Lecture 5 Capacity of Multiuser Channels I-Hsiang Wang - - PowerPoint PPT Presentation

Lecture ¡5 Capacity ¡of ¡Multiuser ¡Channels

I-Hsiang Wang ihwang@ntu.edu.tw 4/10, 2014

From ¡Single-­‑User ¡to ¡Multi-­‑User

• In Lecture 3 we studied various techniques for multiple

access and interference management in cellular systems

• In Lecture 4 we learned about information theory and

investigate the capacity of point-to-point channels

• In this lecture we extend the information theoretic

framework to multi-user channels

• Present new techniques that emerge from the

information theoretic study:

• Success interference cancellation (SIC)
• Superposition coding
• Multi-user diversity
• Opportunistic communication paradigm

2

Plot

• Two scenarios:
• Uplink channel (many-to-one)
• Downlink channel (one-to-many)
• Start with AWGN (no fading)
• Uplink channel: successive interference cancellation (SIC)
• Downlink channel: superposition coding
• Fast Fading
• CSIR only
• Full CSI
• Multi-user Diversity

3

Outline

• Uplink/Downlink AWGN channel
• Uplink/Downlink fading channel
• Multi-user diversity
• Opportunistic beamforming

4

5

Uplink/Downlink ¡ AWGN ¡Channel

Uplink ¡and ¡Downlink ¡Channel

6

y[m] = h1x1[m] + h2x2[m] + w[m] y1[m] = h1x[m] + w1[m] y2[m] = h2x[m] + w2[m]

• Channel gains are fixed over time and known to Tx & Rx
• Uplink noise:
• Downlink noise at User k, k = 1,2:

CN(0, σ2) Uplink

y x1 x2 h1 h2

User 1 User 2 Rx: decodes both users’ data

Downlink

x y1 y2 h1 h2

Tx: encodes both users’ data User 1 User 2

CN(0, σ2

k)

• Point-to-point channel:
• Capacity C
• Multi-user channel
• Each user has its own data
• Two data rates R1 & R2
• Capacity region
• (R1,R2) is achievable ⟺
• both error probability → 0

Capacity ¡Region

7

Achievable if R < C Not Achievable if R > C

C R

R1 R2

C

Achievable if (R1,R2) ∈ C Not Achievable if (R1,R2) ∉ C

C Achievable ⟺ Pe(N) → 0 as N → ∞

Capacity ¡Region ¡of ¡the ¡UL ¡Channel

8

R1 R2 log (1 + SNR1) log (1 + SNR2) log ⇣ 1 +

SNR1 1+SNR2

⌘ log ⇣ 1 +

SNR2 1+SNR1

⌘

CUplink

R1 + R2 ≤ log (1 + SNR1 + SNR2) SNRk := |hk|2Pk σ2 , k = 1, 2 CUplink = [ 8 > < > : (R1, R2) ≥ 0 : 8 > < > : R1 ≤ log (1 + SNR1) R2 ≤ log (1 + SNR2) R1 + R2 ≤ log (1 + SNR1 + SNR2) 9 > = > ;

Non-­‑Achievability ¡Outside ¡Cuplink

• Rk ≤ log(1+SNRk): obvious, since log(1+SNRk) is the

point-to-point capacity as if there is only one Tx

• R1+R2 ≤ log(1+SNR1+SNR2): obvious, since the

maximum received SNR from the two independent Tx is SNR1+SNR2, and therefore the total rate cannot exceed the capacity of the point-to-point channel with this SNR

9

CUplink

Successive ¡Interference ¡Cancellation

10 R1 R2 log (1 + SNR1) log (1 + SNR2) log ⇣ 1 +

SNR1 1+SNR2

⌘ log ⇣ 1 +

SNR2 1+SNR1

⌘

CUplink

A B

Achieving point A

User k encodes its data using a capacity achieving AWGN channel code at rate Rk, k=1,2 Rx first decodes User 2’s data, treating User 1’s signal x1 as Gaussian noise = ⇒ R2 = log ⇣ 1 +

|h2|2P2 |h1|2P1+σ2

⌘ = log ⇣ 1 +

SNR2 1+SNR1

⌘ can be achieved Rx then subtracts x2 from y and get a point-to-point channel for User 1

y x1 x2 h1 h2

User 1 User 2 Rx: decodes both users’ data = ⇒ R1 = log (1 + SNR1) can be achieved Note: smaller R2 can also be achieved

