analysis of uplink power control in cellular mobile systems

A!Aalto UniversityComnet

Wireless Networks Research Seminar-NETS 2020

Analysis of Uplink Power Control inCellular Mobile Systems

U. Oruthota, P. Dharmawansa and O. TirkkonenAalto, Department of Communications and Networking

University of Oulu - 22.04.2013

A! Outline• Introduction

• Fractional power control model

• Objective

• System model

• SINR of a user

• Average inverse interference and bounds

• Average interference of a cell

• Rate approximation

• Selection of FPC parameters

• Conclusion

FPC 2 (18) University of Oulu - 22.04.2013

A! Introduction:• Why Uplink Power Control?• To determine the appropriate transmit power to achieve acceptable

link performance while minimizing inter-cell/intra-cell interferenceand preserving mobile terminal battery power

• 3GPP has approved the use of Fractional Power Control (FPC) whichcompensates a fraction of the path-loss to makes users with a higherpath-loss operate at a lower SINR while minimizing the interferenceto neighbours.

• Baseline Research study:• Coupechoux and Kelif 1 derived an analytical expression for the

average interference caused by neighbouring rings of interfering cellswith uniform user distribution by assuming homogeneity in theangular domain, when seen from the center cell.

• Our Attention:• Analysing the SINR of a user in a cell which suffers from random

interference sources from neighbouring cells in a network withoutangular homogeneity.

1M. Coupechoux and J. M. Kelif, How to set the fractional power control compensation factor in LTE?, SarnoffSymposium, 2011 34th IEEE, May. 2011.


A! FPC Model• We consider simplified per PRB Fractional Power Control

Pt = min{Pm, P0 + αL

}• Pm maximum allowed transmit power of an UE, depends on UE class• P0 cell specific initial power assignment• α fractional power control parameter• L is downlink path loss estimated at UE in dBm

• P0 and α are cell specific parameters assigned by upperlayers.

• Single slope path-loss modelL = K0 + 10λ log10 r

• where K0 is the path-loss at cell radius R = 1km, λ is the path-lossexponent and r is the normalized distance to user from its servingbase station.


A! Objective• In this study:• Derive an approximation for the average rate.

• For this, distributions of inverse interference have to be treated

• Target: Control of fractional control parameters (P0, α) for individualcells to handle their current loads.

• Deliverable:• U. Oruthota, P. Dharmawansa and O. Tirkkonen, “Analysis of Uplink

Power Control in Cellular Mobile Systems,” Accepted forVTC-Spring, June 2013.


A! System Model

d~

BS0

user i

id~

ix~

BSn

ir~

i

R~

1I

2I

user j

jr~

i

• Received SINR at BS0 corresponding to the user j is ofinterest.


A! SINR of User j

• Received SINR at BS0 corresponding to the user j

γj =η

r(1−α)λj

1∑Nn=1 In(ri, θi)

.

• where η = p0kα−1o

• p0 is the cell-specific minimum power in linear domain• k0 is the path loss at reference distance (cell edge) linear domain

• Interference from neighbour n due to user i isIn (ri, θi) = ηrαλi

(d2 + r2i − 2rid cos θi

)−λ/2.• Aggregate interference at BS0 is

∑Nn=1 In(ri, θi)

• Interference caused by the N neighbouring cells is assumed i.i.d.• Assumptions:• 1st tier of cells create dominant interference to the center cell.• Hexagonal cell approximated by circular cell with same radius.• Distances are normalised w.r.t cell radius.• Network is designed such that the cell edge users never reach Pm.


A! Average SINR of User• We are interested in the expected SINR of the user,

conditioned on knowing the path loss of the user• averaged over all possible interference scenarios

• Complicated nature of In(ri, θi) prevents derivation of exactanswers to the statistical properties of γj• Approximately characterize by upper and lower bounds.

• Statistical quantity of interest is

E {γj|rj} =η

r(1−α)λj

J, where J = E

{1∑N

n=1 In(ri, θi)

}


A! Upper Bound• From geometric-arithmetic inequality we have


≤ 1

N

(N∏n=1

In(ri, θi)−1/N

),

• Which leads to

J ≤ 1

NE

{N∏n=1

In(ri, θi)−1/N

}=

1

N

[E

{In(ri, θi)

−1/N}]N

• Upper bound for average inverse interference

JU =1

N

[E

{I(ri, θi)

−1/N}]N

.

Depends on the average on the inverse interference whichcan be modeled as the average interference at path lossexponent −λ/N of a single cell.


