Decentralized Power Control for Random Access with Multi-User Detection

1

Decentralized Power Control for Random

Access with Multi-User Detection

Department of Electronic Engineering

City University of Hong Kong

December 27, 2012

Chongbin Xu, Peng Wang, Sammy Chan, and Li Ping

2

Reference:

C. Xu, Li Ping, P. Wang, S. Chan, and X. Lin, “Decentralized Power Control for Random Access with Successive Interference Cancellation” to appear in IEEE JSAC.

3

Overview

Background and Motivation

Decentralized Power Control Two-User AWGN Channel

K-User AWGN Channel

K-User Fading Channel

Conclusions and Future Work

4

Overview



K-User AWGN Channel



5

Cognitive Radio with Multiple Primary Users

AP

secondary user 1

secondary user 2

primary user 2primary user 1

Consider a cognitive radio with multiple users. The secondary users can transmit only when the primary users are silent. The secondary users access the channel opportunistically whenever a spectrum hole is detected.

6

System Characteristics

1) There is usually no time to establish centralized control such as TDMA.

2) Thus random access is necessary.

3) If the spectrum holes are scarce, the secondary users may accumulate many un-transmitted packets. Thus is highly probable that each user has a packet to transmit whenever a spectrum hole is detected.

4) This is equivalent to the system with packet arrival rate 1.

7

Collision and Throughput

In random-access systems, a collision will occur if k users transmit their packets simultaneously.

Conventionally, the packets involved in a collision are assumed to be unrecoverable. The system throughout is limited by collision probability.

0

0.1

0.2

0.3

0.4

0.5

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

arrival rate of each user λ

pack

et th

roug

hput

T

When k≥1and k∞, the maximum throughput is 36.8% due to high collision probability

Performance of conventional slotted ALOHA

8

How to improve the random access performance?

9

Multi-Packet Reception (MPR)

In many situations, it is possible to recover some or all packets from a collision. This phenomenon is captured by the multi-packet reception (MPR) model. Early work on MPR model is focused on low-rate CDMA-type applications [1][2].

Both [1] and [2] allow multi-user detection (MUD). However, they are based on the traditional ALOHA with only one non-zero transmission power level.

The work in [3] allows multi-level transmission power but it is limited to single user detection (SUD) only.

[1] S. Ghez, S. Verdu, and S. Schwartz, “Stability properties of slotted ALOHA with multipacket reception capability,” IEEE Trans. Autom. Control, vol. 33, no. 7, pp. 640-649, Jul. 1988.[2] L. Tong and V. Naware, “Signal processing in random access,” IEEE Signal Process. Mag., vol. 21, no.5, pp. 29-39. Sep. 2004.[3] Y. Leung, “Mean power consumption of artificial power capture in wireless networks,” IEEE Trans. Commun., vol. 45, no. 8, pp. 957-964, Aug. 1997.

10

Performance of MPR in Fading Channel

[a] M. H. Ngo, V. Krishnamurthy, and L. Tong, “Optimal channel-aware ALOHA protocol for random access in WLANs with multi-packet reception and decentralized channel state information,” IEEE Trans. Signal Process., vol. 56, no. 6, pp. 2575-2588, Jun. 2008. [b] Y. Leung, “Mean power consumption of artificial power capture in wireless networks,” IEEE Trans. Commun., vol. 45, no. 8, pp. 957-964, Aug. 1997.

R = 1 bit/symbol

standard slotted ALOHA

channel multi-level aware [a] SUD [b]

0

0.5

1

1.5

number of active users K

syst

em th

roug

hput

2 4 6 8 10 12 14 16 18 20

The channel aware technique assume only a single non-zero transmission power level.

The multi-level SUD technique is limited to SUD only.

11

Serial Interference Cancellation

Multi-user detection (MUD) has the potential to solve the problem by serial interference cancellation (SIC) [4][5].

In this case, the signals of the users that have been correctly detected are subtracted from the received signal, and there is no interference to the others.

Assume that the signals of users 1, 2, …, k-1 have been correctly detected and subtracted from the received signal. Then the SINR of user k is given by

2

1

kk

ii k

eSINR

e

＋

from remaining interfering users.

[4] T. M. Cover and J. A. Thomas, Elements of Information Theory, New York: Wiley, 2006.[5] D. Tse and P. Viswanath, Fundamentals of Wireless Communications, Cambridge: Cambridge University Press, 2005.

