issues in mimo channel modeling and simulation ali abdi center for communications and signal...

Issues in MIMO Channel Issues in MIMO Channel Modeling Modeling

and Simulationand Simulation

Ali Abdi

Center for Communications and Signal Processing ResearchDepartment of Electrical & Computer Engineering

New Jersey Institute of Technology

Polytechnic University, December 4, 2003

Polytechnic University, Dec. 4, 2003Ali Abdi

OverviewOverview

MIMOModeling

MacrocellsMicrocellsPicocells

Time andFrequencySelectivity

Correlations (space, time, and frequency)

Fade Duration

MIMOSimulation

Ray-based

SpectralSampling

PolynomialEmbedding

Stochastic

NarrowbandWideband


Part IPart I

MIMO Models


MIMO Modeling: MacrocellsMIMO Modeling: Macrocells

• No scatterer around the elevated base station (BS).

• Scatterers around the mobile station (MS) are located on a ring.

pqd

pBS

qBS

pqa

DBS

if

iqx

ipx

iz

D R

x

y

lmdmU

lU

lmb

gUif

lix

mix

iS

v

BSO

UO


MIMO Modeling: Micro & MIMO Modeling: Micro & PicocellsPicocells

• Scatterers around the BS and the MS are located on separate rings.


Advantage of Ring ModelsAdvantage of Ring Models

• Simple to analyze Provide closed-form expressions for space, time, and frequency correlations.

• Capture essential characteristics of MIMO channels using few parameters such as mean angle of arrivals/departures, Doppler, angle spreads, etc.

• Compatible with previous well-accepted models such as Clarke’s.


Advantage of Ring Models Advantage of Ring Models (continued)(continued)

• Suitable for mobile Tx and Rx scenarios.

• Applicable to wideband channels by modifying rings to circular rings.

• Example: wideband macrocell model.y

x

D

R 1

BS p

BS q

O BS

p qd

R 2

S iipx

iqx

izpqa

DB S

if

U m

U l

O U

lmd

m ix

lix

R i Uif

lmb


MIMO Modeling: CorrelationsMIMO Modeling: Correlations

• Closed-form expressions are derived for: (Narrowband) space-time corr.

where is the channel gain between the p-th Tx and

l-th Rx antennas. (Wideband) space-time-frequency corr.

, where is the time-varying transfer function between the p-th Tx and l-th Rx

antennas.

• These correlations are needed to simulate and assess the performance of MIMO systems.

[ ( ) ( )]lp mqE h t h t

[ ( , ) ( , )]lp mqE T f t T f f t t D D

( )lph t

( , )lpT f t


A 3A 34 MIMO Channel4 MIMO Channel

• We use h(t) for the narrowband channels, and T(f,t) for wideband channels.

+

AWGN

+

AWGN

+

AWGN

+

AWGN

11( )h t1

2

3

1

2

3

4

Rec

eive

r

Tra

nsm

itte

r

11( , )T f t

12 ( )h t

43( )h t43( , )T f t

12 ( , )T f t


A 2x2 ChannelA 2x2 Channel

pqd

pBS

qBS

pqa

DBS

if

iqx

ipx

iz

D R

x

y

lmdmU

lU

lmb

gUif

lix

mix

iS

v

BSO

UO


Diffuse and Line-of-Sight Diffuse and Line-of-Sight ComponentsComponents

(Frequency-Flat Fading)(Frequency-Flat Fading)

