On the Achievable User Number of the Downlinks in Cellular-Based Machine-to-Machine Communications

Abstract

In cellular-based machine-to-machine (M2M) networks, supporting large number of machine-type communications (MTC) users (devices) has become a critical challenge in both the uplink and downlink channels. In this paper, we focus on the downlink beamforming using zero-forcing dirty paper coding (DPC). In order to characterize the system's ability of user admission, we consider the achievable user number, which is defined as the number of users whose signal-to-interference plus-noise ratios exceed a target threshold. Due to the complexity of the optimal scheme, we propose two algorithms with random user scheduling and greedy user scheduling in maximizing the achievable user number by dynamical power assignment. Using the joint distribution of effective channel gains, we derive the achievable user number of both the scheduling schemes. An upper bound on the achievable user number of the greedy scheme is then derived which is shown to be tight when there are a large number of users. From numerical results, we show that both of the schemes enjoy a linear increase in the achievable user number as the number of transmitter antennas increases. The performance of the greedy scheduling scheme is close to the optimal scheduling scheme.

1. Introduction

Wireless machine-to-machine (M2M) networks are emerging as new types of communication, where cellular systems are expected to play an important role in wireless M2M networks [1]. In cellular-based M2M networks, supporting large number of machine-type communications (MTC) users (devices) has become a critical challenge in both the uplink [2, 3] and downlink channels. The rate required by each MTC user may be low, however, their number may be large. Thus, the study on efficient user admission policies becomes one of the critical issues in M2M communications. In this paper, we focus on improving the number of active users in the cellular-based M2M communications networks.

In the cellular communications networks, beamforming techniques with multiple antennas at the base station (BS) are used for simultaneous transmission of multiple users. The multi-antenna techniques have received a lot of attention since the heuristic research work by Telatar [4] due to its diversity and multiplexing gain. When perfect channel state information is available at the BS, the optimal downlink beamforming scheme is dirty paper coding (DPC) [5] where the interference known at the transmitter is pre-subtracted at the BS. The optimal beamforming weight vectors and encoding order of DPC can be found using a duality between the downlink and the corresponding uplink channel. In [6, 7], numerical algorithms are proposed based on this duality. However, these algorithms exhibit an undesired iterative feature.

In order to reduce the complexity, a zero-forcing structure is imposed in DPC [5], that is, zero-forcing DPC. Only random user scheduling is considered in [5]. Subsequent works [8, 9] extend zero-forcing DPC in [5] by considering the user selection and propose a greedy user scheduling scheme.

In this paper, we focus on the performance evaluation and algorithms of multi-user systems with zero-forcing DPC. We are interested in the achievable user number, which is defined as the average number of users that are admissible in the system. We say that a user is admissible if the effective signal-to-interference-plus-noise ratio (SINR) achieves a given target SINR threshold. We propose algorithms to maximize the achievable user number. The difference of the zero-forcing DPC considered in this paper from the conventional zero-forcing DPC is that the proposed algorithms should maximize the achievable user number instead of the spectral efficiency (rate). Extensive literatures have been devoted to the spectral efficiency with few exceptions, for example, [10, 11]. The achievable user number is considered in [10] for characterizing the performance of a circuit scenario.

We consider the beamforming vector design and user scheduling of zero-forcing DPC for maximizing the achievable user number. The optimal beamforming vectors for maximizing the achievable user number can be calculated by the Gram-Schmidt process which is a method for orthogonalizing a set of vectors in a successive manner. When there are more users than the number of transmitter antennas, a user scheduling problem exists. The optimal scheduling scheme can be found by solving a combinatorial optimization problem which is not easy to solve. Thus, we propose two suboptimal scheduling algorithms for maximizing the achievable user number, that is, random scheduling and greedy scheduling.

We derive the achievable user number performance of both the random and greedy user scheduling, using the joint distribution of effective channel gains. An upper bound on the achievable user number of the greedy scheme is derived. Various numerical results are present to justify our theoretical analysis. The performance of the greedy scheduling scheme is shown to be close to the optimal scheduling scheme.

The rest of the paper is organized as follows. The system model is presented in Section 2. Some preliminaries of zero-forcing DPC are given in Section 3. The algorithms for maximizing the achievable user number are proposed in Section 4. The achievable user numbers of both random scheduling scheme and greedy scheduling scheme are derived in Section 5. A comprehensive evaluation of the achievable user number of zero-forcing DPC is given in Section 6 with numerical results. We conclude the paper with some remarks in Section 8.

Notations. The superscripts T and H stand for the transpose and Hermitian transpose, respectively. Upper and lower boldfaced letters are used for matrices and column vectors, respectively. Denote by $|x|$ the absolute value of a scalar x, $||\mathbf{x}||$ the 2-norm of vector x, and I the identity matrix of a certain size implicitly given by the context. We denote by $[{\mathbf{w}}_{\mathrm{1}},{\mathbf{w}}_{\mathrm{2}},\dots ,{\mathbf{w}}_N]$ the concatenation of N column vectors, and by $[\mathbf{W},\mathbf{c}]$ the concatenation of a matrix W and a vector c of the same column size. $𝒞𝒩(\mathbf{a},\mathbf{R})$ represents the distribution of circularly symmetric complex Gaussian (CSCG) random vectors with mean vector a and covariance matrix R. We denote the probability of an event X by $\text{Pr}\hspace{0.17em}(X)$ .

2. System Model

Consider the downlink transmission of a single cell system with a base station (BS) and K users in M2M communications. The users may be the group heads of group-based MTC devices. Suppose that the BS is equipped with an antenna array of L elements, while each user is with single antenna. Multiuser downlink beamforming is used for simultaneous transmission for N (≤K) active users in the system. As the number of antennas at the BS is L, a typical value of N is L, that is, $N=L$ , when there are enough candidate users ( $K\ge L$ ). Generally, we assume that $K\ge L$ and $N\le L$ in this paper. Denote the set of indices of all K candidates as $𝒦=\{\mathrm{1,2},\dots ,K\}$ , and the set of indices of the active N users $𝒜=\{k_{\mathrm{1}},k_{\mathrm{2}},\dots ,k_N\}\subseteq 𝒦$ .

