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Spacetime code diversity by phase rotation in multicarrier multiuser systems
EURASIP Journal on Wireless Communications and Networking volume 2013, Article number: 148 (2013)
Abstract
Code diversity using spacetime block codes was developed for singlecarrier and singlereceiver systems. In this paper, the extension of code diversity by phase rotation to multiuser and multicarrier systems is proposed and analyzed. We show that code diversity with reduced feedback is possible in this new scenario and the coding gain has a mild logarithmic decrease with the number of users and the number of subcarriers. In addition, we develop an analytical upper bound for the average error probability whose accuracy is verified by simulation.
1 Introduction
There are many ways of exploiting diversity such as spatial diversity [1], multiuser diversity [2], or multipath diversity [3]. A common technique for exploiting spatial diversity is the use of spacetime coding which obtains improvements in the reliability of wireless communications over fading channels by correlating signals across different transmit antennas [4]. Moreover, if some channel knowledge is available at the transmitter, much better performance can be attained [5, 6]. Besides, it has been shown that maximum likelihood (ML) decoding obtains the best performance, although its complexity becomes prohibitive when the constellation size increases. Sphere decoding [7] can reduce the search space to a lattice in a sphere around the initial estimation. However, the use of simpler decoders is still desirable.
In [8, 9], it is shown that some of the quasiorthogonal spacetime block codes (QOSTBC) [8–11] are actually a family of codes, where each individual code in the family obtains the same characteristics in terms of average performance. Since all of the individual codes in the same family obtain equivalent average performance, it is not important which one is used if we look at this average. However, each of the individual codes within the same family induces a different channel matrix and thus, their instantaneous performance can be very different. For this reason, the authors in [8] introduced the concept of code diversity. Basically, a significant gain can be obtained if we use the adequate code member (within the same family) at each time instant. Moreover, they proved that the gap between the performance obtained with code diversity by using a simple zeroforcing (ZF) decoder and the optimal ML decoder can be reduced, even for meaningful signaltonoise ratio (SNR). There are several mechanisms to obtain code diversity, e.g., spacetime coding [8] or phase rotation [9]. However, code diversity was only proposed for singlereceiver (authors in [8] proposed code diversity for the multiple access channel (MAC), i.e., with a singlereceiver) and singlecarrier systems.
In this paper, the code diversity mechanism by phase rotation is extended to broadcast (BC) multiuser scenarios with multiple receivers and multicarrier systems, keeping the feedback requirements of the same order as for the singlereceiver and singlecarrier case, i.e., a few bits. We first show that it is possible to obtain coding gain for multiuser and multicarrier systems in such a scenario with limited feedback. Once this has been proven, we develop the theory extension and an algorithm which avoids the complexity and unfeasibility of exhaustive search while getting close to its performance, constrained to limited feedback requirements. The main contributions of this paper in comparison to [9] are as follows:

1.
Our analysis is the extension of the code diversity for the case of multiuser and multicarrier systems

2.
The design of an algorithm that provides very close to the optimum performance (obtained by exhaustive search) avoiding the complexity and unfeasibility of exhaustive search. And moreover, with the same feedback data rate as in singlecarrier scenario.

