mimo channel models

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Preface

This report summarizes the achievements of the Master Thesis work carried out

between September and December 2003 at Acreo AB in Norrkping, Sweden.

The report constitutes the final element of a Master of Science exam in Electronics

Design Engineering at the University of Linkping at Campus Norrkping.

I take this opportunity to thank my coordinator Christian Kark at Acreo AB and my

examiner Shaofang Gong at ITN Linkpings University for useful discussions and

suggestions. Im also thankful to Arash Jafari who took his time to read my rapport

and give me his useful comments.

Special thanks are dedicated to my family, friends and boyfriend who have been a

source of encouragement and inspiration to write this thesis.


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Table of Figures

Figure 1. MIMO channel representation........................................................................... 11

Figure 2. Primitive channel model .................................................................................... 13

Figure 3. Scatter cluster example ...................................................................................... 14

Figure 4. One-ring model .................................................................................................. 15

Figure 5. Model D, Power Delay Profile .......................................................................... 17

Figure 6. Impinging waves from Tx toRx......................................................................... 18

Figure 7. Laplacian distribution with AS=30 [14] .......................................................... 20

Figure 8. Relationship among the correlation functions and power spectra ..................... 22

Figure 9. Doppler spread/shift........................................................................................... 24

Figure 10. CDF of modelled I/C vs. measured I/C. ............................................................ 26

Figure 11. Channel model C: a) Impulse response, b) PDP................................................28

Figure 12. Channel model C: a) CDF, b) Spatial correlation of the firs six taps................ 29

Figure 13. Channel model C: a) Spatial correlation of the six middle taps, b) Spatial

correlation of the last two taps........................................................................................ 29

Figure 14. Channel model C: a) Doppler spectra of the first six taps, b) Doppler spectra ofthe middle six taps .......................................................................................................... 30

Figure 15. Channel model C: Doppler spectra of the last two taps..................................... 30

Figure 16. Ptolemy example of C++ code...........................................................................32

Figure 17. ADS with Matlab block .....................................................................................33

Figure 18. ADS with ReadFile block ............................................................................... 34

Figure 19. Channel model in ADS with channel matrixHand power delays ....................35

Figure 20. Impulse response of covariance channel matrixHfor channel model F ........... 36

Figure 21. Impulse response and phase of one-ring model.................................................37

Figure 22. Spatial multiplexing for 2x2 MIMO system...................................................... 38

Figure 23. MIMO system without knowledge of the channel at the transmitter ................ 39


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Tables

Terminology............................................................................................................................. 5

Table 1. Channel model parameters .................................................................................... 14

Table 2. Tap modulation of models D and E ...................................................................... 26

Table 3. Spatial correlation coefficients of model F and one-ring model........................... 37


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1.5 Outline

This diploma work is dividend into different sections. The included sections are

following:

Section 2 describes the background behind wireless LAN, IEEE standard and channel

classification. Short description of the OFDM (Orthogonal Frequency DivisionMultiplexing) that is used in IEEE 802.11a, and its impact on the MIMO

communication channel development is included too.

Section 3 summarizes the theoretical work done on channel models during last couple

of years. It includes model parameters needed to achieve reasonable and good

channel. The channel models are divided into physical and non-physical models.

Section 3.9 presents simulated MIMO channel properties using Matlab program

written by Laurent Schumacher [3].

Section 4 contains several possible implementation methods and their

advantages/disadvantages.

Section 5 contains comparison results of measured and theoretical data for the

channel model F and one-ring model.

Section 6 contains second part of theoretical work applicable to steps before and after

channel modelling. That includes transmission scheme, detection algorithm, coding

and decoding method.


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2.2 MIMO Channel Model Classification

MIMO channel describes the connection between the transmitter (Tx) and receiver

(Rx). In following, only 2 antennas at the Tx and 2 antennas at theRx are considered,

i.e. 2x2 MIMO system.

Figure 1 illustrates a 2x2 MIMO system with theHchannel matrix and the scatteringmedium around (the graphical picture of scatters is depicted in Figure 3 of Section

3.1).

