Capacity of Broadcast MIMO Fading Channel

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Capacity of Broadcast MIMO Fading Channel

• Ivana Stojanovic

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Outline

• A Strategy
• Dirty Paper Encoding
• Part I Sum Capacity of MIMO broadcast channel
• Point to point reciprocity
• Broadcast-Multiple Access Channel duality
• Part II Capacity region of MIMO Broadcast
  channel
• Explicit BC solution via Multiple Access Channel
  (MAC) duality

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General BC channel

• Goal
• send reliably independent information at rate R1
  to receiver Y1 and R2 to receiver Y2
• Capacity not known in general!

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MIMO Broadcast channel
1
N
Point-to-point MIMO SVD H U?VH
of independent channels min(M,N)

• In MIMO (Multiple Input Multiple Output) case
• Each antenna element has different channel gain
• Matrices H1 and H2 may not have same eigenvectors
• Can not decompose the channel into set of
  parallel degraded channels
• Receivers can not cooperate to perform SIC
• Any type of cancellation has to be performed at
  the transmitter

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Channel coding with side information Dirty
Paper Coding (Costa)

• Pre-subtract interference at encoder without
  increasing power!
• Quest is to find appropriate variable U
• Something remarkable happens!!! Capacity is as
  if there is no interference in the channel

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DPC MIMO BC region
x1
Transmit Encoder gt

• Achievable rates
• Achievable region
• brute force to find capacity achieving input
  covariance matrices combinatorial complexity

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MIMO MAC vs. MIMO BC

• We know the capacity region of MIMO uplink
• Uplink MAC (Multiple access channel)
• Is there a dual BC?
• Dual means equivalent in performance

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Part I BC Sum Capacity

• Welfare network
• Rate region (R1,.Rk)
• Profit network
• Sum rate
• Single antenna
• Maximize sum-rate by transmitting to a user with
  the strongest channel
• Multiple antennas
• DPC is strictly larger then any single user
  capacity
• Why?
• Point to point reciprocity
• DL-UL duality

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Part I MIMO Point to point reciprocity

• Original channel
• Reciprocal channel
• ? Capacity is the same!!!!
• Same power constraint
• Independent Gaussian signaling in parallel
• Two channels have the same singular values
• Liner transmit-receive filters
• Left and right eigenvectors can be interchanged

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Part I Uplink Downlink Duality

• MAC Channel
• UL MMSE optimum filter
• maximizes SINR for each user
• BC Channel
• Move UL MMSE receive filter to transmitter
• Perform Costas encoding of inputs
• UL DL sum power constraint
• set of achievable SINRs is the same

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Part I Broadcast Sum Capacity the Proof
BC capacity does not depend on noise correlation
among users -diagonal elements of Qlt1

After noise whitening

• For QQ, MAC users do not have any incentive to
  cooperate
• D is diagonal matrix

For what value of Q MAC sum capacity is equal to
point to point capacity?
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Part II MIMO MAC region
MMSE SIC

• Optimum receiver at BS
• MMSE Successive Interference Cancellation
• Users
• Streams of a given user
• But capacity region is convex hull of union of
  many pentagons
• Decoding order is up to us
• Power splits across streams is arbitrary
• This region is easy to compute via standard
  convex programming

Time sharing
MMSE SIC
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Part II BC capacity region

• BC capacity region Reciprocal MAC capacity
  region
• equal sum power constraint
• user decoding order in MAC is reverse encoding
  order in BC

• Proof find 1-1 BC-MAC transform
• Consider a transform
• Effective MAC channel
• Equivalent MAC covariance matrix
• with

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Conclusions

• MIMO BC Capacity
• Dirty paper coding
• Hard to analyze directly
• Convert MIMO BC to dual MIMO MAC
• MIMO MAC is efficiently solved via convex
  optimization
• Beyond
• Apply MAC/BC Duality to solve your own problems

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QUSTIONS?

• THANK YOU!

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Single Antenna Downlink AWGN channel

• In wireless terms broadcast channel is downlink
• Single antenna AWGN downlink is physically
  degraded
• y2 is only a noisier version of y1

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Single Antenna Downlink AWGN channel

• Capacity of general degraded channel is known
• Capacity achieving strategy
• Superposition coding
• U is auxiliary random variable cloud center
• Successive interference cancellation (SIC)
• Receivers order signals according to their
  quality
• The capacity of a BC channel is no longer
  a
  single number, but rather a region
• Optimize over power allocations