The iterative pilotdatabased estimator presented in this study focuses on phase II where the channel estimator estimates only the relay/cooperative channels. The estimation processing follows Figure 3.
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].
For one OFDM symbol with subcarriers, two consecutive pilot subcarriers are spaced by {N}_{f}. According to the Nyquist theorem, summing {N}_{f} delayed (by K/{N}_{f}) replicas of the input signal is equivalent to filter the pilot positions in the frequency domain, and therefore, the LS estimate in timedomain is madeup of {N}_{f} replicas of the CIR separated by K/{N}_{f}[26]
{\widehat{\stackrel{\u0303}{h}}}_{\left(n\right)}^{\mathsf{\text{LS}}}=\left\{\begin{array}{l}\sum _{m=0}^{{N}_{f}1}{\stackrel{\u0303}{h}}_{\left(nmK/{N}_{f}\right)}+\sum _{m=0}^{{N}_{f}1}{\stackrel{\u0303}{w}}_{\left(nmK/{N}_{f}\right)},\mathsf{\text{for}}n=0,1,\dots ,\frac{K}{{N}_{f}}1\\ 0,\mathsf{\text{remaider}}\end{array}\right.,
(7)
where \stackrel{\u0303}{w} is the noise with noise variance {\sigma}_{n}^{2}.
For one OFDM symbol, the LS estimate is a vector 1\times K where assuming the Nyquist criterion about pilot separation is fulfilled, the last KK/{N}_{f} elements are null
{\widehat{\stackrel{\u0303}{\mathbf{h}}}}_{K}^{\mathsf{\text{LS,pilots}}}=\left[\begin{array}{cc}\hfill {\widehat{\stackrel{\u0303}{\mathbf{h}}}}_{K/{N}_{f}}^{\mathsf{\text{LS}}}\hfill & \hfill {\mathbf{0}}_{K\left(K/{N}_{f}\right)}\hfill \end{array}\right].
(8)
The LS estimate given by (8) is improved by using the MMSE filter that is implemented by \left(K/{N}_{f}\right)\times \left(K/{N}_{f}\right) matrices. For a generic channel, the TD MMSE filter is expressed by {\mathbf{W}}_{\mathsf{\text{MMSE}},\stackrel{\u0303}{h}}={\mathbf{R}}_{\stackrel{\u0303}{h}\widehat{\stackrel{\u0303}{h}}}{\mathbf{R}}_{\widehat{\stackrel{\u0303}{h}}\widehat{\stackrel{\u0303}{h}}}^{1}, where {\mathbf{R}}_{\widehat{\stackrel{\u0303}{h}}\widehat{\stackrel{\u0303}{h}}} is the filter input correlation, \mathsf{\text{E}}\left\{\widehat{\stackrel{\u0303}{\mathbf{h}}}{\widehat{\stackrel{\u0303}{\mathbf{h}}}}^{H}\right\}, which is given by {\mathbf{R}}_{\stackrel{\u0303}{h}\stackrel{\u0303}{h}}+{\sigma}_{n}^{2}{\mathbf{I}}_{K/{N}_{f}} and {\mathbf{R}}_{\stackrel{\u0303}{h}\widehat{\stackrel{\u0303}{h}}} is the filter inputoutput crosscorrelation matrix \mathsf{\text{E}}\left\{\stackrel{\u0303}{\mathbf{h}}{\widehat{\stackrel{\u0303}{\mathbf{h}}}}^{H}\right\}, which is given by {\mathbf{R}}_{\stackrel{\u0303}{h}\stackrel{\u0303}{h}}=\mathsf{\text{diag}}\left(\left[{\sigma}_{1}^{2},{\sigma}_{2}^{2},\dots \mathsf{\text{,}}{\sigma}_{G}^{2},0\mathsf{\text{,}}\dots ,\mathsf{\text{0}}\right]\right). If the channel taps are separated by the sampling interval, the MMSE filter in TD corresponds to a sparse K/{N}_{f} diagonal matrix with nonnull elements whose number is equal to the number of taps occurring only in the diagonal:
{\stackrel{\u0303}{\mathbf{W}}}_{\mathsf{\text{MMSE}}}=\mathsf{\text{diag}}\left(\frac{{\sigma}_{1}^{2}}{{\sigma}_{1}^{2}+\frac{{\sigma}_{n}^{2}}{K/{N}_{f}}},\dots ,\frac{{\sigma}_{G}^{2}}{{\sigma}_{G}^{2}+\frac{{\sigma}_{n}^{2}}{K/{N}_{f}}},0,\dots ,0\right).
