 Research
 Open Access
An iterative pilotdataaided estimator for SFBC relayassisted OFDMbased systems
 Darlene Neves^{1}Email author,
 Carlos Ribeiro^{1, 2},
 Adão Silva^{1} and
 Atílio Gameiro^{1}
https://doi.org/10.1186/16876180201274
© Neves et al; licensee Springer. 2012
 Received: 30 July 2011
 Accepted: 4 April 2012
 Published: 4 April 2012
Abstract
In this article, we propose and assess an iterative pilotdataaided channel estimation scheme for space frequency block coding relayassisted OFDMbased systems. The relay node (RN) employs the equaliseandforward protocol, and both the base station (BS) and the RN are equipped with antenna arrays, whereas the user terminal (UT) is a singleantenna device. The channel estimation method uses the information carried by pilots and data to improve the estimate of the equivalent channels for the path BSRNUT. The mean minimum square error criterion is used in the design of the estimator for both the pilotbased and dataaided iterations. In different scenarios, with only one data iteration, the results show that the proposed scheme requires only half of the pilot density to achieve the same performance of nondataaided schemes.
1. Introduction
Multipleinput and multipleoutput (MIMO)based schemes exploit the benefits from the spatial diversity to enhance the link reliability and achieve high throughput. In some situations however, the integration of multiple antenna elements is unpractical especially in mobile terminals due to the size and power constraints. In order to overcome this shortcoming, virtual antennaarray has emerged as a solution to obtain spatial diversity in a distributed approach. The use of dedicated equipment with relaying capabilities rose as a promising technique to expanded coverage, system wide power savings and better immunity against signal fading [1]. The cooperation is enabled by a relaying protocol [2], e.g., decodeandforward (DF) when the relay has the capability to regenerate and reencode the whole frame; amplifyandforward (AF) where only amplification takes place; and what is designated as equaliseandforward (EF) [3, 4], where more sophisticated filtering operations are used.
A large number of cooperative techniques have been reported in the literature showing the potential of relayassisted scenarios. In order to exploit the full potential of cooperative communication, accurate estimates for the different links are required. Although some work has evaluated the impact of the imperfect channel estimation in cooperative schemes [5–10], new techniques have also been proposed that address the specificities of such systems. While with the DF protocol, channel estimation algorithms developed for pointtopoint links can be used without modifications, the situation is different when employing AF or techniques performing linear filtering at the relay node (RN). In the later, the overall channel from the base station (BS) to the user terminal (UT) is a composite one with an additional source of noise degrading the performance of pointtopoint techniques [11].
This has motivated research on channel estimation considering AF and different scenarios [12–19]. In [12], the overall channels are estimated at the UT through classical estimators based on a predefined amplifying matrix at the RN. The authors of [13] proposed a matrixbased algorithm for channel estimation considering an optimisation problem based on the normalised least mean square (NLMS) cost function. In the same way, the authors of [14] used a similar optimisation problem considering the recursive NLMS. The use of complex polyphase sequences to estimate the channel impulse response (CIR) of the equivalent channel was proposed in [15]. In [16], the authors presented a tensorbased channel estimation algorithm with an iterative scheme based on the structured least square to refine the initial estimation. Transceiver schemes that jointly design the relay forward matrix and the destination equaliser that minimise the MSE have been proposed in [17]. Concerning the twoway relay, the authors of [18] proposed an estimator based on new training strategy to jointly estimate the channels and frequency offset. For MIMO relay channels, the linear mean square error (MSE) estimator and optimal training sequences to minimise the MSE are derived in [19].
