[\phantom{\rule{0.3em}{0ex}}m]+{\xe2}_{k,k}[\phantom{\rule{0.3em}{0ex}}m2M\left]\phantom{\rule{0.3em}{0ex}}\right)\\ +\phantom{\rule{2.77626pt}{0ex}}{\nu}_{k}\left[\phantom{\rule{0.3em}{0ex}}m\right],\end{array}
(8)
where ν_{
k
}[ m] is the additive noise sample, and where C_{
k
} is the channel coefficient on subcarrier k. The channel is assumed to be constant on the duration of the four preamble symbols. We assume additive white Gaussian noise (AWGN) with variance {\sigma}_{n}^{2}. In case of CFO ϕ and STO δ, this becomes
\begin{array}{rcl}{y}_{k}\left[\phantom{\rule{0.3em}{0ex}}m\right]& =& \sqrt{G}\phantom{\rule{2.77626pt}{0ex}}{C}_{k}\left({\xe2}_{k,k,\varphi ,\delta}\left[\phantom{\rule{0.3em}{0ex}}m\right]+\phantom{\rule{0.3em}{0ex}}{\xe2}_{k,k,\varphi ,\delta}[\phantom{\rule{0.3em}{0ex}}m2M]\phantom{\rule{0.3em}{0ex}}\right)\\ +\phantom{\rule{2.77626pt}{0ex}}{\nu}_{k}\left[\phantom{\rule{0.3em}{0ex}}m\right]\end{array}
(9)
with
\phantom{\rule{12.0pt}{0ex}}{\xe2}_{k,{k}^{\prime},\varphi ,\delta}\left[\phantom{\rule{0.3em}{0ex}}m\right]\phantom{\rule{0.3em}{0ex}}=\left(a[\phantom{\rule{0.3em}{0ex}}m+\delta ]{e}^{\frac{j2\pi}{M}(k+\varphi )(m+\delta )}\right)\ast \left(a\left[\phantom{\rule{0.3em}{0ex}}m\right]{e}^{\frac{j2\pi}{M}{k}^{\prime}m}\right)
(10)
3.1 STO estimation
The STO estimator is based on the observation of the amplitude of the received preamble symbols y_{
k
}[ n M/2]  on all subcarriers k for the first part of the preamble n=0,1,…,4 (the second part n=5,6,7 is potentially corrupted by intersymbol interference from the data symbols that follow). Note that even though the preamble is nonzero only for n=0 and n=4, all samples contain some information for the purpose of timing estimation, and we can thus take advantage of the structure of OQAM working at T/2 to utilize the overall information here.
In order to understand the derivation of the STO estimator below, it is interesting to investigate the amplitude of the received preamble y_{
k
}[ m]  on the different subcarriers k for all sample instants m. As an example, the amplitude y_{0}[ m]  for subcarrier k=0 is illustrated in Figure 2 for an ideal channel in the absence of noise. Note the raised cosine filter shape caused by the root raised cosine prototype filter in the filter bank. The STO can be estimated by looking at the difference in amplitude between the received preamble symbols y_{
k
}[ M/2]  and y_{
k
}[ 3M/2] , similarly to the way it is done for earlylate tracking, and as it is illustrated in Figure 2. For instance, when the STO increases, the amplitude of y_{
k
}[ M/2] will decrease while the amplitude of y_{
k
}[ 3M/2] will increase. To cope with frequency selective channels and to increase the precision, y_{
k
}[ M/2]  and y_{
k
}[ 3M/2]  are combined for all even subcarriers k.
The estimation method proposed here is using four amplitude samples per subcarrier: y_{
k
}[ 0] , y_{
k
}[ M/2] , y_{
k
}[ 3M/2] , and y_{
k
}[ 2M] . It is based on the earlylate principle [18] and can be derived by using a few approximations and assumptions:

The four amplitude samples are modeled as linearly dependent on the STO, using a firstorder approximation around δ=0. In particular, the samples y_{
k
}[ 0]  and y_{
k
}[ 2M] , which have a zero slope around δ=0 (see Figure 2), are assumed to be roughly independent of the STO. This approximation is obviously valid only for small STO and makes the method less accurate at high STO. This effect can be partly compensated by using the overall reference function as defined and explained below, which provides a reasonable range to the method.