(9)
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].
Therefore, at subcarrier the element of the pilot vector \mathbf{p} may be expressed as a pulse train equispaced by {N}_{f} with unitary amplitude. The corresponding expression in TD is also given by a pulse train with elements in the instants \left(nmK/{N}_{f}\right) for m\in \left\{0,\dots ,{N}_{f}1\right\}, according to the following expression.
\begin{array}{ll}\hfill {p}_{\left(k\right)}& =\sum _{m=0}^{{N}_{f}1}{\delta}_{\left(km{N}_{f}\right)}\underset{\left(FT\right)}{\stackrel{{\left(FT\right)}^{1}}{\leftrightarrow}}{\stackrel{\u0303}{p}}_{\left(n\right)}=\left\{\begin{array}{c}\frac{1}{{N}_{f}},\mathsf{\text{if}}n=mK/{N}_{f}\\ 0,\mathsf{\text{remaider}}\end{array}\right.\phantom{\rule{2em}{0ex}}\\ \Rightarrow {\stackrel{\u0303}{p}}_{\left(n\right)}=\frac{1}{{N}_{f}}\sum _{m=0}^{{N}_{f}1}{\delta}_{\left(nmK/{N}_{f}\right)}.\phantom{\rule{2em}{0ex}}\end{array}
(10)
The transmitted signal is madeup of data and pilot components. Consequently, at the receiver side the component of the received signal in TD is given by
{\u1ef9}_{\left(n\right)}=\sum _{k=0}^{{N}_{c}1}{\stackrel{\u0303}{h}}_{\left(k\right)}{d}_{\left(nk\right)}+\frac{1}{{N}_{f}}\sum _{m=0}^{{N}_{f}1}{\stackrel{\u0303}{h}}_{\left(nmK/{N}_{f}\right)}+{\xf1}_{\left(n\right)},
(11)
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
According to the scenario presented in Section 2.2, we need to estimate the equivalent channel {h}_{\mathsf{\text{eq}}l,\left(k\right)}={\alpha}_{\left(k\right)}{\mathrm{\Gamma}}_{\left(k\right)}{h}_{\mathsf{\text{ru}}l,\left(k\right)} that depends on {\alpha}_{\left(k\right)} and {\mathrm{\Gamma}}_{\left(k\right)}. The UT is not aware of {\alpha}_{\left(k\right)}{\mathrm{\Gamma}}_{\left(k\right)} since it is dependent on {h}_{\mathsf{\text{br}}ml,\left(k\right)} and the UT is not aware of these channels as well. Nevertheless, the channels {h}_{\mathsf{\text{br}}ml,\left(k\right)} are estimated at the RN, and based on that, {\alpha}_{\left(k\right)}{\mathrm{\Gamma}}_{\left(k\right)} is computed and inserted in the pilot position as explained in Section 2.2. Since the new pilots {\alpha}_{\left(k\right)}{\mathrm{\Gamma}}_{\left(k\right)} are not unitary, the convolution with the received signal results in overlapped replicas of the CIR, as shown in Figure 4. Therefore, it is important to assess the impact of using {\alpha}_{\left(k\right)}{\mathrm{\Gamma}}_{\left(k\right)} as pilots on the estimator performance.
In Figure 5, we present the behaviour of {\alpha}_{\left(k\right)}{\mathrm{\Gamma}}_{\left(k\right)} in terms of amplitude per subcarrier. We considered two values of {E}_{\mathsf{\text{b}}}/{N}_{0}, 2 and 20 dB, where E_{b} corresponds to the energy per bit received at UT and {N}_{0}/2 is the bilateral power spectrum density of the noise that affects the information conveying signals in a pointtopoint link. For these results, we consider the channels according to ITU pedestrian, models A and B [27]. According to the results for {E}_{\mathsf{\text{b}}}/{N}_{0}=20\mathsf{\text{dB}} the {\alpha}_{\left(k\right)}{\mathrm{\Gamma}}_{\left(k\right)} presents amplitude values close to 1 with some negligible fluctuation. However, for {E}_{\mathsf{\text{b}}}/{N}_{0}=20\mathsf{\text{dB}} the result is slightly different to the previous one: {\alpha}_{\left(k\right)}{\mathrm{\Gamma}}_{\left(k\right)} presents an amplitude also close to 1 but the fluctuation is not negligible.
