International Journal of Engineering Trends and Technology (IJETT) – Volume 18 Number2- Dec 2014
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1 Scholar, M. Tech, Digital Communication
Sri Balaji College of Engineering & Technology, Jaipur, India
Abstract— The presented research paper aims to study and analyze the spatially correlated fading channel, Its basic concepts, brief mathematical modelling and simulation of correlation function of a MIMO System. The review work in [1] presents the effect of antenna parameters on the correlation property of a MIMO System and hence the system capacity. In this paper we will specifically analyze the correlation property of the MIMO System under Rayleigh fading and will compare and interpret the simulated results. The simulation results of
MATLAB program [2], have been used to analyze the correlation property.
Keywords — Correlation Coefficients, MIMO, 3GPP.
I.
INTRODUCTION
The types and numbers of independent multiple channels decide the performance of a multiple-input multiple-output
(MIMO) and its entire characterization. With the earlier research and analysis in the field of MIMO channel modelling we come to a conclusion that channel correlation degrades the performance of a MIMO system and hence the capacity also.
Generally the term correlation is defined as a measure of conditional predictability. Similarly Channel correlation is nothing but a measure of likeliness between the channels. If the channels are fully correlated to each other, then their will be no difference between the SISO and MIMO. The capacity of a MIMO system depends on the number of channels (N x
M), and their correlation between each other. In a general scenario, the channel capacity is inversely proportional to channel correlation.
This paper presents a study of spatial correlation property of a
MIMO system for different 3GPP Cases. Rayleigh fading model is used for analysis mainly because of its generality and applicability to the practical situation. A brief introduction of
Primitive Physical channel Model is presented in section 2.
Basic concept of the spatial correlation and the mathematical formulae is presented in section 3. In section 4 we present the brief introduction of the 3GPP standard and their implementation in our perspective. Section 5 briefs about the description of the MIMO capacity MATLAB Simulation. The final simulated results are presented in section 6. The main object of the presented paper is to analyse the spatial correlation property of 2x2 MIMO system and compare the results with interpretation. The conclusion is mentioned in section 7.
II.
PRIMITIVE PHYSICAL CHANNEL
III.
SPATIAL CORRELATION
MODEL
Prior to detail study, the diagram of primitive physical channel model is first depicted, which indicate the block of spatial correlation mapping matrix. Physical channel models characterize an environment on the basis of electromagnetic wave propagation by describing the double-directional multipath propagation between the location of the transmitter
(Tx) array and the location of the receiver (Rx) array. Wave propagation parameters like the complex amplitude, Angle of
Arrival (AoA), Angle of Departure (AoD) and delay of a
Multipath component (MPC) are explicitly modelled by this environment. Depending on the chosen complexity, physical models allow for an accurate reproduction of radio propagation. Physical models are independent of antenna configurations (antenna pattern, number of antennas, array geometry, polarization, mutual coupling) and system bandwidth. A primitive physical channel model is illustrated in Figure 1. It is described by power profiles, spatial correlation functions and fading characteristics. The main input parameters required for the model are the shape of the power delay spectrum, the fading characteristics and the spatial correlation functions at the transmitter and receiver ends
Fig. 1 A primitive physical channel model
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International Journal of Engineering Trends and Technology (IJETT) – Volume 18 Number2- Dec 2014
The spatial correlation depends on the multipath signal environment. Multipath signals tend to leave the transmitter in a range of angular directions (called angles of departure, AOD) rather than a single angular direction. This is the same for the multipath signals arriving at the receiver (called angles of arrival, AOA). Usually, the spatial correlation increases when
AOD and AOA are reduced and vice versa. This is depicted in diagram 2 and 3.
[ X ] 






R
Tx  
R
Rx
1
2





 H iid
(3)
Where R
TX and R
RX
are the receive and transmit correlation matrices respectively, and H iid
is a vector (only here, otherwise it is a matrix) of independent zero mean, unit variance, complex Gaussian random variables, and

RTx  
TxNM 
(4)

RRx
 

RxMN
(5)
Fig. 2 Multipath Signals at Transmitter
Fig. 3 Multipath Signals at Receiver
Prior to the entire discussion we first look for the fundamental
MIMO formulation, which consider the Correlation Matrix.
For 2 × 2 MIMO system, the channel matrix H [3] is:
H  p




K
K  1
H
F

1
K  1
H v




(1) where 
TxNM are the complex correlation coefficients between
N th
and M th
transmitting antennas, and 
RxNM
are the complex correlation coefficients between M th
and N th
receiving antennas. It is generally believed that rich scattering environments are needed for optimal use of multiple antennas.
The reason for the crossover of course is that the multiplexing gain offered by the identically independent distributed channel manifests itself only at sufficiently high SNRs. For ULA [4][5] the complex correlation coefficient at the linear antenna array is expressed as field  f
and envelope  e
:
 e
    f
 
2
 R
XX
   jR
XY
2
(6)
Where D  2  d /  and R
XX and R
XY are the cross-correlation functions between the real parts (equal to the cross-correlation function between the imaginary parts) and between the real part and imaginary part, respectively, with the correlation properties of the fading, and the values of the symmetrical correlation matrices R
TX and R
RX are completely defined by the
PAS and its standard deviation. For the 2 × 2 MIMO channel, transmit and receive correlation matrices are expressed as:
RTx



1

Tx 21

Tx 12
1
 


(7)
 p





K
K  1



 e j  11
1
10 e j  21
1
10 e e j  j  12
22










1
K  1






X
11
1
2
X
21
1
2
X
X
12
22






(2) where X
NM
(N-the receiving and M-the transmitting antenna) are correlated zero-mean, unit variance, complex Gaussian random variables as coefficients of the variable (Rayleigh) matrix H v
, exp  j NM  are the elements of fixed matrix H
F
,
K is the Rician K-factor, and P is the power of each tap. To correlate the X
NM
elements of the matrix X, the Kronecker product of the transmitter and receiver correlation matrices is preformed: and RRx  



1
Rx 21

Rx
1
12
 


(8)
Further, the spatial correlation properties of a MIMO system uses Kronecker product of the spatial correlation matrices R
TX and R
RX to define total correlation.
  
