Channel Coding for Network Communication:

An Information Theoretic Perspective

Zheng Wang

Prof. J. Rockey Luo

Prof. Louis Scharf

Prof. Edwin Chong

Prof. Anton Betten

Advisor:

Committee:

1

Ph.D. Final Defense

05/18/2011

2

Channel Coding

Transmitter Receiver

1, ,w W

Encoder Decoder

ˆ 1, ,w W

1 NX X 1 NY Y

Channel

Noise

|Y XPNX NY

[Definitions] Code Rate: ; log /R W N

Error Probability: ( ) ˆmax Pr |N

ew

P w w w

[Existing Results]

Channel coding theorem:

max ( ; )XP

C I X Y

Channel capacity: [C. Shannon ‘48]

[C. Shannon ‘48] [T. Cover and J. Thomas ‘05]

For any rate , as , . R C N ( ) 0N

eP

For a discrete-time memoryless channel,

Mutual Information

XP Input distribution

( ; )I

Figure. Point-to-point communication system

3

Network Communication [Network systems] Multiple transmitters & receivers interact with each

other to achieve joint or individual communication objectives.

Multiple access channel Broadcast Channel Relay Channel Interference Channel

Objective: Extending information theory toward network communication

scenarios; developing new channel coding results for non-classical

communication models.

[Key assumptions]

Joint determination of key parameters

Infinite communication duration

Stationary channel Information

theory

Network

Communication [Network characteristics]

Bursty traffic

Dynamic users and network activities

4

Outlines

1. Motivations

2. Review of the previous work (before 05/12/2010)

1) Error performance of linear-time complexity block codes

2) Concatenated fountain codes

3. Coding theorems for random access communication

1) Finite-length analysis

2) Random access communication over compound channels

3) Individual user decoding

4. Summary

1 journal published [Comm. Letters ‘09]

1 journal submitted [TIT]；1 conf. paper published [ISIT ‘09]

1 journal submitted [TIT]；1 conf. paper published [ISIT ‘10]

1 journal to be submitted [TIT]；1 conf. paper accepted [ISIT ‘11]

5

Error Exponent Error probability can decay exponentially in channel codeword

length, i.e. . [A. Feinstein ‘54] ( ) ( )N NE R

eP e

[Definition] ( )log

( ) limN

e

N

PE R

N Error Exponent:

Tradeoff between communication rate and error performance

R

E

Upper bounds: [C. Shannon, R. Gallager, and E. Berlekamp ‘67]

Lower bounds: [P. Elias ‘55] [R. Fano ‘61] [R. Gallager 65]

[Existing results]

6

Achievable Error Exponents [Existing results]

Gallager’s exponent:[R. Gallager ‘65]

, 1

0 crit

0 crit,0 1

max ( , ) 0

( , ) max (1, )

max ( , )

X

X

X

x X xP

X X xP

XP

R E P R R

E R P R E P R R R

R E p R R C

11

1

0 |( , ) log ( ) ( | )X X Y X

Y X

E P P X P Y X

, ,1

( ) max (1 ) ,X o

c o XR

P r oC

RE R r E P

r

1

( )

0, ,1

( ) max ,( , )

o

X o

R

r

RP r o L X

C

R dxE R R

r E x P

( , ) max ( , )

X

X L XP

E x P E x P

Forney’s exponent:[G. Forney ‘66]

Blokh-Zyablov exponent:[E. Blokh and V. Zyablov ‘82]

Gallager’s exponent

Forney's exponent

Blokh-Zyablov exponent

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

rate R err

or

exponent

0.4

0.45

,

( , ) log ( ) ( )x X X X

X X

E P P X P X '

'

1/

| | ( | ) ( | )Y X Y X

y

P Y X P Y X

'

Figure. Comparison of error exponent bounds, BSC with cross prob. 0.1

Exponential complexity

Polynomial complexity

7

Linear Complexity Codes

LDPC codes have linear coding

complexity, but with poor error

performance

Approaching Forney’s exponent

with linear complexity, but only

for binary symmetric channels [V.

