# Spatial cross-correlation modeling for propagation channels in indoor distributed antenna systems

- Li Tian
^{1}, - Xuefeng Yin
^{1}Email author, - Xu Zhou
^{1}and - Quan Zuo
^{1}

**2013**:183

https://doi.org/10.1186/1687-1499-2013-183

© Tian et al.; licensee Springer. 2013

**Received: **20 September 2012

**Accepted: **25 June 2013

**Published: **8 July 2013

## Abstract

In this contribution, the spatial cross-correlation of composite channels in a distributed antenna system (DAS) is studied for the case that a nine-antenna DAS is deployed in an indoor environment. As few measurement data of DAS propagation channels are available, the propagation graph modeling based on the electromagnetic wave reverberation theory is used to generate the synthetic impulse responses (IRs) for the composite channels induced by the different groups of antennas in the DAS. The simulations take into account the geographic parameters of the environment, the cluttering scatterers, the realistic visibility conditions, as well as the flexible locations for the antennas, and the user equipment. The uncorrelated scattering assumption is proved to be valid by using the real measurement data from the indoor environments. Based on this assumption, narrowband channel cross-correlation is computed by integrating the small-scale fading cross-correlation within the same delay bins in two channel IRs. The characteristics of the cross-correlation in four cases with different antenna grouping are investigated for the nine-antenna DAS. Part of the modeling results are shown to be consistent with the empirical counterparts for specific DAS constellations calculated using the real measurement data.

## Keywords

## 1 Introduction

A distributed antenna system (DAS) is referred to as the network of spatially separated antenna nodes connected to a common access point. The DAS has several advantages, comparing with the traditional concentrated multiple-input-multiple-output (MIMO) system (CMS). First, it allows the same coverage of an area with less power consumption since the distributed antennas create line-of-sight (LoS) connections to the mobile station, which can significantly reduce the power consumption caused by wall penetration inside buildings. Also, it reduces the delay spreads of the received signals, as well as the complexity of receivers[1, 2]. The locations of antennas can be designed following different principles, e.g., achieving better coverage, enhancing spectral efficiency[3], or increasing the quality of received signals[4].

Recently, the transmission technologies for DAS have attracted a lot of attention[5, 6]. For example, in[5], block diagonalization and zero-forcing dirty-paper coding downlink transmission schemes are extended for a single-cell DAS with multi-antenna-distributed antenna nodes. Only a subset of the antenna nodes in the cell transmits to the user equipment (UE). The aggregate cell spectral efficiency achieved by the proposed schemes per frequency-time resource block is compared for both DAS and CMS architectures. It has been demonstrated that, under the same total power constraint, the cellular DAS can double the ergodic aggregate cell spectral efficiency of the CMS. In addition, the spectral efficiency increases and saturates as the number of antennas per node increases. In[6], a generalized DAS combined with the CMS is introduced, where each distributed node has multiple micro-diversity antennas. All the antennas have a separate feeder to the base station. The signals can be combined using different algorithms.

For the existing theoretical studies of DAS, oversimplified propagation channel models and ideal assumptions are usually adopted for the performance analysis[6]. The impact of real environment on the DAS performance has not been thoroughly investigated. In these simplified channel models, the power of a single channel at each antenna in a distributed antenna node follows the chi-squared distribution with two degrees of freedom (${\chi}_{2}^{2}$), while the mean of the distribution varies independently with log-normal distribution. The average of stochastic power means of the antennas within a node is determined by the path loss from the UE to the node, neglecting the shadow fading attributed to the UE surroundings. The ‘inter-node’ and the ‘intra-node’ fadings are assumed to be independent and identical log-normal and Rayleigh distributions, respectively. According to[6], the independence of the ‘inter-node’ fading is justifiable by the large distance between any two nodes and the distinctive scattering around the antennas in each node. However, as illustrated in[7], propagation environments, particularly the indoor propagation environment, introduce significant dependence among the nodes even when these aforementioned conditions are satisfied. Therefore, field measurements or realistic simulations are essential for establishing DAS channel models in both macroscopic and microscopic view points.

