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Lowcomplexity multiuser MIMO downlink system based on a smallsized CQI quantizer
EURASIP Journal on Wireless Communications and Networking volume 2012, Article number: 36 (2012)
Abstract
It is known that the conventional semiorthogonal user selection based on a greedy algorithm cannot provide a globally optimal solution due to its semiorthogonal property. To find a more optimal user set and prevent the waste of the feedback resource at the base station, we present a multiuser multipleinput multipleoutput system using a random beamforming (RBF) scheme, in which one unitary matrix is used. To reduce feedback overhead for channel quality information (CQI), we propose an efficient CQI quantizer based on a closedform expression of expected SINR for selected users. Numerical results show that the RBF with the proposed CQI quantizer provides better throughput than conventional systems under minor levels of feedback.
1 Introduction
The study of multiuser multipleinput multipleoutput (MUMIMO) has focused on broadcast downlink channels as a promising solution to support high data rates in wireless communications. It is known that the MUMIMO system can serve multiple users simultaneously with reliable communications and that it can provide higher data rates than the pointtopoint MIMO system owing to multiuser diversity [1–3]. In particular, dirty paper coding (DPC) has been shown to achieve high data rates that are close to the capacity upper bound [4, 5]. However, this technique is based mainly on impractical assumption such as perfect knowledge of the wireless channel at the transmitter. To send the channel state information (CSI) back to the transmitter perfectly, considerable wireless resources are required to assist the feedback link between the base station (BS) and the mobile station (MS). This adds a high level of complexity to the communication system, which is not feasible in practice.
Numerous studies have investigated and designed MUMIMO systems that operate reliably under limited knowledge of the channel at the transmitter [6–9]. The semiorthogonal user selection (SUS) algorithm in [6] shows a simple MUMIMO system with zeroforcing beamforming (ZFBF) [10] and limited feedback [11, 12]. Although this system achieves a sumrate close to the DPC in the regime of large number of users, the overall performance is restricted seriously by a quantization error due to the mismatch between the predefined code and the normalized channel. For this reason, antenna combining techniques have been developed that decrease this quantization error using multiple antennas at the MS [7, 8]. However, the SUS algorithm based on the conventional greedy algorithm does not guarantee a globally optimized user set. Furthermore, in earlier research, quantizing the channel quality information (CQI) is not considered.
In this article, we consider a MUMIMO downlink system with minor levels of feedback in which each user sends channel direction information (CDI) quantized by a log_{2} Msized codebook instead of by the large predefined CDI codebook used in SUS. Furthermore, to reduce the feedback overhead for CQI, we propose a smallsized CQI quantizer based on the closedform expression of the CQI of selected users. It is shown that the proposed quantizer provides a point of reference for the quantizing boundaries of CQI feedback and reflects the sumrate growth resulting from multiuser diversity with only 1 or 2 bits. The proposed CQI quantizer operates well with minor levels of feedback.
The remainder of this article is organized as follows. In Section 2, we introduce the system model and propose a lowcomplexity and smallsized feedback multiantenna downlink system which is based on the random beamforming (RBF) scheme in [13]. In Section 3, we present the user selection algorithm in the RBF scheme and we review the SUS algorithm and improve upon its weaknesses. In Section 4, the closed form expression for CQI is proposed when N = M or N ≠ M respectively in order to set up the criteria of quantizing CQI. In Section 5, the numerical results are presented and Section 6 details our conclusions.