The channel fading gain

Average power

Rice factor

)()()( LOSDIF ththth lplplp

]|)([| 2thE lplp

]|)([|)( 2DIF2LOS thEthK lplplp

tfjj

j

gNK

th

UiD

Uili

Uiipi

N

ii

Nlp

lplp

)cos(2)()(2

exp

1lim

1)(

1

DIF

gffxfx

tfjj

K

Kth U

pDlplp

lplplp )cos(2

2exp

1)(LOS gfx


Exact Diffuse Part of the Exact Diffuse Part of the CorrelationCorrelation

mqlpmqlpmqlp ththE )]()([)( DIFDIFDIF,

ffff ffgf

xxxx

UU

UD

mlqp

mqlp

df

fj

j

KK

UUUU

)(

)cos(2

2

exp

)1)(1(

1

gf

xxxx

)cos(2

2

exp][1

lim

)1)(1(

1

1

2

UiD

miliiqipN

ii

N

mqlp

fj

j

gEN

KK

UUi

i

df

NgE

ff )(

][ 2


Small Angle Spread at the Base Small Angle Spread at the Base StationStation

D

ffgf

bffad

ad

UU

UD

lmU

lmU

pqpq

pqpq

mqlpmqlp

df

fj

dj

j

KK

)(

)cos(2

)cos()sin()sin(2

exp

)cos(2exp

)1)(1(

1)(DIF

,

o100 D

)cos(2

)cos(2

)sin()sin()cos(2

)sin()sin()cos(2

lmUlm

m

lmUlm

l

Upqpq

pq

q

Upqpq

pq

p

dR

dR

U

U

UU

UU

bfx

bfx

faad

zx

faad

zx

f

f

ff

ff

D

D


Wave Scattering around the UserWave Scattering around the User

Non-isotropicNon-isotropicscattering inscattering in

a street a street

IsotropicIsotropicscattering inscattering inan open areaan open area


User’s Angle of Arrival & von Mises User’s Angle of Arrival & von Mises PDFPDF

is the mean direction of AOA, seen by the user

controls the width of AOA

),[,)(2

)]cos(exp[)(

0

f

ff UU

U

If

0

),[

- 3 - 2 - 1 0 1 2 3q

0

0.2

0.4

0.6

0.8

1

1.2

p QHqL

0 0.2 0.4 0.6 0.8 1 1.2

- 0.2

- 0.1

0

0.1

0.2

pQHqL


The New Space-Time The New Space-Time CorrelationCorrelation

Let ,2 Dfa ,2 lmlm db d pqpqc 2

21

2222220

0

DIF,

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

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

)(sin

)(

)]cos(exp[

)1)(1(1

)(

abg

bgagb

a

a

pqpqlmlm

lmlmpqpqlmlm

pqpqlm

pqpq

mqlpmqlp

cbaj

bacab

cbaI

I

jc

KK

D

D

D

)cos()cos()cos(exp

)]1)(1[()(LOS,

pqpqlmlm

mqlpmqlpmqlp

cjbjaj

KKKK

abg


Special Cases of the Diffuse Special Cases of the Diffuse PartPart

Clarke’s model: Single Tx-Rx antennas, isotropic scattering

[Abdi99]: Single Tx-Rx antennas, non-isotropic scattering

[Lee70]: Single Tx-multiple Rx antennas, isotropic scattering

[Fulg98]: Multiple Tx-single Rx antennas, isotropic scattering

[Chen00]: Multiple Tx-single Rx antennas, isotropic scattering

)2(0 DfJ

)())cos(44( 02222

0 IfjfI DD

)cos2( 220 glmlm abbaJ

)sin()]cos(exp[ 0 pqpqpqpq cJjc aa D

)sin()sin(2)(sin)]cos(exp[ 22220 gaaa pqpqpqpqpqpq accaJjc DD


The Channel MeasurementsThe Channel Measurements

Locations: suburban and urban areas Data format: 12 pairs of narrowband

inphase and quadrature componentsLength of each record: 47 m or 7 sSpeed of the mobile receiver: fixed at 6.7

m/sCarrier frequency: 910.25 MHzNominal power of the transmitter: 0.2 WSampling frequency: 35156.25 Hz


Correlation Fitting to DataCorrelation Fitting to Data

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1Record #0014

Cor

rela

tion

fm

Empirical Simple model Composite modelClarke`s model

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1Record #0019

Cor

rela

tion

fm

Empirical Simple model Composite modelClarke`s model

0 0.5 1 1.5 2 2.5 3

x 105

-40

-30

-20

-10

0

10

20


Wideband Macrocell ModelWideband Macrocell Model

y

x

D

R 1

BS p

BS q

O B S

pqd

R 2

S iipx

iqx

izp qa

DB S

if

U m

U l

O U

lmd

m ix

lix

R i Uif

lmb


Outdoor Wideband DataOutdoor Wideband Data

• Comparing different characteristics of the circular ring model with the data reported in Pedersen et al. "A Stochastic Model of …,” IEEE Trans. Vehic. Technol., vol. 49, pp. 437-447, 2000.Environment: Typical urban

Location: Aarhus, Denmark

Carrier Frequency: fC =1.8 GHz

BS/user Separation: 300-3000m

BS antenna height: 32m (12 m above the average rooftop level)

No line-of-sight between the BS and user

Sampling Interval: TS =TC/2=122 ns


Comparison with DataComparison with Data

-25 -20 -15 -10 -5 0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

3.5

4

fBS [ ]

f(fB

S)