The estimation methods of the previously referred work were based on pilots or training sequences. However, the channels present in a cooperative scenario can also be estimated or aided using the energy of the transmitted data [20–23]. In [20], a recursive channel estimation method based on the channel coder feedback information and linear interpolation is proposed. In [21] is presented an estimator method that obtain initial estimation based on maximum likelihood and improve it via expectation and maximisation (EM). In [22], the authors proposed an iterative channel estimator based on the EM algorithm to separately estimate the channels B → R (BSRN) and R → U (RNUT), that on the initial phase uses a training sequence and after can use the regenerated data. Although not using directly the regenerated data, in [23] superposition of pilots and data was considered and based on the nonGaussian nature of the dualhop relay link, the authors proposed a firstorder autoregressive channel model and derived a Kalman filterbased estimator.
The works discussed above consider only singleantenna network elements (source, relay and destination). However, in several scenarios, namely in the downlink of cellular systems, it is both feasible and beneficial to consider the BS and the RN (if dedicated) with antenna arrays allowing space diversity [3, 4]. In these cases, there is a need for more complex equalisation at the relay since the use of AF limits the exploitation of space diversity provided by the use of multiple antennas. With this scheme, that we term EF, we obtain similarly to the AF protocol, an equivalent channel with additional sources of distortion that requires improved channel estimation schemes. Unlike the AF case, for which several proposals have been published as we pointed out previously, channel estimation schemes that consider the composite channel of EF have not been reported in the literature. This manuscript address this problem and proposes a channel estimation scheme for the space frequency block coding (SFBC) relayassisted scenario discussed in [4], where both BS and RN are equipped with an antenna array. This manuscript extends the work in [24, 25] by providing detailed derivations, considering additional scenarios and, unlike [24], using the information of the regenerated data to improve the channel estimates. The estimation method at the UT consists of two iterations; in the first one, only pilots are used to estimate the channels and the results are used to perform a first decision on the data symbol. Then, in the next iteration, these symbols are used as virtual pilots to improve the channel estimates to be used in the final symbol decision. The MMSE criterion is used in the design of the estimator for both the pilotbased and dataaided iterations. The results are compared against the pilotbased estimation scheme presented in [24] and they show that, for the same pilot density, the MSE reduces or, alternatively, fewer pilots are needed to achieve the same performance. Therefore, the system's spectral efficiency is improved with only one data iteration.
The remainder of this article is organised as follows. We present in Section 2 the system model and the mathematical description involving the cooperative transmission. In Section 3, we present the proposed estimator scheme. The results in terms of normalised MSE are presented in Section 4. Finally, the main conclusions are outlined in Section 5.
2. System model
The indices and denote time and frequency domain variables, respectively. $E\left\{\cdot \right\}$ is the statistical expectation operator, $\left(\circ \right)$, ${\left(\cdot \right)}^{T}$ and ${\left(\cdot \right)}^{*}$ are the pointwise, transpose and conjugate operations, respectively. $\mathsf{\text{diag}}\left(\cdot \right)$ stands for a diagonal matrix and $\mathsf{\text{FT}}\left(\cdot \right)$ denotes the Fourier transform operation. Variables, vectors or matrices in time domain (TD) are denoted by $\left(\stackrel{\u0303}{}\right)$. All estimates are denoted by $\left(\widehat{}\right)$.
2.1. Channel model
where is the instant when the CIR is evaluated, is total number of paths, ${\beta}_{g}$ and ${\tau}_{g}$ are the complex amplitude and delay of the path . ${\beta}_{g}$ is modelled as a zero mean complex Gaussian variable with variance ${\sigma}_{g}^{2}$ determined by the power delay profile and satisfying ${\sum}_{g=1}^{G}{\sigma}_{g}^{2}=1$. Although the channel is timevariant we assume it quasistatic, i.e. constant during one OFDM symbol interval. In the frequency domain, the channel gains, ${h}_{\left(k\right)},k=0,\dots ,K1$, are therefore also zero mean complex Gaussian variables with unit variance. It is widely known that in typical OFDM systems the subcarrier separation is significantly lower than the coherence bandwidth of the channel. Accordingly, the fading in two adjacent subcarriers can be considered flat and without loss of generality we can assume for generic channel ${h}_{\left(k\right)}={h}_{\left(k+1\right)}$. We also assume $E\left\{{\left{h}_{\left(k\right)}\right}^{2}\right\}=1$.
