Transcript Interference Modelling in Spatially Distributed Shadowed Wireless
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 (IISc), Bangalore, India
Indian Institute of Science, Bangalore
Outline
• Summary of research output • Inter-cell interference modeling • Our two approaches • Results • Conclusions
Indian Institute of Science, Bangalore
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
Indian Institute of Science, Bangalore
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.
Indian Institute of Science, Bangalore
Uplink Interference
Inter-cell interference BS Reference cell Neighboring cell • Mobile stations tx. to base station • Multiple interferers contribute to UL interference • Interference is random – Important to model it correctly 2 2
2
2 1 1 2 2 1 1 2 2 1 1 2 2 2 2
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Wireless Propagation Characteristics
• Path loss ( d ) • Shadowing ( s ) – Lognormal distribution • Fading ( f ) – Rayleigh, Ricean, Nakagami-m P Tx. power
d
0
d
4 Path loss s Shadowing f Fading Rx. power
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Lognormal Probability Distribution
Lognormal Prob. Distribution 1.4
p X
10 / ln10 exp 2
x
(10 log 2 10 2
x
2 ) 2 1.2
1 0.8
p X (x) 0.6
x
e Y
,
Y
N
2 ) 0.4
0.2
0 0 1 2 3 6 7 8 9 4 x 5 • A skewed distribution • Several and varied applications in wireless propagation, finance, health care, reliability theory, optics, etc. 10
Indian Institute of Science, Bangalore 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
Indian Institute of Science, Bangalore Our Approach: Approximate As A Lognormal Model inter-cell interference as a lognormal random variable • 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]
Indian Institute of Science, Bangalore
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
Indian Institute of Science, Bangalore
Our Two Methods to Fix Lognormal Parameters
Lognormal:
p X
10 / ln10 2
x
exp 2 (10 log 10 2
x
2 ) 2
x
e Y
,
Y
N
2 ) Goal: Determine the two parameters μ and σ Developed two methods: • Moment-matching method • MGF-matching method
Indian Institute of Science, Bangalore Moment Matching: Key Results • Match the first two moments of total uplink interference • Advantage: Closed-form expressions possible Moments of actual interference
Indian Institute of Science, Bangalore CCDF Matching: To See Tail Behaviour Ave. # of users/cell= 10 First tier interference Total interference • Lognormal tracks the actual CCDF very well • Better than conventional Gaussian
Indian Institute of Science, Bangalore CDF Matching: To See Head Behaviour Ave. number of users/cell= 10 Total interference • Lognormal significantly better than Gaussian • Gaussian CDF high for small value of interference – Off by 2 orders of magnitude
Indian Institute of Science, Bangalore
With Cell Selection (Handoff Set Size = 2)
Interference with cell-site selection 10 0 10 0 K = 10 10 -1 10 -1 K = 30 K = 30 K = 10 10 -2 10 -2 10 -3 10 -2 10 -1 10 0 Interference 10 1 Simulation F-W method Gaussian 10 2 10 -3 10 0 Simulation F-W method Gaussian 10 1 Interference 10 2 • Moment matching based lognormal approximation is better than Gaussian even with cell site selection – Shown for first-tier interference 10 3
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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
Indian Institute of Science, Bangalore Improved Lognormal Approximation Method • MGF of the total uplink interference from users in cell k •
ψ k
(s): MGF of the interference from an arbitrary user in cell k • Method: Match MGFs at s 1 and s 2 with lognormal’s MGF
Indian Institute of Science, Bangalore 6. Results: CDF and CCDF Matching Accuracy 30 users/cell on average First-tier interference • Lognormal approximation is significantly better than Gaussian • MGF-based lognormal approximation is better than moment-based lognormal approximation
Indian Institute of Science, Bangalore 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
Indian Institute of Science, Bangalore
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
Indian Institute of Science, Bangalore 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 It is a random variable. How do we characterize it? Reference cell Neighboring cell
Indian Institute of Science, Bangalore
System Model With Power Control
Reference cell Interfering cell • 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 x 1 (k), . . . , x Nk (k) :
Indian Institute of Science, Bangalore User Location and Number Modelling • Model as a Poisson Spatial Process – Characterized by an intensity parameter (λ) – Analytically tractable model – Probability that N k users occur within a cell of area A equals Analysis approximation
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Sum of Fixed Number of Lognormals: CDF
• Percentile (CDF) plot comparison Moment matching S-Y method Simulations Mehta et al Interferers [Mehta, Wu, Molisch, & Zhang, IEEE Trans. Wireless 2007]
Indian Institute of Science, Bangalore
Sum of Fixed Number of Lognormals: CCDF
Log scale S-Y Mehta et al Fenton-Wilkinson Simulation • Various approaches exist to accurately characterize the approximating lognormal [Mehta, Wu, Molisch, & Zhang, IEEE Trans. Wireless 2007]
Indian Institute of Science, Bangalore CCDF Matching (Denser User Population) Ave. # of users/cell= 30 First tier interference Total interference • Lognormal approximation is still significantly better • In sync with literature on sums of fixed number of lognormals
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Indian Institute of Science, Bangalore
Sources of Inter-Cell Interference
2 • First tier interference • Second tier interference
2
2 1 2 1 2 2 Must model inter-cell interference accurately • Cell planning and base station deployment • Signal outage probability evaluation • Performance of link adaptation 1 1 2 2 1 1 2 2 2 2
Indian Institute of Science, Bangalore CDF Matching (Denser User Population) Ave. number of users/cell= 30 Total interference • Lognormal better than Gaussian even for denser populations!
• However, inaccuracy does increase
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With Cell Site Selection & Power Control
Reference cell Neighboring interfering cell • 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
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First Tier Interference (Handoff Set Size = 3)
Interference with cell-site selection 10 0 10 0 K = 10 10 -1 10 -1 K = 30 K = 10 K = 30 10 -2 10 -2 10 -3 10 -2 10 -1 10 0 Interference Simulation F-W method Gaussian 10 1 10 2 10 -3 10 0 Simulation F-W method Gaussian 10 1 Interference 10 2 • Lognormal approximation is still better!
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Second Tier Interference (Handoff Set Size = 2)
10 0 10 0 K = 10 K = 30 10 -1 10 -1 K = 10 K = 30 10 -2 10 -2 10 -3 10 -3 Simulation F-W method Gaussian 10 -2 10 -1 Sum of k interferers ;k~poiss(10) 10 0 10 -3 10 -1 Simulation F-W method Gaussian 10 0 Interference 10 1 10 2 • Second-tier cells are further away
Indian Institute of Science, Bangalore
Zero Tier Interference (Handoff Set Size = 2)
Interference with cell-site selection 10 0 10 0 Interference with cell-site selection Simulation F-W method Gaussian Approximation 10 -1 10 -1 10 -2 10 -2 10 -1 10 0 10 1 Sum of k interferers ;k~poiss(30) 10 2 10 -3 10 0 Simulation F-W method Gaussian Approximation 10 1 Sum of k interferers ;k~poiss(30) 10 2 • Even users located within reference cell can cause inter cell interference • Gaussian does well in this case!