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Wireless Personal Communications

, Volume 100, Issue 4, pp 1743–1752 | Cite as

Sum Ergodic Capacity Analysis Using Asymptotic Design of Massive MU-MIMO Systems

  • Ahmad Kamal Hassan
  • Muhammad Moinuddin
  • Ubaid M. Al-Saggaf
Article
  • 98 Downloads

Abstract

This communication attempts to characterize the performance metrics of downlink Massive MU-MIMO systems impaired by cochannel interference and additive noise over a Rayleigh fading environment. We obtain close-form solutions for the probability density function of signal-to-interference-plus-noise ratio (SINR) and the sum ergodic capacity. The proposed work structures SINR in a quadratic form and thereby imposes a condition on its signal and interference power for a large transmit antenna diversity order; the conditional form is then analyzed using a distance correlation metric. Eventually, the sum ergodic capacity is expressed in a close-form by means of a residue theory approach and validated using the Monte Carlo simulation means.

Keywords

Ergodic capacity Residue theory Weighted sum of chi-square Antenna diversity SINR analysis 

1 Introduction

Massive multi-user multiple-input multiple-output (MU-MIMO) refers to a communication system in which a base station (BS) with large antenna array serves several single-antenna users on the same time-frequency resource. Major advantages of Massive MU-MIMO systems have been postulated in [1, 2]. Characterization of performance metrics for massive MU-MIMO is an active research area, however partial or perfect channel state information (CSI) availability is often considered at the transmitter and receiver ([3], and references therein). The proposed work however has the following merits:
  1. 1.

    Channel state information (CSI) is known only at the receiver side [4, 5]. Further, the transmit correlation matrix for each user is distinct.

     
  2. 2.

    Orthogonal set of transmit beamforming vectors is considered for each user [6].

     
  3. 3.

    Blanket transmission is assumed in which all antenna elements of the BS broadcast data [7].

     
  4. 4.

    Distance correlation (\(\xi\)) metric [8] is employed to prove the independence of signal and interference powers in the SINR which eventually leads to mathematically tractable analysis.

     
  5. 5.

    An indefinite quadratic form (IQF) [9] based approach is used to characterize the sum ergodic capacity.

     
It is noteworthy that the existing works on the IQF approach [9, 10, 11] present the cumulative distribution function (CDF) of SINR in close-form, however, the derived expressions are much involved and hence it is challenging to characterize the sum ergodic capacity measure thereof. These works utilize an information theocratic approach to provide close-form expressions for the CDF in terms of the eigenvalues which are a function of a predefined threshold variable besides other functional parameters. In the proposed work, we use a linear algebra property to reformulate the IQF approach such that in the resulting expressions, the eigenvalues are not a function of the predefined threshold variable. In other statistical streams related to the downlink performance metrics, the dependence of signal and interference powers in SINR pose a major hurdle in the characterization of the ergodic capacity measure [12]. Interestingly, a key aspect of the Massive MU-MIMO system [2, 13] indicate that channel vectors become asymptotically orthogonal for large antenna arrays. We investigate yet another dimension and propose that for a ‘large’ number of antenna elements at BS, orthogonal set of beamvectors assists in achieving the independence of signal and interference power given in the SINR metric. We thus utilize the IQF approach and the aforementioned independence criteria of Massive MU-MIMO system to characterize the ergodic capacity measure. We also explicitly show the adequate number of antenna elements required for convergence behavior of Massive MU-MIMO systems in terms of the sum ergodic capacity measure. The said independence is proved by means of the distance correlation metric [8], whereas the close-form expression for the sum ergodic capacity is validated viz. the Monte Carlosimulation means.

Notations Vectors and matrices are indicated by bold letters. \(|{\mathbf {A}} |\) and \(|| {\mathbf {a}} || ^2\) denote determinant of matrix A and norm-2 of vector a respectively. E(.) and u(.) denote the expectation operator and the unit step function. The notation \(()^H\) represents conjugate transposition and \(()^{\frac{H}{2}}\) is the sort representation of \((()^{\frac{1}{2}})^{H}\). For any matrix A, the quadratic form is defined as \(\left\| {{{\varvec{h}}}}\right\| _{\mathbf {A}}^2 \buildrel \bigtriangleup \over = {{{\varvec{h}}}}^H {\mathbf {A}}~ {{{\varvec{h}}}}\).