3.
The application to several spacetime block codes (STBC) codes.
The remainder of this paper is organized as follows. In Section 2, the concept of singlereceiver, singlecarrier code diversity is reviewed. Next, in Section 3, its extension for multiuser and multicarrier systems is developed, including an algorithm for selecting the best single value for phase rotation that gets close performance to exhaustive search. In addition, also in this section the algorithm is analyzed and an upper bound for the global performance is obtained. After that, Section 4 presents some results for QOSTBC and circulant codes and finally, in Section 5, the main conclusions are drawn.
1.1 Notations
Throughout this paper, the following notations will be used: Capital boldface symbols will be used for matrices, boldface symbols for vectors and normalface for scalars. $\mathcal{R}\left\{\mathbf{H}\right\}$ is the rank of matrix H whereas E_{ i }{x} denotes expectation of x with respect to i and ·_{F} represents the matrix Frobenius norm.
2 Code diversity
Code diversity by spacetime coding with limitedrate feedback and low complexity detection was first introduced in [8]. The constellation rotation was introduced in [11, 12] and it has been extensively analyzed for MAC in [13, 14]. The use of channel rotation was firstly proposed in [8]. A phase rotation mechanism to introduce code diversity was later proposed in [9]. Basically, the underlaying idea is that some of the wellknown quasiorthogonal spacetime block codes are actually a family of codes where each individual code within the family exhibits the same characteristics with respect to average capacity and performance.
Let $\mathbf{X}\in {\u2102}^{{N}_{\mathrm{t}}\times T}$ be a spacetime block codeword, where the rows are indexed by transmit antennas (N_{t}) and the columns are indexed by time slots (T). If L distinct symbols are transmitted during one frame of T symbol periods, then the transmission rate for the spacetime block code is L/T. In a system with N_{t} transmit antennas and N_{r} receive antennas, the received signal $\stackrel{~}{\mathbf{R}}\in {\u2102}^{{N}_{\mathrm{r}}\times T}$ is [9]
where $\mathbf{H}\in {\u2102}^{{N}_{\mathrm{r}}\times {N}_{\mathrm{t}}}$ is the channel matrix with entry h_{j,i} representing the channel gain between the i th transmit antenna, and the j th receive antenna; $\stackrel{~}{\mathbf{W}}\in {\u2102}^{{N}_{\mathrm{r}}\times T}$ is the normalized additive white Gaussian noise (AWGN) matrix with zero mean and unit variance entries; N_{0} is the current noise variance; and E_{s} is the average transmitted signal power.
In many of the algebraic constructions of spacetime codes, such as golden code [15], silver codes [16], or many others [17–19], an important feature is the exchangeability of the structure (correlation) of the transmitter with the structure of the receiver. By using this correlation, Equation 1 can be rewritten as
where $\mathbf{r}\in {\u2102}^{{N}_{\mathrm{r}}T\times 1}$ is the received signal vector, $\mathcal{H}\in {\u2102}^{{N}_{\mathrm{r}}T\times L}$ is the induced channel matrix, $\mathbf{c}\in {\u2102}^{L\times 1}$ is the transmitted signal vector, and $\mathbf{w}\in {\u2102}^{{N}_{\mathrm{r}}T\times 1}$ is the AWGN vector. In the absence of noise, the ML estimate for the transmitted symbols c is unique since the induced channel matrix $\mathcal{H}$ is full rank, i.e., $\mathcal{R}\left(\mathcal{H}\right)=L$. However, when this matrix is close to the rank deficient, decoding becomes unstable. Moreover, as shown in [20, 21], the instantaneous performance with simple ZF receivers converges to the optimum ML receivers only in the high SNR regime, while it is quite distant for low SNR due mainly to the noise enhancement in ZF receivers caused by rank deficient channels (especially for deep fades). Since this code diversity^{a} is able to avoid a rank deficient matrix, similar performance of ZF receivers compared to ML can be expected even for SNR that is not so high.
In order to avoid a rank deficientinduced channel matrix, in [9], it is proposed that the phase of the channel gains be modified as
i.e., the phase of the i th antenna is rotated by Φ_{ i }  an arbitrary value from the interval [0,2π), although for limiting the feedback rate and simplicity, it is usually drawn from a small discrete set of phases. It has been shown in [9] that for a specific QOSTBC code, modifications in these phases do not affect the average capacity achieved by the code and obtain the same average performance. Indeed, these phase modifications form the family of codes for such a QOSTBC. Also in [9], it has been shown that the modification of only one phase at one of the antennas is more than enough to get a good performance. Moreover, the number of phases (K) required to obtain significant improvements in performance is small (just K=4 phases are enough). Since each phase will induce a different channel matrix, at every time instant, it will be possible to find a phase (code) performing better than the others; thus, there exists code diversity. In [9], the following algorithm is proposed. At the receiver and after channel estimation, the Kinduced matrices are calculated and the code giving the best performance is sought. The best phase (code) is chosen according to the following criteria:

Firstly, the one which maximizes $\mathcal{R}\left({\mathcal{H}}^{H}\mathcal{H}\right)$ (Let ${d}_{max}=max\mathcal{R}\left({\mathcal{H}}^{H}\mathcal{H}\right)$).

