Lihua Weng Dept. of EECS, Univ. of Michigan

Lihua Weng

Dept. of EECS, Univ. of Michigan

Error Exponent Regions for

Multi-User Channels

Motivation: Downlink Communication

User 1: FTPhigh reliability

Pe = 10 -6

BaseStation

User 2: Multimedialow reliability

Pe=10-3

Motivation (cont.)

• Unequal error protection (ad hoc methods without systematic approach)

• Can reliability be treated as another resource (like power, bandwidth) that can be allocated to different users?

• Formulate this idea as an information theory problem, and study its fundamental limits.

Outline

• Background: Error Exponent

• Error Exponent Region (EER)

• Gaussian Broadcast Channel (GBC)

• Conjectured GBC EER Outer Bound

• Conclusion

• Error exponent: for a codeword of length N, the smallest possible probability of codeword error behaves as

where E(R) is the error exponent (as a function of the transmission rate R)- DMC (Elias55;Fano61;Gallager65; Shannon67)- AWGN (Shannon 59; Gallager 65)

Channel Capacity & Error Exponent: Single-User Channel

• Channel capacity: highest data rate for arbitrarily low probability of codeword error with long codewords

)( eP RENe

Error Exponent

Rcrit C0

random coding

sphere packing

Rcrit C0

random coding

sphere packing

expurgated

minimum distance

Rcrit C0

random coding

sphere packing

expurgated

minimum distance

straight line

Rcrit C0

random coding

sphere packing

expurgated

• We have a tradeoff between error exponent and rate

Capacity: Multi-User Channel

• Channel capacity region: all possible transmission rate vectors (R1,R2) for arbitrarily low probability of system error with long codewords

• Probability of system error: any user’s codeword is decoded in error

capacityregion

capacityboundary

Error Exponent: Multi-User Channel

• Error Exponent: rate of exponential decay of the smallest probability of system error

• For a codeword of length N, the probability of system error behaves as

- DMMAC/Gaussian MAC (Gallager 85)

- MIMO Fading MAC at high SNR (Zheng&Tse 03)

),( syse,

21P RRENe

)),,(log(Plim)R,E(R 21syse,

Single Error Exponent: Drawback

• Multi-user channel – single error exponent- Different applications (FTP/multimedia)

• Our solution

- Consider a probability of error for each user, which implies multiple error exponents, one for each user.

Outline

• Conclusion

Multiple Error Exponents: Tradeoff 1

• We have tradeoff between error exponents (E1,E2) and rates (R1,R2) as in the single-user channel.

0 rate 1

rate 2

Large error exponents

0 rate 1

rate 2

Small error exponents

0 rate 1

rate 2

Multiple Error Exponents: Tradeoff 2

• Fix an operating point (R1,R2), which point from the capacity boundary can we back off to reach A?

• B A : E1 < E2

rate 1

rate 2

• B A : E1 < E2

D A : E1 > E2

rate 1

rate 2

• B A : E1 < E2

D A : E1 > E2

• Given a fixed (R1,R2), one can potentially tradeoff E1 with E2

rate 1

rate 2

Error Exponent Region (EER)

• Definition: Given (R1,R2), error exponent region is the set of all achievable error exponent pairs (E1,E2)

rate 1

rate 2

R2 EER(R1,R2)

• Careful!!!- Channel capacity region: one for a given channel

- EER: numerous, i.e., one for each rate pair (R1,R2)

Outline

• Gaussian Broadcast Channel (GBC)- EER Inner Bound

• Single-Code Encoding• Superposition Encoding

- EER Outer Bound

• Conclusion

Gaussian Broadcast Channel

σ}E{Z

P}E{X2

Single-Code Encoding

CB = {Ck | k=(i-1)*M2+j; i = 1, … ,M1; j = 1, … , M2}

)},(),,(max{

21 MMC

Superposition Encoding

PPPPP ))(1()( 22122111

jiN CCX ,2,1

CB2CB1

1,1 MC

2,2 MC

P12P21

E{|C1,i(k)|2} E{|C2,j(k)|2}

N N)1(

1,1 MC

2,2 MC

N N)1(

Individual and Joint ML Decoding

• Individual ML Decoding (optimal)

1i ,1,2,12,22

1j ,2,2,11,11

)()|( maxarg)|( maxargj :2user

)()|( maxarg)|( maxargi : 1user M

CPCCYPCYP

• Joint ML Decoding

j)|( maxarg)j,i( :2user

i)|( maxarg)j,i( : 1user

,2,12),(

,2,11),(

:1User

,2',1,2,1

- Type 1 error: one user’s own message decoded erroneously, but the other user’s message decoded correctly

Joint ML Decoding (cont.)