M-antennas N-antennasScattering medium

Figure 1. MIMO channel representation

whereM=N=2 represents number of antennas at Tx andRx, respectively.

For the above 2x2 MIMO channel, the input-output relationship can be expressed as

)()(*)()( tntstHty += (1)

wheres(t) is the transmitted signal,y(t) is the received signal, n(t) is additive white

Gaussian noise (AWGN),H(t) is anNbyMchannel impulse response matrix and (*)

denotes convolution.

The thesis is restricted to the frequency flat fading channel, and therefore the

corresponding input output relationship simplifies to

nHsy += (2)

whereHis the narrowband MIMO channel matrix.

The derivation of theHmatrix is the emphasis of this thesis. The channel matrixH

fully describes the propagation channel between all transmit and receive antennas.

Before arriving to channel matrixHthere has to be some additional properties

included, such as power delay, spatial correlation functions and impact of fading,

explained later on.

The MIMO channel without noise and with representation of the channel matrixH

can be expressed as:

=

=L

l

llHH1

)()( (3)

whereL is the number of taps (time bins) of the channel model, )(H is theMxN

matrix of the channel impulse responses.

n

sTx

Processin

g

h 1,1

h 1,2

Rx

Processin

gh 2,2

h 2,1

Channel

H


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For a 2x2 MIMO system the channel matrix is

MxNCH )(

=

2221

1211

hh

hhH (4)

MxNMNl

lH ][= is a complex matrix which describes the linear transformation

between two considered antenna arrays at delay land

l

MNis the complex

transmission coefficient from antennaMat the transmitter to antennaNat the

receiver. The complex transmission coefficients are assumed to be zero mean

complex Gaussian and have the same average power lp . The coefficients are

independent from one time delay to another. The correlation between different pairs

of the complex transmission coefficients is presented in Section 3.5. For all models,

both physical and non-physical, the correlation coefficients are computed using

mathematical formulas from Appendix C.

The models presented in this diploma work are classified in different ways. But before

explaining model structure, the reader should have some knowledge of different

classifications in the area of channel modeling.Wideband Models vs. Narrowband Models: the MIMO channel models can be divided

into the wideband models and the narrowband models directly by considering the

bandwidth of the system. The wideband models treat the propagation channel as

frequency selective, which means that different frequency subchannels have different

channel response. On the other hand, the narrowband models assume that the channel

has frequency non-selective fading and therefore the channel has the same response

over the entire system bandwidth.

Field Measurements vs. Scatter Models: to model the MIMO channel, one approach

is to measure the MIMO channel responses through field measurements. Some

important characteristics of the MIMO channel can be obtained by investigating the

recorded data and the MIMO channel model can be modelled to have similar

characteristics. Models based on MIMO channel measurements were reported in [4].

An alternative approach is to postulate a model (usually involving distributed scatters)

that attempts to capture the channel characteristics. Such a model can often illustrate

the essential characteristics of the MIMO channel as long as the constructed scattering

environment is reasonable. It is the environment of scatters that is in detail studied

here.

Non-physical Models vs. Physical Models: the MIMO channel models can be divided

into the non-physical and physical models. The non-physical models describe MIMO

channel via statistical characteristics obtained from the measured data. Another

category is the physical models that are based on parameter setup and theoreticalresults. In general, these models choose some crucial physical parameters to describe

the MIMO propagation channels. Some typical parameters include Angel of Arrival

(AoA), Angle of Departure (AoD), carrier frequency, antenna spacing.


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3.2 Non-physical MIMO channel model

A statistical model for wireless indoor MIMO channel is conducted at Victoria

University in Melbourne by Jason Gao and Michael Faulkner. The measured data is

characterised and presented as a non-physical model with scatter and wideband

assumptions considering NLOS Rayleigh fading channel.

The non-physical models provide accurate channel characterization for the

environments under study. On the other hand, they give limited insight to the

propagation characteristics of the MIMO channel and depend on the measurement

equipment, e.g. the bandwidth, the configuration and aperture of the arrays, the height

and response of transmit and receive antennas in the measurement.

In order to capture path correlation, a general abstract scattering model is assumed,

based on one-ring model. This model was first employed by [6]. It is an abstract

model, which is non-site-specific and mathematically derivable. Yet, it captures some

of the key aspects of the environment under investigation.

