Simplified 3D Fading Channels Adopted in MIMO Beamforming Schemes

Joy Iong-Zong Chen and Bo Huei Lee

Simplified 3D Fading Channels Adopted in MIMO Beamforming Schemes

Joy Iong-Zong Chen and Bo Huei Lee

—A simplified three-dimension （3D） fading channel model deployed in a multi-input multi-output（MIMO） beamforming system is explored in this article. Both angle of arrival （AoA） and angle of departure（AoD） which impact the overall system performance are examined. The numerical results are given for validating the accuracy of the theoretical derived formulas. Furthermore， the performances of the model with different number of transmitters and receivers are studied and compared. The increment in AoA parameters definitely generates the impact of the system performance when the consideration of simplified 3D channels.

Index Terms—Angle of arrival， characteristic function， multi-input multi-output beamforming system，simplified 3D fading channel.

1. Introduction

It is known that the infrastructure is assigned with 5 layers for the 5th generation （5G） wireless systems. The physical and medium access control layers are definitely impacting the system performance of a 5G cellular system. Extensive angle of departure （AoD） and angle of arrival（AoA）， path loss， and multipath time delay spread measurements are conducted for steerable beam antennas of differing gains and beam widths for a wide variety of transmitter and receiver locations. The paper provides measurements and models that may be used to design future 5G millimeter-waves （MMW） cellular networks and gives insight into antenna beam steering algorithms for these systems［1］. In addition， ［2］ described that the potential capacity gain is highly dependent on the multipath richness，since a fully correlated multi-input multi-output （MIMO）radio channel only offers one subchannel， while a completely decorrelated radio channel potentially offers multiple subchannels depending on the antenna configuration. So far as， a huge numbers of research explored the issue about MIMO. In ［3］， the capacity performance has been investigated for a multiple antennas system which was proposed to switch among different MIMO transmission schemes， including statistical beamforming， double space-time transmit diversity， and spatial multiplexing over spatially correlated channels.

The authors of ［3］， ［4］ claimed that an exact bit error rate （BER） analyzed for different modulation schemes in a correlated Rayleigh MIMO channel was presented. In ［4］， a new MIMO transmission approach that adapts to the changing channel conditions based on the spatial selectivity information was presented. The proposed system switches between different MIMO transmission schemes as a means of approaching the spatially correlated MIMO channel capacity with low-complexity.

All the previously mentioned results are obtained from the assumption with azimuth direction of the propagation channel. However， in fact the evaluation of the MIMO-beamforming signaling should consider the elevation for wave propagation， i.e.， the vertical direction has to be included in discussion of the channel model［5］，［6］. In the previous work， the authors’ work was only to express the current 3GPP （the 3rd generation partnership project）activity around simplified three-dimension beamforming（3D-BF） and FD-MIMO. They have provided the basic concept to enhance the understanding of the current industrial challenges as well as to deeply hint the standards’vision for 3D-BF. In ［7］， the authors took the specifications both azimuth and horizon fading channels of one subscribe into consideration for the interference coordination to maximize both the cell-edge users’ and cell center users’throughputs. An extension of the ITU2D （International Telcommunications Union 2-dimision） channel model to 3D was proposed in ［8］ by adding a distance dependent elevation spread based on observations from ray tracing. Through system-level simulations， we observe that the behavior of 3D MIMO is greatly impacted by the modeling of the 3D channel.

This paper focuses on the consideration of the correlation effect in the transmitting and receiving branches，which are considered as having an arbitrary correlation coefficient and AoA. Consider both transmitter and receiver environments with different antenna numbers， a MIMO system with a 3D-BF receiver combining with maximalratio combining （MRC） operating over correlated fading channel is examined. This paper is organized as follows. In Section 2， an MIMO system with 3D-BF combination is presented first. In Section 3， the channel capacity performance of the MIMO beamforming system with fading 3D channel is evaluated. A validation of the derived theoretical formulas with numerical manifestation is presented in Section 4. Finally， Section 5 concludes the report.

2. MIMO Beamforming System with Simplified 3D Channel

At the investigation of applications， the use of 3D-BF is usually assumed that antennas are arranged in a 2D array where each column contains M antenna elements. There are exactly K antenna elements per antenna port with a pattern AEis given by

where

Fig. 1. Simple deployment of simplified 3D channel model［9］.

Consider an MIMO beamformer scheme employingtransmitter antennas andreceiver antennas is equipped with the receiver signal vectorand the transmit signal vector， respectively. Normally， the radio MIMO channel is anmatrix and can be rewritten as

A simple deployment of 3D channel model is shown in Fig. 1， which was discussed in ［9］. There are two situations，either choosing K M= ， in this situation， the number of antenna ports per column will be equal to M or settingare considered by the 3GPP group TSG-RANWG1（Technique Specification Group-Radio Access Network Working Group 1）. A weighted sum of channels with the K elements is assigned in the element. The channel port is given by

where the sum presented above is performed over all antenna elements in port s. Accordingly， if the real 3D channel is considered， then each antenna port adopts that deployed in ［9］ and is rewritten as （2）.