And secondly, the one which maximizes $\prod _{i=1}^{{d}_{max}}{w}_{i}$, being w_{ i } the nonzero eigenvalues of ${\mathcal{H}}^{H}\mathcal{H}$.
The phases to be used are taken from $\frac{2\mathrm{\pi k}}{K}$, and thus, the receiver feeds back the index $\widehat{k}$ that
This way, only log2(K) bits are enough to improve the performance. It should be remarked that this phase rotation mechanism is only one way of obtaining code diversity. For example, in [8], code diversity relays on the quaternionic design and its goal is the minimization of the cochannel interference in the multiple access channel.
3 Multiuser and multicarrier code diversity
The trivial extension of this theory to multicarrier systems is to perform the computation of $\widehat{k}$ for each subcarrier and thus, the transmitter would use a different phase (code) for each subcarrier n, i.e., ${\widehat{k}}_{n}$. However, the required feedback would be N× log2(K), where N is the number of subcarriers. This is not practical. Moreover, there would be several subcarriers with the same phase. This issue leads to the possibility of compressing the feedback information. However, this way, the complexity would increase and in any case, the amount of feedback data would still be high. Instead of trying to compress the feedback information, it can be useful to find a phase (code) that is good enough for all the subcarriers. It might not be the best one for a particular subcarrier, but the best one for average performance of the whole set of subcarriers.
Furthermore, in case of BC where multiple users are receiving information, there is no trivial extension because the good code for one receiver may be very bad for the others; so it would be interesting to know if there is a code that is reasonably good at the same time for all the subcarriers and all the users in this scenario, and, if any, how to calculate or select it. An example of that scenario could be a multihop relaybased network, where all the relays need to receive the transmitted signal in order to forward to destination.
In order to check this possibility, an exhaustive search has been carried out as explained in the following. A system with N subcarriers, N_{u} users and N_{t} transmit antennas has been set up. We have used the QOSTBC code proposed by Jafarkani et al. [10]  applied to a multicarrier system though  and we have modified the phase of the first antenna as indicated in [9]. The transmitter uses all the K different codes to send the information to the N_{u} receivers, using the same phase (code) in all the subcarriers at one time. At each of the N_{u} receivers, the bit error rate (BER) performance is calculated for each code (phase) at each subcarrier. Next, the average BER across the subcarriers and the N_{u} users are computed. Then, for each time interval, the best average BER performance is stored for final results, i.e., like at each time interval, the optimum code has been used. In Figure 1, obtained with N=256 and a SUI3 channel profile [22], these results are shown. The first curve is the average performance of the original QOSTBC code across the subcarriers and users. The other curves represent the average performance over time and across users and subcarriers when transmitting each time by using the best code (obtained by exhaustive search) for different numbers of users. It should be noted that the same code (phase) is applied for all the subcarriers and that all the users demodulate the signal. Since in the figure the average performance of over all users and subcarriers is shown when the transmitter uses the best instantaneous performanceachieving code, it can be seen that there is a particular code at each time that gives the best average performance for the set of subcarriers and users; thus, the code diversity can be extended to multicarrier and multiuser systems with reduced feedback (one unique code is enough for all subcarriers and all users). Otherwise, if a particular instantaneous best code would not exist, the average global performance would be equal or worse than the STBC curve. Besides, it can also be observed that the code diversity decreases with the number of users. The reason is that as the number of users increases, so does the number of different channel coefficients. Therefore, it becomes very difficult to find a code that shows gain for all the users at the same time. Moreover, also in this figure, the coding gain for a single user can be evaluated. In this case, a high coding gain can be observed in the multicarrier system which makes it interesting for use in realistic scenarios.
As it has just been shown on previous figure, at each instant time, there is a code which provides good performance averaged over all users and subcarriers. For this reason, since an exhaustive search is not possible for a practical implementation, alternatives have to be found.
3.1 Signal model for broadcast channel
In the broadcast channel, the received signal by the u th user at the n th subcarrier can be written as
with ${\mathbf{r}}_{u}^{r}\in {\u2102}^{{N}_{\mathrm{t}}\times {N}_{\mathrm{r}}}$, where ${\mathcal{H}}_{u,n}\in {\u2102}^{{N}_{\mathrm{r}}\times {N}_{\mathrm{t}}}$ are the induced channel matrix by u th user at subcarrier n, ${\mathbf{c}}^{n}\in {\u2102}^{{N}_{\mathrm{t}}\times {N}_{\mathrm{r}}}$ is the desired data to be transmitted and ${\mathbf{w}}_{u}^{n}\in {\u2102}^{{N}_{\mathrm{t}}\times {N}_{\mathrm{r}}}$ is the noise.
For clarity reasons and without loss of generalization, in the following, we are assuming only one antenna at the receiver, i.e., N_{r}=1.
Two important remarks should be highlighted in this model in Equation 5. First, as in BC channel, the desired data c^{n} are common to all users from the transmitter (base station). Second, although channel coefficients in ${\mathcal{H}}_{u,n}$ are different from user to user and (likely) subcarrier to subcarrier, the phase rotation introduced is the same for all users and subcarriers^{b}.
As an example, for the 4×1 QOSTBC codes introduced by Jafarkani [10], the originally induced channel matrix for user u and n th subcarrier will be
As mentioned before and in [9], by only modifying the phase of one antenna is enough. Besides, as indicated above, the idea is to apply the same phase rotation to all the subcarriers and for all the users, thus, the induced channel matrix becomes the following:
where K is the total number of rotations (that implies a feedback of log2K bits per user), and ϕ is the specific code number (ϕ∈{1⋯K}).
3.2 Proposed algorithm
In order to obtain coding gain close to the one obtained in Figure 1 with reduced feedback per user and without resorting to exhaustive search, we propose the following algorithm to obtain ϕ:

1.
Each user (u) estimates the channel for coherent demodulation. With these estimates, it also calculates the product ${p}_{u}^{\varphi ,n}=\prod _{i}{w}_{i,u}^{\varphi ,n}$, where ${w}_{i,u}^{\varphi ,n}$ is the i th eigenvalue of matrix ${\mathcal{H}}_{u,n,\varphi}^{H}{\mathcal{H}}_{u,n,\varphi}$ for u th user, for all subcarriers (n=1⋯N,) and all codes (ϕ=1⋯K).

2.
Each user finds ${m}_{u}=\underset{n}{min}\underset{\varphi}{min}{p}_{u}^{\varphi ,n}$ and ${\varphi}_{u}=\text{arg}\underset{n}{min}\underset{\varphi}{min}{p}_{u}^{\varphi ,n}$.

3.
Each user feeds back the scalar m _{ u } and ϕ _{ u } to the transmitter.

4.
The transmitter selects the code to be used as ϕ ^{∗}=arg maxu m _{ u }

5.
The transmitter broadcasts that the code ϕ ^{∗} will be used, so that all the receivers can adequate reception for the signal.

6.
The code will be used until another user obtains larger m _{ u } than selected.
The idea of maximizing the minimum eigenvalues’ product ${p}_{u}^{\varphi ,n}$ is due to the fact that the coding gain of a STBC or QOSTBC is determined by this product [4]. Since the performance measurement is the BER and it follows an erfc function which is dominated by the worst case, the code that maximizes the minimum coding gain obtains the best global performance because the maximum error is bounded and minimized. With the proposed algorithm, code diversity can be obtained with only a few bits of feedback per user. Besides here, the SNR is not needed to be estimated, as opposed to [8].
Also, other alternatives have been evaluated, such as

Maximizing over maximums (maxmax): ${m}_{u}=\underset{n}{max}\underset{\varphi}{max}{p}_{u}^{\varphi ,n}$ and ${\varphi}_{u}=\text{arg}\underset{n}{max}\underset{\varphi}{max}{p}_{u}^{\varphi ,n}$ and ϕ^{∗}=arg maxu m_{ u }