Guruswami and P. Indyk ‘05]

[Existing results]

[Contribution]

Achieved Forney’s and Blokh-Zyablov exponents with linear coding

complexity over general discrete-time memoryless channels.

Z. Wang, J. Luo, "Approaching Blokh-Zyablov Error Exponent with Linear-Time

Encodable/Decodable Codes," IEEE Communications Letters, Vol. 13, No. 6, pp.

438-440, June 2009.

Gallager’s exponent

Forney's exponent

Blokh-Zyablov exponent

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

rate R

err

or

exponent

0.4

0.45 Figure. Comparison of error exponent bounds,

BSC with cross prob. 0.1

Linear complexity for BSC

8

Fountain Communication w

Encoder 1 2, ,x x

Erasure Channel Decoder 1 2, ,i ix x w

Notification of comm. termination

|Y XP

Channel Erasure 1 2, ,y y1 2, ,i iy y

LT codes [M. Luby ‘02]; Raptor codes [A. Shokrollahi ‘06]

Random fountain codes [S. Shamai, I. Telatar and S. Verdu ‘07]

[Existing results]

Erasure channels

General DMC

Zero error exponent

Positive error exponent

(Almost) Linear complexity

Exponential complexity

[Contribution] Proposed the concatenated fountain codes which can achieve the near-

optimal fountain error exponent with linear coding complexity. The

closed form of the achievable fountain error exponent was derived.

Z. Wang, J. Luo, "Concatenated Fountain Codes," IEEE International

Symposium on Information Theory, Seoul, Korea, June 2009.

Z. Wang, J. Luo, "Fountain Communication using Concatenated Codes,"

submitted to IEEE Trans. on Information Theory.

9

Outlines

1. Motivations

2. Review of the previous work (before 05/12/2010)

1) Error performance of linear-time complexity block codes

2) Concatenated fountain codes

3. Coding theorems for random access communication

1) Finite-length analysis

2) Random access communication over compound channels

3) Individual user decoding

4. Summary

1 journal published [Comm. Letters ‘09]

1 journal submitted [TIT]；1 conf. paper published [ISIT ‘09]

1 journal submitted [TIT]；1 conf. paper published [ISIT ‘10]

1 journal to be submitted [TIT]；1 conf. paper accepted [ISIT ‘11]

10

Multiple Access Comm.

Classical multiple access Random multiple access

Collision

Backlogged traffic (message

is always available) Joint rates determination,

shared with the receiver

Receiver always decodes

Channel information is known at both the transmitters and the receiver

Noisy channel (coding is needed)

Infinite communication

duration

Bursty traffic (message may

not be available) Independent rate determination,

unknown to the receiver

Need for collision detection

Finite communication duration

K K( , )K Kw r

2 2

11

Transmission Model [J. Luo and A. Ephremides, ‘09]

Independent rate determination

11 1( , )w r

Bursty traffic Channel

1| KY X XP

2 2( , )w r

Random Coding Scheme

1

Figure. Transmission model

[Assumptions]

Time-slotted random access system: –slot/codeword length;

coding within one time slot;

Independent rate determination: user choose a communication

rate , unknown to other users and unknown to the receiver;

Discrete-time memoryless channel, known both at the transmitters

and at the receiver;

N

kr

k

User has communication rate options,

i.e. .

Codebook contains codes. The code

contains codewords with length ;

All codeword symbols are independently

generated, according to input distribution

, which is a function of rate ;

Codebook is randomly generated, known

at the receiver. 12

Random Coding Scheme [J. Luo and A. Ephremides, ‘09]

1, , ,k k kh kMr r r r

M

thhMkhNr

e N

k

| khX rP

Figure. Transmission model of user 1, ,k K

Independent rate determination

kRandom Coding

Scheme

k Channel

1| KY X XP

( , )k kw r ( , )k kw rx

[Assumptions]

khr

1 1

(1,1) (1, )

(1,1) (1, )

(2,1) (2, )

( ,1) ( , )

( ,1) ( , )

Nr Nrkh kh

Nr NrkM kM

k k

N

kh kh

N

kh kh

N

kh kh

e e N

kM kM

e e N

x x

x x

x x

x x

x x

Codebook of user k

Code ( ) khr

Codeword

13

Receiving Model [J. Luo and A. Ephremides, ‘09]

Channel

1| KY X XP

Possible for reliable message

recovery?