2.2. Relayassisted (RA)/cooperative scheme
We assume a halfduplex EF relaying protocol which requires two phases. In phase I, the encoded data $d$, with unit variance, are transmitted through the direct link to the UT and the link B → R to the RN. At the RN, linear operations that perform the Alamouti decoding and reencode the soft estimates using the same scheme are performed. It should be emphasised that when the RN is equipped with an antenna array the AF protocol is not the best strategy [4] since it would not allow getting benefits of the space diversity provided by the use of multiple antennas. In such case, and assuming Rayleigh fading, for each data symbol the equivalent channel from source to one antenna element of the relay is the sum of two complex Gaussian random variables. Therefore, a $2\times 2\times 1$ system asymptotically achieves the same diversity as a $2\times 1\times 1$. Consequently, we need to perform an equalisation to decode and combine the received signals on each antenna before Alamouti reencoding at the RN. However, in the considered protocol, no hard decision is performed at the RN, this fact being the reason to refer it as EF.
${q}_{\mathsf{\text{br}},\left(k\right)}$ representing the noise term that is transmitted by the RN.
with ${\sigma}_{\mathsf{\text{br}}}^{2}$ being the variance of the total noise at the input of the RN.
The data symbols are obtained after performing the joint processing which corresponds to combining the softdecision variables received in both phases of the protocol, i.e. via the direct and the cooperative links.
At the BS, the pilots are considered unitary in all positions, i.e. $p=1$. At the RN, the same pilot positions are filled. According to Equations (4) and (5), in order to perform optimal equalisation, we need to estimate ${h}_{\mathsf{\text{eq}}l\mathsf{\text{,}}\left(k\right)}={\alpha}_{\left(k\right)}{\mathrm{\Gamma}}_{\left(k\right)}{h}_{\mathsf{\text{ru}}l,\left(k\right)}$ at the receiver and using $p=1$ will no longer provide the required channels estimates. Therefore, at the RN the pilot positions are filled with ${p}_{\left(k\right)}={\alpha}_{\left(k\right)}{\mathrm{\Gamma}}_{\left(k\right)}$.
3. Proposed pilotdataaided estimator
The superscript indicates in which iteration $\left(i=1,2\right)$ the estimate is obtained. ${\widehat{D}}^{\left(1\right)}$ are the binary decoded data, ${\widehat{d}}^{\left(1\right)}$ represents the data symbols that are obtained after the remodulation and ${\u0125}_{\mathsf{\text{ru}}l}^{\left(i\right)}$ corresponds to the channels estimates. The channels estimates ${\u0125}_{\mathsf{\text{ru}}l}^{\left(1\right)}$ are obtained using only pilot information, whereas for ${\u0125}_{\mathsf{\text{ru}}l}^{\left(2\right)}$ the data regenerated in iteration 1 is used to improve the estimates. In the second iteration, the pilotdatabased estimates ${\u0125}_{\mathsf{\text{ru}}l}^{\left(2\right)}$ are used to perform the SFBC demapping and the output is then fed to the Joint Processing block to produce the final data estimates.
3.1. Pilotbased estimation
3.1.1. The TDMMSE estimator
The initial estimation is obtained via pilots and it is accomplished according to the pilotbased Time Domain Mean Minimum Square Error (TDMMSE) estimator [26]. This method performs in TD the optimal estimation, i.e. the LS estimation and MMSE filtering. The operation in time domain leads to a significant complexity reduction relatively to the conventional frequency domain processing because the MMSE filter corresponds to a sparse diagonal matrix, as was extensively discussed in [26].
where $\stackrel{\u0303}{w}$ is the noise with noise variance ${\sigma}_{n}^{2}$.
The two previous equations may be simultaneously implemented in order to minimise the estimator complexity, thus the final CIR estimate presents nonnull elements and zeros in the remaining [26].
where ${\xf1}_{\left(n\right)}$ corresponds to the complex white Gaussian noise.