Maximizing over the mean of all the products (maxmean): ${m}_{u}={\mathrm{E}}_{n}\left\{\underset{\varphi}{max}{p}_{u}^{\varphi ,n}\right\}$ and ${\varphi}_{u}=\text{arg}\phantom{\rule{1em}{0ex}}{\mathrm{E}}_{n}\left\{\underset{\varphi}{max}{p}_{u}^{\varphi ,n}\right\}$ and c^{∗}=arg maxu m_{ u }

Maximizing over the sum of all the products (maxsum): ${m}_{u}=\sum _{n}\underset{\varphi}{max}{p}_{u}^{\varphi ,n}$ and ${\varphi}_{u}=\text{arg}\phantom{\rule{1em}{0ex}}\sum _{n}\underset{\varphi}{max}{p}_{u}^{\varphi ,n}$ and ϕ^{∗}=arg maxu m_{ u }

Minimizing the maximum product (minmax): ${m}_{u}=\underset{n}{max}\underset{\varphi}{max}{p}_{u}^{\varphi ,n}$, ${\varphi}_{u}=\text{arg}\underset{n}{max}\underset{\varphi}{max}{p}_{u}^{\varphi ,n}$ and ϕ^{∗}=arg minu m_{ u }
The reason why these other alternatives offer worse global performance is the following. The first case, maxmax guarantees the best performance for one user and subcarrier but does not impose anything to the worst case, and exactly the one which dominates the global performance. Similar situation occurs in the maxmean and maxsum, where the maximization is carried out over all the subcarriers (mean and sum, respectively), and thus, the worst case is eliminated for the decision. The minmax rule obtains the worst global performance (see Figure 2) because first, the worst user is ignored and second, in addition, there is a minimization on the best possible candidate. In Figure 2, all these results can be observed using these different criteria or rules for choosing the best code. It can be seen in this figure that the best choice is our proposed criterion, and furthermore, it is close to the exhaustive search approach.
3.3 Analysis of the error probability
For singleuser and singlecarrier systems and restricting ourselves for simplicity to systems with multiple transmit antennas but a single receive antenna, i.e., N_{r}=1, assuming perfect channel state information at the receiver  with Equation 2  the conditional probability is bounded by
This upper bound can be approximated by [9]
being $d=\mathcal{R}\left({\mathcal{H}}^{H}\mathcal{H}\right)$ and w_{ i }, i=1⋯d are the nonzero eigenvalues of ${\mathcal{H}}^{H}\mathcal{H}$. The diversity gain d and the coding gain $\left(\prod _{i=1}^{d}{w}_{i}\right)$ determine the spacetime code performance.
Extending this analysis to the multiuser and multicarrier system using the algorithm above, we find that the average performance over the users and across the subcarriers involves averaging over N_{u} and N uncorrelated i.i.d random variables, i.e., the product of the eigenvalues of the code induced by the channel, which is different for every of the N_{u} receivers and each of the N subcarriers. Since our algorithm selects the maximum, according to [23], it is given by log2 (N+1)× log2(N_{u}+1) times the expectation of the i.i.d process. Besides, our algorithm maximizes the minimum singleuser code performance and, therefore, it comes to the compensation of transmitter diversity by the multiplereceiver diversity. Thus, it gives
where $\mathfrak{F}\left({P}_{e\mathcal{H}}\right)$ involves the maximization of the eigenvalues’ product for different users and the minimization of eigenvalues’ product for all the subcarriers within the same user.
From this equation, several observations can be extracted. Code diversity performance shifts to the right on SNR with the increase in the number of subcarriers and/or number of users following a logarithmic decay. Besides, the diversity due to the different transmit antennas vanishes because the number of transmit antennas is much smaller than N×N_{u}.
3.4 Application to quasiorthogonal spacetime codes
As an application of the above analysis, we propose the use of the QOSTBC introduced by Jafarkani [10] in a system with N_{t}=4 and N_{r}=1 so that
which induces a channel matrix
As mentioned before and in [9], only by the rotation of one antenna is enough for obtaining code diversity. Thus, we can introduce the phase rotation in transmit antenna 1 which yields the following general induced matrix for each code