2 System model and the proposed system
We consider a singlecell MIMO downlink channel in which the BS has M antennas and each of K users has N antennas located within the BS coverage area. The channel between the BS and the MS is assumed to be a homogeneous and Rayleigh flat fading channel that has circularly symmetric complex Gaussian entries with zeromean and unit variance. In this system, we assume that the channel is frequencydependent and the MS experiences slow fading. Therefore, the channel coherence time is sufficient for sending the channel feedback information within the signaling interval. In addition, we assume that the feedback information is reported through an errorfree and nondelayed feedback channel.
The received signal for the k th user is represented as
where ${H}_{k}={\left[{\stackrel{\u0304}{h}}_{k,1}^{T},{\stackrel{\u0304}{h}}_{k,2}^{T},\dots ,{\stackrel{\u0304}{h}}_{k,N}^{T}\right]}^{T}\in {\mathbf{C}}^{N\times M}$ is a channel matrix for each user and ${\stackrel{\u0304}{h}}_{k,n}\in {\mathbf{C}}^{1\times M}$ is a channel gain vector with zeromean and unit variance for the n th antenna of the k th user. $W=\left[{\stackrel{\u0304}{w}}_{1},\phantom{\rule{2.77695pt}{0ex}}\dots ,{\stackrel{\u0304}{w}}_{M}\right]\in {\mathsf{\text{C}}}^{M\times M}$ is a ZFBF matrix for the set of selected users S, ${\stackrel{\u0304}{n}}_{k}\in {\mathbf{C}}^{N\times 1}$ is an additive white Gaussian noise vector with the covariance of I_{ N }, where I_{ N } denotes a N × N identity matrix. $\overline{s}={[{s}_{\pi (1)}\mathrm{,}\phantom{\rule{0.25em}{0ex}}\dots \mathrm{,}\phantom{\rule{0.25em}{0ex}}{s}_{\pi (M)}]}^{T}$ is the information symbol vector for the selected set of users S = {π(1), . . . , π(M)} and $\stackrel{\u0304}{x}=W\phantom{\rule{0.3em}{0ex}}\stackrel{\u0304}{s}={\sum}_{i=1}^{M}{\stackrel{\u0304}{w}}_{i}{s}_{\pi \left(i\right)}$ is the transmit symbol vector that is constrained by an average constraint power, $E\left\{{\parallel \stackrel{\u0304}{x}\parallel}^{2}\right\}=P$. ${\u0233}_{k}$ is the received signal vector at user k.
2.1 Proposed MUMIMO system
In this section, we present a lowcomplexity and smallsized feedback multipleantenna downlink system. The proposed system is based on the RBF scheme in [13] using only one unitary matrix  identity matrix I_{ M }. (This is identical to the per user unitary and rate control (PU^{2}RC) scheme in [14] which uses only one precoding matrix I_{ M }.) For this reason, it is not necessary for each user to send preferred matrix index (PMI) feedback to the BS. In the proposed system, each MS has multiple antennas and an antenna combiner such as the quantizationbased combining (QBC) in [7] or the maximum expected SINR combiner (MESC) in [8] is used. The received signal ${y}_{k,a}^{\mathsf{\text{eff}}}$ after postcoding with an antenna combiner ${\stackrel{\u0303}{\eta}}_{k,a}^{H}\in {\mathbf{C}}^{1\times N}$ is given by
We assume that perfect channel information is available at each MS and that this channel information is fed back to the BS using a feedback link. After computing all M CQIs, the MS feeds back one maximum CQIs to the BS. In this work, CQIs are quantized by the proposed quantizer with 1 or 2 bits.
With the CQIs from K users, the BS constructs the selected user set and sends the feedforward signal through the forward channels. The feedforward signal contains information about which users will be served and which codebook vector is allocated to each selected user. With the feedforward signal, selected users are able to construct proper combining vectors. The proposed RBF system illustrated in Figure 1 is described as follows.