Circular RingMeas. [Pedersen]

Power delay profile Delay distribution Angle distribution at BS

01

2

34

5 02

46

810

0

0.2

0.4

0.6

0.8

1

Frequency (2*Df)User spacing (d/)

1

1,2

1

Space-frequency correlation between T11 & T21

at the MS 01

23

45 0

24

68

10

0

0.2

0.4

0.6

0.8

1

Frequency (2*Df)BS spacing (d/)

1

1,1

2

Space-frequency correlation between T11 & T12

at the BS


Indoor ModelIndoor Model


Indoor Narrowband DataIndoor Narrowband Data

• Comparing different types of correlations of the two-ring model with the data collected at Brigham Young University, 2000-2001.Location: Fourth floor of a five story engineering building on Brigham Young University campus Carrier Frequency: fC =2.45 GHz

Data Format: Narrowband (25 KHz) 10 x10 channel matrix. 124 such matrices collected over 80 ms. Multiple such channel matrices obtained for several Rx and Tx locations in each room

Antenna Spacing: /4 at both the Tx and Rx sides

No line-of-sight between the BS and user


Different Types of Different Types of CorrelationsCorrelations

Parallel Corr. Crossing Corr.

Receive Corr. Transmit Corr.


Comparison with Measured Comparison with Measured CorrelationCorrelation

Parallel Corr. Crossing Corr.

Common Transmit Corr.

Common Receive Corr.Parallel Corr. Crossing Corr.

Tx Corr. Rx Corr.


Comparison with Measured Comparison with Measured CapacityCapacity


Part IIPart II

Fade Duration inMIMO Channels


Outline

MotivationTwo Crossing Problems in MIMO SystemsScalar Crossing • Theory and a numerical example

• Applications (adaptive modulation and Markov modeling)

Vector Crossing • Theory

• Application (block fading model in MIMO channels)

Summary


Motivation

LCR (Level Crossing Rate) & AFD (Average Fade Duration) extensively studied in SISO channels:

• Interleaver optimization

• Adaptive modulation

• Outage analysis in multiuser systems

• Throughput estimation of protocols

• Markov modeling of fading channels

LCR/AFD also studied for receive diversity (SIMO)How about MIMO channels?!


Two MIMO Crossing Problems

Scalar crossing: There is a scalar process x(t) & we count the #

of times it crosses a threshold (a traditional crossing problem)

t

x(t)

t0


Two MIMO Crossing Problems (cont)

Vector crossing: There is a vector process w(t) = [x(t) y(t)]T &

we count the # of times it crosses a multidimensional surface (needs a multidimensional approach)

x(t)

y(t)

t0


Scalar Crossing in MIMO Channels

Total received SNR

+

AWGN

+

AWGN

+

AWGN

+

AWGN

11( )h t

12 ( )h t1

2

3

1

2

3

4

Rec

eive

r

Tran

smitt

er

2,

( ) | ( ) |lpl pt h tg

( )lph t : Channel gain from the p-th Tx to the l-th Rx (a complex Gaussian process)


More on Total SNR in MIMO

Total MIMO SNR is useful for Markov modeling, channel characterization, and system design

Need to calculate ASD (Average Stay Duration) of γ(t) between two thresholds γ1 and γ2

t

γ(t)

γ2

γ1

1 2

1 2

1 2

ASD{ (t),[ , ]}=

Pr[ ]

ICR{ (t),[ , ]}

g g gg g gg g g

Incrossing Rate

Probability


Incrossing Rate of Total SNR

Space-time correlated Rayleigh fading correlated zero-mean complex Gaussian hlp(t)’s

We have used this paper for incrossing of γ(t): A. M. Hasofer, “The upcrossing rate of a class of

stochastic processes,” in Studies in Probability and Statistics. E. J. Williams, Ed., 1974.

For an MN channel, a (2MN-1)-fold integral needs to be solved (very time consuming)

We are developing a simpler technique (not done yet!)