The noise variance is assumed to be constant on all subcarriers (before applying any equalization coefficient). This is usually a valid assumption.

The combination across all subcarriers is performed using maximum ratio combining (MRC), which requires knowledge of the channel coefficients amplitudes. To this end and based on the approximation described above, the samples y_{
k
}[ 0]  and y_{
k
}[ 2M]  are used as estimations of the channel amplitudes.

The channel coefficients are assumed to be constant on the duration of the preamble, which is the case for most applications.
The expression of the estimator is derived below. Based on the linear approximation described above, the amplitude sample y_{
k
}[ 0]  can be written as
\begin{array}{rcl}\left{y}_{k}\right[\phantom{\rule{0.3em}{0ex}}0\left]\phantom{\rule{0.3em}{0ex}}\right& =& \sqrt{G}\left{C}_{k}\right\left{\xe2}_{k,k,\varphi ,\delta}\right[\phantom{\rule{0.3em}{0ex}}0]+\phantom{\rule{0.3em}{0ex}}{\xe2}_{k,k,\varphi ,\delta}[\phantom{\rule{0.3em}{0ex}}2M\left]\phantom{\rule{0.3em}{0ex}}\right\\ +\phantom{\rule{2.77626pt}{0ex}}{n}_{k,0}\end{array}
(11)
\begin{array}{lc}\approx & \sqrt{G}\left{C}_{k}\right+{n}_{k,0}\end{array}
(12)
since {\xe2}_{k,k,0,0}[\phantom{\rule{0.3em}{0ex}}2M]=0 and {\xe2}_{k,k,0,0}\left[\phantom{\rule{0.3em}{0ex}}0\right]=1 due to the normalization of the prototype, and where n_{k,i}=n_{
k
}[ i M/2] denotes the contribution of additive noise on the amplitude samples of interest^{a}. Similarly,
\left{y}_{k}\right[\phantom{\rule{0.3em}{0ex}}2M\left]\phantom{\rule{0.3em}{0ex}}\right\approx \sqrt{G}\left{C}_{k}\right+{n}_{k,4}.
(13)
For the middle points, performing a linear approximation around δ=0, we get
\begin{array}{rcl}\left{y}_{k}\right[\phantom{\rule{0.3em}{0ex}}M/2\left]\phantom{\rule{0.3em}{0ex}}\right& \approx & \sqrt{G}\left{C}_{k}\right\left(\left{\xe2}_{k,k,\varphi ,\delta =0}\right[\phantom{\rule{0.3em}{0ex}}M/2]\right.\\ \left(\right)close=")">+\phantom{\rule{2.77626pt}{0ex}}{\xe2}_{k,k,\varphi ,\delta =0}[\phantom{\rule{0.3em}{0ex}}3M/2]\phantom{\rule{0.3em}{0ex}}{S}_{k,\varphi}\delta \end{array}\n \n \n +\n \n \n \n n\n \n \n k\n ,\n 1\n \n \n ,\n \n \n
(14)
where S_{k,ϕ} is the slope of the amplitude with respect to the STO
{S}_{k,\varphi}={\left(\right)close="">\frac{\partial \left{\xe2}_{k,k,\varphi ,\delta}[\phantom{\rule{0.3em}{0ex}}M/2]+\phantom{\rule{0.3em}{0ex}}{\xe2}_{k,k,\varphi ,\delta}[\phantom{\rule{0.3em}{0ex}}3M/2]\right}{\mathrm{\partial \delta}}}_{}\n \n \delta \n =\n 0\n \n
(15)
Similarly,
\begin{array}{rcl}\left{y}_{k}\right[\phantom{\rule{0.3em}{0ex}}3M/2\left]\phantom{\rule{0.3em}{0ex}}\right& \approx & \sqrt{G}\left{C}_{k}\right\left(\left{\xe2}_{k,k,\varphi ,\delta =0}\right[\phantom{\rule{0.3em}{0ex}}3M/2]\right.\\ \left(\right)close=")">+\phantom{\rule{2.77626pt}{0ex}}{\xe2}_{k,k,\varphi ,\delta =0}[\phantom{\rule{0.3em}{0ex}}M/2]\phantom{\rule{0.3em}{0ex}}+{S}_{k,\varphi}\delta \end{array}\n \n \n +\n \n \n \n n\n \n \n k\n ,\n 3\n \n \n .\n \n \n
(16)
