# Reduced Complexity BER Calculations in Large Scale Spatial Multiplexing Multi-User MIMO Orientations in Frequency Selective Fading Environments

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## Abstract

**:**

## 1. Introduction

## 2. MIMO Spatial Multiplexing Transceiver Model

_{t}antennas at the transmitter and N

_{r}antennas at the receiver (N

_{t}× N

_{r}) then the transmitted signal for the k

^{th}(1 ≤ k ≤ K) user will be given by the equation below [17].

_{t}, N

_{r}) as the dimension of the MIMO channel, then

**x**

_{k}(t) is the dim × 1 transmission signal matrix,

**P**

_{k}is a diagonal dim × dim matrix indicating the transmission power per data stream,

**b**

_{k}(t) is the dim × 1 symbol matrix, c

_{k}is the Gold coding sequence [17], and

**t**

_{k}is the dim × dim precoding matrix. The various data streams in the

**b**

_{k}matrix will also be referred to as modes throughout the rest of this article. The received signal per active user can be described as a superposition of the L multipath components from all active users.

_{l}/

**H**

_{k}

_{,l}are the delay and channel coefficient of the l

^{th}multipath component (1 ≤ l ≤ L) and

**n**

_{k}is the corresponding Gaussian dim × 1 noise. The real and complex parts of each element of the channel matrix are assumed to be normally distributed with zero mean value and a variance equal to 0.5.

^{th}RAKE finger after correlation with the desired spreading sequence and MRC will be given by the Equation below.

**A**

^{H}is the conjugate transpose of matrix

**A**),

**b**

_{k}

_{,0}is the symbol matrix at the current symbol period, and

**b**

_{k}

_{,−1}is the corresponding matrix at the previous symbol period. Assuming independent symbol matrices among the various users of the MIMO, CDMA network, the signal power of the desired user, the Intersymbol Interference (ISI), MAI power, and the total noise power will be given by Equations (4)–(7), respectively.

_{o}is the noise level and Re(x) is the real part of x.

**P**

_{s}

_{,k}matrix.

## 3. BER Calculation in Frequency Selective Fading MIMO Orientations

**P**

_{s,k}can be equivalently written using the Equation below.

_{k,d}is the power allocated to the d

^{th}transmission mode. In the remainder of this article, the i

^{th}entry of matrix

**X**will be denoted as X

_{i}. In this context, in Equations (8) and (9), H

_{k,l,q,v}represents the channel coefficient from the v

^{th}transmit antenna (1 ≤ v ≤ N

_{t}) to the q

^{th}receiver antenna (1 ≤ q ≤ M

_{r}) for the l

^{th}multipath and t

_{k,v}

_{,d}the (v,d) element of the precoding matrix

**t**

_{k}.

_{d}) and standard deviation (σ

_{d}). Under this assumption, Equation (10) can be written by using the formula below.

_{o,d}is the correlation coefficient for an arbitrary pair $\left({\zeta}_{k,d,l,q},{\zeta}_{k,d,{l}^{\prime},{q}^{\prime}}\right)$. Moreover, MAI from the k′

^{th}user can be expressed by Equation (12) below.

_{k}

_{′,d}variable is normally distributed, according to CLT. The equivalent SINR for the k

^{th}user and d

^{th}transmission mode will be given by Equation (13) below (Im(x) is the imaginary part of x).

_{k,d}is Chi-squared distributed [19], since it represents sum of squares of normal RVs. However, an alternate representation based on Gamma distribution will be considered, in order to include the special case of correlated sums.

_{k,d}≥ 0).

_{Z}(z) leads to the pdf of Z.

_{d}denotes the square of γ

_{k,d}, which is normally distributed with mean/standard deviation equal to ${\mu}_{{Y}_{d}}$/${\sigma}_{{Y}_{d}}$, respectively, and X

_{d}is the denominator of Equation (13), then the following is true.

_{d}/θ

_{d}denote the shape/scale parameter of the Gamma distribution. If $u=\sqrt{x}$ then $du=\frac{1}{2\sqrt{x}}dx$. Hence, the equation below is found.

_{d}≈ [a

_{d}], where [x] is the integer part of x. Starting from the known integral, Equation (18) is found, which is shown below.

_{d}

^{th}derivative of Equation (19) with respect to g, ${f}_{{Z}_{d}}\left(z\right)$ can be written as:

_{d}and Y

_{d}are uncorrelated. In the opposite case, the following transformation can be applied.

_{d}and Y

_{d}.

**I**

_{dim}the dim × dim identity matrix, and tr(

**X**) is the trace of matrix

**X**. In this context, singular value decomposition takes place, where the left and right singular matrices (

**U**,

**V**) of the desired signal matrix as well as the corresponding eigenvalues matrix (

**Σ**) are calculated. Note that, even for arbitrary number of interferers, the equivalent parameters of only two users need to be calculated: the desired one and the interferer. Hence, at the n

^{th}channel realization, mean and standard deviation (std) values of all related parameters are calculated considering the first n samples, until convergence of the transmission matrices is achieved. Afterward, total MAI power from K-1 users is calculated using squares of samples generated from a normal distribution, by taking into account calculated correlation.

