Department of Science and Technology Institutionen för teknik och naturvetenskap Linköpings Universitet Linköpings Universitet SE-601 74 Norrköping, Sweden 601 74 Norrköping

Examensarbete LITH-ITN-ED-EX--04/004--SE

MIMO channel models

Aida Botonjić

2004-01-30

LITH-ITN-ED-EX--04/004--SE

MIMO channel models

Examensarbete utfört i Elektronikdesign vid Linköpings Tekniska Högskola, Campus

Norrköping

Aida Botonjić

Handledare: Christian Kark Examinator: Shaofang Gong

Norrköping 2004-01-30

Rapporttyp Report category Examensarbete B-uppsats C-uppsats D-uppsats _ ________________

Språk Language Svenska/Swedish Engelska/English _ ________________

Titel Title MIMO channel models Författare Author Aida Botonjić

Sammanfattning Abstract The objective of this diploma work is to investigate a set of Multiple Input Multiple Output (MIMO) channel models compatible with the emerging IEEE 802.11n standard. This diploma work validates also advanced, innovative tools and wireless technologies that are necessary to facilitate wireless applications while maximizing spectral efficiency and throughput. MIMO channel models can be used to evaluate new Wireless Local Area Network (WLAN) proposals based on multiple antenna technologies. The purpose of this thesis is to investigate means of channel models and their implementation in different environments such as: Matlab, C++ and Advanced Design Systems (ADS). The investigation considers also a comparison between the channel models based on theoretical data and parameter setup to the channel models based on statistical characteristics obtained from measured data. Investigation and comparison of a MIMO channel models consider steering channel matrix H, spatial correlation coefficients, power delay profiles, fading characteristics and Doppler power spectrum.

ISBN _____________________________________________________ ISRN LITH-ITN -ED-EX--04/004--SE _________________________________________________________________ Serietitel och serienummer ISSN Title of series, numbering ___________________________________

Nyckelord Keyword

MIMO, channel model, WLAN, spatila correlation, Doppler Spectrum, fading.

Datum 2004-01-30

URL för elektronisk version http://www.ep.liu.se/exjobb/itn/2004/ed/004

Avdelning, Institution Division, Department Institutionen för teknik och naturvetenskap Department of Science and Technology

To my mother Ismeta, my father Smajil and

my loving boyfriend Almir who always believed in me.

Preface This report summarizes the achievements of the Master Thesis work carried out between September and December 2003 at Acreo AB in Norrköping, Sweden. The report constitutes the final element of a Master of Science exam in Electronics Design Engineering at the University of Linköping at Campus Norrköping. I take this opportunity to thank my coordinator Christian Kark at Acreo AB and my examiner Shaofang Gong at ITN Linköpings University for useful discussions and suggestions. I’m 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.

Abstract The objective of this diploma work is to investigate a set of Multiple Input Multiple Output (MIMO) channel models compatible with the emerging IEEE 802.11n standard. This diploma work validates also advanced, innovative tools and wireless technologies that are necessary to facilitate wireless applications while maximizing spectral efficiency and throughput. MIMO channel models can be used to evaluate new Wireless Local Area Network (WLAN) proposals based on multiple antenna technologies. The purpose of this thesis is to investigate means of channel models and their implementation in different environments such as: Matlab, C++ and Advanced Design Systems (ADS). The investigation considers also a comparison between the channel models based on theoretical data and parameter setup to the channel models based on statistical characteristics obtained from measured data. Investigation and comparison of a MIMO channel models consider steering channel matrix H, spatial correlation coefficients, power delay profiles, fading characteristics and Doppler power spectrum.

Sammanfattning Målet med detta examensarbete är att undersöka Multiple Input Multiple Output (MIMO) kanalmodeller. Dessa modeller skall vara i överensstämmelse med kommande IEEE 802.11n standarden. Inom examensarbetet omfattas också avancerade och innovativa verktyg samt trådlös teknik. Trådlös teknik är nödvändig för att främja trådlösa applikationer samtidigt som man maximerar spektral effektivitet och genomströmning. MIMO kanalmodeller kan användas för att utvärdera kommande Wireless Local Area Network (WLAN) förslag baserat på multiple antennteknik. Undersökning av MIMO kanal modeller och dess implementering i olika miljöer beskrivs i denna rapport. Implementeringsmiljöer som beskrivs här är bland annat Matlab 6.5, C++ och Advanced Design Systems (ADS). Undersökningen omfattar också jämförelse mellan teoretiskt uppbyggda kanalmodeller samt uppmätt data för kanalmodeller. Undersökning och jämförelse av MIMO kanalmodeller omfattar kanal matris H, spatial correlation koefficient, fördröjnings tider, fädande egenskaper och Doppler kraft.

Table of contents TERMINOLOGY .................................................................................................................................. 5

1 INTRODUCTION ....................................................................................................................... 7 1.1 PURPOSE ................................................................................................................................. 7 1.2 LIMITATIONS .......................................................................................................................... 8 1.3 ASSUMPTIONS......................................................................................................................... 8 1.4 METHOD ................................................................................................................................. 8 1.5 OUTLINE ................................................................................................................................. 9

2 BACKGROUND INFORMATION ......................................................................................... 10 2.1 WLAN AND OFDM.............................................................................................................. 10 2.2 MIMO CHANNEL MODEL CLASSIFICATION......................................................................... 11

3 THEORY, PART I..................................................................................................................... 13 3.1 PHYSICAL MIMO CHANNEL MODEL..................................................................................... 13 3.2 NON-PHYSICAL MIMO CHANNEL MODEL ............................................................................ 15 3.3 MIMO MATRIX FORMULATION ........................................................................................... 16 3.4 POWER DELAY PROFILE........................................................................................................ 17 3.5 CORRELATION FUNCTIONS AND PAS DISTRIBUTION............................................................ 18 3.6 TAP TIME AND ANGLE DEPENDENCE ................................................................................... 20 3.7 FADING MULTIPATH CHANNELS .......................................................................................... 21 3.8 DOPPLER SPECTRUM ............................................................................................................ 24

3.8.1 Power Delay Spectrum.................................................................................................. 25 3.8.2 Fluorescent Lights......................................................................................................... 25

3.9 MIMO CHANNEL PROPERTIES .............................................................................................. 27 3.9.1 Capacity ........................................................................................................................ 27 3.9.2 Simulation results.......................................................................................................... 28

4 IMPLEMENTATION RESULTS ............................................................................................ 31 4.1 BACKGROUND ...................................................................................................................... 31 4.2 TRANSLATION OF MATLAB CODE INTO C++ ........................................................................ 31 4.3 ADS SCHEMATIC WITH ‘MATLAB’ BLOCK........................................................................... 33 4.4 ADS SCHEMATIC WITH ‘READFILE’ BLOCK......................................................................... 34 4.5 CHANNEL MODEL IMPLEMENTED IN ADS ............................................................................ 35

5 COMPARISON RESULTS ...................................................................................................... 36

6 THEORY, PART II ................................................................................................................... 38 6.1 SPATIAL MULTIPLEXING SCHEME ......................................................................................... 38 6.2 TRANSMITTER....................................................................................................................... 39 6.3 RECEIVER ............................................................................................................................. 40 6.4 ANTENNA SELECTION ........................................................................................................... 41

7 CONCLUSION .......................................................................................................................... 42

8 FUTURE WORK....................................................................................................................... 43

9 REFERENCES .......................................................................................................................... 44

10 BIBLIOGRAPHY .................................................................................................................. 46

2

APPENDIX A PHYSICAL MODELS A-F ................................................................................. 47 MODEL A......................................................................................................................................... 47 MODEL B ......................................................................................................................................... 48 MODEL C ......................................................................................................................................... 49 MODEL D......................................................................................................................................... 50 MODEL E (1/2)................................................................................................................................. 51 MODEL E (2/2)................................................................................................................................. 52 MODEL F (1/2)................................................................................................................................. 53 MODEL F (2/2)................................................................................................................................. 54

APPENDIX B NON-PHYSICAL MODEL................................................................................. 55 ONE-RING MODEL............................................................................................................................ 55

APPENDIX C POWER DISTRIBUTIONS................................................................................ 56 UNIFORM PAS ................................................................................................................................. 56 TRUNCATED GAUSSIAN PAS........................................................................................................... 57 TRUNCATED LAPLACIAN PAS......................................................................................................... 58

3

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 to Rx......................................................................... 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 of the 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 matrix H and power delays .................... 35

Figure 20. Impulse response of covariance channel matrix H for 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

4

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

5

Terminology Terminology

Abbreviation Explanation

AAU Association of American Universities

AC Analog Converter

ADS Advance Design Systems. Simulation tool.