(1)
Each user computes the direction of the effective channel for QBC in [7] using all code vectors ${\stackrel{\u0304}{c}}_{a}$ (a th row of the identity matrix I_{ M }, 1 ≤ a ≤ M) and normalizes the effective channel.
$$\begin{array}{c}{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}={\stackrel{\u0304}{c}}_{a}{Q}_{k}^{H}{Q}_{k},\phantom{\rule{1em}{0ex}}\left(1\phantom{\rule{2.77695pt}{0ex}}\le a\le M,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}1\phantom{\rule{2.77695pt}{0ex}}\le k\le K\right)\\ {\stackrel{\u0303}{h}}_{k,a}^{\mathsf{\text{eff}}}=\frac{{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}}{\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}\left\right}\end{array}$$(3)where ${Q}_{k}\dot{=}{\left[{\stackrel{\u0304}{q}}_{1}^{T},\phantom{\rule{2.77695pt}{0ex}}\dots ,{\stackrel{\u0304}{q}}_{N}^{T}\right]}^{T}$
$${\stackrel{\u0304}{q}}_{x}\in {\mathbf{C}}^{1\times M}\mathsf{\text{:}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{orthonormalbasisforspan}}\phantom{\rule{2.77695pt}{0ex}}\left({H}_{k}\right)$$$$\left\right\stackrel{\u0304}{x}\left\right\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}\left\right\stackrel{\u0304}{x}{}_{2}:=\sqrt{\stackrel{\u0304}{x}\phantom{\rule{0.3em}{0ex}}{\stackrel{\u0304}{x}}^{H}}\mathsf{\text{:}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{vectornorm}}\left(2\mathsf{\text{norm}}\right)$$ 
(2)
The combining vectors for QBC and MESC in [7, 8] are computed and then normalized to unit vector.
$${\left({\stackrel{\u0304}{\eta}}_{k,a}^{H}\right)}_{\mathsf{\text{QBC}}}={\stackrel{\u0303}{h}}_{k,a}^{\mathsf{\text{eff}}}\left({H}_{k}^{H}\right)\phantom{\rule{2.77695pt}{0ex}}{\left({H}_{k}{H}_{k}^{H}\right)}^{1},\phantom{\rule{1em}{0ex}}\left(1\phantom{\rule{2.77695pt}{0ex}}\le a\le M,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}1\phantom{\rule{2.77695pt}{0ex}}\le k\le K\right)$$(4)$${\left({\stackrel{\u0304}{\eta}}_{k,a}^{H}\right)}_{\mathsf{\text{MESC}}}={\left[{\left(I+{B}_{k}\right)}^{1}\sqrt{\rho}{H}_{k}{\stackrel{\u0304}{c}}_{a}^{T}\right]}^{H}$$(5)where ${B}_{k}=\rho \left[{H}_{k}\left(I{\stackrel{\u0304}{c}}_{a}^{H}{\stackrel{\u0304}{c}}_{a}\right){H}_{k}^{H}\right],\phantom{\rule{1em}{0ex}}\rho =P/M$
$${\stackrel{\u0303}{\eta}}_{k,a}^{H}=\frac{{\stackrel{\u0304}{\eta}}_{k,a}^{H}}{\left\right{\stackrel{\u0304}{\eta}}_{k,a}^{H}\left\right}$$ 
(3)
The expected SINR (CQI) in [6] is computed with every direction of the effective channel. The normalized effective channel of the k th user with the a th effective channel ${\stackrel{\u0303}{h}}_{k,a}^{\mathsf{\text{eff}}}$ is given as follows:
$$\mathsf{\text{CQ}}{\mathsf{\text{I}}}_{k,a}\dot{=}{\gamma}_{k,a}=E\left[\mathsf{\text{SIN}}{\mathsf{\text{R}}}_{k,a}\right]=\frac{\rho {\left\right{\stackrel{\u0303}{\eta}}_{k,a}^{H}\phantom{\rule{0.3em}{0ex}}{H}_{k}\left\right}^{2}{\mathrm{cos}}^{2}\phantom{\rule{2.77695pt}{0ex}}{\theta}_{k,a}}{1+\rho {\left\right{\stackrel{\u0303}{\eta}}_{k,a}^{H}\phantom{\rule{0.3em}{0ex}}{H}_{k}\left\right}^{2}{\mathrm{sin}}^{2}\phantom{\rule{2.77695pt}{0ex}}{\theta}_{k,a}}.$$(6)where ${\theta}_{k,a}=\mathsf{\text{arccos}}\phantom{\rule{2.77695pt}{0ex}}\left(\left{\stackrel{\u0303}{h}}_{k,a}^{\mathsf{\text{eff}}}{\stackrel{\u0304}{c}}_{a}^{H}\right\right),\phantom{\rule{1em}{0ex}}\left(1\phantom{\rule{2.77695pt}{0ex}}\le a\le M,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}1\phantom{\rule{2.77695pt}{0ex}}\le k\le K\right)$
$${\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}={\stackrel{\u0303}{\eta}}_{k,a}^{H}{H}_{k},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\stackrel{\u0303}{h}}_{k,a}^{\mathsf{\text{eff}}}=\frac{{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}}{\left\right{\overline{h}}_{k,a}^{\mathsf{\text{eff}}}\left\right}$$ 
(4)
Each user feeds back CDI and its related CQI to the BS according to the feedback scheme.
3 User selection algorithm
3.1 User selection algorithm in RBF system
In this section, we present the user selection algorithm with the CQI feedback matrix F_{ i }∈ R^{K × M}(1 ≤ i ≤ M), which is made up of CQIs from each user. In the initial feedback matrix F_{1}, the (k, a)th entry CQI_{k,a}represents the CQI feedback of the k th user with the a th effective channel. The CQI_{k,a}that is used for user selection is described in (6).