Due to the symmetry of the prototype, it is easy to show that the slopes at M/2 and 3M/2 are exactly opposite to each other and that the linearization points at M/2 and 3M/2 have the same amplitude: \left{\xe2}_{k,k,\varphi ,\delta}\right[\phantom{\rule{0.3em}{0ex}}M/2]+\phantom{\rule{2.77626pt}{0ex}}{\xe2}_{k,k,\varphi ,\delta}[\phantom{\rule{0.3em}{0ex}}3M/2\left]\phantom{\rule{0.3em}{0ex}}\right=\left{\xe2}_{k,k,\varphi ,\delta =0}\right[\phantom{\rule{0.3em}{0ex}}3M/2]+{\xe2}_{k,k,\varphi ,\delta =0}[\phantom{\rule{0.3em}{0ex}}M/2\left]\phantom{\rule{0.3em}{0ex}}\right. Hence, based on the linearization and on the earlylate principle, a first quantity proportional to the STO can easily be obtained from the samples at subcarrier k:
{\widehat{\delta}}_{k}=\left{y}_{k}\right[\phantom{\rule{0.3em}{0ex}}3M/2\left]\phantom{\rule{0.3em}{0ex}}\right\left{y}_{k}\right[\phantom{\rule{0.3em}{0ex}}M/2\left]\phantom{\rule{0.3em}{0ex}}\right
(17)
=2\delta {S}_{k,\varphi}\left{\u0108}_{k}\right\sqrt{G}+({n}_{k,3}{n}_{k,1}).
(18)
Now, one such quantity can be obtained for each subcarrier k. All theses quantities can then be combined using MRC to form an estimate of the STO. It can be shown that for an ideal channel and for the prototype filter used here, the slopes S_{k,ϕ} are identical for all subcarriers k. Based on this, assuming identical noise variances on all subcarriers and optimizing the weights to minimize the estimation variance under the constraint of an unbiased estimator, it can be shown that the MRC weights corresponding to the different subcarriers must be proportional to \left{\u0108}_{k}\right. Hence, the overall MRC estimate can be written as
\widehat{\delta}=\frac{1}{{A}_{\text{norm}}}\sum _{k=0}^{M1}\left{\u0108}_{k}\right\left(\right{y}_{k}[\phantom{\rule{0.3em}{0ex}}3M/2]\phantom{\rule{0.3em}{0ex}}{y}_{k}[\phantom{\rule{0.3em}{0ex}}M/2]\phantom{\rule{0.3em}{0ex}}\left\right)
(19)
with some normalization coefficient A_{norm}. In practice, the channel amplitudes are not yet available, so the values y_{
k
}[ 0]  and y_{
k
}[ 2M]  are used as estimates of the channel amplitude inside each subcarrier. The estimation is then normalized in order to be independent of the channel coefficient. Finally, only even subcarriers are taken into account as no symbols are sent on odd subcarriers in the chosen preamble. In the end, the estimation is based on the following quantity:
\widehat{z}(\delta ,\varphi )={\u0177}_{\uparrow}{\u0177}_{\downarrow}
(20)
with
{\u0177}_{\downarrow}=\frac{\sum _{{k}^{\prime}=0}^{M/21}\left{y}_{2{k}^{\prime}}\right[\phantom{\rule{0.3em}{0ex}}M/2\left]\phantom{\rule{0.3em}{0ex}}\right\left{y}_{2{k}^{\prime}}\right[\phantom{\rule{0.3em}{0ex}}0\left]\phantom{\rule{0.3em}{0ex}}\right}{\sum _{{k}^{\prime}=0}^{M/21}\left{y}_{2{k}^{\prime}}\right[\phantom{\rule{0.3em}{0ex}}0]\phantom{\rule{0.3em}{0ex}}{}^{2}}
(21)
and
{\u0177}_{\uparrow}=\frac{\sum _{{k}^{\prime}=0}^{M/21}\left{y}_{2{k}^{\prime}}\right[\phantom{\rule{0.3em}{0ex}}3M/2\left]\phantom{\rule{0.3em}{0ex}}\right\left{y}_{2{k}^{\prime}}\right[\phantom{\rule{0.3em}{0ex}}2M\left]\phantom{\rule{0.3em}{0ex}}\right}{\sum _{{k}^{\prime}=0}^{M/21}\left{y}_{2{k}^{\prime}}\right[\phantom{\rule{0.3em}{0ex}}2M]\phantom{\rule{0.3em}{0ex}}{}^{2}}.