Figure. Receiving model

yDecode

Report Collision

ˆ ˆ( , )w r

collision

Pre-determine a communication rate region ,

which is a set of rate vectors;

The receiver decodes if the channel output is

jointly typical with at least one codeword in

this region; otherwise, it reports a collision.

R

1

k

K

r

r

r

r =

R

(1,1) (1, )

(2,1) (2, )

( ,1) ( , )Nr Nrh h

kh kh

N

kh kh

N

kh kh

e e N

x x

x x

x x

Figure. Communication rate region R

Yes

No

[Assumptions]

14

Receiving Model [J. Luo and A. Ephremides, ‘09]

Channel

1| KY X XP

Possible for reliable message

recovery?

Figure. Receiving model

yDecode

Report Collision

ˆ ˆ( , )w r

collision

Yes

No

Figure. Communication rate region R

[Definitions]

Decoding error probability:

Collision miss detection probability:

Achievable communication rate region :

as , for , ; for

, .

( ) ˆ ˆ( , ) Pr ( , ) ( , )|( , )N

dP w r w r w r w r

( ) ( , ) 1 Pr collision|( , ) N

cP w r w r

r R( ) ( , ) 0N

cP w r r R

( ) ( , ) 0N

dP w rN

( , ), w r r R

( , ), w r r R

R

1

k

K

r

r

r

r =

RCollision miss

detection prob. →0

Decoding

error prob.→0

R

Additional element to classical definition

15

Achievable Rate Region [J. Luo and A. Ephremides, ‘09]

[Theorem] : Assume random coding with input distribution for user

at rate . The following rate region is achievable,

where is the mutual information calculated using input

distributions .

|kX rP

; |S SI X Y X

r

1 1| |, ,K KX r X rP P

k

1, , , either 0 or ; |c i S i S Si SR S K r r I X Y X

rr

r

The above achievable communication rate region equals Shannon’s

information rate region without a convex hull operation.

[Example 1]

Identical input distribution

[Example 2]

Symbol collision channel

2r

1r

cR

2r

1r

cR

Shannon’s information

rate region

Shannon’s information rate

region without a convex

hull operation.

16

Finite-Length Analysis

Bursty traffic (message may not

be available)

Finite package/codeword length

[RAC assumptions]

( )lim ( ) 0,N

dN

P

,w r r R

( )lim ( ) 0,N

cN

P

,w r r R

[Asymptotic results]

[Contribution] Obtained the achievable system error probability bound

with finite codeword length.

[Definitions]

System error probability: ( ) ( ) ( )

( , ), ( , ),max max ( , ) max ( , )N N N

es d cP P P

w r r w r r

w r w r，R R

Z. Wang, J. Luo, "Error Performance of Channel Coding in Random Access

Communication," submitted to IEEE Trans. on Information Theory.

Decoding error prob.

Collision miss detection prob.

17

Decoding Scheme

Receiver decodes and outputs the estimated pair

, if both of the following conditions are satisfied:

( , )w r

r R

[Decoding Criteria]

1 11: log ( | ) log ( | ), C p p

N N( , ) ( , )y x w r y x w r

12 : log ( | ) ( )C p

N ( , )

ry x w r y

for all , , R ( , ) ( , )x w r x w r r r

Collision

y ( , )x w r

yy

( )r

y

y ( , )x w r( )

ry( )N

dP

( )N

cP

Typical Sequence Decoding

Maximum Likelihood Decoding

Typicality threshold function

[Theorem]: For -user random access communication over discrete

time memoryless channel . Assume finite codeword length , and

random coding with input distribution for all with ，

. Let be the operation region. There exists a decoding

algorithm whose system error probability is upper bounded by

,

where and will be given later.