In this contribution, a simulation approach is applied to analyzing the characteristics of fading correlation across multiple links in DAS scenarios. The electromagnetic (EM) wave reverberation theory, originally adopted in[8] for generating channel impulse response (IR), is adopted to create the synthetic IRs in DAS scenarios. The spatial cross-correlations are then investigated by taking into account specific DAS configurations in an indoor environment. The validity of the predicted characteristics is evaluated using measurement data.

The organization of this paper is as follows: section 1 gives the general background of the works; in section 2, the approach of DAS channel simulations based on propagation graph are elaborated, and the simulations are validated using measurement data; in section 3, the spatial cross-correlation for DAS channels is defined; the simulation results for an indoor scenario are presented in section 4; experimental confirmation of the characteristics predicted by the simulations is depicted in section 5; finally, conclusive remarks are addressed in section 6.

## 2 Propagation graph modeling for DAS channels

### 2.1 A brief introduction of graph modeling

The propagation graph modeling approach has been applied to predicting channel IRs in[8] by exhaustively searching of the propagation paths connecting the transmitters and receivers. A propagation graph can be generated by taking into account the geometry of the environment, the scatterers’ distribution, the mobility, and visibility of the scatterers, as well as the scatterers’ EM properties.

**D**(

*f*) represents the LoS part of the transmission, and

**R**(

*f*)[1 −

**B**(

*f*)]

^{−1}

**T**(

*f*) is the none line-of-sight (NLoS) component induced by the reverberation of EM waves among scatterers. In Equation 1,

**T**(

*f*),

**R**(

*f*), and

**B**(

*f*) denote the transmission matrices with entries representing the power attenuations and phase changes from individual transmitter (Tx) to scatterers, from scatterers to individual receiver (Rx), and among scatterers, respectively.

**B**

^{ n }(

*f*) refers to the

*n*th bounce inter-reflections of the scatterers. The transfer function of a propagation path that represents a link in Figure1 can be calculated as:

where *A*_{e} is an element in the matrics **D**, **T**, **B**, and **R** according to different kinds of link ends, *g*_{e}(*f*) is the propagation coefficient calculated based on the free-space propagation loss and the reflection coefficient, *τ*_{e} is the delay or time of arrival, and *ϕ* is the random phase rotation, which follows a uniform distribution on the interval [0,2*π*).

The graph modeling approach is adopted in this research because of the following advantages: firstly, an environment can be easily modeled by the location matrices which contain the location information of the transceivers and scatterers; secondly, IRs are calculated with analytical computations based on the EM-wave reverberation theory; thirdly, the models can be generalized for specific frequency bands. Moreover, the complexity of graph modeling approach is limited. For example, IR generation for a graph with 2 transmitters, 2 receivers, and 100 scatterers in a snapshot costs about 8 × 10^{6} flops which can be executed in less than 0.1s using the computing software, e.g., MATLAB^{
®
} with a quad-core computer. In addition, the IR obtained by the graph approach exhibits the specular-to-diffuse transition, which is hard to obtain using conventional ray-tracing-based channel modeling with tractable complexity[9–11].

The effectiveness of applying the graph approach to channel modeling has been investigated in[8] and[12], where the power-delay profiles and the specular-to-diffuse transition of the simulated IRs are compared with those of the real IRs. In[13], the spatial and temporal characteristics of the channel IRs generated by graph in high-speed railway scenarios have been shown to be realistic. In this contribution, the graph modeling approach is further evaluated in an indoor scenario by comparing the statistical characteristics of channel parameters obtained from graph modeling with the parameter values specified in established WINNER II channel models[14]. It will be shown later in section 2.2 that the typical channel characteristics predicted by the graph modeling is consistent with the WINNER II models.

### 2.2 Evaluation of propagation graph modeling

**Parameter settings for graph-based IR simulations**

Description | Value |
---|---|

Carrier frequency | 2.6 GHz |

Bandwidth | 20 MHz |

Heights of Tx antennas | 4.6 m |

Height of scatterers | [ 0,5] m |

Heights of UEs | 1.7 m |

Total UE locations | 1,376 |

Snapshots per location | 150 |

Signal to noise ratio | 30 dB |

Number of Tx antennas | 9 |

Number of UE antennas | 1 |

Groups of Tx antennas | 2 |

Reflection gain | 0.8 |

## 3 Definition of spatial fading cross-correlation

*ρ*is defined as the cross-correlation coefficients of the narrowband

^{a}fading of links ‘1’ and ‘2’ observed at the same UE location, i.e.,

*h*

_{1}and

*h*

_{2}denote the complex narrowband channel coefficients of two links, respectively, as shown in Figure7 and$\stackrel{\u0304}{h}$ is the mean of the channel coefficient.

