CDMA Uplink Power Control Presentation

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(No Transcript)
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Outline

• Model and Motivation for Uplink Power Control
• The cost function, pricing and utility functions
• Optimal user response Reaction function
• Existence and Uniqueness of Nash Equilibrium (NE)
• Conditions for NE to be strictly positive
• Schemes for dropping users
• Update Schemes and Stability
• PUA, RUA, and a sufficient condition for
  stability
• Pricing Strategies at the Base Station
• Centralized pricing and a market based scheme
• Simulations and Summary of Results
• Conclusion

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The Model for CDMA

• Single cell, one base station with M mobiles
• Each user degrades other users performances by
  creating interference
• Downlink power control at the base station,
  centralized
• Uplink power control
• Done by mobiles
• Decentralized in this scheme
• Centralized schemes exist

downlink
uplink
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SIR in CDMA

• Signal to Interference Ratio (SIR)

L Spreading gain, LW/R, such that W Chip
rate, R Total rate hi Channel gain of ith
user hi pi power as received at the pi
Uplink power level of ith user base station ?2
Background noise and interference from
neighboring cells
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Cost Function

• Pricing function, P, is linear in power and
  proportional to channel gain
• Utility, U, is logarithmic in SIR. Parameter ui
  can be user specific, and models users request
  for SIR

,

• The cost function, Ji Pi - Ui is

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Users Reaction Function

• Each user minimizes its cost function Ji given
  ui, ?i, hi, L, and total interference,
• The reaction function is the optimal response of
  user

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Existence and Uniqueness of (NE)

• Nash equilibrium p1,, pm such that
• Ji(pi, p-i) ? Ji(pi, p-i) ? pi ?0 , i1,,M.
• There exists a unique NE.
• M users are indexed as follows
• Two possible cases
• NE is strictly positive with MM users.
• Unique boundary solution where users.
• M1, M2, ,M have zero power level.

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Existence and Uniqueness of (NE)

• NE is strictly positive if there exists MM
  such that
• Equilibrium power levels pi given by

()
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Uniqueness (NE) Proof

• Set of user reaction functions ?i(p-i) are
  linearly independent. If MM , uniqueness
  result follows directly.
• If MltM , () holds for 1? i ? M and for
  users.
• I ?M1, M2, ,M positivity condition
  fails
• Boundary solution is unique.

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Update Algorithms PUA

• Parallel Update Algorithm (PUA)
• Each mobile updates its power level at each
  iteration within the time interval using the
  reaction function ?i(p-i(n))
• Updates are sufficiently frequent

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Update Algorithms RUA

• Random Update Algorithm (RUA).
• Stochastic Modification of PUA.
• Each user i updates its power level randomly at
  each iteration with a predefined probability pi
  gt 0.
• Let

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Update Algorithms
PUA
user 1 user 2 user 3
n
RUA
n
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Stability

• Stability is defined as convergence to the
  equilibrium from ?????any feasible starting
  point

• A sufficient condition applies to both PUA and
  RUA with constant probabilities
• A different condition for RUA with user-dependent
  probabilities
• Proof is based on l?-norm analysis

M Number of users L Spreading gain
Upper lower bounds on update probabilities.
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Stability Proof PUA

• Distance to equilibrium power pi at time n is
  given as ? pi(n)
• Weighted l? -norm of the distance is defined as
• Finally, a sufficient condition for () to be a
  contraction mapping is obtained.

()
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Stability Proof RUA

• By taking the expectation of the distance to
  equilibrium power pi at time n

• Similar to PUA, following an l? -norm analysis

• Sufficient condition is .
• If ?i ? gt 0 ? i, condition is
  , same as in PUA.

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RUA (a.s.) convergence

• RUA converges almost surely (a.s.) under the
  given condition for stability.
• We make use of Borel-Cantelli lemma in the last
  line of the proof.

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Pricing Strategies (a)

• Centralized Pricing Scheme
• All users have the same utility, uiu .
• , proportional pricing.
• Base station adjusts pricing parameter, ki , to
  meet SIR requirements of users.
• Centralized approach (in terms of calculation of
  parameter k).

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Pricing Strategies (a)

• An Admission Control Criterion
• Users have the same utility, uiu1 , and same
  SIR requirement, ? .
• Maximum number of users, M
• to be admitted
• k value ensuring M ? M users achieve SIR level,
  ?

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Pricing Strategies (b)

• Market Based Scheme
• Market based, decentralized scheme
• Base station provides single price, k
  (proportional to channel gain)
• Users quantify their level of request for SIR
  using the utility parameter, ui
• Users adjust ui , and decide on how much to pay
  for the desired SIR level, given the interference
  in the cell

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Pricing Strategies (b)

• Given users min SIR requirement, ?i , ui is
  chosen by user
• For a maximum number of users Mmax , and an
  upper-bound on received power from a user, pmax,
  ui is bounded above

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Simulations

• MATLAB based numerical simulations
• PUA and RUA implemented under delay-free and
  delayed cases.
• Common Parameter values
• Spreading gain, L800
• Background noise, ?210
• Channel gain of users, 0.2lthilt1, uniformly
  distributed
• Stopping criterion for convergence, ?10-5
• ?i ? taken as a variable, but independent of
  i

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Effect of Channel Gains
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Effect of Pricing (1/Utility) Parameter
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Two Groups of Users Fixed-Utility
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Convergence of RUA and PUA
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Convergence of RUA and PUA(with delay)
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Comparison of Convergence Rates
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Robustness Analysis
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Market Based Scheme (Two Groups)
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Summary of Results

• PUA converges faster than RUA for small number of
  users, M ltltL , whereas RUA is superior as M gets
  large.
• The model is robust w.r.t. variations in user
  numbers and channel gains.
• Market based scheme successfully differentiates
  users with different request levels.
• Fairness of the proportional pricing scheme is
  illustrated.
• Increase in pricing k (or decrease in utility, u)
  leads to decrease in power and SIR levels.

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Conclusions and Extensions

• Noncooperative game theory provides a robust and
  versatile model for the CDMA uplink power control
  problem
• Limitation of this model Only one cell with a
  single base station is considered
• One extension is multiple cells with a more
  complex interference pattern
• Another one is multiple base stations in a cell
  and handoffs

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References