The transmitted signal vector from the antenna array to N active users, denoted by x, is given by

$$\begin{array}{cc}& \mathbf{x}=\sum _{n=\mathrm{1}}^N‍{\mathbf{w}}_{k_n}{u}_{k_n},\end{array}$$

(1)

where $u_{k_n}$ 's are the independent data symbols for the active users, and ${\mathbf{w}}_{k_n}$ 's are the corresponding beamforming vectors. The beamforming weight vectors ${\mathbf{w}}_{k_n}$ 's are normalized $L\times \mathrm{1}$ column vectors, that is, $\parallel {\mathbf{w}}_{k_n}\parallel =\mathrm{1}$ . The transmitted powers of each symbol are denoted as $P_{k_n}$ 's, and $P_{k_n}=𝔼[{|u_{k_n}|}^{\mathrm{2}}]$ . It is assumed that the total transmitted power is limited, that is, ${\sum }_n^N‍P_{k_n}\le P$ .

The signal received at the active user $k_i$ in the cell becomes

$$\begin{array}{cc}& y_{k_i}={\mathbf{h}}_{k_i}^H\mathbf{x}+n_{k_i},\end{array}$$

(2)

where ${\mathbf{h}}_{k_i}$ is the $L\times \mathrm{1}$ downlink channel vector from the transmit antenna array to the user $k_i$ , and $n_{k_i}$ is the independent additive white Gaussian noise (AWGN). It is assumed that the noise is CSCG distributed with zero mean and unit variance, that is, $n_{k_i}~𝒞𝒩(\mathrm{0,1})$ .

Substituting (1) into (2) yields

$$\begin{array}{cc}& y_{k_i}={\mathbf{h}}_{k_i}^H\sum _{n=\mathrm{1}}^N‍{\mathbf{w}}_{k_n}{u}_{k_n}+n_{k_i}={\mathbf{h}}_{k_i}^H{\mathbf{w}}_{k_i}{u}_{k_i}+{\mathbf{h}}_{k_i}^H\sum _{n\ne i}‍{\mathbf{w}}_{k_n}{u}_{k_n}+n_{k_i}.\end{array}$$

(3)

The signal-to-interference-plus-noise ratio (SINR) of user $k_i$ , denoted by ${\gamma }_{k_i}$ is given as follows:

$$\begin{array}{cc}& {\gamma }_{k_i}=\frac{{\left|{\mathbf{h}}_{k_i}^H{\mathbf{w}}_{k_i}\right|}^{\mathrm{2}}{P}_{k_i}}{{\sum }_{n\ne i}{\left|{\mathbf{h}}_{k_i}^H{\mathbf{w}}_{k_n}\right|}^2{P}_{k_n}+\mathrm{1}}.\end{array}$$

(4)

The target SINR threshold for user k is denoted as ${\mathrm{\Gamma }}_k$ . We are interested in the achievable user number which is defined as the number of users with SINR higher than the target threshold. Thus, the achievable user number becomes

$$\begin{array}{cc}& M=\sum _i^N‍ℐ\left({\gamma }_{k_i}\ge {\mathrm{\Gamma }}_{k_i}\right),\end{array}$$

(5)

where $ℐ(x)$ is the indicator function, and

$$\begin{array}{cc}& ℐ\left(x\right)=\{\begin{array}{cc}\mathrm{1},\hfill & x\hspace{0.17em}\hspace{0.17em}\text{is}\hspace{0.17em}\hspace{0.17em}\text{true};\hfill \\ \mathrm{0},\hfill & x\hspace{0.17em}\hspace{0.17em}\text{is}\hspace{0.17em}\hspace{0.17em}\text{false}.\hfill \end{array}\end{array}$$

(6)

The aim of this study is to support as much MTC users as possible in a cell, that is, to achieve a higher M.

Throughout the paper, we assume that the channel vectors ${\mathbf{h}}_k$ , $k\in 𝒦$ are available at the BS for both beamforming and scheduling.

3. Preliminaries of Zero-Forcing DPC

In this section, some preliminaries of downlink beamforming with zero-forcing DPC are given.

3.1. Dirty Paper Coding

Assume a specific DPC encoding order [12] π, which means the encoding order of the user n is $\pi (n)$ . The signal received by user $k_i$ can be expanded as

$$\begin{array}{cc}& y_{k_i}={\mathbf{h}}_{k_i}^H{\mathbf{w}}_{k_i}{u}_{k_i}+{\mathbf{h}}_{k_i}^H\sum _{\pi \left(n\right)<\pi \left(i\right)}‍{\mathbf{w}}_{k_n}{u}_{k_n}+{\mathbf{h}}_{k_i}^H\sum _{\pi \left(n\right)>\pi \left(i\right)}‍{\mathbf{w}}_{k_n}{u}_{k_n}+n_{k_i}.\end{array}$$

(7)

The signals intended for the users with encoding order ahead of the user $k_i$ cause no interference to user $k_i$ . That is, the term ${\mathbf{h}}_{k_i}^H{\sum }_{\pi (n)<\pi (i)}‍{\mathbf{w}}_{k_n}{u}_{k_n}$ in (7) is pre-subtracted by DPC at the transmitter. The resulting signal received by user $k_i$ becomes

$$\begin{array}{cc}& y_{k_i}={\mathbf{h}}_{k_i}^H{\mathbf{w}}_{k_i}{u}_{k_i}+{\mathbf{h}}_{k_i}^H\sum _{\pi \left(n\right)>\pi \left(i\right)}‍{\mathbf{w}}_{k_n}{u}_{k_n}+n_{k_i},\end{array}$$

(8)

and the SINR

$$\begin{array}{cc}& {\gamma }_{k_i}=\frac{{\left|{\mathbf{h}}_{k_i}^H{\mathbf{w}}_{k_i}\right|}^{\mathrm{2}}{P}_{k_i}}{{\sum }_{\pi \left(n\right)>\pi \left(i\right)}{\left|{\mathbf{h}}_{k_i}^H{\mathbf{w}}_{k_n}\right|}^2{P}_{k_n}+\mathrm{1}}.\end{array}$$

(9)

For the index specific order, that is, $\pi (n)=n$ , for all $n=\mathrm{1}\cdots N$ , the SINR becomes

$$\begin{array}{cc}& {\gamma }_{k_i}=\frac{{\left|{\mathbf{h}}_{k_i}^H{\mathbf{w}}_{k_i}\right|}^{\mathrm{2}}{P}_{k_i}}{{\sum }_{n>i}{\left|{\mathbf{h}}_{k_i}^H{\mathbf{w}}_{k_n}\right|}^{\mathrm{2}}{P}_{k_n}+\mathrm{1}}.\end{array}$$

(10)

It has been proven that DPC is the capacity-achieving scheme for the multiple-input multiple-output (MIMO) broadcast channel (BC). The optimal beamforming weight vectors and encoding order can be found using a duality between the BC and the corresponding multiple access channel (MAC). Several numerical algorithms have been proposed, for example, [6, 7]. However it has been recognized that these algorithms exhibit an undesired iterative feature.