Equivalent ¡Point-­‑to-­‑Point ¡Channels

• Equivalent channels
• For User 2, the equivalent noise is h1x1+w, with variance
• For User 1, after removing x2, Rx sees a clean point-to-point

channel without interference

11

h2 x2[m] h1x1[m] + w[m] y[m]

|h1|2P1 + σ2

x1[m] h1 w[m] y[m] − h2x2[m]

Time ¡Sharing

12 R1 R2 log (1 + SNR1) log (1 + SNR2) log ⇣ 1 +

SNR1 1+SNR2

⌘ log ⇣ 1 +

SNR2 1+SNR1

⌘

CUplink

A B

Similarly point B can be achieved y x1 x2 h1 h2

User 1 User 2 Rx: decodes both users’ data

To achieve a rate point on AB, say, qA + (1-q)B, the system can take the following two strategies with a prescribed portion of time: Strategy achieving A Decode User 2 first and then decode User 1; q of time Strategy achieving B Decode User 1 first and then decode User 2; (1-q) of time

Comparison ¡with ¡Conventional ¡CDMA

• For each user, treat the other user’s signal as noise
• No successive interference cancellation (SIC)
• Hence a single-user receiver, not a multi-user receiver
• It is strictly suboptimal (achieving point C)

13 R1 R2 log (1 + SNR1) log (1 + SNR2) log ⇣ 1 +

SNR1 1+SNR2

⌘ log ⇣ 1 +

SNR2 1+SNR1

⌘

CUplink

A B C

UL ¡Orthogonal ¡Multiple ¡Access

• Consider time-division access
• User 1 uses the first α of the time
• User 2 uses the rest (1–α) of the time
• Power constraint:
• User 1 can now use power P1/α during its transmission
• User 2 can now use power P2/(1–α) during its transmission
• Achievable rates:
• When α = SNR1/(SNR1+SNR2), the sum capacity is achieved

(i.e., R1+R2 = log(1+SNR1+SNR2) is achieved)

14

( R1 = α log

• 1 + SNR1

α

• R2 = (1 − α) log

⇣ 1 + SNR2

1−α

⌘ α ∈ [0, 1]

Orthogonal ¡MA ¡is ¡Sum ¡Rate ¡Optimal

15

D

Orthogonal multiple access can

• nly achieve the optimal sum

rate at a single point, when

• α = SNR1/(SNR1+SNR2)

[

α∈[0,1]

( (R1, R2) : ( R1 = α log

• 1 + SNR1

α

• R2 = (1 − α) log

⇣ 1 + SNR2

1−α

⌘ ) ( R1 =

SNR1 SNR1+SNR2 Csum

R2 =

SNR2 SNR1+SNR2 Csum

D: Csum = log (1 + SNR1 + SNR2)

Fairness is an issue

R1 R2 log (1 + SNR1) log (1 + SNR2) log ⇣ 1 +

SNR1 1+SNR2

⌘ log ⇣ 1 +

SNR2 1+SNR1

⌘

CUplink

• For a general K-user uplink channel
• Capacity region:
• Sum capacity:
• For example, 3-user uplink channel capacity region:

K-­‑user ¡Uplink ¡Channel ¡Capacity

16

CUplink = [ ( (R1, . . . , RK) ≥ 0 : X

k∈S

Rk ≤ log 1 + X

k∈S

SNRk ! , ∀ S ⊆ [1 : K] ) Csum

Uplink = log

1 +

K

X

k=1

SNRk ! Rk ≤ log (1 + SNRk) , k = 1, 2, 3 R1 + R2 ≤ log (1 + SNR1 + SNR2) R2 + R3 ≤ log (1 + SNR2 + SNR3) R3 + R1 ≤ log (1 + SNR3 + SNR1) R1 + R2 + R3 ≤ log (1 + SNR1 + SNR2 + SNR3)

Capacity ¡Region ¡of ¡the ¡DL ¡Channel

17

SNRk := |hk|2P σ2

k

, k = 1, 2

CDownlink

R1 R2 log (1 + SNR1) log (1 + SNR2) Note: proof of non-achievability outside this region is beyond the scope of this course