In addition， the nth eigenvalue ofandare designated asand， respectively. It is known that by using the eigenvalue decomposition method， the spatial correlation matrices can be rewritten asand， respectively， where the symboldenotes conjugate transpose，andcontain the corresponding entries of the eigenvectors with respect to the related eigenvalues， whileis a diagonal matrix with eigenvalues， which are obtained from spatial correlation matrices.

Once the propagating channel scenario is constructed completely， the received signal intensityfor each spatial correlated channel experienced fading environment can be expressed as

具体措施为：利用网络技术进行电力网格的程序设计，构建数据网格和信息网格，整合电力系统现有的数据和资源，实现数据在不同区域调度中心的交换和计算。电力网格的设置可以实现内网和外网的采用同一操作系统的运作模式，能在很大程度上提高电网运行的效率。

3. 3D-BF Channel Capacity System

In this section， the channel capacity of an MIMO system with 3D-BF transmission is considered. It is characterized as

3D-BF implicitly decomposed as， with. Hence， by substituting the channel matrix into mutual information formula， the new mutual information becomes

This subsection considers the MIMO system which employs 3D-BF with MRC and operates in an indoor environment. It is a critical point to take account of the parameter of AoA for evaluating the system performance，since the deployment is considered in an indoor environment. Thus， the vertical angle will become an important variable to affect the channel capacity of the MIMO beamforming system. Let the transmitted average signals’ power be normalized with the fading parameters of Nakagmi-m fading and represent as，， whererepresent the fading parameters of the corresponding correlated fading channels［9］. The squared value of the intensityiZ is a Gamma distribution， i.e.，and. Moreoverbecomes a special case with the Erlang distribution when all fading parameters，im， are designated as integer numbers， that is， the pdf ofis given as

Equation （13） can be solved by using the equivalent formulas given in ［10］:

whereis the confluent Hypergeometric function.

By applying the well known formula for differentiation of confluent Hypergeometric function［11］， the channel capacity of the MIMO systemcan be evaluated as

4. Numerical Results

The validation of the derived theoretical formulas is illustrated in this section. First the indoor factor is always set as 3.6K= . For simplicity， however， without loss the generality， the correlation coefficients are generated by the Gaussian correlation model of an equally spaced linear array with arbitrary correlation coefficients. It is of interest to note that the correlation matrix followed by the linear array has a Toeplitz form constructed by correlation elements，It is easy to understand that the performance becomes much better when the fading parameter increases. Finally， the impact of AoA parameter on the MIMO beamforming system is illustrated in Fig. 2. The curves are corresponding to differentvalues. It is valuable to see that the correlation coefficient generally relates to the outcome of /dλ which changes the size of AoA. Hence， the channel capacity definitely becomes degradation when the correlation coefficient is promoted.

Fig. 2. Channel capacity for MIMO system over correlated-Nakagami-m fading with m=4，

5. Conclusions

In the report， both AoA and AoD for the MIMO 3D-BF system over the spatial correlated-fading channel were explored. The channel correlation is alternative to present the AoA for numerical analysis. Numerical results were gained from different scenarios with distinct numbers of antennas at transmission and receipt ends. Specifically， the results from the scenario of spatially correlated channels considered to a MIMO system with beamforming schemes showed the outperformance for the situations which has a larger number of receiver antennas. Moreover， it is worthy to note that the performance of the MIMO beamforming scheme will be mostly dominated by the fading parameter，when the correlated-Nakagami-m model （considering the AoA and AoD parameters） was established as the channel environment.

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Joy Iong-Zong Chen received his B.S. degree from the National Taiwan Technical University， Taipei in 1985， his M.S. degree from the Dayeh University，Chunghua in 1995， and his Ph.D. degree from National Defense University，Tao-Yuan in 2001， all in electrical engineering. He is currently a distinguished professor with the Department of Electrical Engineering， Dayeh University. His research interests include green energy surveillance， wireless sensor networks， multiple targets tracking， wireless communications， and OFDM systems.

Bo Huei Lee received his B.S. and M.S. degrees in electronics engineering from the Dayeh University， Chung Hua in 1985 and 1995， respectively. Currently，he works toward to his Ph.D. degree with Dayeh University. His research interests include embedded system，digital signal processing， and wireless communications.

Manuscript received November 7， 2014； revised January 21， 2015.

J. I.-Z. Chen is with the Department of Electrical Engineering， Dayeh University， Changhwa 51505 （Corresponding author e-mail: jchen@mail.dyu.edu.tw）.

B. H. Lee is with the Department of Electrical Engineering， Dayeh University， Changhwa 51505 （e-mail: D0103002@mail.dyu.edu.tw）.

Digital Object Identifier: 10.3969/j.issn.1674-862X.2015.02.010