AoA Angle of Arrival

AoD Angle of Departure

AS Azimuth or Angular Spread

AWGN Additive White Gaussian Noise

BLAST Bell Labs Layered Space-Time architecture

BS Base Station

CDF Cumulative Distribution Function

Coherence bandwidth

The range of frequencies within which the signal impairments of the channel don’t vary significantly

COST European Co-operation in the field of Scientific and Technical research

CP Cyclic Prefix

C/N Carrier to Noise Ratio

Diversity gain Improvements in link reliability obtained by transmitting the same data on independently fading branches

DS Delay Spectrum

Erf Error function

Fading Signal that experiences fluctuations in its amplitude and phase

FFT Fast Fourier Transform

HiperLan/2 High PERformance Local Area Network

HTSG Hattrick TeamSite Guide

IEEE Institute of Electrical and Electronics Engineer

IFFT Inverse Fast Fourier Transform

i.i.d. Independent and identically distributed

6

Interleaving A form of data scrambling that spreads bursts of bit errors evenly over the received data allowing efficient forward error correction

ISI Inter Symbol Interference

IST Information Society Technology

I/C Interference to Carrier Ratio

LOS Line of Sight – direct path between Tx and Rx without any reflections, diffraction and local scattering

METRA Multi Element Transmit Receive Antennas

MIMO Multiple Input Multiple Output

ML Maximum Likelihood

MMSE Minimum Mean-Square Error

MS Mobile Station

NLOS Non Line of Sight – reflected rays between Tx and Rx

OFDM Orthogonal Frequency Division Multiplexing. Modulation scheme which divides the available frequency band into subcarriers (or tones) of smaller bandwidth and thereby drastically simplifies equalization.

PAS Power Azimuth Spectrum

PDF Probability Density Function

PDS Power Delay Spectrum

PDP Power Delay Profile

P/S Parallel to Serial conversion

Rx Receiver

SISO Single Input Single Output

SNR Signal to Noise Ratio

STD Standard Deviation

S/P Serial to Parallel conversion

ULA Uniform Linear Arrays

Tx Transmitter

WLAN Wireless Local Area Network

WSS Wide Sense Stationary

7

1 Introduction Wireless channel modelling has always been the subject to active research, due to continual advancements of wireless technologies. Over the past few decades, there has been rapid development and deployment of cellular phone networks. Recently, there has been development of the so-called wireless LAN technology, as specified in HiperLan/2 for the European standard and IEEE 802.11 for the North American standard. Many of the future wireless services to be provided by the future generation mobile communication systems are likely to be used in low- mobility environments with limited temporal or multipath diversity. This is the reason why most of the communication research that is going on concentrates on realization of indoor environments. This thesis follows that path of MIMO indoor realization. The growing demand of increasing the capacity has pushed researches into investigation of space domains, beamforming, use of space diversity/ “smart antennas” and spectral multiplexing. For this reason conventional techniques with a single antenna fails to provide sufficient diversity. Instead multiple antennas give high-data rates and throughputs. Therefore the solution of using multiple antennas at both transmitter and receiver in indoor environment is growing in radio communication systems. The concept of Multiple Input Multiple Output (MIMO) system is its motivation to achieve higher throughputs within a given bandwidth thanks to space diversity schemes. Early theoretical work of a narrowband random MIMO channel was reported by Telatar [1] and Foschini [2], where they used simple channel model represented by matrix H with M transmit and N receive antennas.

1.1 Purpose The purpose of this thesis is to investigate means of MIMO channel models, their implementation in different environments and a comparison between the channel models based on theoretical data against measured data. Investigation of MIMO channel models should be compatible with the emerging IEEE 802.11n standard. The implementation of the channel models should consider three different environments, such as: Matlab, C++ and Advanced Design Systems (ADS). A comparison should consider the channel models based on theoretical data and parameter setup against the channel models based on the statistical characteristics obtained from measured data.

8

1.2 Limitations The emphasis of the thesis lies in MIMO channel model. However, transmission scheme, detection algorithm and decoding method are shortly presented to get a better overview of the overall channel and communication systems. For the sake of simplicity, the channel models consider only the effect of local scatters. The remote scatters are ignored assuming that the path loss will tend to limit their contribution to the overall channel. In addition, because of local scatters introduce multipath differences that are small compared to the transmit-receive range, the focus is laid on microscopic (Rayleigh) fading only. This thesis is also limited to a frequency-flat fading channel.

1.3 Assumptions In this thesis, the classification is made between physically based and non-physically based models, to easily distinguish between different channel models. Following assumption is not general, the classification is only valid for this thesis work. Assumption for the physically based models is that they rely on some physical parameters and theoretical results. While non-physical models assume a MIMO channel, which is described via statistical characteristics obtained from the measured data.

Further, both physical model and non-physical models assume scatter assumption. Narrowband assumption with both LOS and NLOS component is made for physical models, while non-physical model assumes wideband NLOS Rayleigh fading channel. Note this is not the case in general, this is only assumption made here.

1.4 Method For the investigation of the MIMO channel models, the web research articles using the Internet was a major source.

For the implementation of the MIMO channel models, three different tools were considered, simulation tool and programmable language of Matlab 6.5, programmable language C++ and simulation tool ADS. Programmable language C++ was used in order to build the blocks in ADS.

For the comparison of the MIMO channel models based on the theoretical data and measured data, the simulation tool Matlab 6.5 was used.

9

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 Division Multiplexing) 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.

10

2 Background information

2.1 WLAN and OFDM

In the early days of wireless networks, there were not any standards. The products from one vendor would not work with the products from another vendor. 1997 IEEE 802.11 established it self as the accepted standard for wireless LAN. Up till now the standard have had a chance to evolve from 802.11b to 802.11a and recently 802.11g. This thesis assumes use of IEEE 802.11a standard since it is accounted for 5 GHz high-speed transmissions with OFDM modulation. 802.11a is used in indoor and short outdoor environments, such as office buildings and campus environments. The transmission range is around 15-150 m indoor and 300 m outdoor. IEEE 802.11a achieves as high data rates as 54 Mbps. Orthogonal Frequency Division Multiplexing (OFDM) is the modulation scheme used in IEEE 802.11a. OFDM avoids frequency selective fading by dividing up a wideband channel into multiple narrowband flat fading parallel sub-channels. A subcarrier is placed in each sub-channel where each subcarrier may be modulated separately depending on SNR characteristics. Each subcarrier may be adapted through the time in order to continually optimize the data-carrying capacity of the channel. This increases the symbol duration and mitigates inter symbol interference (ISI) caused due to multipath. In WLAN systems, the signal energy is scattered and reflected from objects in the environment, components of the signal arriving at the receiver are spread out over a longer period of time than is desirable. This causes uneven delays in the signal arrival time. MIMO systems are designed in such a way to smooth out the delays and make the signals to arrive in some form of pattern. In the environment of the WLAN systems, there is always some interference of the signals. The challenge of the MIMO systems is then to provide a high-performance, reliable data link that can operate with restricted receiver power levels, severe channel fading due to multipath reflections and interfering energy from other devices nearby. In earlier works of communication systems this achievements have not been possible. The single input single output (SISO) channels dose not provides such a data transfer reliability. This is one of the reasons why the use of MIMO systems has increased so rapidly in the recent years.

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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 the Rx are considered, i.e. 2x2 MIMO system. Figure 1 illustrates a 2x2 MIMO system with the H channel matrix and the scattering medium around (the graphical picture of scatters is depicted in Figure 3 of Section 3.1).

M-antennas N-antennas Scattering medium

Figure 1. MIMO channel representation

where M=N=2 represents number of antennas at Tx and Rx, respectively. For the above 2x2 MIMO channel, the input-output relationship can be expressed as

)()(*)()( tntstHty += (1) where s(t) is the transmitted signal, y(t) is the received signal, n(t) is additive white Gaussian noise (AWGN), H(t) is an N by M channel 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) where H is the narrowband MIMO channel matrix. The derivation of the H matrix is the emphasis of this thesis. The channel matrix H fully describes the propagation channel between all transmit and receive antennas. Before arriving to channel matrix H there 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 matrix H can be expressed as:

∑=

−=L

lllHH

1)()( ττδτ (3)

where L is the number of taps (time bins) of the channel model, )(τH is the MxN matrix of the channel impulse responses.

n

s

Tx Processin

g

h 1,1

h 1,2

Rx Processin

g

y

h 2,2 h 2,1

Channel ’H ’

12

For a 2x2 MIMO system the channel matrix is

MxNCH ∈)(τ

=⇒

2221

1211

hhhh

H (4)

MxNMN

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

between two considered antenna arrays at delay τl and αlMN is the complex

transmission coefficient from antenna M at the transmitter to antenna N at 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 theoretical results. 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.