Example: Crossing of Total SNR

MIMO channel (macrocell) model, taken from:

A. Abdi and M. Kaveh, “A space-time correlation model for multielement antenna systems in mobile fading channels,” IEEE JSAC, 2002.

pqd

pBS

qBS

pqa

DBS

if

iqx

ipx

iz

D R

x

y

lmdmU

lU

lmb

gUif

lix

mix

iS

v

BSO

UO


Example: Crossing of Total SNR (cont)

21 channelDoppler = 20 Hz

0

0.88

0.99

Spatial correlation


MIMO Application of Scalar Crossing

t

γ(t)

Adaptive Modulation: Use the ASD of SNR in each region, to choose proper power/rate adaptation policy

Markov modeling: Use ICR of SNR to determine the transition probability from one state to another


Vector Crossing in MIMO Channels

Joint dynamic behavior of all the subchannels is of interest

In an MN channel, there are 2MN real space-time correlated processes. Put them into the vector h(t)

Need to calculate the ASD (Average Stay Duration) of the vector process h(t) within a hypercube


Vector Crossing in MIMO Channels (cont)

11Re[ ( )]h t

11Im[ ( )]h t

2

2

0 stayt t T

0t t

0

0

0

ASD{ (t), [ ( ),2 ]}=

Pr[ (t) [ ( ),2 ]]

OCR{ (t), [ ( ),2 ]}

HC t

HC t

HC t

h h

h h

h h

Probability

Outcrossing Rate

Example: M=N=1


A Simple Case Study

Rayleigh fading with no spatial correlation, as the temporal correlation for

each subchannel h(t0) = [1 1 … 1]T, > 0

0 (2 )DJ f

2 2

ASD{ (t), [ ,2 ]}=

[ ( ) ( )]exp[( ) / 2]

2 2 cosh( )D

HC

MNf

h 1

1 2 2( ) (2 ) exp( / 2)y

y z dz


Quantitative Analysis of Block Fading

Question:For how long, all the subchannels staywithin a hypercube of side 2, centered at [1 1 … 1]T ?


Summary of Part II

Two different types of crossing in MIMO channels

Scalar crossing is related to the total SNRAdaptive modulation and Markov modeling

in MIMO channels entail a scalar crossingVector crossing considers the joint

variations of all the subchannelsThe MIMO block fading model was analyzed

using the vector crossing approach


Part IIIPart III

MIMO Simulation


The MIMO ChannelThe MIMO Channel

Propagation medium: Frequency-flat and time-varying multipath Rayleigh channel

+

AWGN

+

AWGN

+

AWGN

+

AWGN

11( )h t

12 ( )h t1

2

3

1

2

3

4

Rec

eive

r

Tran

smitt

er

Channel gains(zero-mean complex Gaussian processes)


The GoalThe Goal

Simulation of space-time correlated hij(t)’s

Simulation of m correlated complex Gaussians Yk(t) needs cross-correlation and cross-spectrum functions

( ) [ ( ) ( )], , 1,...,kl k lC E Y t Y t k l m

2( ) ( ) i fkl klS f C e d


Four Simulation TechniquesFour Simulation TechniquesFour Simulation TechniquesFour Simulation Techniques

Spectral Representation Method needs MIMO cross-spectra

Sampling Theorem MethodSampling Theorem MethodRandom Polynomial MethodRandom Polynomial MethodCirculant Embedding MethodCirculant Embedding Method the last three need MIMO cross-

correlations


Spectral MethodSpectral Method

Correlated bandlimited processes:

Spectral Representation Theorem:

Discrete approximation of order q (# of bins in ):

2( )1

ˆ ( ) rq i f tqrrt e

Y A

1, ,[ ] ( ) ( ) , ,

0, .

r

rr k p l kl kl r rE A A S f df S f df r p

r p

a

a

if

if

2( ) ( )i f tt e d f

Y Z

*[ ( ) ( )] ( ) , ,

0, .k l klE dZ f dZ S f df fx x

otherwise

1( ) [ ( )... ( )]Tmt Y t Y tY

max max[ , ]f f


Sampling MethodSampling Method

Sampling Theorem:

n determines the window size

1( ) ( )ˆ ˆ( ) ( ), ( 1)t

t

n nn uu t tu n n

t a t n T t n T

Y Y

( ) sin[ ( ) ] [ ( ) ]ua t t uT T t uT T

[ / ]tn t T

[( ) ,( 1) ]t tn n T n n T

max max1 (2 ) , :T f f max frequency in the spectrum of

( )ˆ ( )u uTY Y

( )tY

, vector of correlated Gaussian variables


Polynomial MethodPolynomial Method

Linear spline approximation over time interval [a,b]

p is the # of subintervals in [a,b]