3.2. Zero-Forcing DPC

In order to reduce the complexity of the calculation, a zero-forcing structure is imposed to the beamforming weight vectors, resulting in a simplified scheme, zero-forcing DPC scheme. The term ${\mathbf{h}}_{k_i}^H{\sum }_{\pi (n)>\pi (i)}‍{\mathbf{w}}_{k_n}{u}_{k_n}$ in (7) is nulled by the beamforming vector design, which is

$$\begin{array}{cc}& {\mathbf{h}}_{k_i}^H{\mathbf{w}}_{k_n}=\mathrm{0},\hspace{1em}\forall \pi \left(n\right)>\pi \left(i\right).\end{array}$$

(11)

The resulting signal received by user $k_i$ becomes

$$\begin{array}{cc}& y_{k_i}={\mathbf{h}}_{k_i}^H{\mathbf{w}}_{k_i}{u}_{k_i}+n_{k_i},\end{array}$$

(12)

and the SINR

$$\begin{array}{cc}& {\gamma }_{k_i}={\left|{\mathbf{h}}_{k_i}^H{\mathbf{w}}_{k_i}\right|}^2{P}_{k_i}=g_i{P}_{k_i},\end{array}$$

(13)

where $g_i=|{\mathbf{h}}_{k_i}^H{\mathbf{w}}_{k_i}{|}^{\mathrm{2}}$ denotes the effective channel gain of the ith user.

4. Algorithm for Maximizing User Number

For a specific DPC encoding order π and N scheduled users $𝒜=\{k_{\mathrm{1}},k_{\mathrm{2}},\dots ,k_N\}$ , the achieved user number becomes

$$\begin{array}{cc}& M_{\pi ,𝒜}=\sum _i^N‍ℐ\left({\gamma }_{k_i}\ge {\mathrm{\Gamma }}_{k_i}\right),\end{array}$$

(14)

where ${\gamma }_{k_i}$ is given by (13) and ${\mathbf{w}}_{k_i}$ is designed subject to the constraints (11). The optimal beamforming vectors can be found by solving the following optimization problem:

$$\begin{array}{l}\underset{\mathbf{W},𝒫}{\text{maximize}}\hspace{0.17em}\hspace{1em}{M}_{\pi ,𝒜}\\ \text{subject}\hspace{0.17em}\text{to}\hspace{1em}{\mathbf{h}}_{k_i}^H{\mathbf{w}}_{k_n}=\mathrm{0},\hspace{1em}\forall \pi \left(n\right)>\pi \left(i\right),\\ \phantom{\text{subject}\hspace{0.17em}\hspace{0.17em}\text{to}\hspace{1em}\hspace{1em}}\sum _i^N‍P_{k_i}\le P,\end{array}$$

(15)

where $\mathbf{W}=[{\mathbf{w}}_{{\tilde{k}}_{\mathrm{1}}},{\mathbf{w}}_{{\tilde{k}}_{\mathrm{2}}},\dots ,{\mathbf{w}}_{{\tilde{k}}_N}]$ , ${\tilde{k}}_i=k_j,\pi (j)=i$ , and $𝒫=\{P_{k_{\mathrm{1}}},\dots ,P_{k_N}\}$ is the power assignment.

From $\text{Theorem}$ 1 of [13], the optimal solution to the optimization problem in (15) is

$$\begin{array}{cc}& {\mathbf{W}}_{\text{opt}\mathrm{\hspace{0.17em}\hspace{0.17em}}}=\mathbf{Q},\end{array}$$

(16)

where Q is obtained from $\mathbf{Q}\mathbf{R}$ decomposition of the channel matrix $\mathbf{H}=[{\mathbf{h}}_{{\tilde{k}}_{\mathrm{1}}},{\mathbf{h}}_{{\tilde{k}}_{\mathrm{2}}},\dots ,{\mathbf{h}}_{{\tilde{k}}_N}]$ ,

$$\begin{array}{cc}& \mathbf{H}=\mathbf{Q}\mathbf{R},\end{array}$$

(17)

and R is an upper triangular matrix.

The QR decomposition can be implemented using the Gram-Schmidt process which is a method for orthogonalizing a set of vectors in a successive manner. More efficient algorithm for QR decomposition can be carried out by the Householder transformation [14].

In a system where $K>N$ , the optimal achievable user number can be found by

$$\begin{array}{cc}& M=\underset{\pi ,𝒜}{\max \hspace{0.17em}}\hspace{0.17em}\hspace{0.17em}{M}_{\pi ,𝒜}.\end{array}$$

(18)

Due to the successive nature of QR decomposition, the user scheduling process can be easily incorporated into the Gram-Schmidt process as in Algorithm 1.

Algorithm 1: Algorithm of zero-forcing dirty paper coding with user scheduling for maximizing user number.

Input: $h_k,P$ .

Output: $\pi ,𝒜,𝒫,\mathbf{W}$ .

 1: Initialize $𝒜=\varnothing$ , $𝒦=\{\mathrm{1,2},\dots ,K\}$ , $n=\mathrm{1}$ , $\mathbf{W}=$

    $[],\mathbf{H}=[]$ .