• Joint ML Decoding- Type 3 error: both users’ messages are

decoded erroneously

),,,,,(

:1User

',2',1,2,1

• Achievable Error Exponents

)],,,,,(),,,,(min[

112132

PPPPRRE

• Naïve Single-user decoding: Decode one user’s signal by regarding the other user’s signal as noise

Naïve Single-User Decoding

222211

122121

Special Case 1: Uniform Superposition

CB2CB1

E{|C1,i(k)|2} E{|C2,j(k)|2}

1,1 MC

2,2 MC

Special Case 2: On-Off Superposition (Time-Sharing)

E{|C1,i(k)|2}

E{|C2,j(k)|2}

CB1 CB2

1,1 MC

2,2 MC

N N)1( N N)1(

0 1 2 3 4 5 6 7 8

x 10-3

Superposition

EER Inner Bound

R1 = 1

R2 = 0.1

SNR1 = 10

SNR2 = 5

0 1 2 3 4 5 6 7 8

x 10-3

Superposition

Uniform

On-off

0 0.5 10

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

Single-code

Superposition

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

Superposition

Uniform

On-off

EER Inner Bound

R1 = 0.5

R2 = 0.5

SNR1 = 10

SNR2 = 10

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

Superposition

Uniform

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

Superposition

0 0.5 10

P11P12

P21P22

E{|C1,i(k)|2} E{|C2,j(k)|2}

N N)1( N N)1(

Superposition vs. Uniform

P12P21

E{|C1,i(k)|2} E{|C2,j(k)|2}

N N)1( N N)1( k0 k0

P12 P21

E{|C1,i(k)|2} E{|C2,j(k)|2}

N N)1( N N)1( k0 k0

P12 P21

E{|C1,i(k)|2} E{|C2,j(k)|2}

N N)1( N N)1( k0 k0

P12 P21

E{|C1,i(k)|2} E{|C2,j(k)|2}

N N)1( N N)1(

Superposition vs. Uniform (cont.)

},min{)2/1,,,,,(

)2/1,,,(

1111111121313

1111111

EEEPPPPPPRREE

PPPREE

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 100.02

Uniform On-Off 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

Joint ML vs. Naïve Single-User

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

E2sp,jm

E2sp,ns

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

"anticipated" EER using individual ML decoding

Outline

• Gaussian Broadcast Channel (GBC)- EER Inner Bound- EER Outer Bound

• Single-User Outer Bound• Sato Outer Bound

• Conclusion

EER Outer Bound: Single-User

P(Y1,Y2 |X)

E(m1,m2) X

P(Y1,Y2 |X)

E(m1,m2) X

P(Y1 |X) D1

E2 P(Y2 |X)

),( );,( 22

REE susu

EER Outer Bound: Sato

12121 if ),(},min{

PRREEE su

P(Y1,Y2|X)

E(m1,m2) X

P(Y1,Y2|X)

E(m1,m2) X

P(Y1,Y2|X)E(m1,m2) X

D(w'1,w'2)

Z1 Z2'

P(Y1,Y2|X)

E(m1,m2) X

P(Y1,Y2|X)E(m1,m2) X

D(w'1,w'2)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.080

EER Inner & Outer Bounds

R1 = R2 =0.5

SNR1 = SNR2 =10

• This is a proof that the true EER implies a tradeoff between users’ reliabilities

impossiblevalid

Outline

• Conclusion

),,,,(

• Each outer bound is based on single-user error exponent upper bounds. The right hand side of the inequalities depends only on R1 and R2

Review: GBC EER Outer Bound

)},(),,(max{},min{

PRREEE

Gaussian Single-User Channel (GSC) with Two Messages

121, },...,1;,...,1|{ N

ji SMjMiCCB

222 }E{Z Z;XY P;}E{X

X DecoderEncoder(i,j) (i',j')

Background: Minimum Distance Bound

code of distance minimum :)(

)( max)(

mincode spherical

paire ed

SNRNQdxeSNR

CM1,M2

GSC EER Outer Bound - Partition

EddeeP E

CM1,M2

Union of Circles

NNEe l

SNREeeP

Union of Circles

• C = {C1, C2, …, CM}

• A(C,r): area of the union of the circles with radius r

Minimum-Area Code

1. What is the maximum of dmin(C) under the constraint A(C,r) is at most A’?

2. What is the minimum of A(C,r) under the constraint dmin(C) is at least d’?

Intuition: Surface Cap

Conjectured Solution

D = {D1,D2,...,DM}

')(sin min dRd

)( max)( mincode sphere

min CdRdC

D = {D1,D2,...,DM}

')(sin min dRd

)*,(log

1 ),()*,(

NrDArCA

Conjectured GSC EER Outer Bound

),( )],,,([sin

),( )'(sin

)'(sin

SNRRESNRERR

What is the maximum of dmin(C) under the

constraint A(C,r) is at most A’?

)('sin)( 2minmin RdCd

),( )],,,([sin 22

1212 SNRRESNRERRE md

Conjectured GBC EER Outer Bound

),( )],,,([sin 222

11212 SNRRESNRERRE md

R1 = 0.5

R2 = 2.4

SNR1 = 100

SNR2 = 1000

0 1 2 3 4 5 6 7 80

New EER outer bound

Conclusion

• EER for Multi-User Channel- The set of achievable error exponent pair (E1,E2)

• Gaussian Broadcast Channel- EER inner bound : single-code, superposition- EER outer bound : single-user, Sato

• Gaussian Multiple Access Channel- EER is known for some operating points

• MIMO Fading Broadcast ChannelMIMO Fading Multiple Access Channel- Diversity Gain Region

Lihua Weng Dept. of EECS, Univ. of Michigan

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