Figure 4. One-ring model

A circular disc (with radius R ) of uniformly distributed scatters Sis placed around the

mobile unit. The channel parameter h_{NM} connecting transmit elementMand

receive elementNis geometrically constraint. The base station (BS) is assumed to be

elevated and therefore not obstructed by local scattering, while the mobile station

(MS) is surrounded by scatters. Figure 4 illustrates this scenario where Tx is antenna

element at the BS,Rx is the antenna element at the MS.D is the distance between the

BS and MS.R is the radius of the ring of scatters, is the AoA at the BS, is the AS

at BS. Denote the effective scatter on the ring by S() and let be the angle between

the scatter and the array at the MS.In the model, it is assumed that S() is uniformly

distributed over all angles and i.i.d. . It is further assumed that each ray is reflected

only once and that all rays reach the receiver array with the same power [6]. For this

reason, the channel coefficients are modelled by a zero mean complex Gaussianrandom variable.

The path correlation between elements of transmit and receive antenna arrays is

considered and determined by antenna spacing, angle of departure/arrival

(AoD/AoA). When determining the delay power profile the Saleh-Valenzuelas model

[7] was considered, see Section 3.4. The Doppler effects are not considered. As such,

the model is applicable to situations where during data transmission, the transmission

channels can be assumed stationary in time domain. Appendix B lists one-ring model

BS MS


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with 6 clusters and data of Azimuth Spread (AS) and Angle of Arrival/Departure

(AoA/AoD) as a uniform distribution over all angles.

Following sections characterize the properties of both physical and non-physical

models. Their mathematical derivation is the same, except that one-ring model does

not include Doppler effect.

3.3 MIMO Matrix Formulation

The MIMO channel matrixHfor each tap, at one instance of time, in the A-F delay

profile models can be separated into a fixed (constant, LOS) matrix and a variable

Rayleigh matrix. For the case of the one-ring model there is only a part of the

Rayleigh matrix since LOS component is not included.

For 2 x 2 MIMO system, the channel matrixH[5] is:

=

++

+= vF H

KH

K

KPH

1

1

1

(5)

++

+=

2221

1211

2

12

1

1

1

10

110

1

1 2221

1211

XX

XX

Kee

ee

K

KP

jj

jj

whereXNM(N-th receiving andM-th transmitting antenna) are correlated zero-mean,

unit variance, complex Gaussian random variables as coefficients of the variable

(Rayleigh) matrixHv , exp(jNM) are the elements of fixed matrixHF ,Kis the Rician

K-factor, andPis the power of each tap.

To correlate theXNMelements of the matrixX[5], theKroneckerproduct of the

transmit and receive correlation matrices is preformed:

[ ] [ ] [ ] [ ]iidRxTx HRRX2/1

= (6)

whereRTx andRRx are the receive and transmit correlation matrices, respectively, and

Hiidis a vector (only here, otherwise it is a matrix) of independent zero mean, unit

variance, complex Gaussian random variables, and

[ ] [ ]

[ ] [ ]RxMNRxTxNMTx

R

R

=

= (7)

where TxNMare the complex correlation coefficients betweenN-th andM-th

transmitting antennas, and RxMNare the complex correlation coefficients betweenM-

th andN-th receiving antennas. Section 3.5 describes in detail the relationship of

correlation coefficients.


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3.4 Power Delay profile

When determining the power delay profile (PDP) the Saleh-Valenzuelas model was

used [7]. This model is based on indoor measurement results where it was found that

received signal rays (due to multipath) arrive in clusters.

The mathematical representation of the received signal amplitude kl is a Rayleigh-

distributed random variable with a mean-square value that obeys a double exponential

decay law

//22 )0,0( kll eeT

kl

= (8)

where )0,0(2 represents the average power of the first arrival of the first cluster, Tl

represents the arrival time of the lth

cluster, and kl is the arrival time of the kth

arrival

within the lth cluster, relative to Tl. The parameters and determine the inter-cluster

signal level rate of decay and the intra-cluster rate of decay, respectively. The rates of

the cluster and ray arrivals can be determined using exponential rate laws

)(

11)|(

=

ll TT

ll eTTp (9)

)(

,11)|(

=

ll TT

lkkl ep

(10)

where is the cluster arrival rate and is the ray arrival rate.