This can easily be explained according to (3), {\alpha}_{\left(k\right)} depends on the noise variance {\sigma}_{\mathsf{\text{br}}}^{2} and therefore {\alpha}_{\left(k\right)}{\mathrm{\Gamma}}_{\left(k\right)} tends to one for a high signaltonoise (SNR) value, according to the following expression
{\alpha}_{\left(k\right)}{\mathrm{\Gamma}}_{\left(k\right)}=\left(\frac{1}{\sqrt{{\mathrm{\Gamma}}_{\left(k\right)}^{2}+{\mathrm{\Gamma}}_{\left(k\right)}{\sigma}_{\mathsf{\text{br}}}^{2}}}\right){\mathrm{\Gamma}}_{\left(k\right)}\cong 1.
(12)
The results in Figure 5 lead to the conclusion that there are two causes by which factor {\alpha}_{\left(k\right)}{\mathrm{\Gamma}}_{\left(k\right)} at the pilot subcarriers may degrade the estimator performance:

(1)
Pilots with some fluctuation in amplitude:

(2)
Decreasing the amplitude of the pilots
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.
Therefore, the pilots have amplitudes {p}_{{\sigma}_{\alpha \mathrm{\Gamma}}^{2}}=1+z. The performance of a SISO system with pilots values {p}_{{\sigma}_{\alpha \mathrm{\Gamma}}^{2}} is shown in Figure 6 (green line). In these simulations, we used the ITU pedestrian models A and B [27] at a speed of v=10\mathsf{\text{km/h}}, the number of subcarriers was set to 1,024 and the modulation was QPSK. The transmitted OFDM symbol carried pilot and data subcarriers with a pilot separation {N}_{f}=4. The simulations were performed using uncorrelated antenna channels, assuming that the receiver was perfectly synchronised and that the insertion of a long enough cyclic prefix in the transmitter ensured that the orthogonality of the subcarriers is maintained after transmission. For reference, we also include the SISO performance for unitary pilots, {p}_{1}. Since we are focus on the degradation of the estimator performance, the results are presented for a {E}_{\mathsf{\text{b}}}/{N}_{0} range in terms of the normalised MSE, according to
\mathsf{\text{MS}}{\mathsf{\text{E}}}_{h}=\mathsf{\text{E}}\left\{{\left\u0125h\right}^{2}\right\}/\mathsf{\text{E}}\left\{{\lefth\right}^{2}\right\}.
(13)
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 second effect to be evaluated is the decreasing of the amplitude of the transmitted pilots. In order to evaluate this effect, we also consider the previous SISO system. In this case, the transmitted pilots, i.e., {p}_{c}, assume constant values with nonunitary amplitude. Here, we selected three values ascending towards one which correspond to the unitary pilots, {p}_{1}. The results are shown in Figures 7 and 8.
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.
The TDMMSE filter should then be designed as
{\mathbf{W}}_{\mathsf{\text{MMSE,}}{\stackrel{\u0303}{h}}_{\mathsf{\text{eq}}l}}={\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}}{\left({\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}}\right)}^{1}.
(14)
As shown previously in Figure 5, {\alpha}_{\left(k\right)}{\mathrm{\Gamma}}_{\left(k\right)} tends to one for high values of SNR and examining Equation (14), which depends on \stackrel{\u0303}{\alpha \mathrm{\Gamma}}, it is clear that (14) tends to (9) for high values of SNR as well. In order to show this, several simulations were performed for different values of {\mathbf{R}}_{{\widehat{\stackrel{\u0303}{h}}}_{\mathsf{\text{eq}}l}{\widehat{\stackrel{\u0303}{h}}}_{\mathsf{\text{eq}}l}} and noise variance {\sigma}_{\mathsf{\text{br}}}^{2}. In these simulations, we consider channels according to ITU pedestrian models A and B [26]. According to Figure 9, the maximum value out of the main diagonal of the matrix {\mathbf{R}}_{{\widehat{\stackrel{\u0303}{h}}}_{\mathsf{\text{eq}}l}{\widehat{\stackrel{\u0303}{h}}}_{\mathsf{\text{eq}}l}} is close to 40 dB for small values of noise variance.
