Multiuser diversity in correlated Rayleighfading channels
 Behrooz Makki^{1}Email author and
 Thomas Eriksson^{1}
DOI: 10.1186/16871499201238
© Makki and Eriksson; licensee Springer. 2012
Received: 8 July 2011
Accepted: 8 February 2012
Published: 8 February 2012
Abstract
This article studies the effect of scheduling and multiuser diversity on the performance of correlated Rayleighfading channels. More specifically, the powerlimited channel average rate is obtained for quasistatic correlated fading channels. The results are obtained in the cases where there is perfect or imperfect channel state information available at the transmitter. Simulation results show that the average rate reduction due to channels dependencies is ignorable in low correlation conditions. However, the effect of scheduling and multiuser diversity on the average rate reduces substantially as the fading channels dependency increases. Also, for different channels correlation conditions, considerable performance improvement is achieved via very limited number of feedback bits.
1. Introduction
Employment of adaptive modulation and scheduling leads to substantial performance improvement in multiuser systems, normally called multiuser diversity[1–12]. This is the main motivation for the current schedulingbased systems and this article as well. In these methods, the transmitter is provided with some information about the channel quality of different users. This information is then utilized by a scheduler to select the appropriate users, coding, and modulation such that an objective function is optimized. System throughput and fairness between the users are two objective functions mainly considered in the literature. Furthermore, depending on the number of users, channels characteristics and the feedback load resources, the transmitter information about the channels quality can be perfect or imperfect.
Assuming different levels of channel state information (CSI), a large number of scientific reports can be found that have tackled the multiuser diversity problem in different theoretical and practical aspects. For instance, [6–12] investigated the performance of multiuser networks under perfect CSI assumption. These works were later extended by, e.g., [13–19] where the system performance was analyzed in the presence of imperfect CSI available at the scheduler. Furthermore, among different research projects involving in this topic the WINNER+ [20] and the 3rd Generation Partnership Project (3GPP) [21] can be mentioned where multiuser diversity is one of the most important issues.
References [6–19] are all based on the assumption that the fading channels are mutually independent. That is, the network performance is investigated in the case where there is no correlation between the fading channels of different transmission endpoints. However, based on the environmental properties, realistic channels may not be independent [3–5], [22–24]. Therefore, it is important to study the channel performance under correlated channels condition.
In this perspective, this article studies the average rate of correlated Rayleighfading multiuser networks. The results are obtained for quasistatic channels in the cases where there is perfect or imperfect CSI available at the transmitter. It is mainly focused on a system with a single transmitter and two receivers, which allows us to find closedform solutions for the average rate and power allocation criteria. However, some discussions about extending the results to arbitrary number of receivers are also presented and the final conclusions are valid independent of the number of receivers. Assuming imperfect CSI, we evaluate the effect of optimal channel quantization on the system performance. The results show that substantial performance improvement is achieved with a limited number of feedback bits per user. Moreover, the effect of scheduling and multiuser diversity reduces with the channels correlation, although the rate reduction is ignorable in low correlation conditions. The arguments would be interesting for people involved in WINNER+, 3GPP or the ones working on scheduling between close users, for instance scheduling in singlecell networks, e.g., [25–28].
2. System model
where I_{0}(.) is the zerothorder modified Bessel function of the first kind [33]. Finally, note that as the channels have identical pdfs, the maxrate scheduler which transmits to the user with the strongest channel at any given time slot not only optimizes the system total performance but also maintains the longterm fairness between the users. Moreover, although it is the simplest to assume a network with two users, as seen in the following, the results provide valuable insights for the more general cases with arbitrary number of users. Also, extension of the results to arbitrary number of receivers experiencing different fading distributions can be found in the Appendix.