(1)
BS selects the first user π(1) and the first effective channel code (1) simultaneously with the maximum entry from the entries of the initial feedback matrix F_{1}.
$$\pi \phantom{\rule{0.3em}{0ex}}\left(1\right)=\text{arg}\underset{1\le k\le K}{\mathrm{max}}{\mathrm{CQI}}_{k,{\sigma}_{k}},\phantom{\rule{1em}{0ex}}\mathsf{\text{code}}\phantom{\rule{0.3em}{0ex}}\left(1\right)={\stackrel{\u0304}{c}}_{{\sigma}_{\pi \left(1\right)}}$$(7)where ${\sigma}_{k}=\mathrm{arg}\underset{1\le a\le M}{\mathrm{max}}{\mathrm{CQI}}_{k,a}\phantom{\rule{1em}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{1}}\le k\le K,\phantom{\rule{1em}{0ex}}\mathsf{\text{CQ}}{\mathsf{\text{I}}}_{k,a}\in {F}_{1}$

(2)
The (i + 1)th feedback matrix F_{i+1}is constructed by removing the entries of the i th users π(i) and the entries of the i th effective channels code (i) from the i th feedback matrix. After doing this, the BS selects the (i + 1)th user and the effective channel with the maximum entry from the feedback matrix F_{i+1}in (8). This user selection process is repeated until the BS constructs a selected set of users S = {π(1), . . . , π(M)} up to M.
$$\mathsf{\text{let}}\phantom{\rule{2.77695pt}{0ex}}\left(\mathsf{\text{CQ}}{\mathsf{\text{I}}}_{k,a}\in {F}_{i+1}\right)=0$$(8)
when k = π(j) or a = σ_{π(j)}, 1 ≤ j ≤ i
where ${\sigma}_{k}=\mathrm{arg}\underset{1\le a\le M}{\mathrm{max}}\mathsf{\text{CQ}}{\mathsf{\text{I}}}_{k,a}\phantom{\rule{1em}{0ex}}\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{1}}\le k\le K,\phantom{\rule{1em}{0ex}}\mathsf{\text{CQ}}{\mathsf{\text{I}}}_{k,a}\in {F}_{i+1}$
3.2 Modified SUS
In this section, we review the SUS algorithm [6] and modify it to overcome its vulnerable aspects. In the SUSbased MUMIMO system, the codebook design is based on the random vector quantization (RVQ) scheme in [15, 16]. The predefined codebook, $C=\left\{{\stackrel{\u0304}{c}}_{1},\dots ,{\stackrel{\u0304}{c}}_{{2}^{B}\mathsf{\text{CDI}}}\right\}$ of size $L={2}^{{B}_{\mathsf{\text{CDI}}}}$, is composed of L isotropically distributed unitnorm codewords in C^{1×M}, where B_{CDI} denotes the number of feedback bits for a single CDI. In the SUS algorithm, the BS tries to select users up to M out of K users. The BS selects the first user $\pi \left(1\right)=\mathrm{arg}\phantom{\rule{2.77695pt}{0ex}}\underset{k\in {A}_{1}}{\mathrm{max}}\mathsf{\text{CQ}}{\mathsf{\text{I}}}_{k,{\sigma}_{k}}$ which has the largest CQI out of the initial user set A_{1} = {1, . . . , K}. The value of $\mathsf{\text{CQ}}{\mathsf{\text{I}}}_{k,{\sigma}_{k}}\left({\sigma}_{k}=\mathrm{arg}\underset{1\le a\le {2}^{B}\mathsf{\text{CDI}}}{\mathrm{max}}\mathsf{\text{CQ}}{\mathsf{\text{I}}}_{k,a}\phantom{\rule{1em}{0ex}}for\phantom{\rule{1em}{0ex}}1\le k\le K\right)$ is described in (6) according to the antenna combiner. The BS constructs the user set,
where ${\u0125}_{k}={\stackrel{\u0303}{h}}_{k,{\sigma}_{k}}^{\mathsf{\text{eff}}}$ is a quantized effective channel vector of user k, and selects the (i + 1)th user π(i + 1) out of the user set A_{i+1}. In this formulation, the system design parameter ε, which determines the upper bound of the spatial correlation between quantized channels, is the critical parameter for the user selection. When the design parameter is set to a small value or when few users are located within the BS coverage area, user set A_{i+1}can potentially be an empty set for some cases in which i ≤ M, resulting no selection of the (i + 1)th user by the BS.
For this reason, we develop a modified SUS algorithm denoted as SUSepsilon expansion (SUSee). In SUSee, the system increases the design parameter gradually until user set A_{i+1}is not an empty set so as to guarantee the achievement of the multiplexing gain M.
With the modified user set denoted as,
the BS selects the next user π(i + 1). In this formulation, ε^{ee} is an expanded design parameter. With the proposed algorithm, the BS can construct a selected set of users S = {π(1), . . . , π(M)} with cardinality up to M.
4 Proposed CQI quantizer
In the MUMIMO downlink system, the CQI quantizer is also a critical factor determining the size of overall feedback. In this section, we derive the closed form expression of the CQI of selected users in order to quantize CQI with small bits. Then, we propose a CQI quantizer to better reflect the multiuser diversity. The proposed quantizer is derived for QBC because the distribution of the CQI resulting from QBC can be obtained analytically and is more amenable to analysis than MESC.
4.1 N= M: Closed form expression for CQI and the proposed quantizer
4.1.1 CQI quantizer under QBC
In the RBF system, identity matrix I_{ M } is considered as a codebook of log_{2} M bit size. When N = M, the combining vector is given in the shape of the row vector of the pseudo inverse channel matrix.