2 System Model

Consider a single cell downlink Massive MU-MIMO cellular network in which a BS having L transmit antenna elements collectively serve K single antenna user devices. For the kth user, the intended data-symbol \({s}_{k}\) is modulated with a transmit beamforming vector \({\mathbf{w}}_{k}\) of length \(\textit{L} \times 1\) before transmission. The received signal for the kth user is:
$$\begin{aligned} \textit{y}_{k}= {{\mathbf{h}}}_{k}^H {\mathbf {w}} _{k} \textit{s}_{k} + \sum _{i=1, i\ne k}^{K} {{\mathbf{h}}}_{k}^H {\mathbf {w}} _{i} \textit{s}_{i} + \textit{v}_{k}, \end{aligned}$$
(1)
where h\(_{k}\) is an \({L} \times 1\) zero-mean complex circular Gaussian channel vector with covariance matrix R\(_k\) i.e., h\(_{k} \sim \textit{CN} ({\mathbf {0}} , {\mathbf {R}} _k)\), whereas, the first, second and third terms represents desired signal, cochannel interference (CCI), and additive white noise with zero mean and variance \(\sigma _{k}^{2}\) respectively.
The instantaneous SINR for the kth user denoted as \(\varPsi _k\) is therefore given by:
$$\begin{aligned} {\varPsi _{k}} = \dfrac{| {\mathbf {h}}_{k}^H {\mathbf {w}}_{k} |^2}{\sigma _{k}^{2} + Q_N} ,\quad = \dfrac{\left\| \bar{{{\mathbf {h}}}}_{k} \right\| _{\overline{\mathbf{w }}_{k} \overline{\mathbf{w }}_{k}^H}^2}{\sigma _{k}^{2} + Q_N}, \end{aligned}$$
(2)
where \(\bar{{{\mathbf {h}}}_{k}} = {\mathbf {R}} ^{-\frac{H}{2}}_k{\mathbf{h }}_{k}\) is the whitened version of channel \({\mathbf {h}} _{k}\), \(\overline{\mathbf{{w} }}_{k} = {\mathbf {R}} ^\frac{1}{2}_k {\mathbf{w }}_{k}\) is the weight matrix which is represented using the notion of quadratic form, and \(Q_N\) is the weighted sum of N = K-1 chi-squared \(\chi ^2\) random variables representing the CCI of the kth user, i.e.,:
$$\begin{aligned} Q_N = \sum _{i=1, i\ne k}^{K} | {{\mathbf{h}}}_{k}^H {{\mathbf{w}}}_{i} |^2 = \left\| \bar{ {{\mathbf {h}}}}_k \right\| ^2_{\mathbf{A }}, \end{aligned}$$
(3)
where A is an \(L \times L\) Hermitian matrix of rank N defined as:
$$\begin{aligned} {\mathbf {A}} = {\mathbf {R}} ^{\frac{1}{2}}_k \left( \sum _{i=1, i\ne k}^{K} {\mathbf{w}}_{i} {\mathbf{w}}_{i}^{H}\right) {\mathbf{R}} ^{\frac{H}{2}}_k. \end{aligned}$$
(4)
Now, we perform the eigenvalue decomposition of the aforementioned weight matrix, i.e., \({\mathbf{A}} = {\mathbf {U}} \varLambda {\mathbf {U}} ^H\),where \({\mathbf {U}}\) is an \(L \times L\) matrix whose jth column represents the jth eigenvector \(u_j\); for \(j = \{1, 2,\ldots , L\}\) and \(\varLambda\) is a diagonal matrix with entries having corresponding eigenvalues, i.e., \(\varLambda _{jj} = \lambda _j\); for \(j = \{1, 2, \ldots , L\}\). Channel component of the kth user is further transformed by relation \({{\mathbf{g}}}_k = {\mathbf{U}} ^\frac{H}{2} {\bar{\mathbf{h }}}_k\), hence resulting in the following form:
$$\begin{aligned} Q_N = \left\| {{{\mathbf {g}}}}_k \right\| ^2_{{\varLambda }} . \end{aligned}$$
(5)