 2: while $n\le N$ do

 3: Choose a user $k_n$ from 𝒦

 4:   ${\mathbf{w}}_{k_n}=\frac{{\tilde{\mathbf{w}}}_{k_n}}{\parallel {\tilde{\mathit{w}}}_{k_n}\parallel }$ , where ${\tilde{\mathbf{w}}}_{k_n}=(\mathbf{I}-\mathbf{W}{\mathbf{W}}^H){\mathbf{h}}_{k_n}$ .

 5:  $P_{k_n}=\frac{{\mathrm{\Gamma }}_{k_n}}{{|{\mathbf{h}}_{k_n}^H{\mathbf{w}}_{k_n}|}^{\mathrm{2}}}$ .

 6: if ${\sum }_{i=\mathrm{1}}^n\mathrm{‍}{P}_{k_i}\le P$ then

 7:    $\mathbf{H}\leftarrow [\mathbf{H},{\mathbf{h}}_{k_n}]$ .

 8:    $\mathbf{W}\leftarrow [\mathbf{W},{\mathit{w}}_{k_n}]$ .

 9:    $𝒜\leftarrow 𝒜\cup \{k_n\}$ , $𝒦\leftarrow 𝒦\setminus \{k_n\}$ .

 10:    $\pi (n)=n$ .

 11:    $n\leftarrow n+\mathrm{1}$ .

 12:  else

 13:   break.

 14:  endif

  $\mathrm{15}$ : endwhile

 16: return $\pi ,𝒜,𝒫,\mathbf{W}$ .

In Algorithm 1, the powers are assigned dynamically to each active user according to their channel conditions.

The step 4 of Algorithm 1 can be expressed equivalently as

$$\begin{array}{cc}& {\tilde{\mathbf{w}}}_{k_n}=\left(\mathbf{I}-\mathbf{H}{\left({\mathbf{H}}^H\mathbf{H}\right)}^{-\mathrm{1}}{\mathbf{H}}^H\right){\mathbf{h}}_{k_n}.\end{array}$$

(19)

The optimal user scheduling, step 3 of Algorithm 1, can be implemented by an exhaustive search of all possible active user set 𝒜 and encoding order π. The scope of search are $K!/(K-N)!$ possible combinations, which means Algorithm 1 with specified 𝒜 and π should be running $K!/(K-N)!$ times before the optimal user scheduling can be found.

The optimal user scheduling is a combinatorial optimization problem which is obviously not easy to solve. Therefore, two suboptimal user scheduling methods are considered in this study. The first heuristic method is random scheduling, in which a user is randomly scheduled for downlink beamforming. In random scheduling, the step 3 of Algorithm 1 is

$$\begin{array}{cc}& \text{Randomly}\hspace{0.17em}\hspace{0.17em}\text{choose}\hspace{0.17em}\hspace{0.17em}\text{a}\hspace{0.17em}\hspace{0.17em}\text{user}\hspace{0.17em}\hspace{0.17em}{k}_n\hspace{0.17em}\hspace{0.17em}\text{from}\hspace{0.17em}\hspace{0.17em}𝒦.\end{array}$$

(20)

The second method is greedy scheduling. In greedy scheduling, the user requires the least power to achieve SINR threshold is to be selected. Thus, the step 4 of Algorithm 1 becomes

$$\begin{array}{cc}& \text{Choose}\hspace{0.17em}\hspace{0.17em}\text{a}\hspace{0.17em}\hspace{0.17em}\text{user}\hspace{0.17em}\hspace{0.17em}{k}_n\hspace{0.17em}\hspace{0.17em}\text{from}\hspace{0.17em}\hspace{0.17em}𝒦\hspace{0.17em}\hspace{0.17em}\text{with}\hspace{0.17em}\hspace{0.17em}\text{the}\hspace{0.17em}\hspace{0.17em}\text{least}\hspace{0.17em}\hspace{0.17em}{P}_{k_n}.\end{array}$$

(21)

5. Achievable User Number Performance

In this section, we derive the achievable user number of zero-forcing DPC with both random and greedy scheduling for maximizing the achievable user number.

The achievable user number M is a discrete random variable with possible values $\{\mathrm{0,1},\mathrm{2},\dots ,N\}$ , and the probability distribution

$$\begin{array}{cc}& \text{Pr}\hspace{0.17em}\left(\xi =m\right),\hspace{1em}m=\mathrm{0,1},\mathrm{2},\dots ,N.\end{array}$$

(22)

The average achieved user number, denoted by $\stackrel{-}{M}$ , is given by

$$\begin{array}{cc}& \stackrel{-}{M}=\sum _{m=\mathrm{1}}^N‍m\text{Pr}\hspace{0.17em}\left(\xi =m\right).\end{array}$$

(23)

The probability

$$\begin{array}{cc}& \text{Pr}\hspace{0.17em}\left(\xi =m\right)=\{\begin{array}{cc}\text{Pr}\hspace{0.17em}\left(\sum _{i=\mathrm{1}}^m‍P_i\le P,\sum _{i=\mathrm{1}}^{m+\mathrm{1}}‍P_i>P\right),\hfill & \mathrm{0}\le m\le N-\mathrm{1};\hfill \\ \text{Pr}\hspace{0.17em}\left(\sum _{i=\mathrm{1}}^m‍P_i\le P\right),\hfill & m=N,\hfill \end{array}\end{array}$$

(24)

where

$$\begin{array}{cc}& P_n=\frac{{\mathrm{\Gamma }}_n}{{\left|{\mathbf{h}}_n^H{\mathbf{w}}_n\right|}^{\mathrm{2}}}=\frac{{\mathrm{\Gamma }}_n}{g_n}.\end{array}$$

(25)