( ) ( 1)( )ˆ ˆ ˆ( ) [1 ( )] ( )t tp pp t t tx x Y Y Y

[ , ( 1) ]t tt a p T a p T

( ) ( )tt t a p T T x( )T b a p

, vector of correlated Gaussian variables( )ˆ tpY


MIMO Channel Model MIMO Channel Model (macrocell)(macrocell)

A. Abdi & M. Kaveh, “A space-time correlation model for multielement antenna systems in mobile fading channels,” IEEE JSAC, 2002.

pqd

pBS

qBS

pqa

DBS

if

iqx

ipx

iz

D R

x

y

lmdmU

lU

lmb

gUif

lix

mix

iS

v

BSO

UO


MIMO Correlation & MIMO Correlation & SpectrumSpectrum

10

2 2 2 2 2 20

1/ 2

( ) [ ( )] exp[ cos( )]

({ sin ( )

2 cos( ) 2 sin( )[ sin( ) sin( )]

2 [ cos( ) cos( ) sin( )sin( )]} ),

lp mqh h pq pq

lm pq pq

lm lm pq pq lm lm

lm lm pq pq

C I ic

I a b c

ab c a b

i a b c

a

ab g a g b

g b a

D

D

D

2 20

1

22

( ) exp[ cos( )] [ ( ) ]

exp{ [ cos( ) ]( )}

cosh{[ sin( ) ] 1 ( ) },

lp mqh h pq pq m

m

m m

S f ic I f f

i f f

i f f f f

a

g

g

Closed-form MIMO cross-correlation function

Closed-form MIMO cross-spectrum function


Numerical Parameters in Numerical Parameters in SimulationSimulation

2x2 MIMO channel 1000 samples generated for each subchannel, over [0,10] seconds Max Doppler = 0.05 Hz Angle spread at the BS = 8 deg. Isotropic scattering around the MS Parallel Tx/Rx arrays MS moves towards the BS BS element spacing = 17 wavelengths MS element spacing = 0.5 wavelengths q=60 (spectral), n=30 (sampling), p=60 (polynomial)


Simulated and Theoretical Auto and Simulated and Theoretical Auto and Cross-CorrelationsCross-Correlations

0 20 40 60 80 100 120-1

0

1

Cro

ss-c

orr

ela

tion Sim Process h11&h12

Theoretical

0 20 40 60 80 100 120-1

0

1

Cro

ss-c

orr

ela

tion Sim Process h11&h21

Theoretical

0 20 40 60 80 100 120-1

0

1

C

ross

-co

rre

latio

n Sim Process h11&h22Theoretical

Lag

0 20 40 60 80 100 120-1

0

1

Aut

ocor

rela

tion

Sim Process h11Theoretical

0 20 40 60 80 100 120-1

0

1

Aut

ocor

rela

tion Sim Process h12

Theoretical

0 20 40 60 80 100 120-1

0

1

Aut

ocor

rela

tion Sim Process h21

Theoretical

0 20 40 60 80 100 120-1

0

1

Aut

ocor

rela

tion

Sim Process h22Theoretical

Lag

Auto-correlation of the4 subchannels

Cross-correlation between the 4 subchannels


Simulated and Theoretical Simulated and Theoretical Distribution and Level Crossing RateDistribution and Level Crossing Rate

-2 0 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

pdfhistogram

-2 0 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

pdfhistogram

h11 In-Phase Component h11 Quadrature Component

-60 -50 -40 -30 -20 -10 0 10 200

0.2

0.4

0.6

0.8

1

1.2

1.4

level (dB)

mea

n LC

R/f

max

theoretical LCRempirical LCR

PDF’s of real and imaginary parts of the first subchannel,

Level crossing rate of the envelope of 11( )h t 11( )h t


Theoretical Computational Theoretical Computational ComplexityComplexity

Number of Operations

Spectral Sampling Polynomial Embeddin

g

Corr. Matrix generation

48,000 3,810,240 3,572,160 983,040

Matrix Decomposition O(3,840)

O(16,003,008)

O(14,526,784)

O(131,072)

White vector size 240 4488 244 8192

Coloring the white vector 960 77504 59536 32768

Calculating the main expression

360,000 252,000 10,000 315,392


Summary of Part IIISummary of Part III

All the four methods provide good results with proper choices of parameters (q, n, p)

Spectral method requires lower computational effort (so, it is faster)

The Matlab file for the Spectral method is available on:

http://web.njit.edu/~abdi/


A Simulation Demo

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