Title: EE360 Lecture 2 Outline
1EE360 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
2Broadcast Channels
- Synchronization easy.
- Interference signals follow same path as desired
signal (no near-far problem) - Complexity/power at transmitter less restricted
than at receiver.
3Frequency 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
4Time 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
5Code 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
6Hybrid 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
7Examples
- 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
8Broadcast 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.
9Rate 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
10Rate Region Frequency Division
- Frequency Division
- Bandwidth Bi and power Si allocated to each user
is varied.
Equivalent to TD for BitiB and Sitisi.
11Superposition Coding
Best user decodes fine points Worse user decodes
coarse points
12Code 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(No Transcript)
14Fading 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.
15Two-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
16Fading 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
17Ergodic 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.
18CD with successive decoding
- M-user capacity region under CD with successive
decoding and an average power constraint is - The power constraint implies
19Ergodic Capacity Region Boundary
- By convexity, ?m?RM, boundary vectors satisfy
- Lagrangian method
- Must optimize power between users and over time.
20Ergodic 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.
21Water 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
22Time 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
23Optimization
- 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
24CD 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
25Rate Region Rayleigh Fading
Largest Region
Smallest Region
Region in Fading and in AWGN
Capacity Region in Fading
26Outage 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.
27Common 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.
28Independent 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.
29Zero-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.
30Minimum 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.
31Minimum 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.
32Minimum 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.
33Optimal 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.
34Single-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)
35Single-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.
36Single-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.
37Two-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.
38Two-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.
39Two-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.
40Optimal 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).
41Multi-User Water-filling
- Essentially identical to the optimal power
allocation scheme for ergodic capacity (except
the effective noise terms).
42Numerical 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).
43Numerical 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).
44Numerical Results
P 10 mW, B 100 KHz
Independent Rician fading with K1 for both
users (severe fading, but not as bad as Rayleigh
fading).
45Numerical Results
P 10 mW, B 100 KHz
Independent Rician fading with K5 for both users.
46Comparisons
- 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.