where K is the total number of rotations (in our examples K=4 that implies a feedback of 2 bits per user), and ϕ is the specific code number (ϕ∈{1,2,3,4} in our example).
Applying our algorithm, all the subcarriers will use the same code for the transmission to all the users. Therefore, for each user and for each subcarrier, we will have
where $a=\sum _{i=1}^{4}{h}_{i}{}^{2}$ and $b=2\xb7\text{Re}\left({h}_{1}{h}_{4}^{\ast}{h}_{2}{h}_{3}^{\ast}\right)$, and hence the eigenvalues of this matrix become {ϖ_{1},ϖ_{2},ϖ_{1},ϖ_{2}}, with ϖ_{1}=a−b and ϖ_{2}=a+b. Thus, the product is $\left(\prod _{i=1}^{2}{\varpi}_{i}^{2}\right)={(a+b)}^{2}{(ab)}^{2}$. It is well known that a distributes as a noncentral χ^{2} with four degrees of freedom [24]. Regarding b, it is composed by the substraction of two double Gaussian distributions (due to the product of two Gaussian random variables) [25], with the same parameter (channel coefficients are assumed to have the same distributions). Thus, this new random variable distributes as Gaussian. Thus, it yields to ϖ_{1} and ϖ_{2} which are distributed as noncentral χ^{2}, also with four degrees of freedom with noncentrality parameter
where μ_{ i } are the averages of the channel frequency response h_{ i } for each of the transmit antennas (N_{t}=4 with Gaussian distribution), and σ_{ i } are the standard deviations for these h_{ i }. The product of two χ^{2} distributions involves the gamma function [26] whose square approximately exhibits an exponential probability density function. In our case
where
whose expected value is β/2. Thus, the average expected performance over all the users and across all the subcarriers will be
3.5 Application to circulant codes
Another application is the circulant codes [19] based on circulant matrices that exhibit similar performance to other more complex codes but with less complexity. For the case of a 3×3 code, with matrix
with ${\beta}_{1}=exp\left(\frac{2\mathrm{\pi i}}{5}\right)$ and ${\beta}_{2}=exp\left(\frac{2\mathrm{\pi i}}{7}\right)$. Thus, after applying the proposed algorithm, each subcarrier and each user will have the product
with $c={h}_{0}^{2}+{\left({\beta}_{2}{h}_{2}\right)}^{2}+{\left({\beta}_{1}{h}_{1}\right)}^{2}$ and d=h_{0}h_{1}β_{1}+h_{0}h_{2}β_{2}+h_{1}β_{1}h_{2}β_{2}. After some mathematical manipulation, the eigenvalues are ϖ_{1}′=c+d+d^{∗}, ${\varpi}_{2}\prime =\frac{1}{2}d+c\frac{1}{2}{d}^{\ast}+\frac{1}{2}\sqrt{3{d}^{\ast}+6{d}^{\ast}d{3{d}^{\ast}}^{2}}$ and ${\varpi}_{3}\prime =\frac{1}{2}d+c\frac{1}{2}{d}^{\ast}\frac{1}{2}\sqrt{3{d}^{\ast}+6{d}^{\ast}d{3{d}^{\ast}}^{2}}$. Thus, their product is $\prod _{i=1}^{3}{\varpi}_{i}$. Due to the scale parameters β_{1} and β_{2}, c distributes as a gamma distribution with scale (θ) and shape (s) parameters being θ= maxh_{ i }/3 and s=3.5. On the other hand, the real part of d distributes as a standard Gumbel distribution, i.e., location μ≈0 and scale ξ≈1; whereas the imaginary part distributes as a Cauchy distribution with location l=E{h_{ i }}≈0 and scale γ=0.5. Thus, the distribution for the eigenvalues are the following. For ϖ_{1}, a noncentral χ^{2} distribution with 2 degrees of freedom and noncentrality parameter of maxh_{ i }[24]; ϖ_{2} and ϖ_{3} distribute as gamma function with parameters scale ${\theta}_{2}=\sum _{i=1}^{3}{\sigma}_{i}$ and θ_{3}=1; and shape ${s}_{1}=\sum _{i=1}^{3}{\sigma}_{i}$ and s_{2}=1, respectively.
Finally, the product approximately exhibits, as in the QOSTC case, an exponential distribution, this time $f\left(x\right)=\frac{16}{\mathrm{\beta \prime}}{e}^{\left(\frac{4x}{\mathrm{\beta \prime}}\right)}$, where $\mathrm{\beta \prime}={\left(\prod _{i=1}^{3}max{h}_{i}\right)}^{2}{\theta}_{2}{\theta}_{3}$.
4 Numerical results
The following numerical results illustrate the performance that can be achieved with the proposed approach in an orthogonal frequency division multiplexing (OFDM) system. The number of subcarriers has been fixed to N=64 and N=256. One hundred thousand OFDM symbols with QPSK modulation for QOSTBC and circulant codes have been simulated to average results.The channel was the SUI3 model [22].
