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.
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