It is assumed that each receiver has perfect CSI about its corresponding channel gain which is an acceptable assumption in quasistatic condition, e.g., [17, 18, 34–38]. However, the transmitter may be provided with imperfect (Section 3) or perfect (Section 4) CSI about the fading channels. Further, all results are presented in natural logarithm basis, the channel average rate is presented in natsperchanneluse (npcu) and, as stated in the following, the arguments are restricted to Gaussian input distributions. Finally, note that Rayleighfading channels are good models for tropospheric and ionospheric signal propagation as well as the effect of heavily builtup urban environments on radio signals [39, 40]. Also, it is most applicable when there is no dominant propagation along a line of sight between the transmitters and the receivers.
2.1. Average rate with no CSI at the transmitter
3. Average rate in the presence of imperfect CSI at the transmitter
is implemented by each receiver. Here, ${\stackrel{\u0303}{g}}_{i}$'s denote the quantization boundaries and S_{ i } is the ith quantization region. The quantization indices are sent back to the scheduler which selects the user with the higher quantization index (maxrate scheduler). Also, if the channel gains are in the same quantization regions, one of them is selected randomly.
Remark 1: The optimal maxrate scheduler should select the users with the highest SNR. However, as stated in the following, the waterfilling properties imply that higher powers are allocated to the higher quantization regions (see, e.g., (14), (19) and [34–38]). Therefore, the SNR increases with the quantization index and scheduling based on the quantization indices works the same as scheduling based on the SNRs.
is the probability that both channels are in the ith quantization region where one of them is selected randomly.
Here, λ is the Lagrange multiplier satisfying $\stackrel{\u0304}{T}\le T\phantom{\rule{0.3em}{0ex}}\text{and}\phantom{\rule{0.3em}{0ex}}{\u2308x\u2309}^{+}\doteq \text{max}\left(0,x\right).$ Intuitively, using optimal power allocation the power is not wasted on weak channel realization and the saved power is spent on strong gain realizations. Therefore, there will be a quantization index $\stackrel{\u2322}{i}$ where T_{ i } = 0 if $i<\stackrel{\u2322}{i}$and ${T}_{i}>0$ if $i\ge \stackrel{\u2322}{i}$. This point is helpful for simplifying the waterfilling power allocation algorithm.
and finally, (d) is derived by using variable transform $t=\sqrt{x}$, partial integration, defining $\varphi \left(x,y\right)\doteq \xi \left(\sqrt{\frac{2x}{r}},\sqrt{\frac{2y}{r}}\right)$and some calculations.
A simple average rate optimization algorithm: In contrast to transmission power parameters, the powerlimited average rate optimization problem of quantized CSIbased systems, e.g., (13), is not a convex optimization problem in terms of quantization parameters ${\u011d}_{i},\phantom{\rule{0.3em}{0ex}}{\stackrel{\u0303}{g}}_{i}$ ∀_{ i }[34–38]. Therefore, although implementable, gradientbased algorithms are not efficient in determining the optimal quantization parameters. In order to tackle this problem, we propose an iterative algorithm, illustrated in Algorithm 1.
Remark 2: Similar to other techniques for solving nonconvex optimization problems, it can not be guaranteed that the algorithm leads to the globally optimal solution for all channel conditions. However, by extensive testing, it is observed that for many different initial parameter settings and vector generation procedures, the algorithm leads to unique solutions. Furthermore, our experiments show that the algorithm is much more efficient than using greedy search scheme which requires a large number of initial random seeds due to the nonconvexity of (13). Finally, although it may be timeconsuming when the number of optimization parameters increases, the proposed algorithm has been shown to be efficient in many complex optimization problems dealing with local minima issues [42].
In the following, the channel average rate in the presence of perfect CSI available at the transmitter is studied and then the simulation results are presented in Section 5.
4. Average rate in the presence of perfect CSI at the transmitter
where (f) is again based on partial integration. Therefore, the main problem would be to determine the cdf F_{ Z } (z), (20) and then (21).
where (k) follows from the fact that $\Gamma \left(x\right)=\left(x1\right)!$ if x is a positive integer value and (l) is obtained by the definition of the incomplete Gamma function $\Gamma \left(a,x\right)={\int}_{x}^{\infty}{t}^{a1}{e}^{t}\text{d}t.$ Finally, setting n = 2 and 1 in (25) the Equations (20) and (21) are found, respectively.
Here, there are some interesting points to be noted:

Using (20), it can be easily shown that the waterfilling threshold λ^{*} is a decreasing function of the average transmission power constraint T . That is, more realizations of the variable Z, and correspondingly the channel gains, receive powers as the average transmission power constraint increases. Particularly, λ*→ 0 as T → ∞.

Assuming independent fading channels, i.e., setting β = 0 in (3), the auxiliary variable cdf is simplified to${F}_{Z}\left(z\right)=\underset{0}{\overset{z}{\int}}\underset{0}{\overset{z}{\int}}{f}_{{G}_{1}}\left(x\right){f}_{{G}_{2}}\left(y\right)\text{d}x\text{d}y={\left(1{e}^{\frac{z}{\mu}}\right)}^{2}.$(26)
where Ei(x) is the standard exponential integral function $\text{Ei}\left(x\right)={\int}_{x}^{\infty}\frac{{e}^{u}}{u}\text{d}u,x\ge 0.$
Finally, extension of the results to arbitrary number of receivers can be found in the Appendix.
5. Simulation results
which is the ratio of the channel average rate, e.g., (21), and the one for uncorrelated channels, e.g. (28), is demonstrated as a function of the channels correlation.
Channel average rate for different number of quantization regions and correlation coefficients
β  N  

2  3  4  
0  0.491  0.587  0.640 
0.1  0.490  0.586  0.639 
0.5  0.466  0.560  0.611 
0.9  0.401  0.476  0.521 
1  0.374  0.432  0.469 
5.1. Discussions
Theoretical and simulation results emphasize a number of interesting points that can be listed as follows:

For different correlation conditions, considerable performance improvement is achieved via very limited number of feedback bits per user. This point is useful particularly in networks with a large number of users where the feedback load is an important issue. Moreover, the transmitter CSI is more effective when the channels dependency decreases (Figures 1, 2, and Table 1).

The effect of scheduling and multiuser diversity reduces with the channels correlation, although the rate reduction is ignorable in low correlation conditions (Figures 1 and 2). There is an interesting intuition behind this point; In a system with a number of users experiencing independent fading conditions it is more likely that, at any time instant, one of the users experiences good channel quality. Therefore, the data transmission efficiency can be improved by always communicating the best users (multiuser diversity). However, if the channels are not independent, the probability that one of the users has good channel quality while the others experience bad channels, and correspondingly the effect of multiuser diversity, decreases. Therefore, it is expected that for users close to each other, for instance the users in a single cell, e.g., [25–28]], the practical gain due to multiuser diversity would be less than the one theoretically obtained under independent channels assumptions. Note that the conclusion is valid for any number of users. Also, it is interesting to mention that, although channels correlation reduces the forward channel data transmission efficiency, it is very helpful for feedback compression of multiuser channels, as discussed in, e.g., [3–5].

Increasing the channels dependencies, the quantization boundaries converge together (Figure 3). Furthermore, with full correlation between the channels, the results are simplified to the ones obtained for singleuser networks.

