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

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## 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

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

- 3.
Blanket transmission is assumed in which all antenna elements of the BS broadcast data [7].

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

**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

*L*transmit antenna elements collectively serve

*K*single antenna user devices. For the

*k*th 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

*k*th user is:

*\(_{k}\) is an \({L} \times 1\) zero-mean complex circular Gaussian channel vector with covariance matrix*

**h****R**\(_k\) i.e.,

*\(_{k} \sim \textit{CN} ({\mathbf {0}} , {\mathbf {R}} _k)\), 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.*

**h***k*th user denoted as \(\varPsi _k\) is therefore given by:

*N*=

*K*-1 chi-squared \(\chi ^2\) random variables representing the CCI of the

*k*th user, i.e.,:

**A**is an \(L \times L\) Hermitian matrix of rank

*N*defined as:

*j*th column represents the

*j*th 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

*k*th user is further transformed by relation \({{\mathbf{g}}}_k = {\mathbf{U}} ^\frac{H}{2} {\bar{\mathbf{h }}}_k\), hence resulting in the following form:

## 3 Characterization of the Ergodic Capacity

*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:

*L*-dimensional \({\tilde{\mathbf{h}}}_k\) and

*u*(.) is the unit step function (Fourier representation), i.e.,:

*u*(.) function helps in solving the multi-dimensional integrals of the Gaussian PDF [9, 10].

^{1}the conditional PDF of \(\varPsi _k\) is expressed in close-form as:

*u*(.) in the limit of integration, the conditional ergodic capacity has the following form:

## 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 \({\mathbf {i}} _p\).

*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.

## 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

## 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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