Convolving the expression in (11) with the pilots symbols ${\stackrel{\u0303}{p}}_{\left(n\right)}$ we obtain the expression in (7). This convolution corresponds to multiply the subcarriers at frequency ${N}_{f}$ by 1. By design, these are the positions reserved to the pilots thus the data component vanishes.
3.2. TDMMSE estimator for the equivalent channel
 (1)Pilots with some fluctuation in amplitude:

■ As the amplitude of the pilots at the destination is not constant and equal to one, the result of the estimation is a spread of the replicas of the CIR.

 (2)Decreasing the amplitude of the pilots

■ The SNR of the pilots is decreased as well.

In order to quantify how the effects (1) and (2) can degrade the TDMMSE estimator performance, we have evaluated the impact of both of them, separately, in a SISO system, i.e. $1\times 1$, since the compound equivalent channels B → R → U correspond to pointtopoint links.
To evaluate the effect of the amplitude fluctuation, we considered that the pilots (originally with unit amplitude) had their amplitude disturbed by a random Gaussian variable with zero mean and variance equal to ${\sigma}_{\alpha \mathrm{\Gamma}}^{2}=\mathsf{\text{E}}\left\{{\left1{\alpha}_{\left(k\right)}{\mathrm{\Gamma}}_{\left(k\right)}\right}^{2}\right\}$, where ${\sigma}_{\alpha \mathrm{\Gamma}}^{2}$ quantifies how far ${\alpha}_{\left(k\right)}{\mathrm{\Gamma}}_{\left(k\right)}$ would be from the pilots with unitary amplitude.
According to Figure 6, channel model A does not show any difference in performance when the transmitted pilots are ${p}_{{\sigma}_{\alpha \mathrm{\Gamma}}^{2}}$. We point out that channel ITU pedestrian model B is more selective than model A and because of that it presents only 0.2 dB of penalty for low values of ${E}_{\mathsf{\text{b}}}/{N}_{0}$, i.e. $\left[02\right]$ when the transmitted pilots are ${p}_{{\sigma}_{\alpha \mathrm{\Gamma}}^{2}}$.
The results in both figures show a constant shift in the MSE value when the amplitude of the pilots is not unitary. The shift present in all results is not a real degradation. It is caused by the normalisation present in the MSE in (13). In fact, assuming an MSE without normalisation the results are all the same. Transmitting ${p}_{c}$ as pilots, i.e. pilots with constant and nonunitary amplitude, does not bring any noticeable degradation in the TDMMSE performance comparing to transmitting unitary pilots.
The major degradation occurs only when the pilots have some fluctuation in amplitude and solely for low values of ${E}_{\mathsf{\text{b}}}/{N}_{0}$ in highly selective channels.