The waterfilling threshold reduces as the average transmission power or the channels correlation increases (Figure 4). This is intuitively correct because with higher correlations the probability that lower channels gains realizations have the chance of data transmission increases. Therefore, they should receive more power as they have more contribution on the average rate.
Finally, note that the conclusions are valid independent of the fading distributions and the number of receivers.
6. Conclusion
This article studies the average rate of multiuser Rayleighfading channels when there is correlation between the users fading channels. The channel average rate is obtained in both perfect and imperfect transmitter CSI conditions under quasistatic channel assumption. Theoretical and simulation results show that substantial performance improvement is achieved with a limited number of feedback bits per user. On the other hand, while average rate reduction due to channels dependencies is ignorable in low correlation conditions, the effect of scheduling and multiuser diversity on the average rate reduces substantially as the fading channels dependency increases. The results are helpful for scheduling in the cases where the users are close to each other, for instance in singlecell networks. Finally, extending the results to the case of cellular networks is an interesting topic which is left for the future.
Appendix 1: Extension of the results to arbitrary number of receivers
where ${P}_{{j}_{1}...{j}_{i}}^{\prime}$is the probability that there are j_{ w }, w = 1,..., i  1, gains in the quantization region S_{ w } and (j_{ i } + 1) gains in the region S_{ i } . Replacing (30) and (31) in (13) the powerlimited average rate optimization problem can be solved based on the channels distributions. Assuming perfect CSI available at the transmitter, on the other hand, the average rate is obtained by (18) in which the auxiliary parameter Z is redefined as Z = max(G_{1},...,G_{ M } ).
Endnotes
^{a}The noise parameter Z_{ k,t } represents the Gaussian interferences received from the other users/cells as well. ^{b}As discussed in [43], the information theoretic results of quasistatic fading channels match the results of actual codes for practical code lengths, e.g., L_{c}≃ 100 channel uses. ^{c}In an AWGN channel with constant gain g and transmission power T , the maximum rate is obtained by log(1 + gT) [41]. This is particularly because, as there is perfect CSI at the receiver, likelihood decoding can be successfully implemented at the receiver.
Algorithm 1 Average rate optimization
 I.
For a given power constraint T, consider J, e.g. J = 20, randomly generated vectors ${\Lambda}^{j}=\left[{\u011d}_{1},...,{\u011d}_{N},{\stackrel{\u0303}{g}}_{0},{\stackrel{\u0303}{g}}_{1},...,{\stackrel{\u0303}{g}}_{N}\right]$ such that ${\stackrel{\u0303}{g}}_{i1}\le {\u011d}_{i}<{\stackrel{\u0303}{g}}_{i},{\stackrel{\u0303}{g}}_{0}=0,{\u011d}_{N}=\infty .$
 II.For each vector, do the following procedures
 1)
Determine the the probability terms of (7) and (10) based on (15).
 2)
Determine the average rate according to (7) and (14).
 1)
 III.
Find the vector which results in the highest average rate, i.e., Λ ^{ i } where ${\stackrel{\u0304}{R}}^{{j}_{}}\le {\stackrel{\u0304}{R}}^{i},{\forall}_{j}=1,...,J.$
 IV.
${\Lambda}^{1}\leftarrow {\Lambda}^{i}.$
 V.
Generate $b\ll J,$e.g., b = 5, vectors ${\Delta}^{j,\text{new}},j=1,...,b\phantom{\rule{0.3em}{0ex}}\text{around}\phantom{\rule{0.3em}{0ex}}{\Lambda}^{1}.$ These vectors should also satisfy the constraints introduced in I.
 VI.
${\Lambda}^{j+1}\leftarrow {\Lambda}^{j,\text{new}},j=1,...,b.$
 VII.
Regenerate the remaining vectors ${\Lambda}^{j},j=b+2,...,J$randomly such that ${\Lambda}^{j}=\left[{\u011d}_{1},...,{\u011d}_{N},{\stackrel{\u0303}{g}}_{0},{\stackrel{\u0303}{g}}_{1},...,{\stackrel{\u0303}{g}}_{N}\right]$ and ${\stackrel{\u0303}{g}}_{i1}\le {\u011d}_{i}<{\stackrel{\u0303}{g}}_{i},{\stackrel{\u0303}{g}}_{0}=0,{\u011d}_{N}=\infty .$
 VII.
Go to II and continue until convergence.
Declarations
Authors’ Affiliations
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