H
   R
Tx
 R
1
Rx
(9)
Whose elements are correlation coefficients, where
H is the conjugate transpose and the vec   operator rearranges the 2 ×
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International Journal of Engineering Trends and Technology (IJETT) – Volume 18 Number2- Dec 2014
2 matrix H into a column vector of size 4 × 1. Cluster and tap
PAS shape follow Laplacian distribution. The angles of arrival statistics within a cluster were found to closely match the
Laplacian distribution [6][7]. channel keeping the antenna spacing constant (0.5
 ). Table
Mentioned below carries all the cases.
TABLE I
MIMO SETUP FOR DIFFERENT 3GPP CASES p   
2 
1
2  e


(10) where  is the standard deviation of the PAS.
3GPP
Case
1
2
3
4
Spacing between
Nodes (λ)
0.5
0.5
0.5
0.5
No of
Antenna
Nodes
2
2
2
2
Type of Channel
Rayleigh
ITU Pedestrian A
ITU Vehicular A
ITU Pedestrian B
IV.
3GPP STANDARIZATION
The 3 rd
Generation Partnership Project (3GPP) is group of six elite telecommunications standard development organizations
(ARIB, ATIS, CCSA, ETSI, TTA, TTC), known as “Organizational Partners”. The major task of this standard is to provide latest issues, trends, technologies and specifications to the entire members which is a milestone for future 4 th
generation technologies called 3GPP. This project covers the technology associated with cellular telecommunications networks including radio access for indoor and outdoor medium, the fundamental transport network, and hence provides service capabilities and complete system specifications. The specifications also provide hooks for non-radio access to the core network, and for interworking with Wi-Fi networks. There is a contribution of member companies and several research agencies in formation of the organization. In their definition of particular environment this
3GPP cases for MIMO technology are standardized for
Rayleigh, ITU Pedestrian A, ITU vehicular A, ITU Pedestrian
B.
Fig. 4: Simulated Spatial correlation properties for 3GPP case 1
V.
SIMULATION DESCRIPTION
This section describes the content of the MATLAB® implementation of the MIMO radio channel. To demonstrate the correlation property of the MIMO channel with respect to different 3GPP Cases [8], we call the two directories
Correlation_Multiple_Cluster and UMTS_Testbed of
MATLAB
®
scripts. Which
 Derive the spatial correlation properties of a Uniform
Linear Array (ULA) impinged by a Gaussian PAS, where the waves are gathered in a single or in multiple clusters.
 Simulate a MIMO radio channel at link-level in compliance with 3GPP specifications.
The simulated results are shown in section 6.
VI.
SIMULATION RESULTS
This section presents the simulated spatial correlation coefficients of 2x2 MIMO system for different types of
Fig. 5: Simulated Spatial correlation properties for 3GPP case 2
Fig. 6: Simulated Spatial correlation properties for 3GPP case 3
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International Journal of Engineering Trends and Technology (IJETT) – Volume 18 Number2- Dec 2014
Fig. 7: Simulated Spatial correlation properties for 3GPP case 4
VII.
CONCLUSION
The simulated graphs in section 6 clearly justify the statement that the greater the channel correlation, the smaller is the channel capacity. The dashed red markers correspond to the reference values, whereas the blue markers are the outcome of the simulation. The match between reference curves and simulation results using MATLAB program [3] is satisfactory.
Here in the above graphs we can observe that in comparison of ITU Pedestrian A, ITU vehicular A and ITU Pedestrian B,
The Rayleigh channel has greater drop of correlation coefficient for greater value of
 k  1

2 l , where k  l and hence the more amount of channel capacity. In all above channel models ITU vehicular A and ITU pedestrian A ha ve quite similar correlation characteristics.
R EFERENCES
[1] Dimitra Zarbouti, George Tsoulos, and Dimitra Kaklamani, second chapter in “MIMO System Technology for Wireless Communications”
Edited by George Tsoulos, CRC Press 2006. Pages 29–56, eBook
ISBN: 978-1-4200-0592-9, DOI: 10.1201/9781420005929.ch2
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[3] V. Erceg, “Indoor MIMO WLAN Channel Models”, IEEE 802.11-
03/161r2, September 2003.
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Multipath Propagation”, IEEE Journal on Selected Areas
Communication, vol. 5, 1987, pp. 128-137.
[5] F. Adachi, M. Feeny, A. Williamson, and J. Parsons, “Cross-
Correlation between the Envelopes of 900 MHz Signals Received at a
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Oct. 1986.
[6] Q. H. Spencer, “Modeling the Statistical Time and Angle of Arrival
Characteristics of an Indoor Environment”, IEEE Selected Area in
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[8] MIMO Rapporteur, 3GPP Document R1-02-0141 "MIMO Conference
Call Summary", January 2002.
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