In Figure 3, the obtained average BER over N_{u} users and across the N subcarriers by using our proposed algorithm for QOSTBC is plotted jointly with the average BER with exhaustive search from Figure 1 for comparison purposes. It can be observed that our proposed algorithm gets close in performance to the one obtained with exhaustive search, with very reduced feedback and without a lengthy and unfeasible exhaustive search. The most important conclusion from Figure 3 is that the average performance gain due to the code diversity decreases with the logarithm of the number of users (as anticipated in the previous section), which, first, corroborates the analysis and second, allows the usefulness of the scheme in practical scenarios, because with a linear decrease, the advantages of code diversity would disappear with a few users, which is not the case and would not be useful. However, the logarithmic dependence ensures that, even with a large number of users (N_{u}=32), the code diversity gain significantly offsets the multiuser penalty factor. The dependence with the number of subcarriers is also logarithmic, which is again positive. In addition, as was already mentioned, the ZF decoder obtains almost the same performance as more complex decoders such as ML (the BER of ML decoding for the case of 32 users is also plotted as an illustrative example).
In Figure 4, the obtained average BER over N_{u} users and across the N subcarriers by using our proposed algorithm with QOSTC and circulant code is compared to the analytical upper bound expression developed here. It can be observed that this upper bound is accurate enough to describe the performance.
5 Conclusions
In this paper, the extension of the code diversity theory by phase rotation from singleuser and singlecarrier to multiuser and multicarrier systems is presented and analyzed.
We have shown that code diversity can be achieved in a multicarrier and multiuser scenario, resorting to the use of a single code on each transmission interval. We further developed a maxmin algorithm that achieves close diversity performance as optimum search while requiring reduced feedback. The proposed algorithm obtains a tradeoff among complexity, reduced feedback requirements, and performance, which makes it suitable for practical implementation. Also, the performance gain decreases logarithmically with the number of users and subcarriers, which is a desirable property since it allows some code diversity even for a moderate to high number of users or/and subcarriers (it has been shown that for 32 users and 256 subcarriers, the performance gain due to diversity is still significant). In addition, the developed analytical upper bound for the average error probability has been shown to be accurate enough to describe the performance.
Endnotes
^{a} As mentioned before, the use of a specific STB code within the same family with best instantaneous performance for the current channel.
^{b} In order to reduce the feedback.
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Acknowledgements
The first author would like to thank Dr. Robert Calderbank for his help at the beginning of the work. Also we are thankful to the anonymous reviewers for their suggestion that have improved the quality of the manuscript. This work has been partly funded by the Spanish national projects GRE3NSYST (TEC201129006C0303) and COMONSENS (CSD200800010).
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Gil Jiménez, V.P., Gameiro, A. & Armada, A.G. Spacetime code diversity by phase rotation in multicarrier multiuser systems. J Wireless Com Network 2013, 148 (2013). https://doi.org/10.1186/168714992013148
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Keywords
 Orthogonal Frequency Division Multiplex
 Exhaustive Search
 Channel Matrix
 Orthogonal Frequency Division Multiplex Symbol
 Phase Rotation