With the combining vector, the CQI can be represented as the product of an equally allocated power ρ and a norm of effective channel $\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2}$ since there is no CDI quantization error when N = M. The CQI feedback of the k th user with the a th effective channel is described as given by
As shown in (14), the CQI is related to the distribution of entries of the inverse channel matrix. According to [7, 17], $\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2}$ follows Chisquare distribution with variance ${\sigma}^{2}\left(\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2}~{\chi}_{2\left(MN+1\right)}^{2}\right)$ and the cdf is described as
By substituting $\frac{x}{2{\sigma}^{2}}$ with y, X and Y follow the relation X = 2σ^{2}Y. Then, the distribution of Y follows the type (iii) distribution in [[18], Theorem 4].
In that case, the approximated y can be obtained through the study of extreme value theory from order statistics. According to [18, 19], the distribution of Y satisfies following inequality
where ${a}_{{Q}_{a}}=1$, ${b}_{{Q}_{a}}=\mathrm{log}{Q}_{a}$ and Q_{ a } is the number of antennas in the a th user selection process.
When Q_{ a } is large enough, y satisfies the following approximated formulation,
where ${\gamma}_{a:{Q}_{a}}$ in (20) is the approximated value of the CQI when N = M and Q_{ a } is the number of antennas in the a th user selection process. Q_{ a } used under RBF and SUSee system will be presented in the Section 4.3.
4.1.2 CQI quantizer under MESC
While the distribution of the $\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2}$ under QBC can be obtained analytically, it is hard to analyze the distribution of the $\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2}$ under MESC. For this reason, we describe the distribution of the $\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2}$ under MESC using numerical results. According to the numerical results of MonteCarlo simulation, we assume that $\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2}$ has a Chisquare distribution with variance σ^{2} defined by
4.2 1 < N < M: Closed form expression for CQI and the proposed quantizer
In this section, we develop the closed form expression of the CQI of selected users when N ≠ M. In the case of N ≠ M, removing the quantization error between the codeword and the effective channel completely is not possible. To develop the closed form expression of the CQI of selected users, we need to derive the cdf of the CQI. For this reason, we must know the distribution of both the norm of the effective channel $\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2}$ and the quantization error term sin^{2} θ_{k,a}^{.} As explained in Section 4.1, the norm of the effective channel $\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2}$ has a Chisquare distribution $\left(\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2}~{\chi}_{2\left(MN+1\right)}^{2}\right)$. In addition, according to [7], quantization error sin^{2} θ_{k,a}follows the approximated formulation as given by
where
With the distribution of $\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2}$ and sin^{2} θ_{k,a}, we derive the cdf of CQI in the same way as in [[6], Section 5: N = 1]. At first, we derive the distribution of the interference term in Lemma 1 and it is proved in Appendix 1.
Lemma 1: (Interference term)
where
Y ~ Gamma(M  N, 1)
Proof: Appendix 1
As can be seen in Appendix 1, the interference term has a Gamma distribution, Gamma(M  N, 2σ^{2}δ).
Lemma 2: (Information signal term)
where
X ~ Gamma(1, 1), Y ~ Gamma(M  N, 1)
t = 2σ^{2}
Proof: Appendix 2
In Appendix 2, to derive the distribution of $\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2}{\mathrm{cos}}^{2}{\theta}_{k,a}$, we verify that the joint distribution of $\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2}{\mathrm{cos}}^{2}{\theta}_{k,a}$ and $\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2}{\mathrm{sin}}^{2}{\theta}_{k,a}$ is comparable with the joint distribution of I and S. Therefore, the information signal term can be described as the sum of the two Gamma variables X and Y. Furthermore, it is shown that the distribution of ${\gamma}_{k,a}=\frac{\rho \left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2}{\mathrm{cos}}^{2}{\theta}_{k,a}}{1+\rho \left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2}{\mathrm{sin}}^{2}{\theta}_{k,a}}$ is equal to the distribution of $\gamma =\frac{\rho S}{1+\rho I}$.
Lemma 3: (CQI: Expected SINR)
Define
then
Proof: Appendix 3
Since it is proved that the distribution of γ_{k,a}is equal to the distribution of γ, in Lemma 2, the cdf of γ_{k,a}can be derived using the distribution of γ. In Lemma 3, we define γ with two independent Gamma variables X and Y. For this reason, the cdf of γ can be derived using X and Y.