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.
Besides the estimate of the equivalent channel, it is necessary to estimate the factor {\alpha}_{\left(k\right)}^{2}{}_{\mathrm{\Gamma}}^{\left(k\right)}{}_{\mathrm{\Gamma}}^{\mathsf{\text{ru}},\left(k\right)}. This factor is needed to get the variance of the total noise {\sigma}_{\mathsf{\text{t}},\left({h}_{\mathsf{\text{ru}}l\mathsf{\text{,}}\left(k\right)}\right)}^{2} conditioned to the channel realisation, presented in (6). Since we assume \mathsf{\text{E}}\left\{{\left{h}_{\left(k\right)}\right}^{2}\right\}=1, we propose the use of the noise variance unconditioned to the channel realisation, {\sigma}_{\mathsf{\text{t}}}^{2}, referred as the expected value of the variance of the total noise. Also we consider that the channels have identical statistics, i.e. {\sigma}_{\mathsf{\text{bu}}}^{2}={\sigma}_{\mathsf{\text{br}}}^{2}={\sigma}_{\mathsf{\text{ru}}}^{2}, hence {\sigma}_{\mathsf{\text{t}}}^{2} can be expressed numerically by
{\sigma}_{\mathsf{\text{t}}}^{2}\cong \frac{1}{5+2{\sigma}_{\mathsf{\text{u}}}^{2\left(2\right)}}2{\sigma}_{\mathsf{\text{u}}}^{2\left(2\right)}+{\sigma}_{\mathsf{\text{u}}}^{2\left(2\right)}\mathsf{\text{.}}
(15)
3.3. Databased channel estimation
According to our system, the OFDM symbol has subcarriers where the subcarriers carrying pilots symbols are spaced by {N}_{f} and therefore the set of pilot subcarriers is \mathcal{P}=\left\{0,{N}_{f},2{N}_{f},\dots ,K{N}_{f}\right\}. If \mathbf{d} and \mathbf{p} correspond to data and pilot vectors and pilots are multiplexed with data symbols in different subcarriers, \mathbf{d} and \mathbf{p} contain nonzero values in disjoint subcarriers. Consequently, the set of data symbol subcarriers is \mathcal{S}=\left\{1,\dots ,{N}_{f}1,{N}_{f}+1,\dots ,2{N}_{f}1,\dots ,K\right\}. Since in our scenario the BS and RN are equipped with two antennas the pilot subcarriers are arranged such that each antenna has different subsets of subcarriers, i.e. {\mathcal{P}}_{1}=\left\{0,2{N}_{f},\dots ,K{N}_{f}\right\} and {\mathcal{P}}_{2}=\left\{{N}_{f},3{N}_{f},\dots ,K2{N}_{f}\right\}. Thus, the pilot array for one OFDM symbol is represented by \mathbf{P}=\left[{\mathbf{p}}_{1}{\mathbf{p}}_{2}\right]. Similarly, the data symbol array is given by \mathbf{S}=\left[{\mathbf{s}}_{1}{\mathbf{s}}_{2}\right]. The vectors {\mathbf{p}}_{1}\mathsf{\text{,}}{\mathbf{p}}_{2},{\mathbf{s}}_{1} and {\mathbf{s}}_{2} are 1\times K. The nonzero elements of {\mathbf{s}}_{1} and {\mathbf{s}}_{2} correspond to the first and second columns of , where for pairs of symbols at data subcarrier and j+1 the SFBC mapped data symbol matrix, {\mathbf{D}}_{\left(j\right)}, follows the next expression.