Figure 5 shows Model D delay profile with clusters outlined by exponential decay

(straight line on a log-scale)., see Appendix A.

Figure 5. Model D, Power Delay Profile

Source: V. Erceg,Indoor MIMO WLAN Channel Models, 2003.

-50 0 5 10 15 20 25 30 35 400

5

1

1

2

2

3

Delay in nanosec.

dB

Cluster 1

Cluster 2

Cluster 3


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=

dPASDDXXR )()sincos()( (12)

and

=

dPASDDXYR )()sinsin()( (13)

The correlation properties of the fading, and the values of the symmetrical correlation

matricesRTxandRRx, are completely defined by the PAS and its standard deviation.For the 2 x 2 MIMO channel, transmit and receive correlation matrices are expressed

as:

=

1

*1

21

12

Tx

Tx

TxR

and

=

1

*1

21

12

Rx

Rx

RxR

(14)

Further, the spatial correlation properties of a MIMO system uses Kronecker product

of the spatial correlation matricesRTx andRRxto define total correlation

1

)()( RxTxH

RRHvecHvecR==

(15)

whose elements are correlation coefficients, where (.)H is the conjugate transpose and

the vec(.) operator rearranges the 2 x 2 matrixHinto a column vector of size 4 x 1.

Such a model has been experimentally validated in [12].

Cluster and tap PAS shape follow Laplacian distribution [9].

The angle of arrival statistics within a cluster were found to closely match the

Laplacian distribution [13, 14, 15]

/2

2

1)(

= ep (16)

where is the standard deviation of the PAS (which corresponds to the numerical

value of AS). The Laplacian distribution is shown in Figure 7 [14] (a typical

simulated distribution within a cluster, with AS = 30o).

1The structure of the Kronecker product depends whether one wants to simulate a downlink

transmission (as presented here) or uplink TxRx RRR =


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Figure 7. Laplacian distribution with AS=30 [14]

The Laplacian function exhibits the sharp peak in the LOS direction and is confined

within [-180, 180].For the case of the physical models, it was found in [13, 14] that the cluster mean

AoA and AoD have a random uniform distribution over all angles [0, 2]. Due to thecentral limit theorem, when the number of scatters becomes large, the channel

coefficient of matrixHis Gaussian distributed.

For the case of the one-ring model, the references [9, 10] show that the AS increases

with decreasing distance betweenBSandMS, provided that this distance is still much

greater than the radius of the circle within the scatters surrounding theMSare placed,

and that the assumption of scatters distributed aroundBSholds. Consequently, for a

same element separation distanceD,BSelements are less correlated thanMSonce as

they experience a greater AS [10]. On the other hand, it is shown in [9, 11] that the

height of theMShas also an influence on the AS, as the spreading increases with

decreasing antenna height.

3.6 Tap Time and Angle Dependence

The channel impulse response as a function of time and angle is a separable function,

h, [16, 17]

)()(),( hthth = (17)

where tis time and is an angle.


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For the physical models A-F following is valid: In order to avoid frequency selective

fading, the transmission rate is set to be less than the coherence bandwidth of the

channel. And in order to reduce the distortion caused by fast fading, it is important to

set the transmission rate to be more than the channel-fading rate.

The simulations results of the fading are only presented for the case of the frequency

non-selective and slow fading, see Section 3.9. In WLAN technology, the OFDM

modulation is used for IEEE 802.11a standard. OFDM avoids frequency selective

fading by breaking the carrier signal into subcarriers with lower bit rates and thereby

longer symbol duration. The simulation results are presented in Section 3.9.


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Figure 12. Channel model C: a) CDF, b) Spatial correlation of the firs six taps

Figure 13. Channel model C: a) Spatial correlation of the six middle taps, b) Spatial

correlation of the last two taps

a) b)

a) b)


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4 Implementation results

In this section the implementation results of MIMO channel model are presented.