13

3 Theory, Part I In this Section, the summary of literature study is presented. The theory behind the MIMO channel models is needed to understand MIMO channel properties and Matlab program [3]. The theory section is also needed to be able to implement MIMO channel matrix H into ADS (see Section 4) and to be able to compare theoretical results against measured and collected data (see Section 5).

3.1 Physical MIMO channel model A primitive physical channel model is illustrated in Figure 2. A major characteristic of this model is that it does not rely on a geometrical description of the environment under study. It is described by power profiles, spatial correlation functions and fading characteristics.

Figure 2. Primitive channel model 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, explained further down in this section. In order to capture path correlation, a general abstract scattering model is considered, based on multi-cluster model. All the scatters are divided into groups, which are called clusters of scatters. Each of the cluster corresponds to one multipath, see Figure 3. This model represents the physical model D which is further depicted in Figure 5 of Section 3.4.

FIR filter (L taps)

⊗

Fading characteristics

Spatial correlation mapping matrix

Steering matrix

Power delay profile

⊗

⊗

RRx

RTx

Radiation Pattern

Tx RxS/P P/S

14

Figure 3. Scatter cluster example In Figure 3, the scatters are represented with arrival times. It can be seen that the scatters which are closer to Rx have faster arrival time compared to those that are placed far away. This artefact of motion is further studied in Section 3.8 as Doppler shift. There has been a use of A-F models, presented in [5]. These models are representative for small environments, such as residential homes and small offices, i.e. indoor environments. Model F represents larger space either indoor or outdoor. Appendix A lists the tables of all 6 models (A-F) with its model tap delays, corresponding power, Azimuth Spread (AS), Angle of Arrival/Departure (AoA/AoD). Table 1 summarizes the channel model parameters.

Model

Environment

LOS/NLOS

K (dB)

RMS delay spread (ns)

Number

of clusters

A Flat fading NLOS -∞ 0 1 tap B Residential LOS/NLOS 10

(first tap only)

15 2

C Residential/ Small Office

LOS/NLOS 3 (first tap

only)

30 2

D Typical Office NLOS -∞ 50 3 E Large Office NLOS -∞ 100 4 F Large Space/

(Indoors and Outdoors)

NLOS -∞ 150 6

Table 1. Channel model parameters

LO

Cluster 2

Tx Antennas

Rx Antennas

Cluster 3

Cluster 1

Arrival Times

t

15

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 S is placed around the mobile unit. The channel parameter h_{NM } connecting transmit element M and receive element N is 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, α is the AS at BS. Denote the effective scatter on the ring by S(θ) and let θ be the angle between the scatter and the array at the MS. In the model, it is assumed that S(θ) is uniformly distributed over all angles and i.i.d. θ. 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 Gaussian random 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-Valenzuela’s 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

α

16

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 matrix H for 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 matrix H [5] is:

=

++

+= vF H

KH

KKPH

11

1

(5)

++

+=

2221

1211

21

21

11

101

101

1 2221

1211

XX

XX

Kee

ee

KKP

jj

jj

φφ

φφ

where XNM (N-th receiving and M-th transmitting antenna) are correlated zero-mean, unit variance, complex Gaussian random variables as coefficients of the variable (Rayleigh) matrix Hv , exp(jφNM) are the elements of fixed matrix HF , K is the Rician K-factor, and P is the power of each tap. To correlate the XNM elements of the matrix X [5], the Kronecker product of the transmit and receive correlation matrices is preformed: [ ] [ ] [ ]{ }[ ]iidRxTx HRRX 2/1⊗= (6) where RTx and RRx are the receive and transmit correlation matrices, respectively, and Hiid is a vector (only here, otherwise it is a matrix) of independent zero mean, unit variance, complex Gaussian random variables, and

[ ] [ ][ ] [ ]RxMNRx

TxNMTx

RR

ρρ

== (7)

where ρTxNM are the complex correlation coefficients between N-th and M-th transmitting antennas, and ρRxMN are the complex correlation coefficients between M-th and N-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-Valenzuela’s 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 βkl is a Rayleigh-distributed random variable with a mean-square value that obeys a double exponential decay law γτββ //22 )0,0( kll ee T

kl−Γ−= (8)

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

represents the arrival time of the lth cluster, and τkl is the arrival time of the kth arrival within the lth cluster, relative to Tl. The parameters Γ and γ 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)|( −−Λ−

− Λ= ll TTll eTTp (9)

)(

,11)|( −−−

− = ll TTlkkl ep λλττ (10)

where Λ is the cluster arrival rate and λ 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

40

0

5

1

1

2

2

3

Delay in nanosec.

dB

Cluster 1

Cluster 2

Cluster 3

18

3.5 Correlation functions and PAS distribution A cross-correlation function between the waves impinging on two antenna elements mostly depends on the Power Azimuth Spectrum (PAS), the antenna spacing and on the radiation pattern of the antenna elements. In the following, MIMO channel model is presented with the uniform linear array (ULA) and with the omni-directional antenna elements. The PAS distribution in physical models, consider three types of different distributions, Uniform [7][8], Gaussian [7][8] and Laplacian [9], see Appendix C. The non-physical model (one-ring model) is modelled using only Laplacian PAS distribution. For these three distributions, the envelope correlation coefficient is computed as a function of the normalised distance, using the angel of incidence and the Azimuth Spread (AS) as indexing variables. All three distributions consider multicluster model. Which makes them valid to use for all models considered here. It is assumed that the spatial correlation function at the Rx is independent of n. This is reasonable assumption provided that all antennas at the Tx are closely co-located and have the same radiation pattern, so they illuminate the same surrounding scatters and therefore also generate the same PAS at the Rx, i.e. the same spatial correlation function.

Figure 6. Impinging waves from Tx to Rx

Using the notations of [9], with λd standing for the normalised distance between

elements, where d is the element spacing and λ the wavelength, and D=2πλd , one can

easily derive the cross-correlation function between the real and imaginary parts of the complex baseband signals received at two omni-directional antennas separated by the distance d. For ULA [7, 8] the complex correlation coefficient at the linear antenna array is expressed as field ρf(D) and envelope ρe(D):

22

)()()()( DjRDRDD XYXXfe +== ρρ (11)

where λπ /2 dD = , and RXX and RXY 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

φ

Dr = D sinφ

D = 2πd/λ

s

s

Rx Tx

19

∫−

=π

πφφφ dPASDDXXR )()sincos()( (12)

and ∫−

=π

πφφφ dPASDDXYR )()sinsin()( (13)

The correlation properties of the fading, and the values of the symmetrical correlation matrices RTx and RRx, 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

TxTxR

ρρ

and

=

1*1

21

12

Rx

RxRxR

ρρ

(14)

Further, the spatial correlation properties of a MIMO system uses Kronecker product of the spatial correlation matrices RTx and RRx to define total correlation 1)()( RxTx

H RRHvecHvecR ⊗== (15) whose elements are correlation coefficients, where (.)H is the conjugate transpose and the vec(.) operator rearranges the 2 x 2 matrix H into 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]

σθ

σθ /2

21)( −

= ep (16)

where σ 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 ⊗=

20

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, 2π]. Due to the central limit theorem, when the number of scatters becomes large, the channel coefficient of matrix H is Gaussian distributed. For the case of the one-ring model, the references [9, 10] show that the AS increases with decreasing distance between BS and MS, provided that this distance is still much greater than the radius of the circle within the scatters surrounding the MS are placed, and that the assumption of scatters distributed around BS holds. Consequently, for a same element separation distance D, BS elements are less correlated than MS once as they experience a greater AS [10]. On the other hand, it is shown in [9, 11] that the height of the MS has 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] )()(),( θθ hthth = (17) where t is time and θ is an angle.

