EE360 Lecture 2 Outline

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EE360 Lecture 2 Outline

• Announcements
• 3 Papers for possible presentation due April 9.
• Broadcast Channels
• TD, FD, CD Practical Implications
• Capacity of Broadcast Channels with AWGN
• Broadcast Channels with ISI
• Fading Broadcast Channels

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Broadcast Channels

• Synchronization easy.
• Interference signals follow same path as desired
  signal (no near-far problem)
• Complexity/power at transmitter less restricted
  than at receiver.

3
Frequency Division
Total system bandwidth divided into orthogonal
channels assigned to different users.

• Advantages
• Narrowband channels (no ISI)
• Low complexity
• Allows cts. time transmission and channel
  estimation.
• Disadvantages
• Radios must be frequency-agile
• Handoff complicated by continuous transmission
• Dedicated channels (idle ones wasted)
• Difficult to allocate multiple channels per user

FD alone not used in current digital systems
4
Time Division
Time divided into orthogonal slots,
with different timeslots assigned to different
users.

• Advantages
• No need for frequency agility
• Discontinuous transmission facilitates handoff
  and reduces power consumption.
• Easy to allocate multiple channels/user
• Disadvantages
• Synchronization required
• Multipath destroys slot orthogonality
• Typically requires ISI mitigation (short
  timeslots)
• Idle channels may be wasted
• Short transmissions make equalization and dynamic
  resource allocation hard.

TD used (with frequency hopping) in GSM
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Code Division
Orthogonal or semi-orthogonal codes used to
modulate each users signal. Code properties used
to separate users at the receiver.

• Advantages
• With semi-orthogonal codes, no hard limit to of
  users in system (soft capacity - system is
  interference-limited)
• Interference reduction techniques increase
  capacity
• Synchronization not required
• Can allocated multiple channels/user using
  multicode or multirate techniques.
• No near-far problem on downlink
• Disadvantages
• Complexity
• Multipath creates multiple interferers

CD used in IS-95 and many 3G propositions
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Hybrid Techniques

• Multiuser OFDM
• Time and frequency divided into orthogonal slots
• Different users assigned different orthogonal
  slots
• Very flexible technique
• Usual problems with OFDM (peak-to-average
  power,)
• Multicarrier OFDM
• OFDM signal modulated with a CDMA code across
  frequency
• Users separated via CDMA code properties
• Semi-orthogonal codes introduce interference

Multiuser OFDM used in Flarion system
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Examples

• AMPS FDMA/FDD
• GSM (EDGE) TDMA/FDD
• IS-54 and IS-136 TDMA/FDD
• JDC TDMA/FDD
• IS-95 CDMA/FDD
• IMT-2000 CDMA/FDD
• 3G WCDMA/FDD
• 802.11b CDMA/FDD
• 802.11a,g OFDM/FDD

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Broadcast Channel Capacity Region in AWGN

• Model
• One transmitter, two receivers with spectral
  noise density n1, n2 n1ltn2.
• Transmitter has average power S and total
  bandwidth B.
• Single User Capacity
• Maximum achievable rate with asymptotically small
  Pe
• Set of achievable rates includes (C1,0) and
  (0,C2), obtained by allocating all resources to
  one user.

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Rate Region Time Division

• Time Division (Constant Power)
• Fraction of time t allocated to each user is
  varied
• Time Division (Variable Power)
• Fraction of time t and power si allocated to each
  user is varied

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Rate Region Frequency Division

• Frequency Division
• Bandwidth Bi and power Si allocated to each user
  is varied.

Equivalent to TD for BitiB and Sitisi.
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Superposition Coding
Best user decodes fine points Worse user decodes
coarse points
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Code Division

• Superposition Coding
• Coding strategy allows better user to cancel out
  interference from worse user.
• DS spread spectrum with spreading gain G and
  cross correlation r12 r21 G
• By concavity of the log function, G1 maximizes
  the rate region.
• DS without interference cancellation

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Fading Broadcast Channels

• In a fading broadcast channel the effective noise
  of each user varies over time.
• If TX and all RXs know the channel, can optimally
  adapt to channel variations.
• Fading broadcast channel capacity region obtained
  via optimal allocation of power and rate over
  time.