WLOG assume SNR1≥SNR2 Maximum sum rate is achieved when β = 1

= ⇒ Csum

Downlink = log (1 + SNR1)

CDownlink = [

β∈[0,1]

( (R1, R2) ≥ 0 : ( R1 ≤ log (1 + βSNR1) R2 ≤ log ⇣ 1 + (1−β)SNR2

1+βSNR2

⌘ )

Superposition ¡Coding

• Tx sends x = x1+x2, where for k=1,2
• User k’s data is encoded onto xk
• Power of x1 = βP; power of x2 = (1–β)P
• User 1 has a better received SNR
• User 1’s channel is better than User 2
• User 1 can decode whatever User 2 can decode
• Single-user decoding at User 2:
• Decode x2 by treating x1 as noise
• ⟹ can achieve
• SIC Decoding at User 1:
• First decode x2 by treating x1 as noise, and remove it from y1
• Then decode x1 ¡⟹ can achieve

18

x y1 y2 h1 h2

Tx: encodes both users’ data User 1 User 2

R2 = log ⇣ 1 + (1−β)SNR2

1+βSNR2

⌘ R1 = log (1 + βSNR1)

Comparison ¡with ¡Conventional ¡CDMA

• Conventional CDMA: the same as before except that

User 1 does not do SIC

• Strictly suboptimal
• Exercise: how to choose β such that all DL users have

the same received SINR?

19

DL ¡Orthogonal ¡Multiple ¡Access

• Consider time-division access
• User 1 uses the first ¡α ¡of the time with power P
• User 2 uses the rest (1–α) of the time with power P
• Achievable rates:
• Strictly suboptimal (except the two corner points when one of the

users is shut down)

20

( R1 = α log (1 + SNR1) R2 = (1 − α) log (1 + SNR2) α ∈ [0, 1]

K-­‑user ¡Downlink ¡Channel ¡Capacity

• WLOG assume SNR1 ≥ SNR2 ≥ … ≥ SNRK
• Capacity region:
• βk denotes the portion of power allocated to User k’s codeword
• Sum capacity: achieved by sending only to the best user

21

CDownlink = [

β1,...,βK≥0 β1+···+βK=1

8 < :(R1, . . . , RK) ≥ 0 : Rk ≤ log ✓ 1 +

βkSNRk 1+Pk−1

j=1 βjSNRk

◆ , ∀ k ∈ [1 : K] 9 = ;

Csum

Downlink = log (1 + SNR1)

22

Uplink/Downlink ¡ Fading ¡Channel

Setting

• Fast fading: ∀k, {hk[m]} is stationary and ergodic
• Symmetry: ∀k, {hk[m]} is identically distributed
• We shall focus on ergodic sum capacity

23

Uplink

y x1 x2 h1 h2

User 1 User 2 Rx: decodes both users’ data

Downlink

x y1 y2 h1 h2

Tx: encodes both users’ data User 1 User 2 y[m] = h1[m]x1[m] + h2[m]x2[m] + w[m] y1[m] = h1[m]x[m] + w1[m] y2[m] = h2[m]x[m] + w2[m]

Uplink ¡Channel ¡Capacity: ¡CSIR ¡Only

• Without CSIT, the ergodic sum capacity is
• where SNRk := Pk/σ2, k = 1,2,…,K
• Comparison with AWGN capacity:
• due to Jensen’s inequality
• Similar to the point-to-point case: if there is no CSIT,

fading makes things worse

24

Csum

UL, CSIR = E

" log 1 +

K

X

k=1

|hk|2SNRk !#

Csum

UL, CSIR = E

" log 1 +

K

X

k=1

|hk|2SNRk !# ≤ log 1 +

K

X

k=1

E ⇥ |hk|2⇤ SNRk ! = log 1 +

K

X

k=1

SNRk ! = Csum

UL, AWGN

Downlink ¡Channel ¡Capacity: ¡CSIR ¡Only

• Recall channel symmetry:
• ∀k, {hk[m]} is identically distributed
• Due to the symmetry assumption:
• There is a natural ordering of the users
• based on the noise level σk2
• WLOG assume SNR1 ≥ … ≥ SNRK where SNRk := P/σk2
• Sum capacity is achieved by serving the best user only
• By Jensen’s inequality this is strictly worse than the