*E*{·} denotes the expectation operation. Defining the mean-removed channel coefficients$\widehat{{h}_{1}}={h}_{1}-\stackrel{\u0304}{{h}_{1}}$ and$\widehat{{h}_{2}}={h}_{2}-\stackrel{\u0304}{{h}_{2}}$, the covariance$E\left[\phantom{\rule{0.3em}{0ex}}\right({h}_{1}-\stackrel{\u0304}{{h}_{1}}\left){({h}_{2}-\stackrel{\u0304}{{h}_{2}})}^{\ast}\right]$ in Equation 3 can be calculated as:

where *h*(*τ*) denotes the complex-valued spread function of the channel in delay and *T* is the observing time of each snapshot.

*C*

_{1,2}is split into two parts, i.e., the common-delay (cmd) part and cross-delay (crd) part. The narrowband fading cross-correlation coefficient

*ρ*can then be written as:

*ρ*

_{cmd}and crd components

*ρ*

_{crd}calculated in reality are expressed as:

respectively, where *I* is the number of the channel coefficient samples in each UE location and *N* is the number of delay samples in each complex-valued spread function of the channel.

In the case where the uncorrelated scattering (US) assumption is valid, the equality *ρ*_{crd} = 0 holds as the components with different delays are uncorrelated. However, in graph modeling, the cross-correlation among the different delay components is non-negligible because of the limited number of scatterers in the graphs. This implies that if we compute *ρ* by the channel coefficients *h*_{1} and *h*_{2} generated by graph modeling, wrong estimates may arise since *ρ*_{crd} ≠ 0 for the channels constructed by the graphs. Alternatively, we consider to approximate the *ρ* by calculating *ρ*_{cmd} only. The accuracy of this approximation is evaluated in section 3.1 using measurement data.

### 3.1 Validation of the approximation *ρ* ≈ *ρ*_{
cmd
}

*ρ*≈

*ρ*

_{cmd}. Figure8 depicts the map of the measurement environment. During the measurements, the receiver was fixed, and the transmitter was moving along the measurement route marked in red. The measured data were collected in cycles. Here, a cycle is the time duration that the channels between any pair of the transmitter antenna and the receiver antenna are measured once. The channel coefficients obtained in a measurement cycle can be considered as random observations of the same wide-sense-stationary channel, due to the following reasons: (1) the radiation patterns of the antenna elements are nonidentical; (2) the system responses contain random phase noises when using any pair of Tx and Rx antennas; and (3) the channel is not exactly constant during the measurements, as slight differences in small scale such as the movement of the person pushing the Tx trolley and the absorption give rise to the randomness in the observations of the channel. Since the transmitter and the receiver were equipped with 50 and 32 antennas, respectively, for each cycle in total, 50 × 32 = 1,600 channel coefficients are obtained. These channel observations are used to calculate the fading correlation coefficients. Figure9 illustrates the contour plots of

*ρ*and

*ρ*

_{cmd}computed based on the data of 150 cycles, while the contour plot and the CDF of the deviations between

*ρ*and

*ρ*

_{cmd}are depicted in Figure10. It can be observed that most of the deviations between

*ρ*and

*ρ*

_{cmd}are close to zero. More than 97% of the deviations are less than 0.1, and the mean value of

*ρ*-

*ρ*

_{cmd}is about 0.02. Thus, |

*ρ*≈

*ρ*

_{cmd}| can be considered empirically valid in indoor environments.