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Two-User Channel Model
?h1i
n1i
Y1i
x
Xi
Y2i
x
n2i
?h2i
At each time i nn1i,n2i
n1i/?h1i
Y1i
Xi
Y2i
n2i/?h2i
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Fading Capacity Definitions

• Given an average transmit power constraint and
  with perfect transmitter and receiver side
  information
• Ergodic (Shannon) Capacity maximum long-term
  rates averaged over the fading process.
• Shannon capacity applied directly to fading
  channels.
• Delay depends on channel variations.
• Variable transmission rate fluctuating with
  fading conditions.
• Outage Capacity maximum information rate that
  can be achieved in any fading condition during
  non-outage.
• Outage probability constraint satisfied
• Constant transmission rate

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Ergodic Capacity

• Decompose channel into parallel, AWGN broadcast
  channels one for each joint fading state.
• Any point in the capacity region is achievable
  via superposition coding and successive decoding.
• Strongest user in each state decoded last
• Intuitively, more data is transmitted when the
  channel is strong.

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CD with successive decoding

• M-user capacity region under CD with successive
  decoding and an average power constraint is
• The power constraint implies

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Ergodic Capacity Region Boundary

• By convexity, ?m?RM, boundary vectors satisfy
• Lagrangian method
• Must optimize power between users and over time.

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Ergodic Capacity Region Boundary

• Optimal power allocation scheme is multi-user
  water-filling
• Power is optimally distributed between the users
  in a fading state as a function of the priorities
  and noise powers.
• Ergodic capacity takes advantage of channel
  fluctuations by allocating additional power to
  stronger channel states.

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Water Filling Power Allocation Procedure

• For each state n, define p(i)np(1)?np(2)??np(M)
  
• If set Pp(i)0 (remove some
  users)
• Set power for cloud centers
• Stop if ,otherwise
  remove np(i), increase noises np(i) by Pp(i), and
  return to beginning

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Time Division

• For each fading state n, allocate power Pj(n) and
  fraction of time tj(n) to user j.
• Achievable rate region
• Subject to
• Frequency division equivalent to time-division

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Optimization

• Use convexity of region boundary vectors satisfy
• Lagrangian method used for power constraint
• Four step iterative procedure used to find
  optimal power allocation
• For each n the channel is shared by at most 2
  users
• Suboptimal strategy best user per channel state
  is assigned power has near optimal TD
  performance

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CD without successive decoding

• M-user capacity region under CD with successive
  decoding and an average power constraint is
• The best strategy for CDWO is time-division

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Rate Region Rayleigh Fading
Largest Region
Smallest Region
Region in Fading and in AWGN
Capacity Region in Fading
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Outage Capacity

• Two different notions of outage capacity
• Common Outage outage is declared for all users
  simultaneously.
• Independent Outage outage declared independently
  for each user .
• Outage capacity region implicitly defined by
  minimum outage probability required to support
  some rate vector for a given power constraint.
• Weakest channel states are allocated the most
  power because it takes more power to transmit at
  a constant rate in weak states.
• Transmission scheme eliminates all channel
  fluctuation during non-outage.

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Common Outage

• Can determine the minimum power to transmit at a
  constant set of rates in any fading state.
• In states that require less power than some
  threshold level, data is transmitted to all
  users.
• In states requiring more power than the
  threshold, no data is transmitted.
• Threshold level is adjusted to meet the outage
  constraint.

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Independent Outage

• With independent outage cannot use threshold
  approach because can transmit to any subset of
  the users in any fading state.
• Consider the reward from transmitting to a
  specified subset of users in a fading state
  versus the power required
• Reward is non-outage time for users in subset.
• Minimum power is a function of the noise powers.

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Zero-Outage Capacity

• Special case of outage capacity where the outage
  probability is zero.
• Transmitting during poor channel states can
  consume a large portion of the available power.
• Some fading models (i.e. Rayleigh) require
  infinite power to transmit data in certain
  channel states and therefore have no zero-outage
  capacity.