Theorem 1: (Largest order statistic among CQIs for Q_{ a } candidates: using extreme value theory)
For large Q_{ a }
where Q_{ a } : The number of antennas in the a th user selection process
Proof: Appendix 4
In Theorem 1, ${\gamma}_{a:{Q}_{a}}$ is the approximated value of the CQI when 1 < N < M. Since the cdf in Lemma 3 can be changed to follow the type (iii) distribution in [[18], Theorem 4], the closed form expression of CQI_{k,a}can be analyzed using the studies of extreme value theory when N ≠ M. Q_{ a } used under the RBF and SUSee system will be presented in the next section.
4.3 The number of antennas in the a th user selection process
In this section, the number of user candidates in each user selection process are described. At first, Q_{ a } used in RBF is shown as
In contrast to the RBF, the number of user candidates used in the user selection stage under the SUSee algorithm is described as follows:
where ${\alpha}_{a}=\left\{\begin{array}{cc}\hfill 1,\hfill & \hfill a=1\hfill \\ \hfill {I}_{{\epsilon}^{2}}\left(a1,\phantom{\rule{2.77695pt}{0ex}}Ma+1\right),\hfill & \hfill a>1\hfill \end{array}\right.$.
Here, I_{ z }(x, y) is the regularized incomplete beta function which determines the size of the user pool, which varies according to the user selection order [10]. The constant α_{ a } represents the probability that channel vectors of the user pool are in the set of vectors that are semiorthogonal (referred to as εorthogonal in [6]) to all of the CDIs of the formerly selected users. As explained in the Section 3.2, the design parameter ε is expanded in the modified SUSee algorithm and is assumed to be ε^{ee} = ε + 0.05 in the fourth user selection stage according to the numerical results.
4.4 CQI quantization boundary
With the closed form expression of CQI, the quantization boundary of the CQI feedback is determined. In this work, we use 1 or 2 bit size CQI (2 or 4 level) quantizers. In the case of RBF based system, the CQI quantization boundaries are represented in Table 1. The CQI quantization boundaries in SUSee based system are represented in Table 2.
4.5 Complexity analysis
In this section, the complexity of the proposed RBF system is compared to that of a SUSeebased system. The complexity comparison is described in Table 3.
The RBF system is operated under low computational complexity at the BS stage because there is no need for vector computation in the user selection procedure and precoding operation at the beamformer, unlike in SUSee. In SUSee, BS has to let the selected users know their effective channel out of ${2}^{{B}_{\mathsf{\text{CDI}}}}$ effective channels, whereas the BS selects the feedforward information for each selected user out of only M effective channels in RBF. Furthermore, at the MS stage, each user has to compute only M CQIs in RBF, whereas ${2}^{{B}_{\mathsf{\text{CDI}}}}$ CQIs should be computed in SUSee. By decreasing the computational complexity at the BS, selecting users and allocating the desired information to each antenna can be performed more reliably within the signaling interval.
5 Numerical results
The numerical performances of the proposed system are discussed. We compared the numerical results of RBF to the results of three different MUMIMO downlink systems (SUSee with antenna selection (AS) [6, 7], QBC [7] and MESC [8]). The total size of the feedback used by each user is given in Table 4.
First, Figure 2 compares the results between the SUS and the SUSee algorithm under QBC when the system design parameter ε is 0.3. As shown in Figure 2, by adaptively increasing ε in the SUSee algorithm, M users are serviced simultaneously and the sumrate is increased by about 40% when B_{CDI} = 8, K = 30 and P = 15 dB.
Figures 3 and 4 plot the performance of the proposed CQI quantizer. The CQI quantizer shows better performance than the LloydMax quantizer [20, 21] as the number of user increases. Both the proposed quantizer for RBF and the SUSee algorithm can quantize CQI effectively and minimize performance degradation with both 1 and 2 bit CQI feedback. This is attributable to the fact that the proposed CQI quantizers is a function of the number of users and the distribution of the CQI, whereas the conventional quantizer is a function of only the distribution of the CQI. The proposed quantizer for RBF shows better performance than that for SUSee because the exact number of user candidates for SUSee cannot be determined.
In Figure 5, the sumrate results from the numerical simulation and from formulation with a closed form for QBC or MESC are compared. With the closed form expression for CQI in Section 4, the sumrate formulation can be represented as follows:
where ${\gamma}_{a:{Q}_{a}}$ is the expected SINR in (20) and (41, Appendix 4), for N = M and N = M  1 case. As shown in (25), R is the sumrate which grows like M log_{2} log Q due to multiplexing and multiuser diversity gains. According to the assumption of a large user regime in the formulation with a closed form, when the number of users in the system is not large enough, a substantial difference between the numerical results and the expectation based on the closed form can be seen. However, as K increases, the difference decreases to verify the accuracy of the formulation with a closed form.
In Figure 6, RBF shows better performance than SUSeebased systems under minor feedback conditions when N = M or N = M  1. In these numerical simulations, with the QBC or MESC technique, SUSee system uses a 5bit size codebook and with the AS technique, it uses a 6 and 8bit size codebook. Although systems based on the SUSee have 2^{3} times more effective channel vectors for CQI than RBF, the user pool employed in the SUSee algorithm is determined entirely by the formerly selected users. If the previously selected users are not semiorthogonal to the rest of the users, the number of user candidates in the next user selection stage will be highly restricted. Furthermore, if the effective channel vectors of the remaining users in the user selection stage are equal to the effective channel vectors of the previously selected users, these users will not have the opportunity to be serviced because each user feeds back only one CQI. Regardless of the fact that each user can fully remove the interference when N = M, the semiorthogonality between the effective channel of users is a still critical issue of the system. By increasing the system design parameter ε, the effective channel gains for a set of selected users will be increased due to the multiuser diversity. However, the loss resulting from the normalization process in ZFBF matrix W (MoorePenrose pseudoinverse matrix of set of selected users S in [6]) also grows. For these reasons, SUSee does not guarantee that a globally optimized user set solution will be found. In RBF, the selected user set approaches a globally optimized solution because the effective channel vectors are completely orthogonal to each other. Additionally, RBF can guarantee the construction of a user set composed of up to M users, even in a small user regime.
Figures 7 and 8 display the sumrate vs. K curves with power constraint P as 10 or 20 dB. In the figures, the RBF system is operated under 3 or 4 bit feedback conditions, whereas the SUSee system is operated under 9 or 10 bit feedback conditions in Figure 7 and under 7 or 8 bit feedback conditions in Figure 8, respectively. Despite the fact that the numerical results of the RBF performance are about 2.5 bps/Hz below that of SUSee with perfect CSIT in Figure 7, they still show better performance than SUSeebased systems, especially with a small number of users. For the bestcase example, the sumrate results of RBF are 4.5 bps/Hz higher than those of two different MUMIMO systems when K = 8 and P = 20 dB employing 4bit feedback overall. As shown in Figures 7 and 8, while the size of all feedback for RBF with MESC (2 bit CQI) is 6 and 4 bits smaller than that of SUSee with MESC, respectively, the proposed system shows better throughput performance. With RBF (1 bit CQI), the system can achieve a reduction in the feedback overhead of up to 7 bits out of total 10 bits when P = 10 dB in Figure 7. When N is equal or similar to M (N = 4 or 3), the negative effect of a small candidate pool of effective channels for CQI can be offset by the positive effect from fullorthogonality between the effective channel of each user in the proposed user selection scheme.
On the other hand, when N is much smaller than M (N = 1 or 2), removing quantization error entirely is not possible. Therefore, RBF system does not guarantee higher throughput than SUSee. SUSee with QBC or MESC has more codes for antenna combinations than RBF. For this reason, these two systems have additional opportunities to reduce quantization error compared to RBF. In consequence, employing a system which uses large codebook for antenna combinations undoubtedly provides the advantage of increasing the sumrate of the system.
6 Conclusion