The previous results evaluated the effect of the pilot amplitude fluctuations and reduction assuming that the estimator used is the one designed for the conventional pointtopoint links, i.e. the TDMMSE coefficients are the ones obtained with the correlation statistics of (9). Nevertheless, according to our cooperative scheme, we need to estimate the equivalent channel ${h}_{\mathsf{\text{eq}}l}={\alpha}_{\left(k\right)}{\mathrm{\Gamma}}_{\left(k\right)}{h}_{\mathsf{\text{ru}}l,\left(k\right)}$ and its correlation matrix ${\mathbf{R}}_{{\widehat{\stackrel{\u0303}{h}}}_{\mathsf{\text{eq}}l}{\widehat{\stackrel{\u0303}{h}}}_{\mathsf{\text{eq}}l}}$ to use the optimum TDMMSE design. The correlation matrix is defined by $\mathsf{\text{E}}\left\{{\widehat{\stackrel{\u0303}{\mathbf{h}}}}_{\mathsf{\text{eq}}l}{\widehat{\stackrel{\u0303}{\mathbf{h}}}}_{\mathsf{\text{eq}}l}^{H}\right\}={\mathbf{R}}_{\left\{\stackrel{\u0303}{\alpha \mathrm{\Gamma}}\stackrel{\u0303}{\alpha \mathrm{\Gamma}}\right\}}{\mathbf{R}}_{{\stackrel{\u0303}{h}}_{\mathsf{\text{ru}}l}{\stackrel{\u0303}{h}}_{\mathsf{\text{ru}}l}}+{\sigma}_{n}^{2}{\mathbf{I}}_{K/{N}_{f}}$ where the filter inputoutput crosscorrelation, termed ${\mathbf{R}}_{{\stackrel{\u0303}{h}}_{\mathsf{\text{eq}}l}{\widehat{\stackrel{\u0303}{h}}}_{\mathsf{\text{eq}}l}}$, is given by $\mathsf{\text{E}}\left\{{\stackrel{\u0303}{\mathbf{h}}}_{\mathsf{\text{eq}}l}{\widehat{\stackrel{\u0303}{\mathbf{h}}}}_{\mathsf{\text{eq}}l}^{H}\right\}={\mathbf{R}}_{\left\{\stackrel{\u0303}{\alpha \mathrm{\Gamma}}\stackrel{\u0303}{\alpha \mathrm{\Gamma}}\right\}}{\mathbf{R}}_{{\stackrel{\u0303}{h}}_{\mathsf{\text{ru}}l}{\stackrel{\u0303}{h}}_{\mathsf{\text{ru}}l}}$, where both ${\mathbf{R}}_{{\widehat{\stackrel{\u0303}{h}}}_{\mathsf{\text{eq}}l}{\widehat{\stackrel{\u0303}{h}}}_{\mathsf{\text{eq}}l}}$ and ${\mathbf{R}}_{{\stackrel{\u0303}{h}}_{\mathsf{\text{eq}}l}{\widehat{\stackrel{\u0303}{h}}}_{\mathsf{\text{eq}}l}}$ are $\left(K/{N}_{f}\right)\times \left(K/{N}_{f}\right)$ matrices.
According to the MSE results in Figures 6 and 7, transmitting the factor ${\alpha}_{\left(k\right)}{\mathrm{\Gamma}}_{\left(k\right)}$ brings, in the worst case, 0.2 dB of degradation and from the results of Figure 9 the correlation matrix of the equivalent channel has negligible values out of the diagonal elements and therefore there is no need to increase the system complexity by implementing the filter given by (14). Therefore, our cooperative scheme tolerates the use of the TDMMSE estimator without compromising its estimate. The analysis can be applied to any other channel without loss of generality. However, in terms of the overall system performance, better results are expected for less selective channels.
3.3. Databased channel estimation
In an OFDM system, the signal received at the destination is $\mathbf{y}=\left(\mathbf{s}+\mathbf{p}\right)\mathbf{h}+\mathbf{n}$, where is a vector representing the diagonal of the channel matrix and $\mathbf{n}$ represents the additive Gaussian noise. In our $M\times L\times 1$ cooperative system, during phase II follows (2) and is replaced by ${\mathbf{H}}_{\mathsf{\text{ru}}}=\left[{\mathbf{h}}_{\mathsf{\text{ru1}}}{\mathbf{h}}_{\mathsf{\text{ru2}}}\right]$, where ${\mathbf{h}}_{\mathsf{\text{ru1}}}$ and ${\mathbf{h}}_{\mathsf{\text{ru2}}}$ are the diagonals of the $K\times K$ matrices that represent the channel frequency responses (CFRs) of the channels between RN and UT.