These results only consider implementation of physical models A-F.

4.1 Background

Seen from Acreos point of view, there are two possible implementation tools: Matlab

6.5 and Advanced Design Systems (ADS). Both have its advantages and

disadvantages.

It is obvious that the easiest method to implement the channel model is to use Matlab

6.5, since the program written by L. Schumacher [3], is for Matlab. However, all other

work done on algorithms, transmission schemes and detection methods here at Acreo

is done in ADS. So, for Acreo it would be best to even have the channel model

implemented in ADS. For this solution there are three alternatives. The first

alternative is to use a Matlab program written by L. Schumacher [3] and translate itinto the C++ programmable language. The second alternative is to implement a

Matlab block which links between Matlab 6.5 and ADS. The third alternative is to use

the Matlab 6.5 that generates channel matrixHand save it as an ASCII file. This

ASCII file can then be read from ADS and used as a covariance channel matrixH.

4.2 Translation of Matlab code into C++

Matlab 6.5 Converter was used to translate the whole Matlab program [3] written by

L. Schumacher, which consisted of approx. 20 files each 1-2 A4 pages long, into C++

code. The translation went without any problems since behind Matlab code lies a Clanguage. The Matlab converter even linked the files and their functions together. To

be able to use C++ in ADS, the requirement is to build a specific structure of C++ and

use a Make files during the compilation. ADS Ptolemy manual was used and the

structure of C++ got a new appearance, see Figure 16.

After modifying the C++ codes, it was realized that calling the functions and their

parameters is not as smooth as in Matlab. In the Matlab program there is a main

function, which calls other functions together with their multiple parameters of 2-D

and even 3-D matrices. This main function generates the 4-D channel matrixHthat is

needed. However, to be able to generate the channel matrixH, the calculations of

fading matrices and spatial correlation needs to be done. The call of these could not

be done successfully in C++. This appeared to be very time consuming, instead someother alternatives were studied, explained further down.


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Ptolemy example of C++ code

defstar { // Body of the programname {example_MIMO}

domain {SDF}desc { MIMO channel models}version {Source: Indoor MIMO WLAN channel model,Date: 2003-09-19}author {Aida Botonji}

location {MIMO}

inmulti { // Defines multiple inputsname {Fading_Type}type {string}desc {indicats the nature of the Doppler}

}output { // Defines output

name {H}type {FLOAT_MATRIX}desc {4-D matrix}

}defstate { // Defines Parameters

name {ID}type {enum}default {"D"}desc {IEEE 802.11 case to be simulated}enumlist {A, B, C, D, E, F}

//////////Initialisation///////////////////////setup { //Parameter conversion

} // end of setup

code {//Initialisation of size variables//Parameters assessment//Large-scale fading//Set-up of the iteration process

} //end of code

/////////Main loop//////////////////////////////go { //Beginning of simulation

//Defines H channel matrix

}//end of go

////////End of simulation//////////////////////

wrapup {

} //end of wrapup

}//end of defstar

Figure 16. Ptolemy example of C++ code


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4.5 Channel model implemented in ADS

This section presents a channel model implemented in ADS.

Figure 19 illustrates the channel model with the channel matrixHand tap delays. This

model is based on the Jakes model from 1974 [6]. Here is matrixHmultiplied with

tap delays explained in Section 3.4. The delay profile determines the frequency non-

selective nature of the channel. Delay profile is taken directly from the table in

Appendix A and varies from model to model. Other parameters such as impact of

fading, tap power, correlation and generation of actual channel matrixHis done by

the Matlab program written by L. Schumacher [3]. When using this model for

simulations in ADS, the user is required to use one of the above two alternatives to

get the channel matrixHfor the desired channel model.

This schematic (together with the second alternative of the generation of the channel

matrixHin Section 4.4) was used to confirm if the implementation is successful or

not. The comparison between the graphs that ADS generated and those that Matlab

6.5 generated in Section 3.9.2 were identical. This means that the MIMO channel

implemented in ADS gives satisfactory properties comparing to the theoretical

presentation of MIMO channel in Section 3.1-3.8.

The number of delays depends on number of taps a channel model has. When

multiplication is performed, the channel is summed and used as an input to the

antenna array at the receiver.

