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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
Polytechnic University, Dec. 4, 2003Ali Abdi
Part IPart I
MIMO Models
Polytechnic University, Dec. 4, 2003Ali Abdi
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
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if
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Polytechnic University, Dec. 4, 2003Ali Abdi
MIMO Modeling: Micro & MIMO Modeling: Micro & PicocellsPicocells
• Scatterers around the BS and the MS are located on separate rings.
Polytechnic University, Dec. 4, 2003Ali Abdi
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.
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
A 2x2 ChannelA 2x2 Channel
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Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
Indoor ModelIndoor Model
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
Different Types of Different Types of CorrelationsCorrelations
Parallel Corr. Crossing Corr.
Receive Corr. Transmit Corr.
Polytechnic University, Dec. 4, 2003Ali Abdi
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.
Polytechnic University, Dec. 4, 2003Ali Abdi
Comparison with Measured Comparison with Measured CapacityCapacity
Polytechnic University, Dec. 4, 2003Ali Abdi
Part IIPart II
Fade Duration inMIMO Channels
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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?!
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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)
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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!)
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
Example: Crossing of Total SNR (cont)
21 channelDoppler = 20 Hz
0
0.88
0.99
Spatial correlation
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
Quantitative Analysis of Block Fading
Question:For how long, all the subchannels staywithin a hypercube of side 2, centered at [1 1 … 1]T ?
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
Part IIIPart III
MIMO Simulation
Polytechnic University, Dec. 4, 2003Ali Abdi
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)
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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)
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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
Polytechnic University, Dec. 4, 2003Ali Abdi
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/
Polytechnic University, Dec. 4, 2003Ali Abdi
A Simulation Demo
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