{\mathbf{D}}_{\left(j\right)}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}\hfill {d}_{j}\hfill & \hfill {}_{d}^{\left(j+1\right)}\hfill \\ \hfill {d}_{\left(j+1\right)}\hfill & \hfill {d}_{\left(j\right)}\hfill \end{array}\right)\mathsf{\text{,}}j\in \mathcal{S}.
(16)
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}.
As SFBC is used at the RN, the LS estimation based on the data requires a matrix inversion. Considering that two data symbols are encoded in subcarriers and j+1, the LS estimate for the equivalent channels is given by
{\widehat{\mathbf{H}}}_{\mathsf{\text{eq}}l\mathsf{\text{,}}\left(j\right)}^{\mathsf{\text{LS}}}=\sqrt{2}\left({\widehat{\mathbf{D}}}_{\left(j\right)}^{1}{}_{\mathbf{y}}^{\mathsf{\text{ru}}l\mathsf{\text{,}}\left(j\right)}\right),
(17)
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.
The MSE of the estimates in (17) is
\mathsf{\text{E}}\left\{{\lefte\right}^{2}\right\}=\frac{1}{J}\left(\sum _{j\in \mathcal{S}}\mathsf{\text{E}}\left\{{\left{}_{h}^{\left(j\right)}{}_{\u0125}^{\left(j\right)}\right}^{2}\right\}\right),
(18)
where is the size of the data subcarriers set.
For QPSK with unit power, we derive in Appendix an approximate relation between the error probability {P}_{\mathsf{\text{e}}} and the MSE of SISO and MISO channel estimates. Under the assumption that the correlation involving the data and noise are negligible we have
\mathsf{\text{E}}\left\{{\lefte\right}^{2}\right\}\approx \left\{\begin{array}{c}{\sigma}_{n}^{2}\left(1+2{P}_{\mathsf{\text{e}}}\mathsf{\text{SNR}}\right)\mathsf{\text{,forSISOchannels}}\\ \frac{1}{2}{\sigma}_{n}^{2}\left(1+{P}_{\mathsf{\text{e}}}\mathsf{\text{SNR}}\right)\mathsf{\text{,forMISOchannels}}\end{array}\right.,
(19)
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.
Moreover, in (17) we consider that the data subcarriers used in the SFBC coding are adjacent. In fact, when designing the transmitted frame, we insert pilots and therefore not all pairs of subcarriers corresponding to one SFBC codeword will be adjacent. For example, if we consider a pilot spacing of 4, i.e. {N}_{f}=4, there will be pilots at subcarriers 0,4,8,\dots , and the first SFBC codeword will be transported at the adjacent subcarriers 1 and 2, but the second codeword will be transported at the carriers 3 and 5. In order to overcome that, after performing the LS estimation, we set groups of virtual pilots uniformly spaced. This result in {N}_{f}1 groups of LS estimates with virtual pilots equispaced of {N}_{f}1 as well.
{\widehat{\stackrel{\u0303}{\mathbf{h}}}}_{K}^{\mathsf{\text{LS,data}}}=\left[\begin{array}{cc}{\widehat{\stackrel{\u0303}{\mathbf{h}}}}_{\left(K\left(K/{N}_{f}\right)\right)/\left({N}_{f}1\right)}^{\mathsf{\text{LS}}}\hfill & {\mathbf{0}}_{\left(K/{N}_{f}\right)}\hfill \end{array}\right].
(20)
The pilotbased and the databased CIRs estimates are combined according to the next expression. An averaging factor guarantees that the resulting power is normalised to 1 and by design this factor results in {N}_{f}. After combining the CIRs, the MMSE filtering is performed to enhance the estimate.
{\widehat{\stackrel{\u0303}{\mathbf{h}}}}_{K}=\left\{\mathsf{\text{diag}}\left({\stackrel{\u0303}{\mathbf{W}}}_{\mathsf{\text{MMSE}}}\right)\circ \left[\left(\sum _{{N}_{f}1}{\widehat{\stackrel{\u0303}{\mathbf{h}}}}_{K}^{\mathsf{\text{LS,data}}}+{\widehat{\stackrel{\u0303}{\mathbf{h}}}}_{K}^{\mathsf{\text{LS,pilots}}}\right)/{N}_{f}\right]\right\}.
(21)
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.