We denote the domain of (24) as ${𝒟}_m$ . When $\mathrm{0}\le m\le N-\mathrm{1}$ , ${𝒟}_m$ is

$$\begin{array}{cc}& \{\begin{array}{c}{P}_{\mathrm{1}}\le P,\hfill \\ P_{\mathrm{2}}\le P-P_{\mathrm{1}},\hfill \\ \phantom{g}\phantom{g}⋮\hfill \\ P_m\le P-\sum _{i=\mathrm{1}}^{m-\mathrm{1}}‍P_i,\hfill \\ P_{m+\mathrm{1}}>P-\sum _{i=\mathrm{1}}^m‍P_i.\hfill \end{array}⟺\{\begin{array}{c}{g}_{\mathrm{1}}\ge \frac{{\mathrm{\Gamma }}_{\mathrm{1}}}{P},\hfill \\ g_{\mathrm{2}}\ge \frac{{\mathrm{\Gamma }}_{\mathrm{2}}}{P-\left({\mathrm{\Gamma }}_{\mathrm{1}}/g_{\mathrm{1}}\right)},\hfill \\ \phantom{g}\phantom{g}⋮\hfill \\ g_m\ge \frac{{\mathrm{\Gamma }}_m}{P-{\sum }_{i=\mathrm{1}}^{m-\mathrm{1}}\left({\mathrm{\Gamma }}_i/g_i\right)}\hfill \\ g_{m+1}<\frac{{\mathrm{\Gamma }}_{m+1}}{P-{\sum }_{i=\mathrm{1}}^{m-\mathrm{1}}\left({\mathrm{\Gamma }}_i/g_i\right)}.\hfill \end{array}\end{array}$$

(26)

When $m=N$ , ${𝒟}_m$ is

$$\begin{array}{cc}& \{\begin{array}{c}{P}_{\mathrm{1}}\le P,\hfill \\ P_{\mathrm{2}}\le P-P_{\mathrm{1}},\hfill \\ \phantom{g}\phantom{g}⋮\hfill \\ P_m\le P-\sum _{i=\mathrm{1}}^{m-\mathrm{1}}‍P_i.\hfill \end{array}⟺\{\begin{array}{c}{g}_{\mathrm{1}}\ge \frac{{\mathrm{\Gamma }}_{\mathrm{1}}}{P},\hfill \\ g_{\mathrm{2}}\ge \frac{{\mathrm{\Gamma }}_{\mathrm{2}}}{P-\left({\mathrm{\Gamma }}_{\mathrm{1}}/g_{\mathrm{1}}\right)},\hfill \\ \phantom{g\phantom{g}}⋮\hfill \\ g_m\ge \frac{{\mathrm{\Gamma }}_m}{P-{\sum }_{i=\mathrm{1}}^{m-\mathrm{1}}\left({\mathrm{\Gamma }}_i/g_i\right)}.\hfill \end{array}\end{array}$$

(27)

The probability $\text{Pr}(\xi =m)$ can be calculated using the joint distribution of the effective channel gains,

$$\begin{array}{l}\text{Pr}\hspace{0.17em}\left(\xi =m\right)=\int ‍\dots {\int }_{{𝒟}_m}‍f_{g_{\mathrm{1}},g_{\mathrm{2}},\dots ,g_{m+\mathrm{1}}}\left(x_{\mathrm{1}},x_{\mathrm{2}},\dots ,x_{m+\mathrm{1}}\right)\\ \times d{x}_{m+\mathrm{1}}\cdots d{x}_{\mathrm{2}}d{x}_{\mathrm{1}},\end{array}$$

(28)

where the integration domain ${𝒟}_m$ is specified by the right sides of (26) and (27).

For the purpose of analysis, we make the following CSCG assumption as those in many literatures, for example, [15, 16].

Assumption 1.

The elements of each user's channel vector ${\mathbf{h}}_k$ are independent and identically (i.i.d.) distributed and CSCG random variables, $𝒞𝒩(\mathrm{0,1})$ .

Assumption 1 is reasonable in rich scattered environment. The path loss encountered by each user is assumed to be the same, and their channel directions are uniformly distributed. Assumption 1 simplifies the following performance analysis. It is noted that the algorithm in the Section 4 is not based on Assumption 1.

Besides, we assume that all the users have equal target SINR threshold, that is,

$$\begin{array}{cc}& {\mathrm{\Gamma }}_{\mathrm{1}}={\mathrm{\Gamma }}_{\mathrm{2}}=\cdots ={\mathrm{\Gamma }}_K=\mathrm{\Gamma }.\end{array}$$

(29)

The superscripts $R,G$ and U in this section denote the random scheduling, greedy scheduling and upper bound, respectively.

5.1. Achievable User Number with Random Scheduling

In the random scheduling scheme, we have the following lemma [5].

Lemma 1.

The channel gains $g_i$ 's are independently distributed as $g_i~{\chi }_{\mathrm{2}(L-i+\mathrm{1})}^{\mathrm{2}}$ , where ${\chi }_{\mathrm{2}(L-i+\mathrm{1})}^{\mathrm{2}}$ denotes the central Chi-squared distribution with $\mathrm{2}(L-i+\mathrm{1})$ degrees of freedom.

Thus, the probability density function (PDF) of each channel gain $g_i$ is

$$\begin{array}{cc}& f_{g_i}^R\left(x_i\right)=\frac{x_i^{L-i}{e}^{-x_i}}{\left(L-i\right)!},\hspace{1em}i=\mathrm{1,2},\dots ,N,\end{array}$$

(30)

and the cumulative distribution function (CDF)

$$\begin{array}{cc}& F_{g_i}^R\left(x_i\right)=Q\left(x_i,L-i+\mathrm{1}\right),\hspace{1em}i=\mathrm{1,2},\dots ,N,\end{array}$$

(31)

where $Q(·)$ denotes the regularized incomplete gamma function (The regularized incomplete gamma function $Q(x,a)=(\mathrm{1}/\mathrm{\Gamma }(a)){\int }_{\mathrm{0}}^x‍e^{-t}{t}^{a-\mathrm{1}}dt$ , and $\mathrm{\Gamma }(·)$ is the gamma function. We use the MATLAB function gammainc to calculate the value).

Thus, the probability becomes

$$\begin{array}{l}\text{Pr}\hspace{0.17em}\left(\xi =m\right)=\int ‍\cdots {\int }_{{𝒟}_m}‍f_{g_{\mathrm{1}}}^R\left(x_{\mathrm{1}}\right)f_{g_{\mathrm{2}}}^R\left(x_{\mathrm{2}}\right)\cdots f_{g_{m+\mathrm{1}}}^R\left(x_{m+\mathrm{1}}\right)\\ \hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times d{x}_{m+\mathrm{1}}\cdots d{x}_{\mathrm{2}}d{x}_{\mathrm{1}},\end{array}$$

(32)

From (32), (30) and (23), we can calculate the average achievable user number of the random scheduling scheme.