21

3.7 Fading Multipath Channels Multipath fading occurs when the transmitted signal reaches the receiver via multiple paths with different delays and attenuation (or amplification). Characteristics of a multipath medium is: • Time spread introduced in the signal that is transmitted through the channel, and • Time variations in the structure of the medium. Multipath fading is broadly divided into two categories: large (or slow) and small (or fast) scale fading. Large scale fading refers to mean path loss averaged over several wavelengths, whereas small scale fading refers to dramatic changes in amplitude and phase when the receiver moves by as little as half a wavelength. There are several probability distributions that can be considered in attempting to model the statistical characteristics of the fading channel. When there are a large number of scatters in the channel that contribute to the signal at the receiver, as is the case in B-F models and one-ring model, application of the central limit theorem leads to a Gaussian process model for the channel impulse response. It is assumed that the process is zero-mean, that gives, the envelope of the channel response at any time instant has a Rayleigh probability distribution and the phase is uniformly distributed in the interval (0, 2π). That is

2

20

22

00 )( σ

σ

r

er

rp−

= for 00 ≥r (25)

where r0 is the magnitude of the multipath signal. Rayleigh fading is considered a worst-case scenario of fading. In the frequency domain, the reciprocal of the multipath spread is a measure of the coherence bandwidth of the channel. It is the range of the frequencies within which the signal impairments of the channel done not vary significantly, see Figure 8 c). In other words, if two signals are sent that are more than fd apart form each other in frequency, they will experience different channel conditions. Coherence bandwidth is denoted as

m

d Tf 1

≈ (26)

where fd denotes coherence bandwidth and Tm is delay-time of the multipath spread. When a signal is transmitted through the channel, if fd is small in comparison to the bandwidth of the transmitted signal, the channel is said to be frequency-selective. In this case, the signal is severely distorted by the channel and wideband signal could see a large variation in received power over its bandwidth. On the other hand, if fd is large in comparison with the bandwidth of the transmitted signal, the channel is said to be frequency-nonselective.

22

In the time domain, the time variations are evidenced as a Doppler broadening, further explained in Section 3.8. A slow changing channel (slow fading) has large coherence time Td or, equivalently, a small Doppler spread, see Figure 8 a). Td denotes the time period over which the channel’s impulse response is highly correlated.

d

d BT 1

≈ (27)

where Bd is a Doppler spread of the channel. Slow fading is desired fading since the channel conditions are stable and predictable during the time that the symbol is transmitted. Counter to the slow fading there is fast fading where fading conditions are coming and going faster than the symbols are being transmitted. The fading characteristics are defined by shaping a Doppler spectrum, correlation functions and multipath intensity profile. Figure 8 below, illustrates relationship between Doppler spectrum and its Spaced-time correlation function, Scattering function of the channel and Multipath intensity profile. Scattering function in Figure 8 b) provides a measure of the average power output of the channel as a function of the time delay τ and the Doppler frequency λ.

Figure 8. Relationship among the correlation functions and power spectra Source: John G. Proakis, Digital Communications, 2001

Multipath intensity profile Doppler power spectrum

a) b) c)

23

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.

24

3.8 Doppler Spectrum Assumption in this thesis was made as; both transmitter and receiver are stationary. Many may think that the impact of Doppler spectrum should not then have any impact on the channel model. However there are many clusters and their scatter rays in the channel. Their motion, reflection and diffraction give rise to the time variant nature of the channel. The expression of the Doppler spectrum has been derived by Clarke [18]. It is well known that the Doppler spectrum lies within a [-fD, fD] bandwidth, where fD is the maximal Doppler shift/spread, defined as

λvf D = (18)

where v stands for the velocity of the movement. (fD should not be confused with fd which is coherence bandwidth of the channel in frequency domain). Time variations of the channel are evidenced as a Doppler shift of a spectral line. The metric used to measure the impairment caused by the time variant nature of channel is how rapidly the channel fades. The main parameter of the Doppler spectrum is heavily depending on simulation assumptions. On the other hand, in the physical models the shape of the Doppler spectrum can be derived knowing the PAS and the radiation pattern of the receiving antenna [19]. Then the classical U-shaped, see Figure 9 below, Clarke’s Doppler spectrum expression is:

21

1)(−

−∝

DfffP (19)

This formula (19) applies to all physical models, A-E, as both communication ends are surrounded by scatters.

Figure 9. Doppler spread/shift Source: John G. Proakis, Digital Communications, 2001

The Doppler power spectrum is plotted above in the Figure 9 where BDm fff ±= (fD –Doppler spread and fB – spectral broadening).

25

3.8.1 Power Delay Spectrum The Power Delay Spectrum (PDS) has been widely studied as part of the time-domain characterization of wireless radio channels. In accordance with [20], the PDS is accurately modeled by a one-sided exponential decaying function

=−

0)( D

t

etPσ

t>0 (20)

where σD represents the delay spectrum (DS). Values of DS have been proposed for the different environments [12]. For our models, the interesting value is for the indoor environments, which value lies between 35-100ns [17].

3.8.2 Fluorescent Lights Effects of fluorescent lights on signal fading characteristics for indoor channel were presented in [21]. The presence of fluorescent lamps creates an environment where reflections are being introduced, thus creating a fast changing electromagnetic environment. This is an effect that can yield significant variations of received signal power. This effect is included in physical models D and E by modulating several taps in order to artificially introduce an AM modulation by

( ) { }2

0exp (4 (2 1) )l m l

lg t A j l f tπ ϕ

=

= + +∑ (21)

where

( )g t The modulating function

lA Relative harmonic amplitudes

mf The main AC frequency

lϕ A series of i.i.d. phase RV's ~ [0,2 )U π t Time

The interferer to carrier energy ratio is selected using the following random variable:

2I XC

= (22)

where 2~ (0.0203,0.0107 )X G (23) In equation (23), the first figure is the mean of the Gaussian, and the second is the variance (the standard deviation squared).

otherwise

26

Figure 10 shows the cumulative distribution function (CDF) of the modelled I/C (interference-to-carrier ration) in green, and the measured experimental results in blue. This plot shows good agreement with the measured I/C.

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

Source: V. Erceg, Indoor MIMO WLAN Channel Models, 2003 For the models D and E, 3 taps are being modulated by the modulating function ( )g t in accordance with the drawn I/C. The time value of each one of these coefficients is as follows: '( ) ( )(1 ( ))c t c t g tα= + (24) where

( )c t Original tap value '( )c t Modified tap value ( )g t The modulating function

α Normalization constant The value of α is determined such that the total modulation energy (modulation in the modulated taps, compared to the entire channel response) matches the drawn random I/C. The following taps are modulated

Model Cluster Tap numbers D 2 2,4,6 E 1 3,5,7

Table 2. Tap modulation of models D and E

27

3.9 MIMO channel properties In this section, MIMO channel properties (explained in Section 3.1-3.8) are represented using the graphical plots of Matlab program distributed and written by L. Schumacher (together with AAU-Csys, FUND-INFO, and project IST-2000-30148 I-METRA) and is publicly available, see [3].

Before illustrating MIMO properties such as channel impulse response, power profiles, correlation coefficient and Doppler spectra, the capacity equation for MIMO channels is presented. Note that the simulation results only consider a physically based channel model C. However, the simulation results of non-physical model are presented and compared to the applicable physical model in Section 5.

3.9.1 Capacity The capacity of a channel depends completely on the channel realization, noise, and transmitted signal power. Capacity equation is [1] [2]:

ΗΗ+Ι= )det(log *

2 NSNRC MEP b/s/Hz (28)

where (*) means transpose-conjugate, H is the MxN channel matrix and IM is un identity matrix. In the case of two transmitter antennas and two receiver antennas the matrix of channel is 2x2 rank. Capacity grows linearly with m=min (M,N) [12] rather than logarithmically as was described by Foschini [2] and Telatar [1]. The capacity gain is highly dependent on the multipath richness in the radio channel, since a fully correlated MIMO channel only offers one subchannel, while a completely decorrelated channel offers multiple subchannels, depending on the antenna configuration. In this thesis the physical models are assumed partially correlated/decorrelated channels, since that is the case in practice. It has been demonstrated that increasing the number of antennas in the both ends results in a rapid increase in theoretical capacity [12].

28

3.9.2 Simulation results In this section the simulation results of MIMO channel are presented. For the sake of simplicity, only simulation graphs of the model case C is included. This simulation results, of Matlab program [3], have been used to verify the implementation results in Section 4.

There are 5 types of graphs included to illustrate characteristics of the MIMO channel. The plots represent impulse response of the channel matrix H, Power Delay profile (PDP) of the H channel matrix, CDF of the taps, spatial correlation functions for number of paths and Doppler spectra. In the Section 5 this simulated characteristics are compared to measured data conducted at Victoria University in Melbourne, Australia. While here the comparison has been done to the literature references, either the Rayleigh distribution (CDF) or the desired curve (PDP, correlation and Doppler spectrum).