3 Characterization of the Ergodic Capacity

In this section, we aim to present a close-form expression for the sum ergodic capacity, i.e., \({\mathbf{ E}} [\textit{C}_{\varPsi k } ]\). The sum ergodic capacity is defined as [5]:
$$\begin{aligned} {\mathbf {E}} [\textit{C}_{\varPsi } ]&= \sum _{k=1}^{K} {\varvec{{E}}}\left( log_{2}\left( 1 + \mathrm {\varPsi _k} \right) \right) ,\\&= \sum _{k=1}^{K}\int _{0}^{\infty }\log _{2}(1+\psi _{th}) ~\textit{f}_{\varPsi _k}(\psi _{th}) ~d\psi _{th} ,\\&= \frac{1}{\ln (2)} \sum _{k=1}^{K} \int _{1}^{\infty }\ln ({\bar{\psi }_{th}}) ~\textit{f}_{\varPsi _k}({\bar{\psi }_{th}}-1) ~d{\bar{\psi }_{th}} \end{aligned}$$
(6)
where \(\textit{f}_{\varPsi_k}(\psi _{th})\) is the PDF of \(\varPsi _{k}\) for a given threshold level \(\psi _{th}\) and the third equality is obtained by the change of variable by letting \({\bar{\psi }_{th}} = \psi _{th} + 1\).
Next, we purpose that there exists a minimum value of transmit diversity order L beyond which the independence among the channel vectors \({\bar{\mathbf{h }}}_k\) and \({{\mathbf{g} }}_k\) is achieved. Based on such L, we can impose a condition on interference power \(Q_N\) by assigning a threshold level \(q_N\) such that \(Q_N = q_N\). The rational behind this is to adopt a three-step simplification procedure which starts with achieving an analytical expression for the conditional form of PDF, i.e., \(\textit{f}_{\varPsi_k}(\psi _{th}|_{Q_N = q_N})\), followed by obtaining an expression for the conditional sum ergodic capacity \({\mathbf {E}} [\textit{C}_{\varPsi k }|_{Q_N = q_N} ]\), and lastly solving (6) by utilizing the PDF of \(Q_N\), i.e., \(f_{Q_N}(q_N)\). Now, in order to achieve the first step, we start by using the conditional form of the CDF of \(\varPsi_k\) for a given threshold \(\psi _{th}\), i.e., \(\textit{F}_{\varPsi_k}(\psi _{th}|_{Q_N = q_N}) = \mathrm {Pr.}( \varPsi _k < \psi _{th} )\) as:
$$\begin{aligned} \textit{F}_{\varPsi_k}(\psi _{th}|_{Q_N = q_N})&= \mathrm {Pr}\bigg ( \dfrac{\left\| \bar{ {{\mathbf {h}}}}_{k} \right\| _{\overline{\mathbf{{w} }}_{k} \overline{\mathbf{w}}_{k}^H}^2}{\sigma _{k}^{2} + q_N} < \psi _{th} \bigg ) , \\&= \mathrm {Pr} \left( \left( \sigma _{k}^{2} + q_N\right) \psi _{th} - \left\| \bar{ {{\mathbf{h}}}}_{k} \right\| _{\overline{\mathbf{{w}}}_{k} \overline{\mathbf{w}}_{k}^H}^2 > 0 \right) , \\&= \int _{-\infty }^{+\infty } f({\bar{\mathbf{h}}}_k) u\left( \left( \sigma _{k}^{2} + q_N\right) \psi _{th} - \left\| \bar{{{\mathbf{h}}}}_{k} \right\| _{\overline{\mathbf{{w}}}_{k} \overline{\mathbf{w}}_{k}^H}^2 \right) d{\bar{\mathbf{h}}}_k. \end{aligned}$$
(7)
where \(f({\bar{\mathbf{h}}}_k)\) is PDF of L-dimensional \({\tilde{\mathbf{h}}}_k\) and u(.) is the unit step function (Fourier representation), i.e.,:
$$\begin{aligned} f({\bar{\mathbf{h}}}_k)= \frac{1}{\pi ^{L}} e^{-\left\| \tilde{{{\mathbf{h}}}}_k \right\| ^{2}};~ u(y)=\frac{1}{2\pi }\int _{-\infty }^{+\infty } \frac{e^{y(j\omega +\beta )}}{j\omega +\beta } d\omega . \end{aligned}$$
(8)
where \(\beta > 0\) in the u(.) function helps in solving the multi-dimensional integrals of the Gaussian PDF [9, 10].
Thus, using (8) the CDF of \(\varPsi _k\) formulates as:
$$\begin{aligned} \textit{F}_{\varPsi _k}(\psi _{th}|_{Q_N = q_N})&= \frac{1}{2\pi ^{L+1}}\int _{-\infty }^{+\infty } ~\frac{{e^{ \left( \sigma _{k}^{2} + q_N\right) \psi _{th} (j\omega +\beta )}}}{(j\omega +\beta )} \\&\quad\times \int _{-\infty }^{+\infty } e^{-\left\| {\bar{\mathbf{h}}}_k \right\| _{I + \overline{\mathbf{{w}}}_{k} \overline{\mathbf{w}}_{k}^H(j\omega +\beta ) }^2}~d {\bar{\mathbf{h}}}_kd\omega . \end{aligned}$$
(9)
Next, by taking differentiation of the above expression, the conditional \(\textit{f}_{\varPsi _k}(\psi _{th})\) is:
$$\begin{aligned} \textit{f}_{\varPsi _k}(\psi _{th}|_{Q_N = q_N})&= \frac{\left( \sigma _{k}^{2} + q_N\right) }{2\pi ^{L+1}}\int _{-\infty }^{+\infty } ~{e^{ \left( \sigma _{k}^{2} + q_N\right) \psi _{th} (j\omega +\beta )}} \\& \quad \times \int _{-\infty }^{+\infty } e^{-\left\| {\bar{\mathbf{h}}}_k \right\| _{I + \overline{\mathbf{w}}_{k} \overline{\mathbf{w}}_{k}^H(j\omega +\beta ) }^2}~d {\bar{\mathbf{h}}}_kd\omega , \\&= \frac{ \sigma _{k}^{2} + q_N}{2\pi ^{}}\int _{-\infty }^{+\infty } \frac{e^{ \left( \sigma _{k}^{2} + q_N\right) \psi _{th} (j\omega +\beta )}}{1+ \left\| \overline{\mathbf{{w}}}_{k} \right\| ^{2} (j\omega +\beta )} d\omega . \end{aligned}$$
(10)
where the second equality is obtained viz. the solution of complex Gaussian integral and also by employing a property of determinant for unit rank matrix \(\overline{\mathbf{{w}}}_{k} \overline{\mathbf{w}}_{k}^H\) as:
$$\begin{aligned} \frac{1}{\pi ^{L}} \int _{-\infty }^{+\infty } e^{-\left\| \bar{\mathbf{{h} }}_k \right\| _{I + \overline{\mathbf{{w}}}_{k} \overline{\mathbf{w}}_{k}^H (j\omega +\beta ) }^2}d\bar{ \mathbf{{h}}}_k &= \frac{1}{|I+ \overline{\mathbf{{w}}}_{k} \overline{\mathbf{w}}_{k}^H(j\omega +\beta )|}, \\& = \frac{1}{ 1+ \left\| \overline{\mathbf{{w}}}_{k} \right\| ^{2} (j\omega +\beta )}. \end{aligned}$$
(11)
Now, by employing the residue theory,1 the conditional PDF of \(\varPsi _k\) is expressed in close-form as:
$$\begin{aligned} \textit{f}_{\varPsi _k}(\psi _{th}|_{Q_N = q_N}) =\frac{\left( \sigma _{k}^{2} + q_N\right) }{\left\| \overline{\mathbf{{w}}}_{k} \right\| ^{2}} e^{-\frac{\left( \sigma _{k}^{2} + q_N\right) \psi _{th}}{\left\| \overline{\mathbf{{w}}}_{k} \right\| ^{2}}} u\left( \frac{\left( \sigma _{k}^{2} + q_N\right) \psi _{th}}{\left\| \overline{\mathbf{{w}}}_{k} \right\| ^{2}}\right) . \end{aligned}$$