12

Power Requirement for SIC

The following is an illustration of the power requirement for a two-user SIC system. We can recover both packets during an collision.

user 1 decoded first:

2

0

log 1e

RN

0N

2e

1e

user 1 user 2

user 2 decoded second:

1

2 0

log 1e

Re N

transmission power

13

Power Control and Feasible Region

Power control is crucial for MUD. We refer to the closure of power profiles that can support reliable transmissions of all users as feasible power region.

The following is an example of the feasible region for a two-user system with R ≥ 1 bit/symbol, ideal coding and SIC.

E1 = (2R – 1)N0

E2 = (2R – 1)∙(E1 + N0)

1

2

0

(21)(

)

R

e

eN

2

1

0

(21)

(

)

R

e

eN

2 0(2 1)Re N

10

(21)

Re

N

00

1

2

ee

1e

2e

1E 2E

1E

2E

e1=e 2 is the worst-case situation.

14

Feasible Region

When R>1, equal power line is not in the feasible region. If centralized power control is possible, we can allocate the two users with power inside the feasible region. In particular, the power pair (E1, E2) or (E2, E1) leads to

the minimum information theoretic sum-power.

However, how to allocate powers without centralized control?

1

2

0

(21)(

)

R

e

eN

2

1

0

(21)

(

)

R

e

eN

2 0(2 1)Re N

10

(21)

Re

N

00

1

2

ee

1e

2e

1E 2E

1E

2E

E1 = (2R – 1)N0

E2 = (2R – 1)∙(E1 + N0)

15

Objectives

We aim to improve the throughput of random-access systems by allowing MUD at the receiver and decentralize power control at the transmitters.

We will limit our discussions to the simple slotted ALOHA systems. It is expected that the results can be extended to systems with more sophisticated random access protocols such as CSMA.

We will discuss the decentralized power control problems in two-user AWGN channel, K-User AWGN channel, and K-User fading Channel.

16

Overview



K-User AWGN Channel



Problem Formulation

17

To improve the performance by optimizing the power distribution f(e).

1

2

0

(21)(

)

R

e

eN

2

1

0

(21)

(

)

R

e

eN

2 0(2 1)Re N

10

(21)

Re

N

00

1

2

ee

1e

2e

1E 2E

1E

2E

e1

e2

e

f(e)

e

f(e)

18

Discrete Power Levels

1

0, 0;

( ), 0.nn

nE

E n

We define a discrete set {En}

with 0 = E0 < E1 < … En < …,

where En is the minimum power

level that guarantees successful decoding of one user when the interference power level from the other user is En–1.

0E 1E 2E 3E0E

1E

2E

3E

4E

4E

1e

2e

1

2

ee

2

1

boun

dary

()

e

e

1

2

boundary

()

e

e

3 2( , )E E

4 3( , )E E2 3( , )E E

3 4( , )E E

2 1boundary e E

11

boun

dary

eE

1 2( , )E E

2 1( , )E E

19

Theorem 1

1

0, 0;

( ), 0.nn

nE

E n

The support of the optimal distribution is a subset of the discrete set {En}.

This theorem greatly simplifies the optimization problem.

0E 1E 2E 3E0E

1E

2E

3E

4E

4E

1e

2e

1

2

ee

2

1

boun

dary

()

e

e

1

2

boundary

()

e

e

2 1( , )E E

3 2( , )E E

4 3( , )E E

1 2( , )E E

2 3( , )E E

3 4( , )E E

2 1boundary e E

11

boun

dary

eE

20

Optimization Problem

Based on Theorem 1, the design problem can be formulated as follows.

20 1{ }

0

0

min imize

subject to 1

0 1, .

n

N

nnp

N

nn

N

n nn

n

p p

p

E p e

p n

Notes: (1) p0 is the probability that a user does not transmit.

(2) pn2 is the probability that both users using the same powers, and so

transmission will fail.

p0 0E 1E 2E 3E0E

1E

2E

3E

4E

4E

1e

2e

1

2

ee

pn2

21

Proof of Theorem 1

Any transmit power E' (E0, E1) is unnecessary since E1 is the minimum power for reliable transmission without interference.

1

2

0

(21)(

)

R

e

eN

2

1

0

(21)

(

)

R

e

eN

2 0(2 1)Re N

10

(21)

Re

N

00

1

2

ee

1e

2e

1E 2E

1E

2E

1

0, 0;

( ), 0.nn

nE

E n

22

Proof of Theorem 1 (Continued)

Provided that the probability of (E0, E1) is zero, any E' (E1, E2) is also unnecessary

since E2 is the minimum power for reliable transmission when the interfering packet

has power E1.