The following pages present the plots of 2x2 MIMO channel, model C, using the Matlab program [3]. The carrier frequency used is 5.25 GHz (fd = 6 Hz) with one wavelength (λ= 6 cm) spacing at the transmitter and half wavelength spacing at the receiver. 16 384 blocks of 32 samples have been simulated at 0 speed (note: Stationary Receiver), and a sample has been stored per block. Dashed red curves/markers correspond to the reference values, whereas the blue curves/markers are the outcome of the simulation. In the Doppler plot, the green curve represents the Welch periodogram [22, p. 256].

The match between reference curves and simulation results using Matlab program [3] is satisfactory. The achieved tap power distribution in Figure 11 b) fit the PDPs defined as Rayleigh distribution in [23]. In Figure 12, the spatial correlation coefficient of the simulated impulse response match the mathematical formulas presented in Section 3.5. In Figure 13 and 14, the red vertical lines are drawn at df± . The upper blue line is set at the maximum of Doppler spectrum, and lower blue line lies 10 dB below. Ideally, the Doppler spectrum should meet the crossing of the red and blue lines. This would be the case, if the jitter would be removed from the sampled spectra presented in [24].

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

a) b)

29

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)

30

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

middle six taps

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

a) b)

31

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 Acreo’s 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 it into 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 matrix H and save it as an ASCII file. This ASCII file can then be read from ADS and used as a covariance channel matrix H.

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 C language. 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 matrix H that is needed. However, to be able to generate the channel matrix H, 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 some other alternatives were studied, explained further down.

32

Ptolemy example of C++ code defstar { // Body of the program name {example_MIMO} domain {SDF} desc { MIMO channel models} version {Source: Indoor MIMO WLAN channel model, Date: 2003-09-19} author {Aida Botonjić} location {MIMO} inmulti { // Defines multiple inputs name {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

33

4.3 ADS schematic with ‘Matlab’ block

Following presents schematics from ADS for the second implementation alternative. The implementation method described here is to link Matlab 6.5 together with ADS using pre-defined ADS block. This means to have the Matlab program written by L. Schumacher [3], as it is and use ADS block which links the Matlab 6.5 with the ADS and generates desired channel matrix H, see Figure 17.

This implementation method had a big disadvantage. That is, the simulation took long time, 30h. Even the generation of the matrices is not as smooth as in Matlab. In comparison to Matlab where one could work with as big matrices as wonted, in ADS this was not possible. ADS is limited to 3-D matrices. This means that generation of the channel matrix H, which is of size 4-D, has been carried out with errors. The schematics in Figure 17 generated H matrix without a imaginary part. This of course does not give a right image of the channel matrix H which is a complex matrix. Luckily an updated version of ADS is coming after New Years Eve 2003 and improvements have been made on the simulation speed when using Matlab together with ADS. Also, a generation of bigger matrices than 3-D is going to be possible. This schematic was left as it was and some other alternatives were explored, described further down.

Figure 17 depicts ADS schematic. In the schematic, the Matlab block opens, internally from ADS, a Matlab program written by L. Schumacher [3], and performs all calculations using Matlab. Then it generates a desired output, in this case channel matrix H, to ADS. The complex channel matrix H is stored in a block called ‘NumericSink’. This block can show only data on the screen or it can plot desired form of graphs, such as linear, Smith-chart, and so on. Each ADS schematic requires a controller called ‘DF’ in to order perform simulations. This block indicates the simulation setup. In reality this schematic consists of 14 inputs. But here are only 4 input parameters illustrated, for the sake of simplicity.

Figure 17. ADS with Matlab block

Matlab block NumericSink

Controller

Input parameters

34

4.4 ADS schematic with ‘ReadFile’ block Figure 18 depicts channel model implemented with delays and matrix H multiplied together with incoming data, arbitrary bits. Before using this schematic the generation of matrix H must be done in the Matlab 6.5 using the Matlab program written by L. Schumacher [3]. The channel matrix H is then stored into the ASCII file. This ASCII file is read from ADS using a ‘ReadFile’ block. Channel matrix H is then multiplied first with arbitrary bits and then with the power delay in order to separate data in time.

Figure 18. ADS with ‘ReadFile’ block

Arbitrary bits

Delay Channel H

ReadFile

35

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 matrix H and tap delays. This model is based on the Jake’s model from 1974 [6]. Here is matrix H multiplied 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 matrix H is 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 matrix H for the desired channel model.

This schematic (together with the second alternative of the generation of the channel matrix H in 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 matrix H and power delays

Delays

Channel H

Arbitrary bits

36

5 Comparison results In this section the comparison results between the measured data conducted at Victoria University in Melbourne by Jason Gao and Michael Falukner and theoretical data generated by the Matlab program written by L. Schumacher [3] are presented.

When investigating and comparing the physical models against one-ring model the conclusion is drawn that the best suitable model for measured data is a case F model since model F includes 6 clusters, same as for one-ring model. The following graphs consider only one-ring model compared against the F model.

The MIMO channel impulse responses are generated using Matlab 6.5. The Fourier transform is used to transform the simulated impulse responses into the frequency domain, see Figure 20 and 21. Figure 20 represents impulse response simulated for case F channel model. The impulse response is an averaged sum for absolute magnitude of channel matrix H (10-

3 = – 40 dB). Figure 21 illustrates impulse response for the one-ring model. The graph represents impulse response that is accounted for hardware-induced errors, such as insertion loss and leakage in the switches and cabling. It can be seen that the magnitude for the one-ring model is much lower than for the model F. This may be due to the interference of people moving around in the corridors at the time when the data was collected. But also since impulse response for model F is averaged over time while one-ring model is measured for a certain time period.

Figure 20. Impulse response of covariance channel matrix H for channel model F

Impulse response of h11 Impulse response of h12

Impulse response of h21 Impulse response of h22

37

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

It is known from theoretical study [25] that the lower the spatial correlation within the indoor environment, the greater the achieved capacity. The spatial power correlation coefficient for the one-ring is not treated at the BS since antennas separated with 1.5λ is decorrelated. Model F assumes partly correlated Tx and Rx antennas because of 0.5 – 1 λ antenna separation. Antenna spacing on the order of 0.4λ – 0.6 λ is adequate for independent fading [26], see also Section 6.4. Table 3 represents the spatial correlation coefficients for the channel model F and one-ring model at the MS antennas. Numbers in brackets are theoretical results for the channel model F while numbers without brackets are estimated correlation coefficients for the one-ring model. It can be seen that deviation is marginal.

Cluster |R12|2 |R21|2

1 1.00 (1.00) 1.00 (1.00) 2 0.86 (0.82) 0.55 (0.54) 3 096 (0.97) 0.88 (0.90) 4 0.74 (0.78) 0.39 (0.45) 5 0.87 (0.90) 0.65 (0.69) 6 0.67 (0.68) 0.22 (0.25)

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

The Doppler power spectrum in the one-ring model is not included due to stationary realization of MS and BS antennas. It follows that the comparison results does not consider Doppler power spectrum. The reader should notice that since the data presented here considers different scenarios, the power correlation coefficients and impulse response vary. The data for one-ring model is collected for smaller rooms and for a 12m long corridor. Model F considers much bigger space than the space where one-ring model was measured. There may be several reasons why the simulated and measured results do not match perfectly: insufficient amount of statistics, Wide Sense Stationary (WSS) assumption not 100% fulfilled, … This needs further investigations.

38

6 Theory, Part II This section presents the second part of literature study. It is explained how the channel ‘H’ is transmitted and received and how the algorithms for detection should be developed. Focus of MIMO systems is to combine the signals at the receiver in such a way that the quality, bit error rate BER and data rate, is dramatically improved. To achieve these improvements, both transmitter and receiver must be designed in a special way. The transmission of the channel H at the transmitter and algorithms for detection and decoding at the receiver are discussed in this section.

6.1 Spatial multiplexing scheme The wireless channel constitutes a hostile propagation medium, which suffers from fading (caused by destructive addition of multipath components) and interference from other users. Diversity is a powerful technique to combat fading and interference. Spatial diversity has become very popular in recent years since it can be provided without loss in spectral efficiency, see section 6.4. Because of its popularity and spectral efficiency, here is only transmission of spatial multiplexing considered. Spatial multiplexing transmits independent parallel data streams through multiple antennas at both transmitter and receiver, see Figure 22.

Figure 22. Spatial multiplexing for 2x2 MIMO system In a spatial multiplexing system the data stream to be transmitted is demultiplexed into lower rate streams which are then simultaneously sent from the transmitter antennas after coding and modulation. As soon as signals leave antennas at the transmitter they are mixed together in the wireless channel, since they use the same frequency spectrum. Each receiver antenna observes a superposition of the transmitted signals. The receiver then separates them into constituent data streams and remultiplexes them to recover the original data steam. This occurs for 2x2 MIMO system, as two unknowns are resolved from a linear system of two equations. Clearly, the separation step determines the computational complexity of the receiver. In the following, both transmitter and receiver architecture is discussed.