(12)
Next, by adjusting the argument of the above PDF, i.e., \({\bar{\psi }_{th}} = \psi _{th} + 1\) as in (6) and absorbing u(.) in the limit of integration, the conditional ergodic capacity has the following form:
$$\begin{aligned} {\mathbf {E}} [\textit{C}_{\varPsi k }|_{Q_N = q_N} ]&= \frac{1}{\ln (2)} \sum _{k=1}^{K} \frac{\left( \sigma _{k}^{2} + q_N\right) }{\left\| \overline{\mathbf{{w}}}_{k} \right\| ^{2}}~ e^{\frac{\left( \sigma _{k}^{2} + q_N\right) }{\left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2}}} \int _{1}^{\infty }\ln ({\bar{\psi }_{th}}) e^{-\frac{\left( \sigma _{k}^{2} + q_N\right) {\bar{\psi }_{th}}}{\left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2}}} d{\bar{\psi }_{th}}, \\&= \frac{1}{\ln (2)} \sum _{k=1}^{K} e^{\frac{\left( \sigma _{k}^{2} + q_N\right) }{\left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2}}} \int _{1}^{\infty } \frac{1}{\bar{\psi }_{th}} {e^{-\frac{\left( \sigma _{k}^{2} + q_N\right) {\bar{\psi }_{th}}}{\left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2}}}} d{\bar{\psi }_{th}}.~~~ \end{aligned}$$
(13)
where the second equality is obtained from Proposition 8.213-15 [14] and from the definition of the exponential integral function \(\left( E_1(\mu ) = \int _{1}^{\infty } \frac{e^{-\mu x}}{x} dx\right)\).
In what follows, we remove condition on the above expression. Note that \(Q_N\) in (5) is in a standard form on which the procedure of IQF [9, 11] can be directly applied in order to obtain the desired PDF. Alternatively, authors in [15] also report the PDF of \(Q_N\) with and without multiplicity of eigenvalues. Thus, considering the case of distinct eigenvalues without repetition, both [9, 15] will provide a similar close-form expression for the PDF of \(Q_N\) reproduced herein as:
$$\begin{aligned} f_{Q_N}(q_N) = \sum _{l=1}^{L}\frac{\lambda _l^{L-2}}{\prod _{i=1,i\ne l}^{L} (\lambda _l -\lambda _i)} e^{-\frac{q_N}{\lambda _{l}}} \textit{u}\bigg (\frac{q_N}{\lambda _{l}}\bigg ). \end{aligned}$$
(14)
Hence, we utilize the aforementioned expression to remove the condition in (13) as follows:
$$\begin{aligned} {\mathbf {E}} [\textit{C}_{\varPsi k } ]&= \frac{{1}}{\ln (2)} \sum _{k=1}^{K} \sum _{l=1}^{L}\frac{\lambda _l^{L-2} e^{\frac{\sigma _{k}^{2} }{ \left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2} }}}{\prod _{i=1,i\ne l}^{L} (\lambda _l -\lambda _i)} \int _{1}^{\infty } \frac{e^{-\frac{\sigma _{k}^{2} {\bar{\psi }_{th}} }{ \left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2} }}}{{\bar{\psi }_{th}}} \\&\quad\times \int _{0}^{\infty } e^{-\frac{1}{ \left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2} }\left( {\bar{\psi }_{th}} + \frac{ \left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2} }{ \lambda _l } -1 \right) q_N} ~d{q_N} d{\bar{\psi }_{th}} , \\&= \frac{{1}}{\ln (2)} \sum _{k=1}^{K} \sum _{l=1}^{L}\frac{\lambda _l^{L-2} }{\prod _{i=1,i\ne l}^{L} (\lambda _l -\lambda _i)} \left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2} e^{\frac{\sigma _{k}^{2} }{ \left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2} }} \\&\quad\times \int _{1}^{\infty } \frac{e^{-\frac{\sigma _{k}^{2} {\bar{\psi }_{th}} }{ \left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2} }}}{{\bar{\psi }_{th}}\left( {\bar{\psi }_{th}} + \frac{ \left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2} }{ \lambda _l } -1 \right) } d{\bar{\psi }_{th}} , \\&= \frac{1}{\ln (2)} \sum _{k=1}^{K} \sum _{l=1}^{L}\frac{\lambda _l^{L-1}}{\prod _{i=1,i\ne l}^{L} (\lambda _l -\lambda _i)} \frac{\left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2} e^{\frac{\sigma _{k}^{2} }{ \left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2} }} }{\left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2} - \lambda _l } \\&\quad\times \int _{1}^{\infty } \left[ \frac{e^{-\frac{\sigma _{k}^{2} {\bar{\psi }_{th}} }{ \left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2} }}}{\bar{\psi }_{th}} -\frac{e^{-\frac{\sigma _{k}^{2} {\bar{\psi }_{th}} }{ \left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2} }}}{ {\bar{\psi }_{th}} + \frac{ \left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2} }{ \lambda _l } -1 } \right] d{\bar{\psi }_{th}}, \end{aligned}$$
(15)
where the second equality is obtained by noting the limit of \({\bar{\psi }_{th}}\) and thus applying Proposition 3.310 [14] in the inner integral, while the third equality is from partial fraction expansion.
Eventually, we use Proposition 3.352-2 [14] on (15) to obtain the following close-form expression for the sum ergodic capacity:
$$\begin{aligned} {\mathbf {E}} [\textit{C}_{\varPsi k } ]&= \frac{1}{\ln (2)} \sum _{k=1}^{K} \sum _{l=1}^{L}\frac{\lambda _l^{L-1}}{\prod _{i=1,i\ne l}^{L} (\lambda _l -\lambda _i)} \frac{\left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2} }{\left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2} - \lambda _l } \\&\times \left[ e^{\frac{\sigma _{k}^{2} }{ \left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2} }} E_1\left( {\frac{\sigma _{k}^{2} }{ \left\| \overline{\mathbf{{w} }}_{k} \right\| ^{2} }} \right) - e^{\frac{\sigma _{k}^{2} }{ \lambda _l }} E_1\left( {\frac{\sigma _{k}^{2} }{ \lambda _l }} \right) \right] .~~ \end{aligned}$$
(16)
Note that the aforementioned expression can be easily evaluated using standard mathematical tools such as MATLAB, MATHEMATICA, and MAPLE.