Figure 19. Channel model in ADS with channel matrixHand power delays

Delays

ChannelH

Arbitrary bits


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copy of the parallel to serial converted output of theN-point IFFT is then prepended.

The resulting OFDM symbols are launched simultaneously from the individual

transmitting antennas. The CP is essentially a guard interval which serves to eliminate

interference between OFDM symbols and turns linear convolution into circular

convolution such that the channel is diagonalized by the FFT. In the receiver the

individual signals are passed through OFDM demodulator which first discard the CP

and then perform anN-point FFT. The outputs of the OFDM demodulator are finally

separated and decoded.

Transmitting and receiving diversity are both similar and different in many ways.

While receiving diversity needs merely multiple antennas which fade independently,

and is independent of coding/modulation schemes, transmitting diversity needs

special modulation/coding schemes in order to be effective. Also, receive diversity

provides array gain, whereas transmitting diversity does not provide array gain when

the channel is unknown in the transmitter.

6.3 Receiver

In the following a detection and decoding algorithm at the receiver are presented. Thereceiver components are illustrated in the above Figure 23.

Detection algorithm, so called V-BLAST, is most promising one to be used in future

realizations of MIMO systems. V-BLAST does not jointly decode all the transmit

signals, it first decodes the strongest signal then it subtracts this strongest signal

from the received signal, proceed to decode the strongest signal of the remaining

transmit signal, and so on. The optimum detection order in such a nulling and

constellation strategy is from the strongest to the weakest signal. Assuming that the

channelHis known, the main steps of the V-BLAST algorithm can be summarized as

follows:

Nulling: An estimate of the strongest transmit signal is obtained by nulling outall the weaker transmit signals, say using zero forcing (ZF) criterion. ZF

basically inverts the channel transfer matrix H. Then the transmitted data

symbol vector s is obtained as rHs 1 = . Slicing: The estimated signal is detected to obtain the data bits. Cancellation: These data bits are remodulated and the channel is applied to

estimate its vector signal contribution at the receiver. The resulting vector is

then subtracted from the received signal vector and the algorithm returns to

the nulling step until all transmit signals are decoded.

For a more in depth treatment of the V-BLAST algorithm the interested reader is

referred to [27, 28].

The optimum decoding method is maximum likelihood (ML) where the decoder

compares all possible combinations of symbols which could have been transmitted

with what is observed.

The ML decoder yields the best performance in terms of error rate. However, this

decoder also has the highest computational complexity which moreover exhibits

exponential growth in the number of transmit antennas. For 2x2 MIMO system ML

decoder works very well.


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7 Conclusion

This thesis investigates two types of channel models, physically based models and

non-physically based models. Six types (A-F) of physical models for different

scenarios are presented based on physical parameters using Matlab program written

by L. Schumacher [3]. One type of non-physical model, one-ring model, is presented

obtained from measured data conducted at Victoria University by Jason Gao andMichael Falukner.

Conclusion of implementation of the physically based channel model is that the ADS

tool gives more variations than the Matlab 6.5 does. In ADS the channel can be much

easier integrated into the whole communication chain. Additional properties can be

integrated independently leaving user with the choice to affect or not to affect the

channel model and its characteristics.

Conclusion of the comparison between the physical model F and non-physical model

onering model is: both channels follow Rayleigh distribution very well and the

deviation in spatial correlation coefficients is marginal. The impulse response initiates

that the statistical data has lower magnitude than the case F model of channel matrix

Hdue to the fact that impulse response for model F is an average value while for one-

ring model it is measured under a certain period of time. The interference of people

walking in corridors/rooms causes also a visible interference in one-ring model.

Looking at the results of Uniform, Gaussian and Laplacian PAS one can notice that

correlation coefficient decreases with increasing angular spread (AS) and with

decreasing angle of incidence of the signals, from broadside to end [29].

The main motivation for using OFDM in a MIMO channel due to the fact that OFDM

modulation turns a frequency-selective MIMO channel into a set of parallel

frequency-flat MIMO channels. This significantly reduces receiver complexity inwireless broadband systems.