|YP X N

|X rP r 1, ,k k kMr r r

1, ,k K R

| |

,

( ) 1, ,| |s ,

| |,

1, , ,

exp ( , , , )

max ,max exp ( , , , )max

max max exp ( , , , )

S S

S S

S SS S

m

N S Kie

i

S K

NE S

NE SP

NE S

X r X r

r r =r

r

X r X r'r' r' r

X r X r'r r' r' =r

r r r

r P P

r P P

r P P

R

R

R

R RR

K

| |( , , , )iE S X r X r'r P P| |( , , , )mE S X r X rr P P

18

Error Probability Bound

( ) ( ) ( )

( , ), ( , ),max max ( , ) max ( , )N N N

es d cP P P

w r r w r r

w r w r，R R

[Definition]

19

Error Exponent Bound

[Definitions]

Decoding error exponent:

Collision miss detection exponent:

System error exponent:

( )

(

1min lim log ( )N

d dN

E PN

, ),

,w r r

w rR

( )

( ),

1min lim log ( )N

c cN

E PN

,

,w r r

w rR

Z. Wang, J. Luo, "Achievable Error Exponent of Channel Coding in Random

Access Communication," IEEE International Symposium on Information

Theory, Austin, TX, June 2010.

[Contribution] Obtained the achievable system error exponent bound.

( )1lim log min ,N

s es d cN

E P E EN

[Corollary]: For -user random access communication over discrete-time

memoryless channel . Given the input distribution for all with

, . Let be the operation region. There exists

a decoding algorithm with system error exponent bounded by

|YP X X|rP

1, ,k k kMr r r

r

1, ,k K

K

20

Error Exponent Bound

1, , , , 1, , , ,min min min ( , , , ), min min ( , , , )

S S S Ss m i

S K S KE E S E S

X|r X|r X|r X|r

r r r r r r r rr P P r P P

R R R

|0 1 0 1

1

| | | |

( , , , ) max max log ( )

( ) ( | ) ( ) ( | )

k

S

k k

S S

m k X r ks

k S Y k S

s

s

X r k Y X r k Y

k S k S

E S r P X

P X P Y P X P Y

X X

X|r X |r

X

X X

r P P

X X

|0 1 0 1

1

| | | |

( , , , ) max max log ( )

( ) ( | ) ( ) ( | )

k

S

k k

S S

i k X r ks

k S Y k S

s ss

s

X r k Y X r k Y

k S k S

E S r P X

P X P Y P X P Y

X X

X|r X |r

X

X X

r P P

X X

R

21

RAC over Compound Channel

Channel

Channel

|YPX

Receiver （Channel estimation）

[Assumption]

Channel information is known at both transmitters and the receiver.

Bursty traffic with fractional channel access

[Contribution] Derived the achievable system error probability bound

for random access system where channel information is not perfectly

known at the receiver.

Z. Wang, J. Luo, “Coding Theorems for Random Access Communication over

Compound Channels," IEEE International Symposium on Information Theory,

Saint Petersburg, Russia, Aug. 2011.

Continuous message transmission

Receiver （Channel estimation）

Channel

Transmitter 1

22

Compound Channel [Csiszar ‘81]

conditional probabilities with cardinality .

[Definition] A compound discrete-time memoryless channel consists of

a family of discrete-time memoryless channels, characterized by a set of

(1) ( )

| |, , H

Y YP PX X H

K( , )K Kw r

2

1 1( , )w r

2 2( , )w ry

Decode

Report Collision

ˆ ˆ( , )w r

collisionChannel

(1) ( )

| |, , H

Y YP PX X

Figure. Random access communication over compound channels

In each time slot, one channel realization is randomly chosen from

, and remains static within each slot duration.

Transmitters and the receiver know the channel compound set, but

do not know the actual channel realizations.