According to Equations (2)(4), the extra sources of distortion imply that the accuracy of the initial estimates present some penalties relatively to the case of a pointtopoint link. Therefore, in order to improve their accuracies a databased LS estimation is carried out using the virtual pilots, i.e. the regenerated data symbols $\widehat{d}$.
where ${\widehat{\mathbf{H}}}_{\mathsf{\text{eq,}}\left(j\right)}^{\mathsf{\text{LS}}}={\left[\begin{array}{cc}\hfill {\widehat{\mathbf{h}}}_{\mathsf{\text{eq1,}}\left(j\right)}\hfill & \hfill {\widehat{\mathbf{h}}}_{\mathsf{\text{eq2,}}\left(j\right)}\hfill \end{array}\right]}^{T}$, and ${}_{\mathbf{y}}^{\mathsf{\text{ru,}}\left(j\right)}$ follows (2).
It is important to note that although we have two subcarriers, we obtain a single estimate for each antenna, i.e. if there was no noise, we would obtain the average of the equivalent channels in subcarriers and $j+1$.
where is the size of the data subcarriers set.
where SNR is the signaltonoise (SNR) ratio assuming that the noise power per subcarrier is ${\sigma}_{n}^{2}$ and the average received signal power (including pilots) is normalised to 1, i.e. $\mathsf{\text{SNR}}=1/{\sigma}_{n}^{2}$. Equation (19) shows that even for a moderate probability of symbol error (e.g. $0.01$) the increase is quite small. Therefore, we can anticipate that even with first data iteration being very inaccurate still there is potential for improving the channel estimates using data.
3.4. Complexity analysis of the dataaided estimation
The computational complexity of the dataaided iteration is related to the SFBCdecoding and the LS estimation. The merge of both operations requires $5J+{\mathsf{\text{log}}}_{2}\left(J\right)$ multiplications and $2J+J{\mathsf{\text{log}}}_{2}\left(J\right)$ additions per OFDM symbol whereas, according to [26], the pilotbased iteration requires $L+\left(K{\mathsf{\text{log}}}_{2}\left(K\right)\right)/2$ multiplications and $L{N}_{f}+K{\mathsf{\text{log}}}_{2}\left(K\right)$ additions per OFDM symbol, as well. By analysing only the number of multiplications we found that, despite the effective gains in terms of MSE performance or spectral efficiency, the complexity of the dataaided estimator is about twice of the pilotbased scheme.
4. Simulation results
4.1. Simulation parameters
In order to evaluate the performance of the presented channel estimation method, we considered the scenario described in Section 2.2 and in the simulation we used the ITU pedestrian channel models A and B [27] at a speed of $v=10\mathsf{\text{km/h}}$. The number of subcarriers set to 1,024 and modulation is QPSK. The transmitted OFDM symbol carried pilot and data subcarriers with a pilot separation ${N}_{f}$. We used the same pilot pattern at the BS and RN and since they were double antenna arrays we allocated different set of pilot subcarriers to perform the estimation. Hence for both the BS and RN, the pilot subcarriers were spaced by $2{N}_{f}$ for each antenna. Since BS and RN are both equipped with an antenna array the resulting MSE of the direct channels B → U (DL) and the relay channels R → U (RL) are obtained by averaging the individuals MSEs.
The simulations were performed assuming uncorrelated antenna channels, the receiver was perfectly synchronised and the insertion of a long enough cyclic prefix in the transmitter ensured that the orthogonality of the subcarriers is maintained after transmission.