(22)
Note that this quantity is a function of both the STO δ and the CFO ϕ as emphasized in the notation. It is represented in Figure 3 as a function of the STO when there is no CFO (ϕ=0), for a prototype filter with overlapping factor K=4 and for an ideal channel in the absence of noise. It appears clearly that it is approximately linear on a significant range of STO values and can therefore be used efficiently to perform the STO estimation. In theory, the function can even be used if it is not linear, as long as it is a known onetoone relationship with the true STO. In this paper, both methods are considered. We start with the more general one, assuming a known onetoone relationship between the STO and the value of the quantity (20). In order to analyze this relationship, we define the socalled reference function. This reference function will be denoted by z(δ,ϕ) and is defined as the value of \widehat{z}(\delta ,\varphi ) for an ideal channel and in the absence of noise (the effect of noise will be investigated in more detail in Section 3.1.3). In other words, \widehat{z}(\delta ,\varphi ) represents the actual measured value computed with (20) to (22), while z(δ,ϕ) represents the theoretical value that would be obtained on an ideal channel and in the absence of noise. If a reasonable estimate \widehat{\varphi} of the CFO has been obtained (for instance using the technique explained in the next subsection), the STO can be estimated as
\widehat{\delta}=arg\underset{\Delta}{min}\leftz\right(\Delta ,\widehat{\varphi})\widehat{z}(\delta ,\varphi \left)\right)
(23)
In the second part, we consider a linear approximation of the reference function which provides a simpler but less precise estimation.
3.1.1 General version
Let us first analyze the reference function z(δ,ϕ). As previously stated, this reference function is defined as \widehat{z}(\delta ,\varphi ) on an ideal channel and in the absence of noise. Figure 3 illustrates this reference function z(δ,ϕ) for three values of the CFO. It is unbiased and exhibits a very good linearity except for large STO (close to ±M/2). The slope of the curve however depends on the CFO. This is further illustrated in Figure 4 which represents z(1,ϕ) as a function of the CFO ϕ. A larger slope is of course preferable as it makes the estimate less sensitive to additional noise. So, the estimation method performs better when the CFO is small although the difference is not very large, as can be seen in Figure 3.
The principle of the estimation, as described in (23) is to compute a reference function in advance and identify which value of the STO corresponds to the observed value of the quantity (20). Note that z(δ,ϕ) does not have to be recalculated for each estimation. It can be precalculated and stored in memory. Therefore, in a practical implementation, the minimization of (23) does not require a long search over a large set of values; it simply corresponds to a lookup table. The estimation method is thus of low complexity; it amounts to the computation of one closedform expression (20) followed by a lookup table. Regarding the memory needed, the STO is discrete, but the CFO is not. The reference function should be precalculated for a number of CFOs and interpolated for the others. The larger that number, the more precise the reference function (and hence the STO estimation) will be and the larger the memory usage as well. As can be seen in Figure 4, z(δ,ϕ) = z(δ,−ϕ) which can help reduce the memory usage.