(1) ( )

| |, , H

Y YP PX X

[Assumptions]

Yes

No

( )

| | |ˆ ˆ( , , ) 1 Pr collision | ( , , ) Pr ( , ) ( , ) | ( , , )

N

c Y Y YP X X Xw r P w r P w r w r w r P

23

Error events

( )

| | | |ˆ ˆ( , , ) Pr ( , ) ( , ) | ( , , ) ( , , ), ( , )N

d Y Y Y YP X X X Xw r P w r w r w r P w r P r P R

| | | |

( ) ( ) ( )

| |( , , ),( , ) ( , , ),( , )

max max ( , , ), max ( , , )Y Y Y Y

N N N

es d Y c YP P P

X X X X

X Xw r P r P w r P r P

w r P w r PR R

[Definitions]

Decoding error prob.:

Collision miss detection prob.:

System error prob.:

,

For all , , we have 1, , , ; |Y

Y i S Si SS K r I X Y X

|X|X r P

r P R

[Operation region]

| | ( , , ), ( , )Y Y X Xw r P r P R Output correct estimation instead of collision

function of |YP X

|YP X

R

|( , )YP Xr

( , , )w r

24

Decoding for Compound Channel

Receiver decodes and outputs a message and rate pair if the

following condition is satisfied:

( , )w r

[Decoding Criteria]:

|

| |

| | |

|

|

| ( ,

1 1log Pr | , log Pr | , ,

for all , , , and , , ,

( ,

with , 1log Pr | , ( )

Y

Y Y

Y Y Y

Y

Y

Y P

P PN N

P P P

P

PP

N

( , ) ( , )

( , ) )

( , ) ( , ) ( , ) ( , ) ( , )

)

( , )

X

w r X w r X

X X X y

X

y X

w r X r

y x y x

w r w r w r w r w r

r

w ry x y

R

RR

|, , YP( )Xw r

y

|YP X

( , , )w r

yR

25

Error Probability Bound

[Theorem]: Consider -user multiple random access communication

over compound discrete-time memoryless channel . Let

K

(1) ( )

| |, , H

Y YP PX X

be the input distribution for all users and all rates

, and be the operation region. Assume finite codeword

length . There exists a decoding algorithm, whose system error

probability is upper bounded by

where and are given in the next

page.

|PX r

1, ,k K

1, ,k k kMr r r RN

|

|

|

|||

, ,

,( ) 1, ,

|, ,

|, ,,

, ,

exp ( , , , )

max ,

max exp ( , ', , ' )max

max max exp ( , ', , ' )

Y S S

Y

Y S S

Y S SYY S S

m Y Y

N S Ki Y Ye

i Y Y

NE S

NE SP

NE S

X

X

X

XXX

|X |X

r P r =r

r P

|X Xr' P' r' =r

|X Xr' P' r' =rr P

r P r r

r P P

r P P

r P P

R

R

R

RRR 1, ,S K

|( , ', , ' )i Y YE S |X Xr P P( , , , )m Y YE S|X |X

r P P

26

Error Exponent Bound [Corollary]: Consider -user multiple random access communication

over compound discrete-time memoryless channel given in the previous

theorem. The system error exponent is bounded by

where and are given by

K

|0 1 0 1

1

| | | |

( , , , ) max max log ( )

( ) ( | ) ( ) ( | )

k

S

k k

S S

m Y Y k X r ks

k S Y k S

s

s

X r k Y X r k Y

k S k S

E S r P X

P X P Y P X P Y

X X

|X |X

X

X X

r P P

X X

| |0 1 0 1

1

| | | |

( , , , ) max max log ( )

( ) ( | ) ( ) ( | )

k

S

k k

S S

i Y Y k X r ks

k S Y k S

s ss

s

X r k Y X r k Y

k S k S

E S r P X

P X P Y P X Y

X

|X X

X

X

X X

r P P

X P X

|( , , , )i Y YE S|X X

r P P( , , , )m Y YE S|X |X

r P P

| | | |( , ),( , ) ( , ) ,( , )

min min ( , , , ), min ( , , , )Y Y Y Y

s m Y Y i Y YP P P P

E E S E S

X X X X

|X |X |X |Xr r r r

r P P r P PR R R

27

Individual User Decoding K( , )K Kw r

2

1 1( , )w r

( , )k kw ry

Decode

Report Collision

collisionChannel

Yes

No |YPX

Figure. Random access communication with individual user decoding

[Assumption]