AWGN downlink sum capacity

25

x y1 y2 h1 h2

Tx: encodes both users’ data User 1 User 2

= ⇒ Csum

DL, CSIR = E

⇥ log

• 1 + |h1|2SNR1

⇤

Impact ¡of ¡Multiple ¡Users

• Under fast fading without CSIT:
• The ergodic sum capacity of multiuser UL/DL channels is smaller

than that without fading

• Similar to the point-to-point case
• As K (# of users) increases, this capacity loss behaves

differently in the uplink and the downlink

• In uplink, the loss vanishes as K → ∞
• In downlink, the loss remains as K → ∞
• Explored in Homework 3

26

Downlink ¡Channel ¡Capacity: ¡Full ¡CSI

• User symmetry assumption: σk = σ, ∀k = 1,…,K
• Downlink channel sum capacity is achieved by sending
• nly to the instantaneously best user.
• Optimization problem:
• Solution:

27

max

P (h)≥0 E

 log ✓ 1 + maxk∈[1:K] |hk|2P(h) σ2 ◆ s.t. E [P(h)] = P P ∗(h) = ✓ ν − σ2 maxk∈[1:K] |hk|2 ◆+ , ν satisfies E "✓ ν − σ2 maxk∈[1:K] |hk|2 ◆+# = P

Uplink ¡Channel ¡Capacity: ¡Full ¡CSI

• Full CSI assumption:
• At each time m, all users know the instantaneous realization of all

channel gains {hk[m] | k = 1,2,…,K}

• In other words, a user can know not only its own channel but also
• thers’ channels
• User symmetry assumption: Pk = P, σk = σ, ∀k = 1,…,K
• Program in finding uplink sum capacity under full CSI:
• Consider an L-parallel uplink AWGN channel
• Relax individual power constraints to a total power constraint
• Solve the new problem under finite L, and take L → ∞
• By channel and user symmetry, argue that the found solution is

also feasible under individual power constraints

28

Power ¡Allocation ¡Problem

• Original problem:
• Relaxed problem:
• We will solve the relaxed problem first, and verify that the

solution found there is also feasible for the original one

29

Individual power constraint max

Pk(h)≥0 k∈[1:K]

E " log 1 + PK

k=1 |hk|2Pk(h)

σ2 !# s.t. XK

k=1 E [Pk(h)] = KP

max

Pk(h)≥0 k∈[1:K]

E " log 1 + PK

k=1 |hk|2Pk(h)

σ2 !# s.t. E [Pk(h)] = P, k = 1, 2, . . . , K Total power constraint

Parallel ¡Uplink ¡Channel

• L parallel K-user uplink channel:
• Channel gains for the l-th sub-channel: {hk,l | k = 1,2,…,K}
• Power allocated to the l-th sub-channel: {Pk,l | k = 1,2,…,K}
• Optimization problem (total power constraint):

30

total power constraint max

Pk,l≥0 k∈[1:K], l∈[1:L]

1 L

L

X

l=1

log 1 + PK

k=1 |hk,l|2Pk,l

σ2 ! s.t.

K

X

k=1

1 L

L

X

l=1

Pk,l = KP

Optimal ¡Allocation ¡in ¡Parallel ¡UL ¡(1)

• Rewrite the total power constraint as
• For a fixed partition {Pl | l ∈ [1:L]} of the total power

LKP, the sum rate of the l-th sub-channel is maximized if all of Pl is allocated to the best user: (

• )

31

K

X

k=1

1 L

L

X

l=1

Pk,l = KP ⇐ ⇒

L

X

l=1

K X

k=1

Pk,l ! | {z }

Pl

= LKP max

Pk,l≥0 k∈[1:K]

log 1 + PK

k=1 |hk,l|2Pk,l

σ2 ! = log 1 +

• maxk∈[1:K] |hk,l|2

Pl σ2 !

Pl := PK

k=1 Pk,l

Optimal ¡Allocation ¡in ¡Parallel ¡UL ¡(2)

• How to determine the best partition {Pl | l ∈ [1:L]} of the

total power LKP?