5.2. Achievable User Number with Greedy Scheduling

In the greedy scheduling scheme, it can be shown that

$$\begin{array}{cc}& g_{\mathrm{1}}\ge g_{\mathrm{2}}\ge \cdots \ge g_N.\end{array}$$

(33)

Therefore, the integration domain with ordered effective channel gains, denoted by ${𝒟}_m^o$ , becomes

$$\begin{array}{cc}& \{\begin{array}{c}{g}_{\mathrm{1}}\ge \frac{{\mathrm{\Gamma }}_{\mathrm{1}}}{P}\\ g_{\mathrm{1}}\ge g_{\mathrm{2}}\ge \frac{{\mathrm{\Gamma }}_{\mathrm{2}}}{P-\left({\mathrm{\Gamma }}_{\mathrm{1}}/g_{\mathrm{1}}\right)},\\ ⋮\\ g_{m-\mathrm{1}}\ge g_m\ge \frac{{\mathrm{\Gamma }}_m}{P-{\sum }_{i=\mathrm{1}}^{m-\mathrm{1}}\left({\mathrm{\Gamma }}_i/g_i\right)},\\ g_{m+\mathrm{1}}<\frac{{\mathrm{\Gamma }}_{m+\mathrm{1}}}{P-{\sum }_{i=\mathrm{1}}^m\left({\mathrm{\Gamma }}_i/g_i\right)},\end{array}\hspace{1em}\mathrm{0}\le m\le N-\mathrm{1},\end{array}$$

(34)

$$\begin{array}{cc}& \{\begin{array}{c}\begin{array}{c}{g}_{\mathrm{1}}\ge \frac{{\mathrm{\Gamma }}_{\mathrm{1}}}{P},\\ g_{\mathrm{1}}\ge g_{\mathrm{2}}\ge \frac{{\mathrm{\Gamma }}_{\mathrm{2}}}{P-\left({\mathrm{\Gamma }}_{\mathrm{1}}/g_{\mathrm{1}}\right)},\\ ⋮\\ g_{m-\mathrm{1}}\ge g_m\ge \frac{{\mathrm{\Gamma }}_m}{P-{\sum }_{i=\mathrm{1}}^{m-\mathrm{1}}\left({\mathrm{\Gamma }}_i/g_i\right)},\end{array}\end{array}\hspace{1em}m=N.\end{array}$$

(35)

The probability $\text{Pr}(\xi =m)$ becomes

$$\begin{array}{l}\text{Pr}\hspace{0.17em}\left(\xi =m\right)=\int ‍\cdots {\int }_{{𝒟}_m^o}‍f_{g_{\mathrm{1}},g_{\mathrm{2}},\dots ,g_{m+\mathrm{1}}}^G\left(x_{\mathrm{1}},x_{\mathrm{2}},\dots ,x_{m+\mathrm{1}}\right)\\ \hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times d{x}_{m+\mathrm{1}}\cdots d{x}_{\mathrm{2}}d{x}_{\mathrm{1}},\end{array}$$

(36)

where the joint distribution of the ordered effective channel gains in the greedy scheduling scheme is given in the Lemma 2.

Following the derivation in [9], we can have the lemma.

Lemma 2.

The channel gains $g_i$ are jointly distributed as

$$\begin{array}{l}{f}_{g_{\mathrm{1}},g_{\mathrm{2}},\dots ,g_{m+\mathrm{1}}}^G\left(x_{\mathrm{1}},x_{\mathrm{2}},\dots ,x_{m+\mathrm{1}}\right)\\ \hspace{0.17em}\hspace{1em}\hspace{1em}=\frac{K!}{(K-m-\mathrm{1})}{\left[{\int }_{y=\mathrm{0}}^{x_{m+\mathrm{1}}}‍{\varphi }_{m+\mathrm{1}}(y,x_m,\dots ,x_{\mathrm{1}})dy\right]}^{K-m-\mathrm{1}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\times \prod _{i=\mathrm{1}}^{m+\mathrm{1}}‍{\varphi }_i(x_i,\dots ,x_{\mathrm{1}}),\end{array}$$

(37)

where

$$\begin{array}{l}{\varphi }_i\left(x_i,x_{i-\mathrm{1}},\dots ,x_{\mathrm{1}}\right)=\frac{\mathrm{1}}{\mathrm{\Gamma }\left(L-i+\mathrm{1}\right)x_i^{L-i}}\\ \hspace{1em}\hspace{1em}\times {\int }_{v_{i-\mathrm{1}}=x_i}^{x_{i-\mathrm{1}}}‍{\int }_{v_{i-\mathrm{2}}=v_{i-\mathrm{1}}}^{x_{i-\mathrm{2}}}‍\cdots {\int }_{v_{\mathrm{1}}=v_{\mathrm{2}}}^{x_{\mathrm{1}}}‍d{v}_{\mathrm{1}}\cdots d{v}_{i-\mathrm{1}},\end{array}$$

(38)

$$\begin{array}{ll}& {\varphi }_{\mathrm{1}}\left(x_{\mathrm{1}}\right)=\frac{\mathrm{1}}{\mathrm{\Gamma }\left(L\right)}{x}_{\mathrm{1}}^{L-\mathrm{1}}{e}^{-x_{\mathrm{1}}}.\end{array}$$

(39)

Proof.

The derivation follows from that of Proposition  1 in [9] with a notable exception that we proof the important Claim  1 in [9]. We give an equivalent of Claim  1 in [9] in Lemma 3.

Lemma 3.

Let h and q denote independent L-dimensional random column vectors with i.i.d., CSCG entries with zero mean and unit variance. Let $Y:={\mathbf{h}}^H\mathbf{h}$ and $X:={\mathbf{h}}^H\mathbf{Q}\mathbf{h}$ , where $\mathbf{Q}=\mathbf{I}-\mathbf{q}({\mathbf{q}}^H\mathbf{q}{)}^{-\mathrm{1}}{\mathbf{q}}^H$ (cf. (19)). Then, the CDF of X, given Y, is given by

$$\begin{array}{cc}& F_{X\mid Y}\left(x\mid y\right)=\{\begin{array}{cc}{\left(\frac{x}{y}\right)}^{L-\mathrm{1}},\hfill & \text{for}\hspace{0.17em}\hspace{0.17em}x\in \left[\mathrm{0},y\right],\hfill \\ \mathrm{0},\hfill & \text{elsewhere}.\hfill \end{array}\end{array}$$

(40)

Proof.