The receiver is only interested in decoding for user ;

The channel is known both at the transmitters and the receiver. 1, ,k K

1, , , , either 0,

or , , such that,

( ; | )

k

k

i r SSi S

S K k S r

S S k S

r I X Y X

rR

[Theorem J. Luo and A. Ephremides, ’09]

Consider the random access system where the receiver is only interested in

decoding for user . The following rate region is achievable k

Transmitter 1

ˆ ˆ( , )w rˆ ˆ( , )k kw r

28

Two-User case

1r

2r

1 2( ; | )I X Y X

2 1( ; | )I X Y X

1( ; )I X Y

[Example] 2-user random access system; the receiver is only interested in

decoding for user 1. The input distribution is the same for all rates.

[Decoding]

Figure. Achievable rate region 1R

Decoder 1: Jointly decode for both users.

Decoder 2: Decode for user 1 while regarding user 2 as interference.

Decoding for user 1, while regarding user

2 as interference

Joint decoding for both users

Receiver outputs an estimate for user 1 only if

both decoders give the same estimate for user 1;

one decoder outputs an estimate, while the other reports a collision.

Partitioning

29

-User case K

: 1, , ,

'

\

, ' , , ' 1, , , , ',

( ; | ), ,

D

D D K k D

D D

i DD D Di D

D D D D K k D D

r I Y D D

r X X r

R = R

R RR

[Lemma]: Any operation region contained inside the achievable

rate region can be partitioned into the following sub-regions kR R

kR

[ -Decoder]

Given a user subset , and a rate region , the decoder is

only interested in decoding for users in , while regards the signals from

users not in as interference.

1, ,D K DRD

D

, DD R

DR

[ , ]D Dr = r r D D

Messages Interference

Dr D Dw ,r

30

-Decoder , DD R

Receiver decodes and outputs a message and rate pair for users in

together with an estimate for user not in if the following condition is

satisfied:

D D( , )w r

[Decoding Criteria]

D D

( ,

1 1log Pr | , log Pr | , ,

for all , , , and , , ,

1with , , log Pr | , ( )

D D D D

D D

D D

D D D D D D D D D DD D D

D D D DD D

N N

N

( , ) ( , )

( , ) )

( , ) ( , ) ( , ) ( , ) ( , )

( , )

y

y

w r w r

D

D w r r

y x r y x r

w r r w r w r w r r w r r

w r r r y x r y

R

R R[Definitions]

Decoding error prob.:

Collision miss detection prob.:

System error prob.:

( ) ˆ ˆ( , , ) Pr ( , ) ( , ) | ( , , ) ( , , ),N

d D D D D D D D D D D DD D DP w r r w r w r w r r w r r r R

( ) ( , , ) 1 Pr collision | ( , , )

ˆ ˆ Pr ( , ) ( , ) | ( , , ) ( , , ),

N

c D D D DD D

D D D D D D D D DD D

P

w r r w r r

w r w r w r r w r r r R

( ) ( ) ( )

( , , ), ( , , ),( , ) max max ( , , ), max ( , , )

D D D D D DD D

N N N

es D d D D c D DD DP D P P

w r r r w r r rw r r w r r

R RR

31

Error Performance

,

( )

, '

, ',

exp ( , , )

max ,

max exp ( , , ')( , ) max

max max exp ( , , ')

D D D

D

D D D

D D D DD D D

mD

N D DiDes D

iD

D D

NE D

NE DP D

NE D

r r r

r

r' r r

r r' r rr r r

r r

r r

r r

R

R

R

R RR

R

[Lemma]: Consider a -user random multiple access communication

system over a discrete-time memoryless channel with an -

decoder. The system error probability is bounded by

K

|YP X , DD R

|0 1 0 1

\

( , ) max max log ( )k

D

mD k X r ks

Yk D D k D

E D r P X

X

r,r

\ \

1

| |

\ \

( ) ( | , ) ( ) ( | , )k k

D D D D

s

s

X r k D X r k DD Dk D D k D D

P X P Y P X P Y

X X

X r X r

|0 1 0 1

\

( , ') max max log ( )k

D

iD k X r ks

Yk D D k D

E D r P X

X

r,r

\ \

1

| | '