A! Lower Bound• From Jensen’s inequality,

J ≥ 1∑Nn=1E {In(ri, θi)}

=1

N[E {In(ri, θi)}]−1 ,

• Lower bound for average inverse interference

JL =1

N[E {I(ri, θi)}]−1 .

Depends on the average interference generated by a singlecell, E {I(r, θ}.


A! Average Interference/Cell• Average interference experience at BS0 due to uniformly

distributed users of a single cell (ri ∈ [0, 1] and θi ∈ [0, 2π))

I(α, λ) = E {I(ri, θi)}

=2η

π

∫ 1

0

rαλ+1i

∫ π

0

dθi

(d2 + r2i − 2rid cos θi)λ2

dri.

• Average Interference is

I(α, λ) = 2η

∞∑k=0

ak(λ)B[b1, 1]

dλ+k2F1[a1, b1, ; c1;−1/d]

• where ak(λ) = (λ/2)k(1/2)k(1)kk!

4k with (z)k = z(z + 1)) . . . (z + k − 1)

denoting the Pochhammer symbol. Parameters of the Gausshypergeometric function 2F1[a1, b1; c1; z1] are a1 = 2k + λ,b1 = αλ + k + 2 and c1 = b1 + 1. B[a, b] is a beta function.


A! Interference Boundaries• Average Interference is bounded by

JL ≤ J ≤ JU

• where

JU =1

NηN+1

[I

(α,− λ

N

)]N• and

JL =1

N

[I (α, λ)

]−1.

• Average of the upper and lower bound provides a goodapproximation.

J ≈ 1

2N

{1

ηN+1

[I

(α,− λ

N

)]N+[I (α, λ)

]−1}.


A! Fitness of Approximation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 160

80

100

120

140

160

180

200

220

Ave

rag

e I

nve

rse

In

terf

ere

nce

(d

Bm

)

α

Actual at λ=2

Approx. at λ=2

Actual at λ=3.76

Approx. at λ=3.76

• Actual and approximate average inverse interference for typical cellular network(N = 6) with α for P0 = −78dBm and λ = [2, 3.76].


A! Rate Approximation (1)• Achievable average rate of a cell,

R = E

{log2

(1 + ηr

−(1−α)λj


)}.

• Random variables (ri, θi), i = 1, .., N and (rj) areindependent. Jensen’s inequality leads to

R ≤ 2

∫ 1

0

rj log2

(1 + ηr

−(1−α)λj E

{1∑N

n=1 In(ri, θi)

})drj

=1

ln(2)

[ln(J + 1)− B[b3, 1]2F1[a3, b3; c3;−1/J ]

J+ α

]• here J = ηJ , α = (1− α)λ/2, a3 = 1, b3 = 1/α + 1, and c3 = b3 + 1.• Upper bound on average rate depends on J for which we do not have

a closed form solution. Approximation of J may be used.


A! Rate Approximation (2)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.8

1

1.2

1.4

1.6

1.8

2

2.2

α

Ave

rag

e r

ate

(b

ps/H

z)

Actual average

Jensen's bound

Approximation

• Actual average rate with the derived approximation, free space path-loss. Jensen’supper bound is also simulated for the comparison.


A! Select FPC parameters (1)• Typical cellular system, FPC is implemented with a single

(P0, α) pair for the whole network.

• Different cells may have different distributions of servicesto provide to the users, and accordingly different fairnessrequirements.

• React to this, it would be preferable to tune the powercontrol parameters on a per-cell basis.

• P0 and α can be independently selected in each cell,according to the current user distribution and their needs

• Network planning may set the tolerable interference levels.Each cell selects P0 and α keeping interference constant.


A! Select FPC parameters (2)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-200

-180

-160

-140

-120

-100

-80

-60

-40

α

Ave

rag

e I

nte

rfe

ren

ce

/Ce

ll (d

Bm

) fo

r λ

=2

P0=-38 dBm

P0=-48 dBm

P0=-58 dBm

P0=-68 dBm

P0=-78 dBm

P0=-88 dBm

IFIXED

=-127 dBm

• Overall interference is evenly distributed among the N interference originatingcells and the average interference for different P0 values are depicted for freespace.


A! Conclusion• New approximation for average inverse interference

provided• Approximation is tight at low values of α irrespective of λ.

• Slight discrepancy is visible at large values of α when λ increases.

• Resulting rate approximation is rather tight at small valuesof α and Jensen’s bound becomes tighter when α goes toone.

• Parameters P0 and α may be separately selected for eachcell depending on current user population and their needs,keeping the interference constant.


analysis of uplink power control in cellular mobile systems

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