Capacity of Fading Broadcast Channels with Rate Constraints

1
Capacity of Fading Broadcast Channels with Rate
Constraints

• Chris T. K. Ng, Andrea J. Goldsmith
• Dept. of Electrical Engineering
• Stanford University

2
Introduction

• For a fading channel, multiple definitions of
  capacity exist.
• Each definition imposes a different set of
  constraints on how the transmitter can adapt to
  the time-varying channel.
• We propose a unifying framework for calculating
  the capacity region of fading broadcast channels.
• It provides a methodology for obtaining capacity
  regions under existing constraints.
• New types of constraints are defined, and the
  corresponding capacity region is derived under
  the same framework.
• There are limitations in practical systems not
  addressed by the existing capacity definitions.
• New constraints are introduced to allow for a
  more realistic and accurate modeling of the
  operation of a wireless channel.

3
Capacity definitions for fading channels

• Fading channels
• Time-varying channel gain.
• Adapt transmission rates/power for different
  fading states.
• Ergodic (Shannon) capacity
• Long-term average of maximum achievable rate.
• Small or zero rate when channel is in an
  unfavorable state.
• Large delay.
• Zero-Outage capacity
• Constant rate across all fading states.
• Large energy is needed to invert the channel.

4
Capacity with constraints

• Outage capacity
• Transmission rate is zero with some outage
  probability.
• Otherwise transmission rate is constant.
• Minimum rate capacity
• Maintain a minimum rate across all fading states.
• Excess power can be freely allocated to any
  states to maximize average rates.
• Capacity regions comparisons
• More restrictive constraints lead to smaller
  capacity region.
• Ergodic gt minimum rate, outage gt zero-outage.

5
Rate constraints parameters under unifying
framework

• Maximum rate constraint Rm
• Can be used to represent system bottlenecks
  (e.g., limited processing rate).
• Minimum rate constraint Rn
• Shortage probability q
• With probability q the minimum rate constraints
  are relaxed.
• Different from outage probability since
  transmission rates need not be zero when in
  shortage.
• For simplicity, a common shortage mode is
  assumed all users declare shortage at the same
  time.

6
Capacity definitions under rate constraint
parameters

• Types of capacity regions represented by the rate
  constraint parameters

7
System Model

• Discrete-time flat fading Gaussian broadcast
  channel
• Perfect channel state information at transmitter
  and all receiver average power constraint for
  the transmitter.
• Received signal
• Zero-mean Gaussian random variable z with
  variance nB.
• Time-vary noise density is referred to as fading
  state.
• Superposition coding with successive decoding,
  each user j has rate
• Consider the case with two users (M2).

8
Rate constraints for the AWGN channel

• No fading
• Noise density n1, n2 are constants.
• Assumes n2 gt n1 (reverse subscripts otherwise).
• Capacity region with the rate constraints
• First consider zero shortage probability.
• Effects of imposing the rate constraints
• Power allocation is not simply determined by the
  total power, but also by the maximum and minimum
  rate constraints.
• The achieved rates expression will be extended to
  formulate the optimal power allocation for a
  fading channel.

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Optimal power allocation between users

• Consider for a given fading state.
• Noise densities n1, n2 are known and constant.
• Assume a certain amount of power is allocated for
  this fading state.
• Will address power allocation for different
  fading states later.
• Within a fading state, how to divide up the
  available power between the users?
• Need to satisfy the given maximum and minimum
  rate constraints.

10
Rate constraints power allocation boundaries

• Represent rate constraints as power allocation
  boundaries.
• Each additional constraint reduces the feasible
  power region.

11
Achievable rates possible maximum points

• All possible maximum points have the same form
  for the achieved rates in terms of the available
  total power P.
• Allows formulation of optimal power allocation
  strategy across fading states.

12
Capacity region with rate constraints for a
fading channel

• Fading channel
• Noise density n1, n2 are time-varying.
• Consider first shortage probability q is zero.
• Minimum rate constraints need to be held at all
  times.
• Power allocation is done in 2 stages
• Allocate power P(n) to each fading state.
• Allocate power for each state between users.

13
Power allocation between users in a fading channel

• Suppose the power P(n) allocated to a fading
  state n is given.
• Allocate power between users as in the AWGN case.
• Leads to a common form of rates achieved in terms
  of the total power P(n) for all possible maximum
  points.
• This common rate expression is used to optimize
  power allocation across fading states.

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Optimal power allocation across fading states

• Minimum power required by a fading state
• Determined by the minimum rate constraints.
• Maximum power limit for a fading state
• No allocation of additional power once maximum
  rates are achieved.
• Allocation of excess power between the maximum
  and minimum limits.
• Optimal power allocation strategy is
  water-filling.
• Water-filling level depends on the effective
  noise of the fading states, and the average total
  power constraint.

15
Water-filling power allocation for the excess
power

• Water-filling power allocation
• Depends on the effective noise n of the fading
  states.
• Determined by the constant parameters A, B, C, D
  of the common rate expression RP(P(n), n).
• Some fading states have the same effective noise
  and water-filling parameters.
• Three distinct water-filling equations

16
Combined optimal power allocation strategy

• Power allocation illustrated in the direction of
  n1.
• Water-filling level depends on the average total
  power available.

17
Shortage probability

• Calculate optimal power allocation for two cases
• Case 1) All rate constraints are present.
• Case 2) Minimum rate constraints are removed.
• Select the set of shortage fading states that
  results in maximum power savings.
• Use power allocation 1) for non-shortage states,
  and 2) for shortage states.
• Water-filling levels for non-shortage and
  shortage states need to be jointly optimized.
• For small shortage probability q, often it is
  optimal to suspend transmission when in shortage.
• Reduces to outage probability.

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Optimal power allocation for shortage states

• Optimal power allocation is obtained from power
  allocation for zero-shortage, and power
  allocation without minimum rate constraints.

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Numerical results Imbalanced fading states

• Symmetrical two-user fading broadcast channel 40
  dB SNR difference.

20
Numerical results More balanced fading states

• More balanced fading states 20 dB SNR difference.

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Conclusion

• Unifying framework for calculating capacity
  region of fading broadcast channels.
• Existing capacity types ergodic, outage, minimum
  rate.
• New constraints maximum rate, shortage
  probability.
• Optimal power allocation between users
• Evaluate a finite set of extreme points.
• Optimal power allocation across fading states
• Minimum and maximum power allocation limits.
• Excess power allocation determined by
  water-filling.
• Water-filling level depends on effective noise of
  the fading states.
• Shortage capacity
• Removing the minimum rate constraints.
• Jointly optimize power allocation over shortage
  and non-shortage.