• Problem becomes:
• Water-filling solution:

32

P∗

l =

✓ ν − σ2 maxk∈[1:K] |hk,l|2 ◆+ , ν satisfies

L

X

l=1

P∗

l = LKP

max

Pl≥0 l∈[1:L]

1 L

L

X

l=1

log 1 +

• maxk∈[1:K] |hk,l|2

Pl σ2 ! s.t.

L

X

l=1

Pl = LKP

Optimal ¡Allocation ¡in ¡Parallel ¡UL ¡(3)

• Optimal power allocation in L parallel K-user uplink

channel under the total power constraint:

• where
• Take L → ∞, we obtain the solution of the power

allocation problem of the uplink fading channel under total power constraint

• where

33

P ∗

k,l =

( P∗

l ,

if k = arg maxj∈[1:K] |hj,l|2 0,

• therwise

P ∗

k (h) =

( P∗(h), if k = arg maxj∈[1:K] |hj|2 0,

• therwise

P∗

l =

✓ ν − σ2 maxk∈[1:K] |hk,l|2 ◆+ , ν satisfies

L

X

l=1

P∗

l = LKP

P∗(h) = ✓ ν − σ2 maxk∈[1:K] |hk|2 ◆+ , ν satisfies E [P∗(h)] = KP

Solution ¡to ¡the ¡Original ¡Problem

• Recall the original vs. the relaxed problem
• Note: solution to the relaxed problem is
• Due to channel symmetry,
• are equal ∀k ⟹
• ∀k ⟹ feasible in the original problem!

34

P ∗

k (h) =

( P∗(h), if k = arg maxj∈[1:K] |hj|2 0,

• therwise

where P∗(h) = ✓ ν − σ2 maxk∈[1:K] |hk|2 ◆+ , ν satisfies E [P∗(h)] = KP

max

Pk(h)≥0 k∈[1:K]

E " log 1 + PK

k=1 |hk|2Pk(h)

σ2 !# s.t. E [Pk(h)] = P, k = 1, . . . , K Original Problem s.t. XK

k=1 E [Pk(h)] = KP

Relaxed Problem

E [P ∗

k (h)]

E [P ∗

k (h)] = P

UL ¡Capacity ¡with ¡Full ¡CSI: ¡Summary

• Solution:

35

max

Pk(h)≥0 k∈[1:K]

E " log 1 + PK

k=1 |hk|2Pk(h)

σ2 !# s.t. E [Pk(h)] = P, k = 1, 2, . . . , K P ∗

k (h) =

( P∗(h), if k = arg maxj∈[1:K] |hj|2 0,

• therwise

where P∗(h) = ✓ ν − σ2 maxk∈[1:K] |hk|2 ◆+ , ν satisfies E "✓ ν − σ2 maxk∈[1:K] |hk|2 ◆+# = KP

Remarks

• Sum capacities and optimal power allocation solutions of

the DL and the UL channels are essentially the same

• UL total power constraint: KP
• DL total power constraint: P
• Full CSIT requirement in UL:
• We begin with the assumption that all users know all the channels
• However, to attain the optimal power allocation, each user only

needs to know its own channel and whether it is the best channel

• Amount of feedback to each user is not increasing with K !

36

37

Multi-­‑User ¡Diversity

A ¡Key ¡Feature ¡of ¡Wireless ¡Channel

• Time variation!
• Multi-path fading
• Large-scale channel variations (path loss, shadowing)
• Time-varying interference

38

Mobile environment Channel strength Dynamic range Time

Traditional ¡Design ¡Approach

39

Dynamic range Time Fixed environment Channel strength Mobile environment Channel strength Dynamic range Time

• Compensates for channel fluctuations

Example: ¡CDMA ¡Systems

• Two main compensating mechanisms:
• Channel diversity
• Interference management
• Channel diversity
• Frequency diversity via Rake receiver
• Macro-diversity via soft handoff
• Tx/Rx antenna diversity
• Interference management
• Intra-cell: power control
• Inter-cell: interference averaging

40

What ¡Drives ¡this ¡Approach?