## 4 Modeling spatial cross-correlation of fading in a DAS

We are interested at the spatial cross-correlation of the fadings of two composite channels when nine antennas are grouped into two clusters. Figure2 depicts an indoor environment where a DAS with nine antennas is deployed. The corresponding locations of transmitters and scatterers are depicted in Figure3. The antennas are allocated in different parts of the room in nine positions. These antennas can be flexibly combined into two groups. An example illustrated in Figure3 shows two groups of antennas marked in different colors. The graph modeling method is used to simulate the IRs of the composite channels when the UE is located at each position in the environment. The parameter settings for the simulations are shown in Table1. In order to have enough number of random realizations of the fading coefficients, for an individual UE position, 150 channel IRs are generated with the UE moving randomly in 150 locations in the vicinity of this position within *λ*/4. This vicinity range is determined accordingly such that random realizations are in the same WSS condition. Then the fading correlation is calculated based on the simulated IRs.

From these results, it is apparent that the locations of antennas in the DAS lead to different fading cross-correlation distribution in the area covered by the DAS. Thus, it is necessary to design the DAS and its transmission algorithms by considering the empirical fading correlation in the environments of interest. A common belief for achieving better coverage and low-power consumption is to allocate antenna symmetrically for the DAS[1, 2, 16]y. However, the investigation results illustrated here show that the improvement by utilizing the DAS is marginal due to the non-negligible cross-correlation attributed to the symmetric antenna constellations.

## 5 Experimental evaluation

The spatial correlation in most of the DASs can be ignored because the spacing of the antennas are much larger than the coherent distance. However, in section 4, the simulation results show that significant fading cross-correlation can be observed in the case where two transmitting antennas are distributed in symmetric locations w.r.t. the UE. In this section, we use measurement data collected in the same campaign described in section 3.1 to validate this observation.

*ρ*are around 0.3 which are non-negligible in computing the diversity gain of the DAS.

In addition, the empirical statistics of the DAS channel correlation are extracted based on the measurements. Figure14b depicts the empirical and simulated CDFs of the absolute values of fading correlation coefficients for symmetric links only and all links. It can be observed from Figure14b that 90% of the correlation coefficients for the measured symmetric links are larger than 0.3, while the correlation coefficients of all links are much less and negligible. Similar results are observed for the simulations, even though the correlation coefficients are underestimated in the simulations. It is our conjecture that the underestimation is due to the settings of random phase in Equation 2 for graph modeling approach which might not hold in realistic environment.

## 6 Conclusions

The spatial cross-correlation of channel fadings in different links of a DAS has been investigated via simulations using stochastic propagation graphs. We first proposed an approach for computing the fading cross-correlation, i.e., it is approximated by integrating the cross-correlation coefficients of the small-scale fadings in the same delay bins of two channels. The effectiveness of this approach has been evaluated using the measurement data. Then, the applicability of random propagation graphs in channel modeling was validated. It has been shown that the statistics of path loss and delay spreads extracted using graphs is close to those specified by the WINNER II models. Using the proposed methods, the narrowband fading cross-correlation of the composite channels in DASs for a hall-alike indoor environment has been investigated. The results obtained demonstrated that the highest cross-correlation coefficients can be observed when the locations of antennas in different groups are overlapped. When antennas belonging to different groups are located in two well-separated regions, the cross-correlation becomes smaller. Moreover, significant fading cross-correlation can be observed in the cases where the distributed antennas belonging to different groups are deployed symmetrically w.r.t. the location of the UE. These results have been shown to be consistent with the observations obtained in real measurements.

## Endnote

^{a} Here, narrowband means that signal bandwidth times delay spread is much smaller than 1.

## Declarations

### Acknowledgments

This work is supported by the fundamental project of the Science and Technology Commission of Shanghai Municipality (10ZR1432700, Multidimensional power spectrum characterization and modeling for wide-band propagation channels), the China Education Ministry “New-teacher” Project (20090072120015, Time-Variant Channel Characterization, Parameter Estimation and Modeling), Central Higher Education Fundamental Research Project (Polarization characteristics of propagation channel), the fundamental project of the Science and Technology Commission of Shanghai Municipality (13510711000, System design and demo-construction for cooperative networks of high-efficiency 4G wireless communications in urban hot-spot environments), and Central Higher Education Fundamental Research Project (ROF based distributed MIMO systems). The authors wish to acknowledge Dr. Junhe Zhou from Tongji University for his enlightening advices and valuable comments on the paper, and Elektrobit, Finland for kindly providing the measurement data. Part of the works in this paper has been presented in the COST IC1004 + iPLAN Joint Workshop on “Small Cell Cooperative Communications”.

## Authors’ Affiliations

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