Indian Institute of Science (IISc), Bangalore, India Interference Modelling in Spatially Distributed Shadowed Wireless Systems Neelesh B. Mehta ECE Department,

Indian Institute of Science (IISc), Bangalore, India

Interference Modelling in Spatially Distributed Shadowed Wireless Systems

Neelesh B. Mehta

ECE Department, IISc

Project 602 duration: April 2008 to March 2010

Indian Institute of Science, Bangalore

Outline

• Summary of research output

• Inter-cell interference modeling

• Our two approaches

• Results

• Conclusions


Summary of Output: Conference Publications

• Sarabjot Singh and Neelesh B. Mehta, “An Alternate

Model for Uplink Interference in CDMA Systems with

Power Control,” National Conference on Communications

(NCC), Guwahati, India, Jan. 2009.

• Neelesh B. Mehta, Sarabjot Singh, and Andreas F.

Molisch, “An Accurate Model For Interference From

Spatially Distributed Shadowed Users in CDMA Uplinks,”

IEEE Global Telecommunications Conf. (Globecom),

Honolulu, USA, Nov.\ 2009


Summary of Output: Journal Publications

• Sarabjot Singh, Neelesh B. Mehta, Andreas F. Molisch,

and Abhijit Mukhopadhyay, “Moment-Matched Lognormal

Modeling of Uplink Interference with Power Control and

Cell Selection,” IEEE Trans. on Wireless

Communications, March 2010.

• Neelesh B. Mehta, Sarabjot Singh, Abhijit Mukhopadhyay,

and Andreas F. Molisch, “Accurately Modeling the

Interference From Spatially Distributed Shadowed Users

in CDMA Uplinks,” To be submitted to IEEE Trans. on

Communications, 2010.


Uplink Interference

• Mobile stations tx. to base station

• Multiple interferers contribute to UL

interference

• Interference is random

– Important to model it correctly

Reference cell Neighboring cell

Inter-cell interference

1

1

2

2

2

2

2

1

2

2

2

2

2

1

2

1

1

2

BS


Wireless Propagation Characteristics

• Path loss (d)

• Shadowing (s)

– Lognormal distribution

• Fading (f)

– Rayleigh, Ricean, Nakagami-m

ss ff

4

0

d

dP

Path loss Shadowing Fading

Rx. power

Tx. power


Lognormal Probability Distribution

• A skewed distribution

• Several and varied applications in wireless propagation,

finance, health care, reliability theory, optics, etc.

2 210

2

2

(10log )10 / ln10( ) exp

22

, ( , )

X

Y

xp x

x

x e Y N

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Lognormal Prob. Distribution

x

pX(x)


Conventional Model: Gaussian Approximation

• Problem: Closed-form tractable expressions for

probability distribution of sum are not known

• Conventional solution: Model as a Gaussian RV

– [Chan, Hanly’01; Tse,Viswanath’05]

• Two justifications given:

– Central limit theorem

– Less randomness in the presence of power control and

cell site selection


Our Approach: Approximate As A Lognormal

• Related literature supports this approach– Works much better given number of summands

– [Mehta et al'07, Fenton-Wilkinson’60, Schleher‘77, Schwartz-

Yeh‘82, Beaulieu-Xie’04]

• ‘Permanence' of lognormal sums– [W. A. Janos ‘70, R. Barakat’76]

Model inter-cell interference as a lognormal random variable

Model inter-cell interference as a lognormal random variable


Unique Feature of Our Problem: Several Sources of Randomness

• User locations are random within a cell– Use Poisson point process model

• Number of users is also random

• Interferer’s transmit power is random

– Power control

– Cell site selection


Our Two Methods to Fix Lognormal Parameters

2 210

2

2

(10log )10 / ln10( ) exp

22

, ( , )

X

Y

xp x

x

x e Y N

Goal: Determine the two parameters μ and σGoal: Determine the two parameters μ and σ

Lognormal:

Developed two methods:

• Moment-matching method

• MGF-matching method


Moments of actual interference Moments of actual interference

Moment Matching: Key Results

• Match the first two moments of total uplink interference

• Advantage: Closed-form expressions possible


CCDF Matching: To See Tail Behaviour

• Lognormal tracks the actual CCDF very well

• Better than conventional Gaussian

Ave. # of users/cell= 10First tier interference

Total interference

Com

plem

enta

ry C

DF


CDF Matching: To See Head Behaviour

• Lognormal significantly better than Gaussian

• Gaussian CDF high for small value of interference

– Off by 2 orders of magnitude

Ave. number of users/cell= 10

CD

F

Total interference


With Cell Selection (Handoff Set Size = 2)