## 4. Results

_{t},M

_{r},L,PG) has been considered. In Figure 1, three MIMO cases are depicted (three group of curves from left to right): (4,4,6,32), (4,4,2,32) and (8,8,6,32). In this particular set of simulations, pdf curves of γ

_{k}

_{,d}from MC simulations considering the first transmission mode are compared with the ones derived using curve fitting with normal distribution. As is evident from Figure 1, for large MIMO orientations, normal distribution can accurately model the desired user signal.

_{t}Kbps. Mean BER simulation values for the BPSK modulation were calculated for two MIMO cases (4 × 4, 8 × 8) (denoted as S), and compared to the theoretical ones (denoted as Th). Users may vary from 5 to 31 with a step of 2. Hence, total throughput varies from 2.4 Mbps to 14.88 Mbps in the 4 × 4 orientation and from 4.8 Mbps to 29.76 Mbps in the 8 × 8 orientation. In Figure 2, results are depicted considering reduced diversity order MIMO cases (i.e., L × M

_{r}). In particular, in each group of curves, two resolvable multi-path components are taken into account. Hence, the diversity order can be either 8 (4 × 4 system) or 16 (8 × 8 system). As can be observed, significant variations may occur among the calculated BER values with the proposed approach and the ones derived from simulations. However, in both groups of curves, convergence is improved for an increased number of interfering users, since, in this case, CLT is more accurate for the MAI estimation.

^{−3}/7.1 × 10

^{−3}for four multipath components, respectively. Hence, the calculation error is 3%. For six multi-paths and 23 users, corresponding values are 5.3 × 10