41

Dynamic range Time Fixed environment Channel strength Mobile environment Channel strength Dynamic range Time

• Main application is voice, with tight latency constraints
• Need a consistent channel

Opportunistic ¡Communication

• A completely different view!
• Transmit more when and where the channel is good
• Exploits fading to achieve higher long-term throughput,

but no guarantee that “the channel is always there”

• Appropriate for data with non-real-time latency

requirements (file downloads, video streaming)

42

Single-­‑User ¡Fading ¡Channel

43

–5 5 10 15 SNR (dB) 20

AWGN CSIR Full CSI

C (bits /s / Hz) –10 –15 –20 7 6 5 4 3 2 1

CFull CSI > CAWGN ≅ CCSIR CAWGN > CFull CSI ≅ CCSIR

Single-­‑User ¡Channel: ¡Low ¡SNR ¡Regime

44

–10 –5 5 10 0.5 –15 –20 3 2.5 2 1.5 1

awgn

SNR (dB)

CSIR Full CSI

C CAWGN

Power gain due to dynamic power allocation

Hitting ¡Peaks ¡over ¡Time

45

Optimal Best Only: Near-Optimal

σ2 |h[m]|2 m ν σ2 |h[m]|2 m

Interpretation: at low SNR, one only transmits when the channel is at its peak! ⟹ primarily a power gain at low SNR!

Multi-­‑User ¡Fading ¡Channel

46

2 4 6 5 –5 –10 –15 –20 10 15 20 8

AWGN CSIR Full CSI

Csum(bits /s / Hz) SNR (dB) K = 16 K = 2 K = 4 K = 1

AWGN

For UL, SNR := KP/σ2 For DL, SNR := P/σ2

Increase in spectral efficiency with number of users K(∀K>1) at all SNR’s, not just low SNR

Multi-­‑User ¡Channel: ¡Low ¡SNR ¡Regime

47

1 5 –5 –15 –20 –25 –30 10 2 3 4 5 6 7 CSIR Full CSI SNR (dB) Csum CAWGN K = 16 K = 4 K = 2 K = 1 –10

Multi-­‑User ¡Gain

• Let us compare the single-user and the multi-user cases:
• Point-to-point capacity
• Multi-user downlink capacity

48

P ∗(h) = ✓ ν − σ2 |h|2 ◆+ , ν satisfies E "✓ ν − σ2 |h|2 ◆+# = P

Cpoint-to-point = E  log ✓ 1 + |h|2 P ∗(h) σ2 ◆ CDownlink = E  log ✓ 1 + max

k∈[1:K]|hk|2 P ∗(h)

σ2 ◆

P ∗(h) = ✓ ν − σ2 maxk∈[1:K]|hk|2 ◆+ , ν satisfies E "✓ ν − σ2 maxk∈[1:K]|hk|2 ◆+# = P

Multi-­‑User ¡Opportunistic ¡Communication

49

…

• Dedicate full power to serve only the best user + the

peak value is higher than the mean ⟹ multi-user gain!

• Hitting peaks not only over time (at low SNR), but also
• ver users (at all SNR)

User 1 User 2 User K

Multi-­‑User ¡Diversity

• In a large system with users fading independently:
• Likely to have a user with very good channel at any time
• Different users peak at different times
• The more random the channel is, the higher the rate is

50

5 10 15 20 25 30 35 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 Number of users Sum capacity at SNR = 0 dB (bits /s / Hz) AWGN Rayleigh fading Rician fading

Multi-­‑User ¡vs. ¡Classical ¡Diversity

• Both due to the existence of independently faded paths
• Classical diversity: over time, frequency, antennas in a link
• Multi-user diversity: over multiple users in the network
• Classical diversity is to compensate channel fluctuation
• Multi-user diversity aims to exploit channel fluctuation
• Classical diversity increases reliability
• Suitable for application with stringent latency constraints (voice)
• Multi-user diversity increases total throughput (long-term)
• Suitable for application with long latency constraints (data)

51

Issues ¡in ¡System ¡Implementation

• Fairness:
• Multi-user diversity offers a system-wide benefit (sum capacity ↑)
• How to share this benefit among all users in a fair way (in an

asymmetric environment)?