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Minimum Rate Capacity

• Combines the concepts of ergodic and zero-outage
  capacity
• A minimum rate is maintained in all fading
  states.
• Long-term rates in excess of the minimum rates
  are maximized.
• Minimum rate allows delay-constrained data to be
  transmitted at all times.
• Channel variation exploited by transmitting data
  without delay constraints at the maximum possible
  long-term rates.

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Minimum Rate Capacity Region

• All achievable average rates such that the
  minimum rate constraints R (R1,,RM) are
  satisfied in all fading states for all users.
• Ergodic and zero-outage capacity regions are
  special cases of minimum rate capacity
• Ergodic capacity R (0,,0).
• Zero-outage capacity R on the boundary of the
  zero-outage region
• Minimum rate capacity region boundary lies
  between the boundaries of the ergodic and
  zero-outage capacity regions.

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Minimum Rate Capacity Region

• As R approaches the boundary of the zero-outage
  capacity region, the minimum rate capacity region
  decreases because more power is required to
  maintain the minimum rates.

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Optimal Coding and Power Allocation

• Superposition coding with interference
  cancellation in the standard decoding order (i.e.
  strongest user last) is optimal.
• Power allocation is broken down into two steps
• First allocate the minimum power to achieve the
  minimum rates in all fading states.
• Then optimally allocate the excess power to
  maximize the ergodic rate in excess of the
  minimum rates.

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Single-User Channel with Min Rates

• Power constraint P, minimum rate R
• P(n) minimum power to achieve R
• P(n) additional power in state n
• Total power P(n) P(n) P(n)

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Single-User Channel with Min Rates

• Optimal power allocation scheme is
• where the water-filling level satisfies the
  excess power constraint P EP(n).
• Water-filling with effective noise (n P(n))
  instead of n.
• Minimum power P(n) is effectively another source
  of noise.

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Single-User Channel with Min Rates

• Without minimum rates all 3 states are allocated
  power.
• With minimum rates, the distribution of noises
  becomes more skewed towards the better states .
• As a result, only the two best states are
  allocated additional power.

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Two-User BC with Minimum Rates

• Power constraint P, minimum rates (R1, R2)
• where µ1 and µ2 are the priorities of the users
  and R1(n) and R2(n) are the rates assuming
  superposition coding.

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Two-User Power Allocation

• Can allocate minimum power, as in the single-user
  case, but now have to deal with interference
  between the users.
• Power allocated to the stronger user is seen as
  interference by the weaker user.

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Two-User Power Allocation

• Assuming n1 lt n2,
• Power allocated to user 1 increases R1, but some
  power must also be allocated to user 2 to offset
  the additional interference.
• Power allocated to user 2 increases R2 (but not
  by as much because n1 lt n2), but does not affect
  R1.
• Must balance these effects, along with the
  priorities µ1 and µ2, when allocating power.

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Optimal Power Allocation Scheme

• The optimal allocation of the excess power is
• where n1 and n2 are effective noises
• and the water-level satisfies power constraint
• P EP1(n) P2(n).

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Multi-User Water-filling

• Essentially identical to the optimal power
  allocation scheme for ergodic capacity (except
  the effective noise terms).

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Numerical Results
P 10 mW, B 100 KHz
Symmetric channel with 40 dB difference in noises
in each fading state (i.e. user 1 is 40 dB
stronger in 1 state, and vice versa).
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Numerical Results
P 10 mW, B 100 KHz
Symmetric channel with 20 dB difference in noises
in each fading state (i.e. user 1 is 20 dB
stronger in 1 state, and vice versa).
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Numerical Results
P 10 mW, B 100 KHz
Independent Rician fading with K1 for both
users (severe fading, but not as bad as Rayleigh
fading).
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Numerical Results
P 10 mW, B 100 KHz
Independent Rician fading with K5 for both users.
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Comparisons

• Minimum rate capacity is much smaller than the
  ergodic capacity region when one user is
  significantly better than the other user
• Rician K1 fading.
• 40 dB difference in noise level example.
• If the zero-outage capacity region is much
  smaller than the ergodic capacity region, the
  minimum rate capacity region will also generally
  be considerably smaller than the ergodic capacity
  region.