Tx1

Tx 2

Rx1

Rx2

Spatial Demulti-plexing

Spatial Multi- plexing

39

6.2 Transmitter There are two alternatives for the transmitting channel. The first alternative is that the transmitter has partial or full knowledge of the channel and uses this knowledge to increase diversity gain. The second alternative is that the transmitter has no knowledge of the channel and uses a coding technique to achieve diversity. In both categories it is presumed that the receiver has knowledge of the channel in order to separate data from the multiple spatial channels. This information is obtained by sending training or pilot symbols to estimate the channel. For the first alternative, the transmitter has to gain knowledge of the channel. There are two possibilities, (1) knowledge of the channel at the receiver is turned around and used at the transmitter (no feedback is necessary) or (2) knowledge of the channel at the receiver is fed back to the transmitter using a handshaking protocol. When the transmitter has no knowledge of the channel an algorithm should be selected accordingly. A transmitter encodes the bits over space and frequency and transmits these bits over multiple spatial channels. The receiver then separates the symbols from the channel and decodes the bits, see Figure 23. In the following, it is assumed that the transmitter has no knowledge of the channel since that is the usual case. Transmitter IFFT &

Add CP 802.11a Encoder Space- Modulation Spatial & puncturing Frequency (BPSK, QPSK Mapping Interleaver 16&64-QAM) IFFT & Add CP Receiver OFDM-Modulator FFT & Remove CP

Antenna Soft Space- Depuncture selection V-BLAST decision Frequency &

FFT & Interleaver Decoder(V) Remove CP

OFDM-Demodulator

Figure 23. MIMO system without knowledge of the channel at the transmitter Source: Intel Technology Journal, High-Throughput Wireless LAN Air Interface

Interleaving and coding at the transmitter can be used to reduce the carrier-to-noise (C/N) ration requirement for accurate detection. Coding provides redundancy by sending multiple copies of the signal on orthogonal symbol paths. Interleaving provides robustness to the link by spreading errors out in time, avoiding large amounts of contiguous data loss that would cut a wireless link. As illustrated in Figure 23, assuming N subcarriers (or tones) the individual data streams are passed through OFDM modulator, which performs an IFFT on blocks of length N followed by a parallel to serial conversion. A cyclic prefix (CP) containing a

40

copy of the parallel to serial converted output of the N-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 an N-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. The receiver 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 channel H is 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 out all 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 1ˆ −= .

• 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.

41

Received signal is expressed as: nsHr +⋅= (29) where s is a codeword expressed in vector form and n is a noise. Optimum ML decoding rule is:

∑=

−=M

mmmssHrs

1

2

ˆminargˆ (30)

where the minimization is performed over all possible codeword vectors s. Note that if the size of the scalar constellation used is q (e.g. q=4 for QPSK), the receiver has to perform an enumeration over a set of size qM (M is the number of transmit antennas). For high order modulation such as 64-QAM this complexity can become prohibitive even for a small number of transmit antennas. For example, for 64-QAM and M=3 the receiver has to enumerate over 262 144 different vectors on the symbol rate.

6.4 Antenna selection Diversity is a common technique to combat small scale fading caused by multipath effect. Common forms of diversity are time diversity (due to Doppler spread), frequency diversity (due to delay spread) and spatial (or antenna) diversity. Spatial diversity involves placing two receive antennas at such a distance from each other that it is statistically unlikely for both antennas to experience the identical channel conditions from the transmitter. The objective is that when one antenna is in a deep fade, the other antenna still has a strong signal. Since the spatial diversity is used at both the receiver and the transmitter, antennas should be spaced by more than one coherence distance apart. The coherence distance is the minimum spatial separation of antennas for independent fading and depends on the angular spread of the multipaths arriving at (or departing from) an antenna array. For example, if the multipath signals arrive from all directions in the azimuth, antenna spacing on the order of 0.4λ – 0.6λ is adequate for independent fading [26]. Impact of antenna selection at both transmitter and receiver is also essential in the sense of correlation function. For increasing antenna spacing, it follows that correlation function will decrease its spread. The recommendation in choice of antenna direction is to choose omnidirectional antennas at both transmitter and receiver.

42

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 and Michael 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 one–ring 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 H due 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 in wireless broadband systems. In order to gain full advantage of spatial multiplexing the Signal to Noise Ration (SNR) should be reasonably good.

43

8 Future work

The simulations of the fading, for the physical models, have been done for the case of frequency non-selective and slow fading. There should be some simulations for frequency selective and fast fading also to indicate upper and lower limits, respectively, for signaling rate to avoid the worst fading effects.

The one-ring model, that was investigated here, postulate a scattering environment and derive a MIMO channel mode involving scattering parameters. Still, an accurate description of the cross correlation between pairs of channel coefficients is one of the important modeling aims and should be further investigated and maybe corrected.

The non-physical MIMO channel model (one-ring model) assumes NLOS scenario. However, in some circumstances, the LOS exists and therefore the model should even reflect impact of LOS component.

Until now, no outdoor MIMO channel models have been reported based on the MIMO channel measurements. The outdoor scenarios are very different from the indoor scenarios. For instance, in the indoor scenarios, the Doppler shift is small while the outdoor scenarios may have relatively large Doppler shift.

44

9 References

[1] E. Telatar, Capacity of multiantenna Gaussian channels, AT&T Bell Laboratories, 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_terms.html

[4] H. Bölcskei 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.

[8] F. Adachi, M. Feeny, A. Williamson, and J. Parsons, Cross-Correlation between the Envelopes of 900 MHz Signals Received at a Mobile Radio Station Site, IEEE Proceedings Pt. F., vol. 133, pp. 506-512, Oct. 1986.

[9] J. Salz and J. Winters, Effect of Fading Correlation on Adaptive Arrays in Digital Mobile Radio, IEEE Transaxtions on Vehicular Technology, vol. 43, pp. 1049-1057, Nov. 1994.

[10] K. I. Pedersen, P. E. Mogensen, and B. H. Fleury, Spatial Channel Characteristics in Outdoor Environments and their Impact on BS Antenna System Performance, in Proceedings of IEEE Vehicular Technology Conference VTC 1998, Ottawa, Canada, vol. 2, pp. 719-723, 1998.

[11] U. Martin, Spatio-Temporal Radio Channel Characteristics in Urban Macrocells, IEEE Proceedings on Radar, Sonar and Navigation, vol. 145, no. 1, pp. 42-49, Feb. 1998.

[12] M. Pettersen, P. H. Lehne, J. Noll, O. Røstbakken, E. Antonsen, R. Eckhoff, Characterization of the Directional Wideband Radio Channel in Urban and Suburban Areas, IEEE Proc. Vechicular Technology Conference VTC’99 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 characteristics of 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

45

with clustering and correlation properties, in prog. IEEE Veh. Technol. Conf., vol. 1, Sept. 2002, pp. 629-633.

[17] A. S. Y. Poon and M. Ho, Indoor multiple antenna channel characterization from 2 to 8 GHz, submitted to ICC 2003 Conference.

[18] G. German, Q. Spencer, L. Swindlerhurst, and R. Valenzuela, Wireless indoor channel modeling: Statistical agreement of ray tracing simulations and channel sounding measurements, in proc. IEEE Axoustics, Speech, and Signal Proc. Conf., vol. 4, 2001, pp. 2501-2504.

[19] R. H. Clarke, A Statistical Theory of Mobile Radio Reception, Bell Labs System Technical Journal, vol. 47, pp. 957-1000, July- August 1968.

[20] P. Petrus, J. H. Reed, T. S. Rappaport, Effects of Directional Antennas at the Base Station on the Doppler Spectrum, IEEE Communication Letters, vol. 1, no. 2, pp. 40-42, March 1997.

[21] Commission of the European Communities, Digital Land Mobile Radio Communications – CPST 207 Final Report, ECSC-EEC-EAEC, 1989.

[22] N. Tal, Time variable HT MIMO channel measurements, IEEE 802.11-03/r0, July 2003.

[23] M. Kunt, Traitement numérique des signaux, Presses Polytechniques Romandes, 1984.

[24] L. Schumacher, K. I. Pedersen, and P. E. Mogensen, From antenna spacings to theoretical capacities – guidelines for simulating MIMO systems, in Proc. PIMRC conf., vol. 2, Sept. 2002, pp. 587-592.