See Appendix A.

From (37), (36) and (23), we can calculate the average achievable user number of the greedy scheduling scheme.

Inspired by the distribution in Lemma 1, an upper bound of the achievable user number can be found using the following lemma.

Lemma 4.

The best performance of the greedy user scheduling can be achieved when the channel gains $g_i$ 's are independently distributed as the maximum of $K-i+\mathrm{1}$ random variables with distribution ${\chi }_{\mathrm{2}(L-i+\mathrm{1})}^{\mathrm{2}}$ .

Thus, using order statistics, the CDF of each channel gain $g_i$ is

$$\begin{array}{cc}& F_{g_i}^U\left(x_i\right)={\left[F_{g_i}^R\left(x_i\right)\right]}^{K-i+\mathrm{1}},\end{array}$$

(41)

and the PDF

$$\begin{array}{cc}& f_{g_i}^U\left(x_i\right)=\left(K-i+\mathrm{1}\right){\left[F_{g_i}^R\left(x_i\right)\right]}^{K-i}{f}_{g_i}^R\left(x_i\right).\end{array}$$

(42)

The probability becomes

$$\begin{array}{l}\text{Pr}\hspace{0.17em}\left(\xi =m\right)=\int ‍\cdots {\int }_{{𝒟}_m}‍f_{g_{\mathrm{1}}}^U\left(x_{\mathrm{1}}\right)f_{g_{\mathrm{2}}}^U\left(x_{\mathrm{2}}\right)\cdots f_{g_{m+\mathrm{1}}}^U\left(x_{m+\mathrm{1}}\right)\\ \hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times d{x}_{m+\mathrm{1}}\cdots d{x}_{\mathrm{2}}d{x}_{\mathrm{1}}.\end{array}$$

(43)

From (43), (42) and (23), we can calculate the upper bound on the average achievable user number of the greedy scheduling scheme.

6. Simulations

In this section, various numerical results are presented. In Section 6.1, the theoretic results derived from Section 5 are shown. In Section 6.2, we compare the performance of the proposed schemes with other beamforming schemes. In Section 6.3, the performances of the random and greedy scheduling schemes are compared with the optimal scheme. In Section 6.4, the improvements of the scheme with dynamic power assignment over that with equal power assignment are presented. The performance characterizations of the algorithms with dynamic power assignment are given in Section 6.5.

In the following simulations, samples of each user's channel vector ${\mathbf{h}}_k$ are drawn from CSCG distribution as specified in Assumption 1. The noise power spectral density is set at 1. $\mathrm{1}{\mathrm{0}}^{\mathrm{5}}$ channel realizations are used to calculate the mean value of achievable user number.

6.1. Theoretic Results

In Figure 1 the achievable user number performance of the schemes with dynamic power assignment is plotted for different SINR thresholds $\mathrm{\Gamma }$ , with $K=\mathrm{4}$ , $L=\mathrm{2}$ , $N=\mathrm{2}$ , and $P=\mathrm{3}$  dB. The analytical result of the random scheduling is obtained from (32), and that of the greedy scheduling from (36). The analytical lines in Figure 1 agree with those from simulations.

Figure 1

Achievable user number versus SINR threshold, with $K=\mathrm{4}$ , $L=\mathrm{2}$ , $N=\mathrm{2}$ , and $P=\mathrm{3}$  dB.

The upper bound of the greedy scheduling scheme with dynamic power assignment is shown in Figure 2, with $L=\mathrm{4}$ , $N=\mathrm{4}$ , and $P=\mathrm{6}\hspace{0.17em}$ dB. The upper bound line is obtained from (43) in Section 5. Different number of users $K=\mathrm{8}$ and $K=\mathrm{24}$ are considered. The upper bound becomes tighter when there are larger number of users.

Figure 2

Achievable user number versus SINR threshold, with $L=\mathrm{4}$ , $N=\mathrm{4}$ , and $P=\mathrm{6}$  dB.

These results justify our analysis in Section 5.

6.2. Comparison with Other Beamforming Schems

In Figure 3, the achievable user numbers of different beamforming schemes are shown. Four active users are considered with $K=\mathrm{6}$ , $L=\mathrm{4}$ , and $P=\mathrm{6}\hspace{0.17em}$ dB. The performance of zero-forcing DPC (ZF DPC) scheme is compared with orthogonal beamforming (OBF) [17, 18] and zero-forcing beamforming (ZFBF) schemes with dynamic power assignment. In ZFBF scheme with greedy scheduling, we select the first four users with largest channel gains. As expected, the zero-forcing DPC scheme outperforms both OBF and ZFBF schemes.

Figure 3

Achievable user number of different beamforming schemes, with $K=\mathrm{6}$ , $L=\mathrm{4}$ , $N=\mathrm{4}$ , and $P=\mathrm{6}$  dB.

6.3. Comparison with the Optimal Scheduling

The achievable user numbers of both random and greedy scheduling schemes are presented in Figures 4 and 5 compared with the optimal scheduling scheme. The active user set and DPC encoding order of the optimal scheduling scheme are found by an exhaustive search.

Figure 4

Achievable user number versus SINR threshold, with $K=\mathrm{6}$ , $L=\mathrm{3}$ , $N=\mathrm{3}$ , and $P=\mathrm{5}$  dB.

Figure 5

Achievable user number versus SINR threshold, with $K=\mathrm{6}$ , $L=\mathrm{4}$ , $N=\mathrm{4}$ , and $P=\mathrm{6}\hspace{0.17em}$ dB.

In Figure 4 three active users are considered with $K=\mathrm{6}$ , $L=\mathrm{3}$ , and $P=\mathrm{5}\hspace{0.17em}$ dB. The greedy scheme achieves a significant portion of the user number compared with the optimal scheme. In Figure 5 four active users are considered with $K=\mathrm{6}$ , $L=\mathrm{4}$ , and $P=\mathrm{6}\hspace{0.17em}$ dB. It is shown that the greedy scheme achieve performance close to the optimal scheme.