4 Performance Evaluation

The aim of this section is to show the effectiveness of close-form expression derived in (16). Transmit correlation matrices are considered unique for each user and they are based on correlation coefficient \(\rho\) such that \({\mathbf {R}} _{i,j} =\rho ^{|i-j|}\) and \(0<\rho <1\). The total number of channel realizations is equal to 15,000. Further, a Gram–Schmidt orthogonalization procedure is used to formulate an orthonormal set of beamforming vectors which are thereby used for each user. Moreover, the signal-to-noise-ratio (SNR) for all users is set to 0 dB.

In Fig. 1, we show graphically the adequate number of transmit antenna elements that correspond to the asymptotic value, i.e., \(L \rightarrow \infty\), for a given set of users K. Specifically, for number of users set to 2, 4, and 8, it is seen that the analytical expression (16) and Monte Carlo simulations converge at L values of 2, 4, and 8 respectively. The initial disagreement at low transmit antenna diversity for K values of 4 and 8 is due to the dependence of numerator and denominator terms expressed in (2), these terms however approach the said independence for the corresponding asymptotics of L. To this end, we use a distance correlation coefficient which is proposed as a statistics tool in [8] to show the independence of two variables which in our case are the signal power (\({\mathbf {s}} _p\)) and interference power (\({\mathbf {i}} _p\)) in the SINR. The distance correlation coefficient (\(\xi\)) is in the range \(0 \le \xi ({\mathbf {s}} _p, {\mathbf {i}} _p) \le 1\), where the terminal values of \(\xi ({\mathbf {x}}, {\mathbf{y}} ) = 1\) and \(\xi ({\mathbf {x}}, {\mathbf{y}} ) = 0\) represents total dependence and complete independence respectively. In Fig. 2, we chose an eight user scenario and test the independence of \({\mathbf {s}} _p\) and \({\mathbf {i}} _p\) in (2) for each of the eight users. It is observed that for all users, the \(\xi\) coefficient has a larger value initially which indicates a high dependence, the \(\xi\) value for all users however converges to \(\xi \approx 0.02\) at L = 8 and beyond which indicates excellent level of independence in \({\mathbf {s}} _p\) and \({\mathbf {i}} _p\).

Lastly, in Fig. 3 we show the effect of SNR on the sum ergodic capacity for different values of K while setting L = 128. It is noteworthy that the nature of beamvectors highly influence the sum ergodic measure, herein we have used an orthonormal set of beamforming vectors generated by means of the Gram-Schmidt orthogonalization procedure. It is shown that the system is noise-limited at low SNR value, hence an increase in the total number of users increase the sum ergodic capacity with CCI not effecting adversely. On the other hand, the system is interference-limited at high SNR value which point towards high impact of CCI and therefore the sum ergodic capacity decreases with an in increase in K. A high degree of match is apparent in the Monte Carlo simulation setup and the derived analytical results across the SNR range in the horizontal depiction and for varying number of total users K in the vertical illustration.
Fig. 1

Comparison of the analytical and Monte Carlo simulation results for the sum ergodic capacity with increasing transmit diversity L and users K

Fig. 2

Independence test of numerator and denominator terms of (2) for each user in 8 user scenario

Fig. 3

Comparison of the analytical and Monte Carlo simulation results for the sum ergodic capacity with respect to SNR for L = 128

5 Conclusion

In this paper, we have analyzed a Massive MU-MIMO system without CSI availability at the transmitter side. Specifically, independence of transmit power and interference power is examined using a distance correlation test which enabled us to achieve a weighted and quadratic conditional form of SINR. Furthermore, for a particular transmit antenna diversity range, we have utilized an indefinite quadratic form approach to achieve a novel close-form expression of the sum ergodic capacity.

Footnotes

  1. 1.
    A special case of residue theory is generalized from [9] as
    $$\begin{aligned} \frac{1}{2\pi } \int _{-\infty }^{\infty }\frac{e^{j \omega p}}{a+ j \omega }~d\omega = e^{-a p} u(a p),\quad a > 0 . \end{aligned}$$

Notes

Acknowledgements

This project was funded by the Center of Excellence in Intelligent Engineering Systems (CEIES), King Abdulaziz University, under Grant No. (CEIES-16-12-02). The authors, therefore, acknowledge the technical and financial support of CEIES.

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Ahmad Kamal Hassan
    • 1
    • 2
  • Muhammad Moinuddin
    • 2
    • 3
  • Ubaid M. Al-Saggaf
    • 2
    • 3
  1. 1.Faculty of Electrical EngineeringGhulam Ishaq Khan Institute of Engineering Sciences and TechnologyTopiPakistan
  2. 2.Center of Excellence in Intelligent Engineering Systems (CEIES)King Abdulaziz UniversityJeddahSaudi Arabia
  3. 3.Department of Electrical and Computer EngineeringKing Abdulaziz UniversityJeddahSaudi Arabia

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