Assessed scenarios
Scenario #  Links statistics 

1  ${E}_{\mathsf{\text{b}}}/{{N}_{\mathsf{\text{0}}}}^{\mathsf{\text{B}}\to \mathsf{\text{R}}}={E}_{\mathsf{\text{b}}}/{{N}_{\mathsf{\text{0}}}}^{\mathsf{\text{R}}\to \mathsf{\text{U}}}={E}_{\mathsf{\text{b}}}/{{N}_{\mathsf{\text{0}}}}^{\mathsf{\text{B}}\to \mathsf{\text{U}}}$ 
2  ${E}_{\mathsf{\text{b}}}/{{N}_{\mathsf{\text{0}}}}^{\mathsf{\text{B}}\to \mathsf{\text{R}}}={E}_{\mathsf{\text{b}}}/{{N}_{\mathsf{\text{0}}}}^{\mathsf{\text{B}}\to \mathsf{\text{U}}}+10\mathsf{\text{dB}}$ 
3  ${E}_{\mathsf{\text{b}}}/{{N}_{\mathsf{\text{0}}}}^{\mathsf{\text{B}}\to \mathsf{\text{R}}}={E}_{\mathsf{\text{b}}}/{{N}_{\mathsf{\text{0}}}}^{\mathsf{\text{R}}\to \mathsf{\text{U}}}\mathsf{\text{=}}{E}_{\mathsf{\text{b}}}/{{N}_{\mathsf{\text{0}}}}^{\mathsf{\text{B}}\to \mathsf{\text{U}}}+10\mathsf{\text{dB}}$ 
4.2. MSE channel estimation performance
Figures 12 and 13 present the results relative to Scenario # 2, for channel models A and B, respectively. In both cases, the pilotbased estimates of the RL and DL present approximately the same performance. This is due to the fact that in the case that the links between BS and RNs are highly reliable, most of the data information is successfully detected at the RN, which has a positive impact on the relays links. We can observe that the proposed pilotdata estimator for ${N}_{f}=16$ achieves approximately the same performance of the pilotbased one for ${N}_{f}=4$; therefore, requiring only 1/4 of the pilot subcarriers used by the pilotbased method.
Figures 14 and 15 present the results relative to Scenario # 3 for channel models A and B, respectively. These results show that in such scenario both links B → R and R → U have higher quality conditions over the direct one. In this case, the noise variances have a minor effect on the pilotbased estimates and due that the RL performance overreaches the DL one. The proposed scheme nevertheless can improve the RL performance. For ${N}_{f}=8$, the proposed estimator presents a performance close to the pilotdata performance considering only 1/2 of the pilots used by the pilotbased estimator, i.e. ${N}_{f}=4$. In this scenario, the MSE of the pilotdatabased estimator for a given ${N}_{f}$ is quite close to the one achieved considering the pilotbased estimator with pilot separation of ${N}_{f}/2$.
5. Conclusion
We proposed a pilotdatabased estimation algorithm for an OFDMbased cooperative scenario where spatial diversity provided by SFBC is complemented with the use of a halfduplex RN using the EF protocol. The proposed method consists of two iterations and uses the MMSE criterion to design the estimator for both the pilotbased and dataaided iterations. The dataaided estimation component is carried out using the regenerated data symbols as virtual pilots. In different scenarios, the results have shown that for the same pilot density the MSE is reduced approximately by 3 dB or alternatively requires half of pilot density to achieve the same performance therefore improving the overall system spectral efficiency with only one data iteration.
It is clear from the presented results that the proposed pilotdatabased method has significant interest for application in next generation wireless networks for which cooperation is anticipated.
Appendix
Throughout this section, we use the following definitions:

The received power is given by$\sum _{j\in \mathcal{S}}{\left{h}_{\left(j\right)}\right}^{2}\mathsf{\text{E}}\left\{{\left{s}_{\left(j\right)}\right}^{2}\right\}={\sigma}_{j}^{2}\sum _{j\in \mathcal{S}}{\left{h}_{\left(j\right)}\right}^{2},\mathsf{\text{where}}\mathcal{S}\mathsf{\text{isthesetofdatasubcarriers}}\mathsf{\text{.}}$

The power at the pilot subcarriers is$\sum _{p\in \mathcal{P}}{\left{h}_{\left(p\right)}\right}^{2}1=\sum _{p\in \mathcal{P}}{\left{h}_{\left(p\right)}\right}^{2},\mathsf{\text{where}}\mathcal{P}\mathsf{\text{isthesetofpilotsubcarriers}}\mathsf{\text{.}}$

The noise variance per subcarriers is represented by ${\sigma}_{n}^{2}$ and therefore the total power is given by $K{\sigma}_{n}^{2},\mathsf{\text{where}}K\mathsf{\text{isthenumberofsubcarries}}$.