In order to gain full advantage of spatial multiplexing the Signal to Noise Ration

(SNR) should be reasonably good.


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9 References

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Tech. Memo., June 1995.

[2] G.J. Foschini and M.J. Gans, On limits of wireless communications in a fading

environment when using multiple antennas, Wireless Pers. Commun., vol. 6, pp. 331-335, Mar.1998.

[3] L. Schumacher, WLAN MIMO Channel Matlab program, download information:

http://www.info.fundp.ac.be/~lsc/Research/IEEE_80211_HTSG_CMSC/distribution_te

rms.html

[4] H. Blcskei and A. J. Paulraj,Multiple-Input Multiple-Output (MIMO) Wireless

Systems, chapter in "The Communications Handbook", 2nd edition, J. Gibson, ed.,

CRC Press, pp. 90.1 - 90.14, 2002.

[5] V. Erceg, Indoor MIMO WLAN Channel Models, IEEE 802.11-03/161r2, September

2003.

[6] W. C. Jakes,Microwave Mobile Communications, New York: Wiley, 1974.

[7] A. A. M. Saleh and R. A. Valenzuela,A statistical model for indoor multipath

propagation, IEEE J. Select. Areas Comm., vol.5, 1987, pp. 128-137.

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Envelopes of 900 MHz Signals Received at a Mobile Radio Station Site, IEEE

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[9] J. Salz and J. Winters, Effect of Fading Correlation on Adaptive Arrays in Digital

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[10] K. I. Pedersen, P. E. Mogensen, and B. H. Fleury, Spatial Channel Characteristics in

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[11] U. Martin, Spatio-Temporal Radio Channel Characteristics in Urban Macrocells,

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[12] M. Pettersen, P. H. Lehne, J. Noll, O. Rstbakken, E. Antonsen, R. Eckhoff,

Characterization of the Directional Wideband Radio Channel in Urban and Suburban

Areas, IEEE Proc. Vechicular Technology Conference VTC99 Fall, pp. 1454-1459,

Amsterdam, The Netherlands, September 1999.

[13] L. Schumacher, J. P. Kermoal, F. Frederiksen, K. I. Pedersen, A. Algans, and P.

Mogensen,MIMO Channel Characterization, IST-1999-11729 METRA Deliverable

D2, February 2001. Available at http://www.ist-metra.org/deliverables

[14] Q. H. Spencer, et al.,Modeling the statistical time and angle of arrival characteristicsof an indoor environment, IEEE J. Select. Areas Commun., vol. 18, no. 3, March 2003,

pp. 347-360.

[15] R. J-M. Cramer, R. A. Scholz, and M. Z. Win,Evaluation of an ultra-wide-band

propagation channel, IEEE Trans. Antennas Propagat., vol. 50, no. 5, May 2002, pp.

561-570.

[16] Chi-Chin Chong, David I. Laurenson and Stephen McLaughlin, Statistical

Characterization of the 5.2 GHz wideband directional indoor propagation channels


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47

Appendix A Physical models A-F

Model A

Tap

index1

Excess

delay

[ns]

0

Power

[dB]0

AoAAoA

[]45

AS

(receiver)

AS

[]40

AoDAoD

[]45

AS

(transmitter)

AS

[]40


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49

Model C

Tapindex

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Excess

delay

[ns]

0 10 20 30 40 50 60 70 80 90 110 140 170 200

Cluster 1Power

[dB]0 -2.1 -4.3 -6.5 -8.6 -10.8 -13.0 -15.2 -17.3 -19.5

AoA AoA

[]290.3 290.3 290.3 290.3 290.3 290.3 290.3 290.3 290.3 290.3

AS

(receiver)

AS

[]24.6 24.6 24.6 24.6 24.6 24.6 24.6 24.6 24.6 24.6

AoDAoD

[]13.5 13.5 13.5 13.5 13.5 13.5 13.5 13.5 13.5 13.5

AS

(transmitter)