3.1.2 Linear approximation
In order to reduce the memory usage even more, the reference function can be approximated linearly:
\stackrel{~}{z}(\delta ,\varphi )=z(0,\varphi )+z(1,\varphi )\delta
(24)
It is clear from Figure 3 that this approximation is quite accurate for moderate values of the STO. For large STOs, the approximation error becomes more significant however. Using this approximation, the complexity of the STO estimation reduces even further:
\widehat{\delta}=\frac{{\u0177}_{\uparrow}{\u0177}_{\downarrow}z\left(0,\widehat{\varphi}\right)}{z\left(1,\widehat{\varphi}\right)}
(25)
3.1.3 Effect of the noise
When AWGN is added to the channel, all the amplitude samples y_{
k
}[ i M/2]  are corrupted by noise. Now, since the noise on the initial y_{
k
}[ i M/2] samples is Gaussian, the probability density function of the amplitude samples y_{
k
}[ i M/2]  is a Rice distribution. In particular, it also means that the average effect of the noise is not zero. On average, the respective contributions of the noise on y_{
↓
} and y_{
↑
} do not cancel each other, and the estimate \widehat{z}(\delta ,\varphi ) deviates from the reference function z(δ,ϕ). The overall effect is illustrated in Figure 5 which represents the average value of the estimate \widehat{z}(\delta ,\varphi ) in the presence of noise as a function of the STO δ and when the CFO ϕ=0. Two SNR cases (15 and 25 dB, respectively) are presented, and the result is compared to the reference function z(δ,ϕ) in the absence of noise. Once again, the effect is negligible for small STOs and more significant at high STOs. This generates an estimation error that gets larger for higher STOs. However, it is interesting to observe that the average effect of the noise at high (positive) STO is to decrease the estimate \widehat{z}(\delta ,\varphi ), which is the opposite of the nonlinear behavior of the reference function z(δ,ϕ) that tends to deviate above the linear slope. The overall result is that the average estimate \widehat{z}(\delta ,\varphi ) exhibits an even better linear behavior than the reference function z(δ,ϕ) as can be seen on Figure 5. In order to explain this, a complete analytical derivation of the noise distribution for \widehat{z}(\delta ,\varphi ) would be long and tedious, so we restrict ourselves to a qualitative justification which is provided in the Appendix.
3.1.4 Effect of the multipath channel
The frequency selectivity of the channel also has an influence on \widehat{z}(\delta ,\varphi ), not only for large STO but for the entire range. For instance, the bias \widehat{z}(0,\varphi ) might not be zero anymore depending on the channel impulse response taps. The longer the channel, the larger the divergence with the reference function z(δ,ϕ) can be.
To improve the estimation, it is possible to use some basic information about the channel. The idea is to assume some statistical channel model and try to take its effect into account in the reference function. A new reference function z_{mult}(δ,ϕ) is used in that case, that is simply replacing z(δ,ϕ) which was calculated for an ideal channel. This new reference function z_{mult}(δ,ϕ) is defined as the expectation of \widehat{z}(\delta ,\varphi ) in (20) in the absence of noise and averaged over the possible realizations of the channels, according to the chosen model. In practice, it is difficult to obtain the true expectation; so, the practical computation of z_{mult}(δ,ϕ) comes down to computing it for a certain number of realizations and compute the average.
Just as previously, this new reference function is computed in advance without the knowledge of the true channel realization, but some channel model needs to be available. Obviously, the accuracy of the model has a direct impact on the performance of this method. Several results are presented below in the simulation section.
3.1.5 Complexity
Even though a detailed complexity analysis would depend on the chosen implementation, and hence is outside the scope of this paper, a few comments can be made on the issue of complexity. As mentioned above, the proposed method relies on a closedform expression, and does not require a min or max search over a potentially large number of candidates, which helps reduce the complexity significantly. The method also assumes that the frequencydomain samples of the preamble are available, so the method is for instance very well suited to an architecture where the analysis filter bank is implemented separately and applies to all received symbols, including the preamble.