If there is any distinction among pilot and data subcarriers the SNR is$\mathsf{\text{SNR}}=\frac{{\sigma}_{j}^{2}\mathsf{\text{E}}\left\{\sum _{k}{\left{h}_{\left(k\right)}\right}^{2}\right\}}{K{\sigma}_{n}^{2}}.$

If ${\sigma}_{j}^{2}=1,\mathsf{\text{E}}\left\{{\left{h}_{\left(k\right)}\right}^{2}\right\}=1$ and $\mathsf{\text{E}}\left\{\sum _{k}{\left{h}_{\left(k\right)}\right}^{2}\right\}=K$ the SNR is given by $\mathsf{\text{SNR=}}1/{\sigma}_{n}^{2}$.
SISO channel
where ${d}_{\left(k\right)}$ and ${\widehat{d}}_{\left(k\right)}$ are the transmitted and the regenerated data, respectively, and for QPSK ${\left{d}_{\left(k\right)}\right}^{2}=1$ and ${d}_{\left(k\right)}^{*}=1/{d}_{\left(k\right)}$.
${}_{\widehat{d}}^{\left(k\right)}$  ${\left\epsilon \right}^{2}$  Error probability  

$\left(1+j\right)/\sqrt{2}$  0  0  $1{P}_{\mathsf{\text{e}}}$ 
$\left(1+j\right)/\sqrt{2}$  $2/\sqrt{2}$  2  $~{P}_{\mathsf{\text{e}}}/2$ 
$\left(1j\right)/\sqrt{2}$  $2\left(1+j\right)/\sqrt{2}$  4  $~0$ 
$\left(1j\right)/\sqrt{2}$  $2j/\sqrt{2}$  2  $~{P}_{\mathsf{\text{e}}}/2$ 
MISO channel
Since our scenario is a cooperative $2\times 2\times 1$ and the LS data estimation is used to estimate the channels R → U which is $2\times 1$, we need provide expression for the squared norm of the error in this case as well.
${}_{\widehat{d}}^{\left(k\right)}$  ${}_{\widehat{d}}^{\left(k+1\right)}$  ${\left\epsilon \right}^{2}$  Error Probability  

$\left(1+j\right)/\sqrt{2}$  $\left(1+j\right)/\sqrt{2}$  $0$  $0$  $1{P}_{\mathsf{\text{e}}}$ 
$\left(1+j\right)/\sqrt{2}$  $\left(1+j\right)/\sqrt{2}$  $1j$  $2$  ${P}_{\mathsf{\text{e}}}/4$ 
$\left(1j\right)/\sqrt{2}$  $\left(1+j\right)/\sqrt{2}$  $2j$  $4$  $0$ 
$\left(1j\right)/\sqrt{2}$  $\left(1+j\right)/\sqrt{2}$  $1j$  $2$  ${P}_{\mathsf{\text{e}}}/4$ 
$\left(1+j\right)/\sqrt{2}$  $\left(1+j\right)/\sqrt{2}$  $1j$  $2$  ${P}_{\mathsf{\text{e}}}/4$ 
$\left(1+j\right)/\sqrt{2}$  $\left(1j\right)/\sqrt{2}$  $2j$  $4$  $0$ 
$\left(1+j\right)/\sqrt{2}$  $\left(1j\right)/\sqrt{2}$  $1j$  $2$  ${P}_{\mathsf{\text{e}}}/4$ 
$\left(1j\right)/\sqrt{2}$  $\left(1j\right)/\sqrt{2}$  $4j$  $16$  $0$ 
Declarations
Acknowledgements
The authors acknowledge the support of the European project CODIVFP7/ICT/2007/215477, the Portuguese projects CADWINPTDC/EEATEL/099241/2008 and CROWNPTDC/EEATEL/115828/2009, and the Portuguese Foundation for Science and Technology (FCT) grant for D. Neves.
Authors’ Affiliations
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