^{−3}/5.4 × 10

^{−3}. Therefore, the calculation error is further reduced to 2%.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Step 1: n ← 0. Set d_{u} ← 1, i_{u} ← 2, U← d_{u}$\cup $i_{u}, ε = 10^{−3}.For every k$\in $U: t_{k,n} ←$\left(1/\sqrt{dim}\right){I}_{dim}$, P_{k,n} ← ${t}_{k,n}^{\mathrm{H}}{t}_{k,n}$Step 2: n ← n + 1, ${r}_{k,l,n}={\left({H}_{k,l}{t}_{k,n}\right)}^{\mathrm{H}}$ and ${A}_{k,n}\leftarrow {\left({\displaystyle \sum _{l=1}^{L}{r}_{k,l,n}{H}_{k,l}}\right)}^{\mathrm{H}}\left({\displaystyle \sum _{l=1}^{L}{r}_{k,l,n}{H}_{k,l}}\right)$Step 3: ${A}_{k,n}\leftarrow {U}_{k,n}{\Sigma}_{k,n}{V}_{k,n}^{\mathrm{H}},\text{}{t}_{k,n+1}\leftarrow {V}_{k,n}$Step 4: ${Y}_{d}\leftarrow {\left({\displaystyle \sum _{l=1}^{L}{\displaystyle \sum _{q=1}^{{N}_{r}}{\left|{\displaystyle \sum _{v=1}^{{N}_{t}}{H}_{l,q,v}{t}_{{d}_{u},v,d}}\right|}^{2}}}\right)}^{2}{P}_{{d}_{u},d},\text{}1\le d\le dim$For arbitrary values of l, q, l′, q′ calculate: ${\mu}_{{Y}_{d},n}\leftarrow L{N}_{r}\langle {\zeta}_{{d}_{u},d,l,q}\rangle $, ${\sigma}_{n}\leftarrow \mathrm{std}\left({\zeta}_{{d}_{u},d,l,q}\right)$ ${\rho}_{o,n}\leftarrow E\left\{\left({\zeta}_{{d}_{u},d,l,q}-\langle {\zeta}_{{d}_{u},d,l,q}\rangle \right)\left({\zeta}_{{d}_{u},d,{l}^{\prime},{q}^{\prime}}-\langle {\zeta}_{{d}_{u},d,{l}^{\prime},{q}^{\prime}}\rangle \right)\right\}/{\sigma}_{n}^{2}$ ${\sigma}_{{Y}_{d},n}\leftarrow {\sigma}_{n}\left\{\sqrt{L{N}_{r}\left(1+{\rho}_{o,n}\left(L{N}_{r}-1\right)\right)}\right\}$ ${\zeta}_{{i}_{u},d,l,q}\leftarrow \left(\overline{{\displaystyle \sum _{v=1}^{{N}_{t}}{h}_{{d}_{u},l,q,v}{t}_{{d}_{u},v,d}}}\right){\displaystyle \sum _{{l}^{\prime}=1}^{L}{\displaystyle \sum _{{d}^{\prime}=1}^{dim}\left({\displaystyle \sum _{v=1}^{{N}_{t}}{h}_{{d}_{u},{l}^{\prime},q,v}{t}_{{i}_{u},v,{d}^{\prime}}}\right)}{b}_{o,{i}_{u},{d}^{\prime}}\sqrt{{P}_{{i}_{u},d}}}$ For the real and complex part of ${\zeta}_{{i}_{u},d,l,q}$ calculate: ${\mu}_{MA{I}_{{i}_{u},d,n}}\leftarrow \left(K-1\right)L{N}_{r}\langle {\zeta}_{{i}_{u},d,l,q}\rangle $, ${\sigma}_{MA{I}_{{i}_{u},d,n}}\leftarrow \mathrm{std}\left({\zeta}_{{i}_{u},d,l,q}\right)$ ${\tilde{\rho}}_{o,n}\leftarrow E\left\{\left({\zeta}_{{i}_{u},d,l,q}-\langle {\zeta}_{{i}_{u},d,l,q}\rangle \right)\left({\zeta}_{{i}_{u},d,{l}^{\prime},{q}^{\prime}}-\langle {\zeta}_{{i}_{u},d,{l}^{\prime},{q}^{\prime}}\rangle \right)\right\}/{\sigma}_{MA{I}_{{i}_{u},d,n}}^{2}$ ${\sigma}_{MA{I}_{{i}_{u},d,n}}\leftarrow {\sigma}_{MA{I}_{{i}_{u},d,n}}\left\{\sqrt{\left(K-1\right)L{N}_{r}\left(1+{\tilde{\rho}}_{o,n}\left(\left(K-1\right)L{N}_{r}-1\right)\right)}\right\}$ ${x}_{{i}_{u},d,n}\leftarrow \left\{{\displaystyle \sum _{l=1}^{L}{\displaystyle \sum _{q=1}^{{N}_{r}}\left(\overline{{\displaystyle \sum _{v=1}^{{N}_{t}}{h}_{{d}_{u},l,q,v}{t}_{{d}_{u},v,d}}}\right)}{\displaystyle \sum _{{l}^{\prime}=1}^{L}{\displaystyle \sum _{{d}^{\prime}=1}^{dim}\left({\displaystyle \sum _{v=1}^{{N}_{t}}{h}_{{d}_{u},{l}^{\prime},q,v}{t}_{{i}_{u},v,{d}^{\prime}}}\right)}{b}_{o,{i}_{u},{d}^{\prime}}\sqrt{{P}_{{i}_{u},d}}}}\right\}$ $\tilde{{\rho}_{n}}\leftarrow \frac{\mathrm{cov}\left\{Re\left({x}_{{i}_{u},d,n}\right),Im\left({x}_{{i}_{u},d,n}\right)\right\}}{\sqrt{\mathrm{cov}\left\{Re\left({x}_{{i}_{u},d,n}\right),Re\left({x}_{{i}_{u},d,n}\right)\right\}\mathrm{cov}\left\{Im\left({x}_{{i}_{u},d,n}\right),Im\left({x}_{{i}_{u},d,n}\right)\right\}}}$ ${P}_{k,n+1}$←${t}_{k,n+1}^{2}{A}_{k,n}{t}_{k,n+1}$ If $\left|\mathrm{tr}\left({P}_{k,n+1}\right)-\mathrm{tr}\left({P}_{k,n}\right)\right|\ge \epsilon \cdot \mathrm{tr}\left({P}_{k,n}\right)$ go to Step 2 Step 5:$\chi \sim N\left({\mu}_{MA{I}_{{i}_{u},d,n}},{\sigma}_{MA{I}_{{i}_{u},d,n}}^{2}\right),z\sim N\left({\mu}_{MA{I}_{{i}_{u},d,n}},{\sigma}_{MA{I}_{{i}_{u},d,n}}^{2}\right)$$y\leftarrow \tilde{{\rho}_{n}}\chi +\left(\sqrt{1-{\tilde{{\rho}_{n}}}^{2}}\right)z,\text{}{X}_{d}\leftarrow {\chi}^{2}+{y}^{2}\text{\hspace{1em}}{\rho}_{{X}_{d},{Y}_{d}}\leftarrow \frac{\mathrm{cov}({Y}_{d},{X}_{d})}{{\sigma}_{{Y}_{d}}{\sigma}_{{X}_{d}}}$ Calculate α _{d}, θ_{d} from the generated samples of X_{d}Calculate BER from Equation (21) |

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**MDPI and ACS Style**

Gkonis, P.K.; Kaklamani, D.I.
Reduced Complexity BER Calculations in Large Scale Spatial Multiplexing Multi-User MIMO Orientations in Frequency Selective Fading Environments. *Electronics* **2019**, *8*, 727.
https://doi.org/10.3390/electronics8070727

**AMA Style**

Gkonis PK, Kaklamani DI.
Reduced Complexity BER Calculations in Large Scale Spatial Multiplexing Multi-User MIMO Orientations in Frequency Selective Fading Environments. *Electronics*. 2019; 8(7):727.
https://doi.org/10.3390/electronics8070727

**Chicago/Turabian Style**

Gkonis, Panagiotis K., and Dimitra I. Kaklamani.
2019. "Reduced Complexity BER Calculations in Large Scale Spatial Multiplexing Multi-User MIMO Orientations in Frequency Selective Fading Environments" *Electronics* 8, no. 7: 727.
https://doi.org/10.3390/electronics8070727