In this article, we propose a lowcomplexity multiantenna downlink system based on a smallsized CQI quantizer. First, in the proposed system, each user feeds back a CDI and its related CQI collected from M CQIs that are computed according to the every codeword from a codebook of log_{2} M bit size instead of using a large codebook. In addition, using the extreme value theory, the closed form expression of the expected SINR of selected users is derived. With this formulation, a CQI quantizer is proposed in order to maintain the smallsized feedback system and reflect the sumrate growth resulting from multiuser diversity. In this work, the sumrate throughput of the RBF system is obtained by MonteCarlo simulation and is compared to that of a conventional MUMIMO system based on SUS. Numerical results show that, in the proposed system, the sumrate can approach the result of SUSee with perfect CSIT, outperforming all other systems which are based on SUSee under minor amounts of feedback. Furthermore, the results show that performance degradation due to CQI quantization is negligible under the proposed lowbit quantizer. Considering the fairness level of the system, the data rates are distributed quite uniformly among M selected users for RBF, whereas the data rates are weighted too much on the first and second selected users in the SUSee algorithm. Finally, the complexity at the BS is reduced as there is no need for precoding multiplication and vector computation in the user selection procedure.
Appendix 1
Proof of Lemma 1
Using the distribution of $\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2}$ and sin^{2} θ_{k,a}, the distribution of the interference term is derived. The cdf of $\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2}{\mathrm{sin}}^{2}{\theta}_{k,a}$ is described as follows.
Appendix 2
Proof of Lemma 2
In Lemma 2, we define both the interference term and the information signal term such as ${I}_{k}=\phantom{\rule{2.77695pt}{0ex}}\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2}{\mathrm{sin}}^{2}{\theta}_{k,a}$ and ${S}_{k}=\phantom{\rule{2.77695pt}{0ex}}\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2}{\mathrm{cos}}^{2}{\theta}_{k,a}$.
At first, we develop the relation between the joint distribution of (I_{ k }, S_{ k }) and that of $\left(\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2},{\mathrm{sin}}^{2}{\theta}_{k,a}\right)$. The relation between the joint distribution of (I_{ k }, S_{ k }) and that of $\left(\left\right{\stackrel{\u0304}{h}}_{k,a}^{\mathsf{\text{eff}}}{}^{2},{\mathrm{sin}}^{2}{\theta}_{k,a}\right)$ are as given by
where
where
Then, after defining I = δtY and S = t(X + (1  δ)Y) where t = 2σ^{2}, the relation between the joint distributions of (I, S) and that of (X, Y ) are derived as follows.
where
By comparing the equations (31) and (34), we can verify that the joint distribution ${f}_{{I}_{k},{S}_{k}}\left(u,v\right)$ is the same as the joint distribution f_{I,S}(u, v). Therefore, the information signal term S_{ k } follows the distribution of S = t(X +(1  δ)Y) which is described as the sum of two Gamma variables X and Y.
Appendix 3
Proof of Lemma 3
To derive the cdf of γ_{k,a}, we define the γ using S and I in Lemma 2 $\left(\gamma =\frac{\rho S}{1+\rho I}=\frac{\rho tX+\rho t\left(1\delta \right)Y}{1+\rho \delta tY}\right)$. The distribution of γ is described as follows.
when δx + δ  1 ≥ 0
when $x\ge \frac{1\delta}{\delta}$
Appendix 4
Proof of Theorem 1
In this section, we define the relation $\gamma \prime =\frac{\gamma +1}{t\rho}.\left(\gamma =t\rho \gamma \prime 1\right)$ By substituting γ' with $\frac{\gamma +1}{t\rho}$ the cdf is changed to follow type (iii) distribution in [[18], Theorem 4].
where
Therefore, γ' can be analyzed using the studies of extreme value theory in order statistics. According to [18, 19], the distribution of γ' satisfies the following inequality
where
When Q_{ a } is large enough, γ' satisfies the following approximated formulation,
Abbreviations
 SINR:

signal to interference plus noise ratio
 SUS:

semiorthogonal user selection
 SUSee:

semiorthogonal user selection epsilon expansion
 RBF:

random beamforming
 CQI:

channel quality information
 CDI:

channel direction information
 CSI:

channel state information
 MUMIMO:

multiuser multipleinput multipleoutput
 DPC:

dirty paper coding
 BS:

base station
 MS:

mobile station
 ZFBF:

zeroforcing beamforming
 PU^{2}RC:

per user unitary and rate control
 PMI:

preferred matrix index
 RVQ:

random vector quantization
 QBC:

quantizationbased combining
 MESC:

maximum expected SINR combiner.
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Acknowledgements
This study was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (2011000316) and the Korea Communications Commission (KCC) under the R&D program supervised by the Korea Communications Agency (KCA) (KCA20110891304003).
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Song, J., Lee, J., Kim, S. et al. Lowcomplexity multiuser MIMO downlink system based on a smallsized CQI quantizer. J Wireless Com Network 2012, 36 (2012). https://doi.org/10.1186/16871499201236
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Keywords
 Mobile Station
 Antenna Selection
 User Selection
 Effective Channel
 Multiuser Diversity