\ \

( ) ( | , ) ( ) ( | , ' )k k

D D D D

s ss

s

X r k D X r k DD Dk D D k D D

P X P Y P X P Y

X X

X r X r

32

Individual User Decoding

[Theorem] Consider a -user random multiple access system over a discrete-

time memoryless channel , with the receiver only interested in recovering

the message from user . Assume the receiver chooses an operation region

contained inside the achievable rate region. Let be an arbitrary

partitioning of the operation region. System error probability of the single-

user decoder is upper-bounded by

where is the system error probability of the -decoder.

K

|YP X

k

kR R

( ) ( )

: 1, , ,

min ( , )N N

es es D

D D K k D

P P D

R

( ) ( , )N

es DP D R ( , )DD R

[Single-user Decoder]

The receiver outputs an estimate for user if all the -decoders

that do not report collisions have the same estimation for user .

( , )DD Rkk

[Definitions] Decoding error prob.:

Collision miss detection prob.:

System error prob.:

( ) ˆ ˆ( , ) Pr ( , ) ( , ) | ( , ) ( , ),N

d k k k k k k kP w w r w r w w R r r r r

( ) ˆ ˆ( , ) 1 Pr collision | ( , ) Pr ( , ) ( , ) | ( , ) ( , ),N

c k k k k k k k kP w w w r w r w w R r r r r r

( ) ( ) ( )

( , ), ( , ),max max ( , ), max ( , )N N N

es d k c kw R w R

P P w P w

r r r r

r r

33

Summary [Objective] Extending information theory to network communication;

developing new channel coding results for non-classical

communication models.

[Contributions]

Fountain communication (before 05/12/2010)

1. Proved the achievability of the best known constructive error

exponent with linear coding complexity for DMC;

2. Proposed the concatenated fountain codes which can achieve a

positive fountain error exponent for any rate within fountain

capacity, with linear coding complexity for DMC;

Random multiple access communication (after 05/12/2010)

1. Finite-length error performance analysis of the new channel

coding approach proposed in [J. Luo and A. Ephremides ‘09];

2. Random access communication over compound channels;

3. Error performance for individual user decoding.

34

Publications [Journal papers] Z. Wang, J. Luo, "Approaching Blokh-Zyablov Error Exponent with Linear-Time

Encodable/Decodable Codes," IEEE Communications Letters, Vol. 13, No. 6, pp. 438-440, June

2009.

Z. Wang, J. Luo, "Fountain Communication using Concatenated Codes," submitted to IEEE Trans.

on Information Theory.

Z. Wang, J. Luo, "Error Performance of Channel Coding in Random Access Communication,"

submitted to IEEE Trans. on Information Theory.

Z. Wang, J. Luo, “Channel Coding for Random Multiple Access Communication over Compound

Channel," to be submitted to IEEE Trans. on Information Theory.

[Conference papers] Z. Wang, J. Luo, “Coding Theorems for Random Access Communication over Compound Channel,"

IEEE International Symposium on Information Theory, Saint Petersburg, Russia, Aug. 2011.

Z. Wang, J. Luo, "Achievable Error Exponent of Channel Coding in Random Access

Communication," IEEE International Symposium on Information Theory, Austin, TX, June 2010.

Z. Wang, J. Luo, "Concatenated Fountain Codes," IEEE International Symposium on Information

Theory, Seoul, Korea, June 2009.

35

Acknowledgement

My deepest gratitude to my advisor Dr. J. Rockey Luo, and my

committee members Dr. Louis L. Scharf, Dr. Edwin K. P. Chong and

Dr. Anton Betten.

Great thanks to all, without mentioning their names, who have

supported and helped me during the past several years.

Sincere thanks to my parents, and the other relatives and friends in

China.

Special thanks to my fiancé Sean Zhang.