23

Proof of Theorem 1 (Continued)

The above reasoning is generalized to show that E' (En, En+1) is unnecessary for any n.

Example: Packet Throughput Comparison

24

Packet throughput T of a 2-user AWGN channel with ideal coding. R = 1 bit/symbol

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

arrival rate of each user λ

pack

et th

roug

hput

T

0 0.2 0.4 0.6 0.8 1

ML - SICconventional

25

Overview



K-User AWGN Channel



26

K-User Systems

To optimize the power distribution in a K-user system, we need to analyze the general K-user feasible region, which is a tedious issue and we will not pursuit it further. Instead, we will discuss a simple and sub-optimal solution.

We refer to a collision involving k (2 ≤ k ≤ K) users as a type-k collision. For the sub-optimal solution, we will only consider type-2 collisions.

In this sub-optimal solution, the system throughput is given by T = T1 + T2,

where T1 is the throughput related to transmissions without collisions and T2

the throughout related to type-2 collisions.

27

Throughput CalculationDenote by pn the probability of transmission power taking value En. We

calculated T1 and T2 as follows. For convenience, we assume full load for all

users. The discussions can be extended to the general loading case.

T1 is the throughput related to transmissions without collisions, calculated

by

T2 the throughout related to type-2 collisions, calculated by

Thus, we have

11 0 0(1 )KT Kp p

2 2 2 22 0 0 0

2 (1 )KK nn

T C p p p

1 2

1 2 2 20 0 0 0 0

(1 ) ( 1) (1 )K Knn

T T T

Kp p K K p p p

28

Throughput Optimization

1 2 2 20 0 0 0 0

(1 ) ( 1) (1 )K Knn

T Kp p K K p p p

1

2

1{ ,... }

01

1

minimize

subject to 1

0 1, .

N

N

nnp p

N

nn

N

n nn

n

p

p p

E p e

p n

To optimize the throughput of the proposed scheme, we discretize the value of p0, and solve the following optimization problem for each discretized p0.

Repeat:

29

Example: Performance Comparison

Performance in a 3-user system with ideal coding. R = 1 bit/symbol.

-6 -4 -2 0 2 4 6average power ē

syst

em th

roug

hput

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

full search

ML-SUD

ML-SIC by only considering type-2 collisions

30


Performance comparison among various schemes in AWGN channels with ideal coding and different K. R = 1 bit/symbol.


syst

em th

roug

hput

2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

conventionalML-SUDML-SIC

31

Overview



K-User AWGN Channel



32

Fading Channels

Consider a K-user system with fading. The received signal is given by

where the channel gains of all users, {gk}, are assumed to be independent and

identically distributed.

Assume that each user k knows its instantaneous channel gain gk. Our aim is

to optimize the conditional distributions {fT(eT|g)}, with eT the transmit

power, such that the system throughput is maximized.

The basic assumption above is that each user knows its own channel gain. This can be accomplished in different ways. A possible general solution is that the receiver will transmit a beacon signal, which will be used by the transmitters for channel estimation.

1

K

k k kky g e x

33

Channel-Aware ALOHA

As a reference, a channel-aware ALOHA scheme is proposed in [6] based on the following special form of fT(eT|g),

where ET is a predetermined power value and p(g) the transmission

probability when the channel gain is g.

In the following, we will show that the system throughput can be significantly enhanced by jointly optimizing the transmission power levels and the related transmission probabilities.

[6] M. H. Ngo, V. Krishnamurthy, and L. Tong, “Optimal channel-aware ALOHA protocol for random access in WLANs with multi-packet reception and decentralized channel state information,” IEEE Trans. Signal Process., vol. 56, no. 6, pp. 2575-2588, Jun. 2008.

( | ) ( ) ( ) (1 ( )) ( )T T T T Tf e g p g e E p g e

34

Support in Fading ChannelsGiven g, the support of the optimal conditional transmit power distribution fT(eT|g) is a subset of {E0/g, E1/g, E2/g, …..}.

Let (g) be the PDF of g. The received power distribution fR(eR) is given by

With {pn} available, the throughput can be calculated similarly as in AWGN

channels.

We next discuss the optimization of {pn(g)} based on the discretization of g.