AS

[] 24.7 24.7 24.7 24.7 24.7 24.7 24.7 24.7 24.7 24.7

Cluster 2Power

[dB]-5.0 -7.2 -9.3 -11.5 -13.7 -15.8 -18.0 -20.2

AoAAoA

[]332.3 332.3 332.3 332.3 332.3 332.3 332.3 332.3

ASAS

[]22.4 22.4 22.4 22.4 22.4 22.4 22.4 22.4

AoDAoD

[]56.4 56.4 56.4 56.4 56.4 56.4 56.4 56.4

ASAS

[]22.5 22.5 22.5 22.5 22.5 22.5 22.5 22.5


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52

Model E (2/2)

Cluster 3Power

[dB]-7.9 -9.6 -14.2 -13.8 -18.6 -18.1 -22.8

AoAAoA

[]80.0 80.0 80.0 80.0 80.0 80.0 80.0

ASAS

[]37.4 37.4 37.4 37.4 37.4 37.4 37.4

AoDAoD

[]61.9 61.9 61.9 61.9 61.9 61.9 61.9

ASAS

[]38.0 38.0 38.0 38.0 38.0 38.0 38.0

Cluster 4Power

[dB]-20.6

AoAAoA

[]182.0

ASAS

[]40.3

AoDAoD

[]275.7

ASAS

[]38.7


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Model F (1/2)

Tap

index1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Excess

delay

[ns]

0 10 20 30 50 80 110 140 180 230 280 330 400 490 600

Cluster 1Power

[dB]-3.3 -3.6 -3.9 -4.2 -4.6 -5.3 -6.2 -7.1 -8.2 -9.5 -11.0 -12.5 -14.3 -16.7 -19.9

AoAAoA

[]315.1 315.1 315.1 315.1 315.1 315.1 315.1 315.1 315.1 315.1 315.1 315.1 315.1 315.1 315.1

AS

(receive)

AS

[]48.0 48.0 48.0 48.0 48.0 48.0 48.0 48.0 48.0 48.0 48.0 48.0 48.0 48.0 48.0

AoDAoD

[]56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2 56.2

AS

(transmit)

AS

[]41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6 41.6

Cluster 2Power

[dB] -1.8 -2.8 -3.5 -4.4 -5.3 -7.4 -7.0 -10.3 -10.4 -13.8 -15.7

AoAAoA

[]180.4 180.4 180.4 180.4 180.4 180.4 180.4 180.4 180.4 180.4 180.4

ASAS

[]55.0 55.0 55.0 55.0 55.0 55.0 55.0 55.0 55.0 55.0 55.0

AoDAoD

[]183.7 183.7 183.7 183.7 183.7 183.7 183.7 183.7 183.7 183.7 183.7

ASAS

[]55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2

Cluster 3Power

[dB]-5.7 -6.7 -10.4 -9.6 -14.1 -12.7 -18.5

AoAAoA

[]74.7 74.7 74.7 74.7 74.7 74.7 74.7

ASAS

[]42.0 42.0 42.0 42.0 42.0 42.0 42.0

AoD

AoD

[] 153.0 153.0 153.0 153.0 153.0 153.0 153.0

ASAS

[]47.4 47.4 47.4 47.4 47.4 47.4 47.4


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Truncated Laplacian PAS

Truncated Laplacian PAS model is defined as:

=

+

=

c

kL

N

k kk

kk

k

kL

kL

L eQ

PAS1 ,0

,0

,2

,

,

)]([

)]([

2)( ,

2

0

The normalisation condition is given by

12

exp1,

, =

=

cN

mk kL

kkLQ

The cross-correlation function is then given by:

+

+

+=

==)2cos(

2)2sin(2

2exp

2

)2cos(

)2(2

)(

24)()(

,

,,

1

,0

2

2

,

2

1 ,

,

0,

k

kL

k

kL

k

kL

m

k

kL

mN

k kL

kL

LXX

mmm

m

m

DJQDJDR

c

[ ][ ] [ ]

++++

+

++

=

=

+

=

k

kL

k

kL

k

kL

m

k

kL

mN

k kL

kL

LXY

mmm

m

m

DJQDR

c

)12(cos

2)12(sin)12(

2exp

2

)12(sin

)12(2

)(

24)(

,

,,

0

,0

2

2

,

)12(

1 ,

,

,

mimo channel models

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