3.2 CFO estimation
The CFO estimation used here is a direct application of the one presented in [19] for OFDM. Similar CFO estimation methods have also been used for FBMC/OQAM systems in [12, 13, 20] although for different preamble schemes. The CFO ϕ is estimated by looking at the phase difference between the received preamble symbols y_{
k
}[ 0] and y_{
k
}[ 2M] on each even subcarrier k. The estimated CFO will be denoted by \widehat{\varphi}:
\widehat{\varphi}=\frac{1}{4\pi}\angle \left(\sum _{{k}^{\prime}=0}^{M/21}{y}_{2{k}^{\prime}}^{\ast}\left[\phantom{\rule{0.3em}{0ex}}0\right]{y}_{2{k}^{\prime}}\left[\phantom{\rule{0.3em}{0ex}}2M\right]\right)
(26)
With the preamble considered in this paper, the distance of 2T between y_{
k
}[ 0] and y_{
k
}[ 2M] is quite large. This improves the precision of the estimation but also limits the range of CFOs that can be estimated correctly. More precisely, this only allows correct estimation of CFOs in the range of ϕ∈ [ −0.25,0.25]. A CFO of ϕ=0.30 would be estimated as \widehat{\varphi}=0.20. Because of the noise, the practical range of this estimator is of course much smaller than [ −0.25,0.25] and depends on the SNR of the channel.
To cope with this problem, a heuristic adjustment has been used. It is taking into account the sign of the phase difference between y_{
k
}[ M/2] and y_{
k
}[ 3M/2]. This phase difference will be denoted by {\widehat{\varphi}}_{s}. When the CFO is large and there is a risk of ambiguity, {\widehat{\varphi}}_{s} is taken into account. When the CFO is small on the other hand, {\widehat{\varphi}}_{s} is neglected since it is more susceptible to noise in this case than \widehat{\varphi}. Hence, the estimated CFO \stackrel{~}{\varphi} is
\stackrel{~}{\varphi}=\left\{\begin{array}{ll}0.5+\widehat{\varphi}& \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\text{if}\left\widehat{\varphi}\right>0.15\phantom{\rule{1em}{0ex}}\mathrm{and}\phantom{\rule{1em}{0ex}}\mathit{\text{sign}}\left(\widehat{\varphi}\right)=1\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathrm{and}\phantom{\rule{1em}{0ex}}\mathit{\text{sign}}\left({\widehat{\varphi}}_{s}\right)=1\\ \widehat{\varphi}0.5& \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{1em}{0ex}}\left\widehat{\varphi}\right>0.15\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}\mathit{\text{sign}}\left(\widehat{\varphi}\right)=1\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\mathrm{and}\phantom{\rule{1em}{0ex}}\mathit{\text{sign}}\left({\widehat{\varphi}}_{s}\right)=1\\ \widehat{\varphi}& \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\text{otherwise}\end{array}\right.
(27)
with
{\widehat{\varphi}}_{s}=\angle \left(\sum _{{k}^{\prime}=0}^{M/21}{y}_{2{k}^{\prime}}^{\ast}[\phantom{\rule{0.3em}{0ex}}M/2]{y}_{2{k}^{\prime}}[\phantom{\rule{0.3em}{0ex}}3M/2]\right)
(28)
The threshold for using {\stackrel{~}{\varphi}}_{s} is set on \left\stackrel{~}{\varphi}\right=0.15. This value was chosen to assure correct CFO estimation for CFOs in the range of ϕ∈ [ −0.25,0.25] even when the SNR is low. It is the result of a tradeoff but does not come from any specific theoretical justification.
Note that in the method proposed here, the CFO is estimated before the STO. Hence, the CFO estimation is sensitive to the actual STO (as it could not be compensated yet). This is mainly due to the interference between the preamble symbols. For δ=0, there is no interference from one preamble symbol to the other on y_{
k
}[ 0] and y_{
k
}[ 2M]. However, when the STO increases, the interference increases which modifies the observed phases and degrades the CFO estimation.