Capacity Analysis on Downlink MIMO OFDMA Systems

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Capacity Analysis on Downlink MIMOOFDMA Systems
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Project Topic Maximization of capacity on the
downlink of multi-user MIMO OFDM systems under a
fixed total power constraint, assuming perfect
CSI at the transmitter, over a slow fading
Rayliegh channel.
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• Preview
• Generic MIMO multiuser system model description.
• Derive the capacity of MIMO enabled OFDMA system
  under a generic multiuser multicarrier framework.
• Prove the optimality of OFDMA in the downlink for
  independent decoding receivers for any adaptive
  modulation scheme.
• Describe 2 suboptimal (but less complex)
  subcarrier allocation criteria for OFDMA MIMO
  systems.
• Here we Go!!

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• Channel
• Slow Rayliegh fading
• Hkn channel gain matrix on subcarrier n for
  user k.
• AWGN iid noise, vkn
• EvknvknH No.I , (No noise power)
• Receiver
• Rykn EyknyknH Hkn(Hkn)Hqn No.I
• Assume independent decoding.

• Generic System Model
• System Parameters
• T total number of transmit antennas
• R number of receive antennas for each user
• K total number of users in system
• N total number of subcarriers
• Q total power budget
• Transmitter
• xkn xkn(1), ,xkn(T)T
• Assume independent signals for all users,
• ExknxknH I , ExinxjnH 0 / i?j

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Generic System Model
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• Capacity Analysis
• Assume perfect CSI at the transmitter
• ckn capacity of user k on subcarrier n

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• Capacity Analysis (cont..)
• Goal
• Maximize total capacity under total power
  constraint
• Optimization problem
• (1)
• (Bonus embedded subcarrier allocation!)

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• Capacity Optimization
• Optimization in (1) can be decoupled for
  different subcarriers.
• Solve sub-problem for each carrier
  individually
• Sub-problem
• (2)

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• Capacity Optimization (cont)
• Replacing Hkn(Hkn)H, in (2) ,by its
  eigen-decomposition we get

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• Capacity Optimization (cont)
• The feasible region of (2) is a bounded
  polyhedral
• C(n) is a convex function of g1n,,gkn ,
  bounded by S
• max of C(n) occurs at one of the K vertices of S
• max occurs only when one element of g1n,,gkn
  is nono-zero

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• Capacity Optimization (cont)
• each subcarrier is assigned to only one user

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• Capacity Optimization (cont)
• So, max of C(n) occurs at the element
  0,..,gknn,0..,0) of S corresponding to user kn
  with highest ckn, i.e.
• Let kn be the selected user on subcarrier n
• gt
• Substituting the above result in expression of
  C(n) we get

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• Capacity Optimization (cont)
• So for each subcarrier, the optimal solution is
  to transmit with entire power qn to only one
  user the one with the highest ckn(qn) for that
  specific subcarrier

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• Capacity Optimization (cont)
• So the optimization problem in (1) can be
  rewritten as
• (3)

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• Capacity Optimization (cont)
• (3) is a convex optimization problem,
• use KKT conditions to find the optimal solution
• (4)
• Let q (q1,,qN) be the optimal solution,
  
• then using the KKT, q must satisfy the
• following conditions (5)

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• Capacity Optimization (cont)
• Deriving (4) w.r.t qn , setting the derivative
  to 0 (i.e using (5a) ), and then using (5c) and
  (5d), we get
• (6)
• Optimal distribution of power across subcarriers
  is given by
• (7)

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Capacity Optimization (cont)
Conclusion(3)
Thus optimal power distribution over subcarriers
reduces to a multi-dimensional water-filling
solution! (no closed form solution exists though!)

• If TR1
• gt and

1 dimensional water-filling
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Capacity Optimization (cont) Summary of results

• The optimal scheme for capacity maximization on
  the
• downlink when independent decoding is used, is
  OFDMA.

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Optimal power loading is equivalent to optimal
subcarrier allocation, and is given , for
subcarrier n, by
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Optimal power distribution over subcarriers
reduces to a multi-dimensional water-filling
solution!
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• Complexity Analysis
• So how to jointly determine the optimal power
  and subcarrier allocation?
• Brute Force Method
• perform exhaustive search over all users and all
  subcarriers.
• one must perform optimal power loading using (6)
  and (7) for each of the KN subcarrier
  allocations, then select best.
• Complexity is exponential in N and polynomial in
  K!

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• Complexity Analysis (cont..)
• Q Is there a less complex procedure?
• A Yes!! Make the subcarrier allocation process
  independent from the power loading process.
• Q How?
• A Use Product criterion or Sum criterion.

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• Product Criterion Method
• As shown previously kn is given as
• (8)
• But for large SNR we can simplify
• kn as follows
• After user to subcarrier
• allocation use (6) (7) to
• perform optimal power allocation.

Product Criterion
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• Sum Criterion Method
• Q But what about Low SNR region?
• A Use the Sum criterion instead.
• for small values of SNR, (8) can be
  approximated as

Sum Criterion
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• Complexity Analysis
• Use Product criterion for high SNR region, and
  Sum criterion for Low SNR region.
• Both Sum criterion and Product criterion yield
  sub-optimal solutions, but with a huge reduction
  in complexity (KN for suboptimal criteria as
  opposed to KN for the optimal criterion).
• What about performance?

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Complexity Analysis Low SNR region
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Complexity Analysis High SNR region
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Complexity Analysis Affect of Increasing number
of users
I did say sub-optimal!!
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• Wrapping up!
• We showed that OFDMA maximizes total system
  capacity in the downlink.
• We derived the optimal subcarrier allocation
  criterion and optimal power loading criterion.
• We proposed 2 suboptimal subcarrier allocation
  criteria.
• Sum criterion approximates optimal solution at
  low SNR
• Product criterion approximates optimal solution
  at high SNR
• Simulation results verify expectations.

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• Thanks For Listening!!
• and staying awake )