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EE360 Lecture 2 Outline

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Title: EE360 Lecture 2 Outline


1
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

2
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
5
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
6
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
7
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

8
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.

9
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

10
Rate Region Frequency Division
  • Frequency Division
  • Bandwidth Bi and power Si allocated to each user
    is varied.

Equivalent to TD for BitiB and Sitisi.
11
Superposition Coding
Best user decodes fine points Worse user decodes
coarse points
12
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

13
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14
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.

15
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
16
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

17
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.

18
CD with successive decoding
  • M-user capacity region under CD with successive
    decoding and an average power constraint is
  • The power constraint implies

19
Ergodic Capacity Region Boundary
  • By convexity, ?m?RM, boundary vectors satisfy
  • Lagrangian method
  • Must optimize power between users and over time.

20
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.

21
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

22
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

23
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

24
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

25
Rate Region Rayleigh Fading
Largest Region
Smallest Region
Region in Fading and in AWGN
Capacity Region in Fading
26
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.

27
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.

28
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.

29
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.

30
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.

31
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.

32
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.

33
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.

34
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)

35
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.

36
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.

37
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.

38
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.

39
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.

40
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).

41
Multi-User Water-filling
  • Essentially identical to the optimal power
    allocation scheme for ergodic capacity (except
    the effective noise terms).

42
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).
43
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).
44
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).
45
Numerical Results
P 10 mW, B 100 KHz
Independent Rician fading with K5 for both users.
46
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.
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