[25] MIMO Rapporteur: 3GPP TSG R1-02-0181, MIMO discussion summary, Jan. 2002.

[26] K. Pedersen, J. B. Andersen, J. P. Kermoal, and P. Mogensen, A stochastic multiple-input-multiple-output radio channel model for evaluation of space-time coding algorithms, IEEE Vehicular Tech. Conf. VTC 2000, Sep 2000, vol. 2, pp. 893-897.

[27] H. Bölcskei, Principles of MIMO-OFDM Wireless Systems, chapter in CRC Handbook on Signal Processing for Communications, M. Ibnkahla, Ed., 2003.

[28] G. D. Golden, G. J. Foschini, R.A. Valenzuela, and P. W. Wolniansky, Detection algorithm and initial laboratory results using the V-BLAST space-time communication architecture, Electronics Letters, vol. 35, no. 1, pp. 14-15, 1999.

[29] G. J. Foschini, G. D. Golden, R.A. Valenzuela, and P. W. Wolniansky, Simplified processing for high spectral efficiency wireless communication employing multi-antenna arrays, IEEE J. Sel. Areas Comm., vol. 17, no. 11, pp. 1841-1852, 1999.

[30] J. Fuhl, A. F. Molisch, E. Bonek, Unified Channel Model for Mobile Radio Systems with Smart Antennas, IEEE Proc. Radar, Sonar and Navigation, vol. 145, no. 1, pp. 32-41, February 1998.

46

10 Bibliography Agilent Technologies, Agilent Ptolemy Simulation, Agilent Technologies, USA, May 2003. Agilent Technologies, User’s Guide, Agilent Technologies, USA, September 2002. A. Hemani, Channel Estimation, Phase Tracking and Equalization – Algorithm design report, SocTRix WG3, 2003. B. Bangerter, E. Jacobsen, M. Ho, A. Stephens, A. Maltsev, A. Rubtsov, and A. Sadri, High-Throughput Wireless LAN Air Interface, Intel Technology Journal, vol. 07, August 2003. B. O’Hara, A. Petrick, IEEE 802.11 Handbook: A Designer’s Companion, Standards Information Network IEEE Press, New York, USA, 1999. D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, From Theory to Practice: An Overview of MIMO Space-Time Coded Wireless Systems, IEEE Journal on Selected Areas in Comm., vol. 21, no. 3, April 2003. H. Suzuki, A statistical model for urban radio propagation, IEEE Transaction on Communications, vol. COM-25, no. 7, pp. 673-680, July 1977. Jan Skansholm, C++ direct, Studentlitteratur, Sweden, 2000, 2nd edition. John G. Proakis, Digital Communications, McGraw-Hill Higher Education, USA, 2001, 4th edition. J. P. Kermoal, L. Schumacher, P. E. Mogensen, and K. I. Pedersen, Experimental Investigaion of Correlation Properties of MIMO Radio Channels for Indoor Picocell Scenarios, Kai Yu and Björn Ottersten, Models for MIMO Propagation Channels, A Review, Wiley Journal on Wireless Communications and Mobile Computing, KTH Stockholm, Sweden, IR-S3-SB-0223, 2002. Q. Li, M. Ho, J. Lung, and D. Cheung, On the tap angular spread and Kronecker structure of the WLAN channel model, IEEE 802.11-03/584r0, July 2003. P.J. Smith, M. Shafi, Waterfilling methods for MIMO systems, in Proc. 3rd Australian Communication Theory Workshop, Canberra, Australia, 2002, AusCTW 2002. The MatlabTM curriculum series, The student edition of MatlabTM MS-DOS personal computers, Prentice Hall, New Jersey, USA, 1992.

47

Appendix A Physical models A-F

Model A

Tap index 1

Excess delay [ns]

0

Power [dB] 0

AoA AoA [°] 45

AS (receiver)

AS [°] 40

AoD AoD [°] 45

AS (transmitter)

AS [°] 40

48

Model B

Tap index 1 2 3 4 5 6 7 8 9

Excess delay [ns]

0 10 20 30 40 50 60 70 80

Cluster 1 Power [dB] 0 -5.4 -10.8 -16.2 -21.7

AoA AoA [°] 4.3 4.3 4.3 4.3 4.3

AS (receiver)

AS [°] 14.4 14.4 14.4 14.4 14.4

AoD AoD [°] 225.1 225.1 225.1 225.1 225.1

AS (transmitter)

AS [°] 14.4 14.4 14.4 14.4 14.4

Cluster 2 Power [dB] -3.2 -6.3 -9.4 -12.5 -15.6 -18.7 -21.8

AoA AoA [°] 118.4 118.4 118.4 118.4 118.4 118.4 118.4

AS AS [°] 25.2 25.2 25.2 25.2 25.2 25.2 25.2

AoD AoD [°] 106.5 106.5 106.5 106.5 106.5 106.5 106.5

AS AS [°] 25.4 25.4 25.4 25.4 25.4 25.4 25.4

49

Model C

Tap index 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 1 Power [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

AoD AoD [°] 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 2 Power [dB] -5.0 -7.2 -9.3 -11.5 -13.7 -15.8 -18.0 -20.2

AoA AoA [°] 332.3 332.3 332.3 332.3 332.3 332.3 332.3 332.3

AS AS [°] 22.4 22.4 22.4 22.4 22.4 22.4 22.4 22.4

AoD AoD [°] 56.4 56.4 56.4 56.4 56.4 56.4 56.4 56.4

AS AS [°] 22.5 22.5 22.5 22.5 22.5 22.5 22.5 22.5

50

Model D

Tap index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Excess delay [ns]

0 10 20 30 40 50 60 70 80 90 110 140 170 200 240 290 340 390

Cluster 1 Power [dB] 0 -0.9 -1.7 -2.6 -3.5 -4.3 -5.2 -6.1 -6.9 -7.8 -9.0 -11.1 -13.7 -16.3 -19.3 -23.2

AoA

AoA [°] 158.9 158.9 158.9 158.9 158.9 158.9 158.9 158.9 158.9 158.9 158.9 158.9 158.9 158.9 158.9 158.9

AS (receiver)

AS [°] 27.7 27.7 27.7 27.7 27.7 27.7 27.7 27.7 27.7 27.7 27.7 27.7 27.7 27.7 27.7 27.7

AoD AoD [°] 332.1 332.1 332.1 332.1 332.1 332.1 332.1 332.1 332.1 332.1 332.1 332.1 332.1 332.1 332.1 332.1

AS (transmitter)

AS [°] 27.4 27.4 27.4 27.4 27.4 27.4 27.4 27.4 27.4 27.4 27.4 27.4 27.4 27.4 27.4 27.4

Cluster 2 Power [dB] -6.6 -9.5 -12.1 -14.7 -17.4 -21.9 -25.5

AoA AoA [°] 320.2 320.2 320.2 320.2 320.2 320.2 320.2

AS AS [°] 31.4 31.4 31.4 31.4 31.4 31.4 31.4

AoD AoD [°] 49.3 49.3 49.3 49.3 49.3 49.3 49.3

AS AS [°] 32.1 32.1 32.1 32.1 32.1 32.1 32.1

Cluster 3 Power [dB] -18.8 -23.2 -25.2 -26.7

AoA AoA [°] 276.1 276.1 276.1 276.1

AS AS [°] 37.4 37.4 37.4 37.4

AoD AoD [°] 275.9 275.9 275.9 275.9

AS AS [°] 36.8 36.8 36.8 36.8

51

Model E (1/2)

Tap index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Excess delay [ns]

0 10 20 30 50 80 110 140 180 230 280 330 380 430 490 560 640 730

Cluster 1 Power [dB] -2.6 -3.0 -3.5 -3.9 -4.5 -5.6 -6.9 -8.2 -9.8 -11.7 -13.9 -16.1 -18.3 -20.5 -22.9

AoA

AoA [°] 163.7 163.7 163.7 163.7 163.7 163.7 163.7 163.7 163.7 163.7 163.7 163.7 163.7 163.7 163.7

AS (receive)

AS [°] 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8 35.8

AoD AoD [°] 105.6 105.6 105.6 105.6 105.6 105.6 105.6 105.6 105.6 105.6 105.6 105.6 105.6 105.6 105.6

AS (transmit)

AS [°] 36.1 36.1 36.1 36.1 36.1 36.1 36.1 36.1 36.1 36.1 36.1 36.1 36.1 36.1 36.1

Cluster 2 Power [dB] -1.8 -3.2 -4.5 -5.8 -7.1 -9.9 -10.3 -14.3 -14.7 -18.7 -19.9 -22.4