6.4. Improvement over Equal Power Assignment Schemes

In the equal power assignment schemes, $P_{\mathrm{1}}=P_{\mathrm{2}}=\cdots =P_N=(P/N)$ .

In Figure 6 the performance of the dynamic power assignment schemes is compared with that of the equal power assignment schemes for an increasing SINR threshold $\mathrm{\Gamma }$ , with $K=\mathrm{12}$ , $L=\mathrm{8}$ , $N=\mathrm{8}$ , and $P=\mathrm{9}\hspace{0.17em}$ dB. As expected, the dynamic power assignment schemes outperform the equal power assignment schemes. In Figure 6, the achievable user number of greedy scheduling with dynamic power assignment is more than twice of that of greedy scheduling with equal power assignment at SINR threshold 10 dB.

Figure 6

Achievable user number versus SINR threshold, with $K=\mathrm{12}$ , $L=\mathrm{8}$ , $N=\mathrm{8}$ , and $P=\mathrm{9}$  dB.

In Figure 7 the performance is compared for different number of transmitter antennas, with $K=\mathrm{16}$ , $N=\mathrm{4}$ , $P=\mathrm{6}$  dB and $\mathrm{\Gamma }=\mathrm{10}$  dB. It is shown that the dynamic power assignment schemes of both greedy and random scheduling need less transmitter antennas than the equal power assignment schemes to achieve the best user number, that is, $N=\mathrm{4}$ . The number of antennas for the dynamic power assignment scheme with greedy scheduling to achieve a user number of $\mathrm{4}$ is $L=\mathrm{10}$ , while that for the equal power assignment scheme is $L=\mathrm{14}$ .

Figure 7

Achievable user number versus number of transmitter antennas, with $K=\mathrm{16}$ , $N=\mathrm{4}$ , $P=\mathrm{6}$  dB and $\mathrm{\Gamma }=\mathrm{10}$  dB.

6.5. Performance Characterizations

The performances of the dynamic power assignment schemes with both greedy and random scheduling are investigated.

In Figure 8, the performance is shown for an increasing number of antennas L, with $K=\mathrm{16}$ , $P=\mathrm{6}$  dB and $\mathrm{\Gamma }=\mathrm{10}$  dB. The number of scheduled users is not restricted. Both of the schemes enjoy a linear increase as the number of transmitter antennas increases.

Figure 8

Achievable user number versus number of transmitter antennas, with $K=\mathrm{16}$ , $P=\mathrm{6}$  dB and $\mathrm{\Gamma }=\mathrm{10}$  dB. The number of scheduled users is not restricted.

The achievable user number is shown in Figure 9 versus number of scheduled users, with $K=\mathrm{16}$ , $L=\mathrm{16}$ , $P=\mathrm{10}$  dB and $\mathrm{\Gamma }=\mathrm{10}$  dB. The achievable user number of both schemes is saturated when $N\ge \mathrm{11}$ due to total power constraint, while that of the greedy scheme is higher than the random scheme.

Figure 9

Achievable user number versus number of scheduled users, with $K=\mathrm{16}$ , $L=\mathrm{16}$ , $P=\mathrm{10}$  dB and $\mathrm{\Gamma }=\mathrm{10}$  dB.

The achievable user number is plotted in Figure 10 versus total number of users, with $L=\mathrm{16}$ , $P=\mathrm{12}$  dB and $\mathrm{\Gamma }=\mathrm{10}$  dB. As expected, the random scheme can not benefit from multiuser diversity of the system. It can be seen that the greedy scheduling scheme achieves a diversity gain of about $\mathrm{25}\%$ when $K=\mathrm{60}$ over the random scheme.

Figure 10

Achievable user number versus total number of users, with $L=\mathrm{16}$ , $P=\mathrm{12}$  dB and $\mathrm{\Gamma }=\mathrm{10}$  dB.

7. Discussion

In an M2M network, large number of MTC users challenge the admission capability of the communication system. However, the data rate required by each user may be small. Thus, small portion of the system's bandwidth will be allocated to each user, and different users can be allocated with different carriers. In a multi-carrier scenario, the algorithm proposed in this paper can be incorporated with the subcarrier allocation process to maximize the achievable user number. During the subcarrier allocation process, each user is assigned with one subcarrier. After that, the beamforming weight vectors can be calculated by the proposed algorithm in Section 4.

Denote the number of carriers as Q. After the subcarrier allocation process, subcarrier q is allocated with a candidate set of users ${𝒦}_q$ . The achievable user number of subcarrier q with candidate user set ${𝒦}_q$ becomes

$$\begin{array}{cc}& M_q\left({𝒦}_q\right).\end{array}$$

(44)

The maximum achievable user number can be found by solving the following problem

$$\begin{array}{l}\underset{{𝒦}_{\mathrm{1}},\dots ,{𝒦}_Q}{\text{maximize}}\hspace{0.17em}\hspace{1em}\sum _{q=\mathrm{1}}^Q‍M_q\left({𝒦}_q\right),\\ \text{subject}\hspace{0.17em}\hspace{0.17em}\text{to}\hspace{1em}\bigcup _q{𝒦}_q=𝒦.\end{array}$$

(45)

The optimal subcarrier allocation remains an open question. However some other suboptimal schemes can be considered. It is noted that the greedy method adopted in the user scheduling can also be used in the subcarrier allocation process. The achievable user number of each subcarrier can be calculated one after another.

8. Conclusion

In order to characterize a system's ability of user admission, we considered the achievable user number, that is, the number of users whose SINRs exceed a threshold. The downlink beamforming using zero-forcing DPC was considered. The algorithms for maximizing the achievable user number were proposed, and the achievable user number of both random and greedy scheduling schemes were derived using the joint distribution of effective channel gains. An upper bound on the achievable user number of the greedy user scheduling scheme was derived. Various numerical results were presented. It was shown that the upper bound becomes tighter when there are larger number of users. The performance of the greedy scheduling scheme is close to the optimal scheduling scheme. Achievable user number was shown to be one useful metric in understanding the performance of a system, especially in M2M communications, where large number of users challenge the user admission.