• Slow and limited fluctuations:
• Channels are less random ⟹ multi-user diversity gain ↓
• How to retain the benefit even in a rather static environment?
• Channel measurement and feedback:
• Tracking channel is crucial in getting multi-user diversity
• Overhead has to be considered

52

Proportional ¡Fair ¡Scheduler

• At each time slot,
• Each user will request a data rate from the base station
• A scheduler decides which user to transmit and at what rate
• To obtain multi-user diversity:
• Transmit to the best user + stronger user requests higher rates
• ⟹ Most likely will select the statistically strongest all the time
• Highly unfair!
• Solution: schedule the user with the highest ratio Rk/Tk,

where

• Rk := current requested rate of user k
• Tk := average throughput of user k in the past tc time slots

53

Proportional ¡Scheduler: ¡Two-­‑User

54

50 100 150 200 250 300 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 Time slots Requested rates in bits / s / Hz 50 100 150 200 250 300 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time slots Requested rates in bits /s / Hz

Statistically Symmetric Asymmetric

• The statistically stronger user gets an higher avg. rate
• But the statistically weaker user still gets served fairly!
• The algorithm serves each user when it is near its peak

within the latency time-scale tc

Multi-­‑User ¡Diversity ¡in ¡Practice

• Fixed environment has limited fluctuation
• High mobility environment has a lot of fluctuation, but it is

difficult to get this gain because the system cannot track the channel!

55

Fixed environment: 2Hz Rician fading with κ = 5 Low mobility environment: 3 km/hr, Rayleigh fading High mobility environment: 120 km/hr, Rayleigh fading

2 4 6 8 10 12 14 16 100 200 300 400 500 600 700 800 900 1000 1100

Low mobility environment Fixed environment Number of users Total throughput (kbps) High mobility environment latency time scale tc = 1.6s Average SNR = 0dB

Inducing ¡Randomness

• Scheduling algorithm exploits the nature-given channel

fluctuations by hitting the peaks

• Not enough fluctuations ⟹ multi-user diversity gain ↓
• Why not purposely induce fluctuations?

56

Dumb ¡Antennas

• Multiply the information bearing signal at each Tx

antenna by a random complex gain

• α(t): portion of power allocated to the first antenna
• θ(t): phase shift

57

User k x(t) h1k(t) h2k(t) √α (t) √1– α(t) e jθ(t)

Slow ¡Fading ¡→ ¡Fast ¡Fading

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Before After

Opportunistic ¡Beamforming

• Dumb antennas create a beam in

random time-varying directions

• In a large system, there is likely to

be a user near the beam at any

• ne time
• By transmitting to that user, close

to true beamforming performance is achieved

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Slow Fading: Opportunistic Beamforming

Performance ¡Improvement

• Opportunistic beamforming with dumb antennas

increases the performance of the fixed environment significantly

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Overall Performance Improvement

Fixed environment: 2Hz Rician fading with κ = 5 Mobile environment: 3 km/ hr, Rayleigh fading

Dumb, ¡Smart, ¡and ¡Smarter ¡Antennas

• Smart antennas (space-time code in Lecture 3)
• Improve reliability of point-to-point links
• Reduce multi-user diversity (less fluctuations)
• Dumb antennas
• Add fluctuations to point-to-point links
• Increase multiuser diversity gains
• Smarter antennas
• With full CSI, antennas can actually form beams pointing to users
• Coherent beamforming

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Dumb ¡vs. ¡Smarter ¡in ¡Slow ¡Fading ¡

62

• As # of users grow, performance of opportunistic

beamforming → that of coherent beamforming

Comparison

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Table 6.1 A comparison between three methods of using transmit antennas.

Dumb antennas (Opp. beamform) Smart antennas (Space-time codes) Smarter antennas (Transmit beamform) Channel knowledge Overall SNR Entire CSI at Rx Entire CSI at Rx, Tx Slow fading performance gain Diversity and power gains Diversity gain only Diversity and power gains Fast fading performance gain No impact Multiuser diversity ↓ Multiuser diversity ↓ power ↑

Summary

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Conventional multiple access Opportunistic communication Guiding principle Averaging out fast channel fluctuations Exploiting channel fluctuations Knowledge at Tx Track slow fluctuations No need to track fast ones Track as many fluctuations as possible Control Power control the slow fluctuations Rate control to all fluctuations Delay requirement Can support tight delay Needs some laxity Role of Tx antennas Point-to-point diversity Increase fluctuations Power gain in downlink Multiple Rx antennas Opportunistic beamform via multiple Tx antennas Interference management Averaged Opportunistically avoided

Conventional Multiple Access vs. Opportunistic Communication