AoA AoA [°] 251.8 251.8 251.8 251.8 251.8 251.8 251.8 251.8 251.8 251.8 251.8 251.8

AS 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

AoD AoD [°] 293.1 293.1 293.1 293.1 293.1 293.1 293.1 293.1 293.1 293.1 293.1 293.1

AS AS [°] 42.5 42.5 42.5 42.5 42.5 42.5 42.5 42.5 42.5 42.5 42.5 42.5

52

Model E (2/2)

Cluster 3 Power [dB] -7.9 -9.6 -14.2 -13.8 -18.6 -18.1 -22.8

AoA AoA [°] 80.0 80.0 80.0 80.0 80.0 80.0 80.0

AS AS [°] 37.4 37.4 37.4 37.4 37.4 37.4 37.4

AoD AoD [°] 61.9 61.9 61.9 61.9 61.9 61.9 61.9

AS AS [°] 38.0 38.0 38.0 38.0 38.0 38.0 38.0

Cluster 4 Power [dB] -20.6 -20.5 -20.7 -24.6

AoA AoA [°] 182.0 182.0 182.0 182.0

AS AS [°] 40.3 40.3 40.3 40.3

AoD AoD [°] 275.7 275.7 275.7 275.7

AS AS [°] 38.7 38.7 38.7 38.7

53

Model F (1/2)

Tap index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Excess delay [ns]

0 10 20 30 50 80 110 140 180 230 280 330 400 490 600 730 880 1050

Cluster 1 Power [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

AoA AoA [°] 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

AoD AoD [°] 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 2 Power [dB] -1.8 -2.8 -3.5 -4.4 -5.3 -7.4 -7.0 -10.3 -10.4 -13.8 -15.7 -19.9

AoA AoA [°] 180.4 180.4 180.4 180.4 180.4 180.4 180.4 180.4 180.4 180.4 180.4 180.4

AS AS [°] 55.0 55.0 55.0 55.0 55.0 55.0 55.0 55.0 55.0 55.0 55.0 55.0

AoD AoD [°] 183.7 183.7 183.7 183.7 183.7 183.7 183.7 183.7 183.7 183.7 183.7 183.7

AS AS [°] 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2 55.2

Cluster 3 Power [dB] -5.7 -6.7 -10.4 -9.6 -14.1 -12.7 -18.5

AoA AoA [°] 74.7 74.7 74.7 74.7 74.7 74.7 74.7

AS AS [°] 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

AS AS [°] 47.4 47.4 47.4 47.4 47.4 47.4 47.4

54

Model F (2/2)

Cluster 4 Power [dB] -8.8 -13.3 -18.7

AoA AoA [°] 251.5 251.5 251.5

AS AS [°] 28.6 28.6 28.6

AoD AoD [°] 112.5 112.5 112.5

AS AS [°] 27.2 27.2 27.2

Cluster 5 Power [dB] -12.9 -14.2

AoA AoA [°] 68.5 68.5

AS AS [°] 30.7 30.7

AoD AoD [°] 291.0 291.0

AS AS [°] 33.0 33.0

Cluster 6 Power [dB] -16.3 -21.2

AoA AoA [°] 246.2 246.2

AS AS [°] 38.2 38.2

AoD AoD [°] 62.3 62.3

AS AS [°] 38.0 38.0

55

Appendix B Non-physical model

One-ring model

Tap index n

Cluster 1

AoA AoA [°] 44.89

AS (receive)

AS [°] 0.82

AoD AoD [°] 346.07

AS (transmit)

AS [°] 26.42

Cluster 2

AoA AoA [°] 61.14

AS AS [°] 39.63

AoD AoD [°] 289.25

AS AS [°] 16.81

Cluster 3

AoA AoA [°] 48.17

AS AS [°] 9.13

AoD AoD [°] 294.77

AS AS [°] 15.01

Tap index n

Cluster 4

AoA AoA [°] 45.51

AS AS [°] 31.82

AoD AoD [°] 314.90

AS AS [°] 38.06

Cluster 5

AoA AoA [°] 30.65

AS AS [°] 14.22

AoD AoD [°] 305.93

AS AS [°] 0.21

Cluster 6

AoA AoA [°] 27.15

AS AS [°] 33.39

AoD AoD [°] 302.40

AS AS [°] 1.08

56

Appendix C Power distributions

Uniform PAS Uniform PAS model is defined as:

∑=

∆+−−

∆−−=

cN

k kk

kkkUU QPAS

1 ,0

,0, )]([

)]([)(

φφφεφφφε

φ

where ε(φ) is the step function and Nc is the number of clusters. The first step is to normalise the PAS such that it can be regarded as a probability distribution. The constants QU,k are derived such that PASU (φ) fulfils the requirements of a probability distribution function:

∑ ∫∫=

∆+

∆−−

==c kmk

mk

N

kkUU dQdPAS

1,

0

0

1)(φφ

φφ

π

π

φφφ

where ∆φ stands for the half-domain definition of the PAS (domain assumed symmetric).

Derivation of probability distribution leads then to ∑=

=∆cN

kkkUQ

1, 12 φ

The cross-correlation function for real-real and imaginary-imaginary parts are derived in the same way:

][][)()( yyExxEDRDR YYXX === The cross-correlation function between the real-real part is:

)2sin()2cos(2

)(4)()()sincos()(

1,0

2

1,0, k

mk

mN

kkUUXX mm

mDJ

QDJdPASDDRc

φφφφφπ

π

∆+== ∑∑∫∞

==−

where Jm is the Bessel function of the first kind and mth order. The cross-correlation function between real-imaginary part is:

][][)()( xyEyxEDRDR YXXY −==−=

])12sin[(])12sin[(12

)(4)()sinsin()(

0,0

)12(

1,, k

mk

mN

kkUUXY mm

mDJ

QdPASDDRc

φφφφφπ

π

∆+−+

== ∑∑∫∞

=

+

=−

From these resolutions, both field ρf(D) and envelope ρe(D) correlation coefficient are defined

as: 22

)()()()( DjRDRDD XYXXfe +== ρρ

57

Truncated Gaussian PAS Truncated Gaussian PAS model is defined as:

∑=

−−

∆+−−

∆−−=

ckG

N

k kk

kk

kG

kGG e

QPAS

1 ,0

,02)(

,

,

)]([)]([

2)( ,

2

20

φφφεφφφε

πσφ σ

φφ

The normalisation constant QG,k are derived such that

12,

, =

∆∑=

cN

mk kGkG erfQ

σφ

Using above definition of Truncated Gaussian PAS and normalisation condition the cross-correlation functions are easily derived:

−

∆−−

−

∆

ℜ−+= ∑∑∞

==2

2

22

)2exp()2cos()()()(

,,

,,2

,2

1,02

1,0,

kGkG

k

kGkG

k

kGm

km

N

kkGGXX

jmerf

jmerf

mmDJQDJDRc

σσ

φ

σσ

φ

σφ

+−

∆−−

+−

∆

ℜ

+−−= ∑∑

∞

=+

=)

21(2

2

)21(2

2)

21(2exp])12sin[()()(

,,

,,2

,2

1,0)12(

1,,

mjerf

mjerf

mmDJQDR

kGkG

k

kGkG

k

kGm

km

N

kkGGXY

c

σσ

φ

σσ

φ

σφ

58

Truncated Laplacian PAS Truncated Laplacian PAS model is defined as:

∑=

−−

∆+−−

∆−−=

ckL

N

k kk

kkk

kL

kLL e

QPAS

1 ,0

,0,2

,

,

)]([)]([

2)( ,

20

φφφεφφφε

πσφ σ

φφ

The normalisation condition is given by

12

exp1,

, =

∆−∑

=

cN

mk kL

kkLQ

σφ

The cross-correlation function is then given by:

∆−∆

∆−+

+

+= ∑∑

∞

==)2cos(2)2sin(2

2exp2

)2cos(

)2(2

)(2

4)()(

,

,,

1,0

2

2

,

2

1 ,

,0,

kkL

k

kL

k

kL

mk

kL

mN

k kL

kLLXX

mmm

m

m

DJQDJDR

c

φσ

φ

σφ

σφ

σ

σ

[ ][ ] [ ]

∆++∆++

∆−−

+

++

= ∑∑

∞

=

+

=

kkL

k

kL

k

kL

mk

kL

mN

k kL

kLLXY

mmm

m

m

DJQDR

c

φσ

φ

σφ

σφ

σ

σ)12(cos2)12(sin)12(

2exp2

)12(sin

)12(2

)(2

4)(

,

,,

0,0

2

2

,

)12(

1 ,

,,