• Moment matching based lognormal approximation is better than

Gaussian even with cell site selection

– Shown for first-tier interference

10-2

10-1

100

101

102

10-3

10-2

10-1

100

Interference

CD

F

Simulation

F-W methodGaussian

K = 10

K = 30

100

101

102

103

10-3

10-2

10-1

100

InterferenceC

CD

F

Interference with cell-site selection

SimulationF-W methodGaussian

K = 10

K = 30


Further Improvement Using MGF Matching

• Key idea: Match moment generating function

instead of moments

• Advantage: Gives the parametric flexibility to

match both portions of distribution well

• Technical enabler: Can evaluate MGF relatively

easily when users are distributed as per a Poisson

spatial process

– Benefit from the extensive theory on Poisson processes


Improved Lognormal Approximation Method

• MGF of the total uplink interference from users in cell k

• ψk(s): MGF of the interference from an arbitrary user in cell k

• Method: Match MGFs at s1 and s2 with lognormal’s MGF• Method: Match MGFs at s1 and s2 with lognormal’s MGF


6. Results: CDF and CCDF Matching Accuracy

• Lognormal approximation is significantly better than Gaussian

• MGF-based lognormal approximation is better than moment-based lognormal approximation

CCDF

First-tier interference

CDF

30 users/cell on average


Conclusion

• Goal: Model inter-cell interference in uplink of CDMA

systems

• Showed: Lognormal is better than the conventional

Gaussian

• New methods: To determine parameters of approximating

lognormal

– First method :Based on moment-matching

– Second improved method: MGF-based moment matching


Extensions

Two model generalizations:

• Extend the femto cells

– Multiple femto cells within a macrocell

• Hybrid macrocell/microcell cellular layouts

Two other improvements:

– Include peak power constraints

– Better cell area approximation techniques


Inter-Cell Interference in CDMA Uplinks

• Spreading codes diminish interference but do not annul it

• Sum of signals from many users served by other BSs

• Undergoes shadowing/fading

Reference cell Neighboring cell

It is a random variable. How do we characterize it? It is a random variable. How do we characterize it?


System Model With Power Control

• Fading-averaged inter-cell interference

• Path loss and shadowing model:

• Interference power (with power control) at BS 0 from users served by BS k, located at x1(k), . . . , xNk(k) :

Reference cellInterfering cell


User Location and Number Modelling

• Model as a Poisson Spatial Process– Characterized by an intensity parameter (λ)

– Analytically tractable model

– Probability that Nk users occur within a cell of area A equals

Analysis approximation


Sum of Fixed Number of Lognormals: CDF

Moment matching

SimulationsMehta et al

S-Y method

CD

F Sig

nal

Interferers

[Mehta, Wu, Molisch, & Zhang, IEEE Trans. Wireless 2007]

• Percentile (CDF) plot comparison


Sum of Fixed Number of Lognormals: CCDF

• Various approaches exist to accurately characterize the approximating lognormal

Simulation

Fenton-Wilkinson

Mehta et al

S-YLog scale

Com

plem

enta

ry

CD

F

[Mehta, Wu, Molisch, & Zhang, IEEE Trans. Wireless 2007]


CCDF Matching (Denser User Population)

• Lognormal approximation is still significantly better

• In sync with literature on sums of fixed number of

lognormals26

Ave. # of users/cell= 30First tier interference

Total interference

Com

plem

enta

ry C

DF


Must model inter-cell interference accurately• Cell planning and base station deployment

• Signal outage probability evaluation

• Performance of link adaptation

Must model inter-cell interference accurately• Cell planning and base station deployment

• Signal outage probability evaluation

• Performance of link adaptation

Sources of Inter-Cell Interference

• First tier interference

• Second tier interference

1

1

2

2

2

2

2

1

2

2

2

2

2

1

2

1

1

2


CDF Matching (Denser User Population)

• Lognormal better than Gaussian even for denser

populations!

• However, inaccuracy does increase28

Ave. number of users/cell= 30

Total interference

CD

F


With Cell Site Selection & Power Control

• Serving base station chosen by a user need not be the

geographically closest one

– Due to shadowing

• Depends on soft handoff set size

– The number of neighboring base stations a user tracks

Reference cell Neighboring interfering cell


First Tier Interference (Handoff Set Size = 3)

• Lognormal approximation is still better!

100

101

102

10-3

10-2

10-1

100

CC

DF

Interference


K = 30

K = 10

10-2

10-1

100

101

102

10-3

10-2

10-1

100

Interference

CD

F



K = 10

K = 30

CD

F

CC

DF


Second Tier Interference (Handoff Set Size = 2)

• Second-tier cells are further away

10-3

10-2

10-1

100

10-3

10-2

10-1

100

Sum of k interferers ;k~poiss(10)

CD

F

Simulation

F-W methodGaussian

K = 30

K = 10

10-1

100

101

102

10-3

10-2

10-1

100

InterferenceC

CD

F

Simulation

F-W methodGaussian

K = 30

K = 10

CD

F

CC

DF


Zero Tier Interference (Handoff Set Size = 2)

• Even users located within reference cell can cause inter-cell

interference

• Gaussian does well in this case!

10-1

100

101

102

10-2

10-1

100


CD

F


Simulation

F-W method

Gaussian Approximation

100

101

102

10-3

10-2

10-1

100


CC

DF


Simulation

F-W method

Gaussian Approximation

CD

F

CC

DF

Documents

Indian Institute of Science (IISc), Bangalore, India Interference Modelling in Spatially Distributed Shadowed Wireless Systems Neelesh B. Mehta ECE Department,