0( | ) ( ) ( / )T T n T nn

f e g p g e E g

0( ) ( ),where ( ) ( )R R n R n n nn g

f e p e E p p g g dg

35

Throughput Optimization in Fading Channels

We discretize the value of g according to M+1 thresholds {g(m)|m = 1,…, M+1} and assume that the received power distributions when g [g(m), g(m+1)) are the same, i.e.,

pn(g) = pn(m) for g(m) ≤ g < g(m+1).

We optimize {pn(m)} to maximize the system throughput.

( )

2

1{ }

( )

( ) ( )

1

( )

0

( ) ( )

0

maximize

subject to 0 1, 0,1,..., , 1,...,

, 0,1,...,

1, 1,...,

/

mn

nnp

mn

M m mn nm

N mnn

N m mn nn

p

p n N m M

p p q n N

p m M

p E g

1

M

me

36


Performance comparison among various schemes in fading channels with ideal coding and different K. R = 1 bit/symbol.

0

0.5

1

1.5


syst

em th

roug

hput

2 4 6 8 10 12 14 16 18 20

CA-SUD in [6]ML-SIC: M = 10ML-SIC: M = 25

conventional

ML-SIC: M = 40

37

MUD for Practically Coded Systems

We have focused on ideal coding and assume successive interference cancellation (SIC) for MUD in the previous discussions.

We now consider practically coded systems, where IDMA is a simple scheme for MUD with relatively low receiver complexity.

hk

hK

user-1

ENC

user-k

user-K

. ... ..

1

K

k kk

h

r x

x1

xk

xK

1h1

ENC k

ENC K

chip-by-chip Processing

11-1

k

k-1

K

K-1

DEC

DEC

DEC

38

MUD Feasible Region with LDPC CodingThe following is the feasible region of a (3, 6) regular LDPC coded systems with iterative receiver for BER ≤ 10-5. Please note the followings.

1) When powers for both users are low, near equal powers are not feasible.

2) When powers for both users are high, equal powers are feasible.

1

2

3

4

5

6

7

8

01 2 3 4 5 6 7 80

power of user 1

pow

er o

f us

er 2

unfeasible

unfe

asib

le

feasible

feasible

39

Analysis of the Optimal Support

Q Qpoint Q ( , )E E

2 3( , )E E

3 2( , )E E

0

2 1( )e e

1 2( )e e

1e

2e

11

eE

2 1e E

1 2( , )E E

2 1( , )E E

1E2E 3E 4E 5E

1E

2E

3E

4E5E

0

In this case, the support of the optimal power distribution f(∙) is a subset of {E0,

E1, …, En, …, EQ} (for a feasible region with monotonically increasing

boundary function).

40

Example: LDPC Coding

Performance comparison among various schemes in fading channels with LDPC coding and different K. R = 1 bit/symbol

0

0.5

1

1.5


syst

em th

roug

hput

2 4 6 8 10 12 14 16 18 20

CA-SUD in [6]ML-SUD

conventional

ML-SIC

41

Overview



K-User AWGN Channel



42

Conclusions

We have developed a decentralized power control scheme for random access systems with MUD.

We proved that the support of the optimal power distribution f is discrete. Based on this finding, we designed f. Numerical results demonstrate that significant performance gain can be obtained by the proposed scheme.

We have limited our focus to slotted ALOHA-type random access schemes. It is expected that the results can be extended to systems with more sophisticated random access protocols such as CSMA.

43

Q & A

Thank you!

44

Introduction: Decentralized Power Control

We study the decentralized power control for ALOHA-type random access scheme with MUD.

2 0(2 1)Re N

00

1 2( , )E E

1 1( , )E E

1e

2e

A

B

1E 2E

1E

2E

Example: f1(2) = f2

(2) = f (2) = 0.5(e) + 0.5(e – E1)

(∙) is the Dirac delta function

45

Merging of Probabilities

For given f, we define a new distribution f[n] constructed as follows.

1

[ ] 0 ( )

( )( )

n

n l l nl

n

e E e Ef e

f e e E

1

where ( )l l

l E e Ef e de

46

Proof of Theorem 1

The distribution pair (f[n+1], f[n]) has a better performance than (f[n], f[n]), n.

a sample of f[n+1] can be equivalently obtained through the following steps:

• Step 1: Draw a power value e1 according to f[n];

• Step 2: If En < e1 < En+1, reduce e1 to En; otherwise, keep e1 unchanged.

Denote by white circles {Ai} power pairs drawn from (f[n], f[n]) while black

circles {Ai'} represent those after the power change in Step 2, which are

also samples drawn from (f [n+1], f[n]).

47

Example: SUD based OptimizationWe also consider optimizing the power distribution for systems with SUD (denoted by ML-SUD for multiple-level transmission and SUD).

The conventional slotted ALOHA can be regarded as a special case of ML-SUD, where the distribution is optimized over two levels E0 = 0 and E1 only.

power changing ∆ in dB

pack

et th

roug

hput

T

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-4 -2 0 2 4 6 8 10 12

conventional

ML-SUDML-SIC

48

Optimization Problem

Based on Theorem 1, the design problem can be formulated as follows.

20 1{ }

0

0

min imize

subject to 1

0 1, .

n

N

nnp

N

nn

N

n nn

n

p p

p

E p e

p n

Notes: (1) p0 is the probability that a user does not transmit.

(2) pn2 is the probability that both users using the same powers, and so

transmission will fail.

p0

0E 1E 2E 3E0E

1E

2E

3E

4E

4E

1e

2e

1

2

ee

pn2

49

Proof of Theorem 1

0E 1E 2E 3E0E

1E

2E

3E

4E

4E

1e

2e

5E

5E

1e

2e

[4]1( )f e

1

2

ee

2A

1A'1A

'2A

1e

e

[3]1( )f e

[3]2

()

fe

The power change leads to the following three possibilities.a) A1 fails while A1' succeeds. Such

events result in increased throughput; b) A2 succeeds while A2' fails. Such

events cannot happen as f[n](e2) = 0,

e2(En–1, En);

c) In all other situations, both power pairs fail or succeed simultaneously.

50

Example: Power Constraint

System throughput of a four-user Rayleigh fading channel with ideal coding under different power constraints. R = 1 bit/symbol.

-4 -3 -2 -1 0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

transmission power ET

syst

em th

roug

hput

CA-SUDML-SIC

51

Example: Power Distributions

The received power distributions of the ML-SIC scheme in the previous figure.

received power levels

prob

abil

ity

received power distribution at ET = 0 dB

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

received power distribution at ET = 3 dB

E0 E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12

52

K-Users Slotted ALOHA Systems

Denote by pi the probability of transmission power taking value Ei. We

calculated T1 and T2 as follows.

T1 is the throughput related to transmissions without collisions.

T2 the throughout related to type-2 collisions.

1 11 0 01

10 0

(1 ) (1 )

(1 ) (1 )

K k k K k kK kk

K

T C C p p

K p p

2 2 2 22 0 02 0

2 2 2 20 0 0

2 (1 ) (1 )

( 1) (1 ) (1 )

K k k K k kK k ik i

Kii

T C C p p p

K K p p p

53

Optimization for Type-2 Collisions

1 2

10 0

2 2 2 20 0 0

(1 ) (1 )

( 1) (1 ) (1 )

K

Kii

T T T

K p p

K K p p p

2

1

Q 0

Q Q 0

Q

minimize

subjectoto 1

0 1,

0 1.

ii

ii

i ii

i

p

p p

E p E p e

p i

p

To optimize the throughput of the proposed scheme, we discretize the value of p0, and solve the following optimization problem for each discretized p0.

Repeat:

54

MUD Feasible Region with LDPC Coding

1

2

3

4

5

6

7

8

01 2 3 4 5 6 7 80

2

1

1

2

innerbound

outer bound

1

2

1power of user 1 e

2po

wer

of

user

2 e

11

eE

2 1e E

outer bound

inner bound

The feasible region of a (3, 6) regular LDPC coded systems with iterative receiver.

55

Example: LDPC Coding

Performance comparison among various schemes in fading channels with LDPC coding and iterative receiver.

0

0.5

1

1.5


syst

em th

roug

hput

2 4 6 8 10 12 14 16 18 20

CA-SUDML-SUDOuterBound

conventional

InnerBound

56

Power Requirement for SIC

With SIC, its possible to recover two or more packets involved in a collision even for high-rate transmissions.

For example, if the SINR threshold for successful detection is given by SINRth, the following transmit power profile can lead to successful detection

of all users for MUD.

21 2

2

1

2

th ii

k th ii k

K th

e SINR e

e SINR e

e SINR

decoded first

decoded last

…

Documents

Decentralized Power Control for Random Access with Multi-User Detection