Error Exponent Regions for MultiUser Channels Lihua Weng

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Error Exponent Regions for Multi-User Channels Lihua Weng Dept. of EECS, Univ. of Michigan

Error Exponent Regions for Multi-User Channels Lihua Weng Dept. of EECS, Univ. of Michigan

Motivation: Downlink Communication 2

Motivation: Downlink Communication 2

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

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. 3

Outline • Background: Error Exponent • Error Exponent Region (EER) • Gaussian Broadcast Channel

Outline • Background: Error Exponent • Error Exponent Region (EER) • Gaussian Broadcast Channel (GBC) • Conjectured GBC EER Outer Bound • Conclusion 4

Channel Capacity & Error Exponent: Single-User Channel • Channel capacity: highest data rate for

Channel Capacity & Error Exponent: Single-User Channel • Channel capacity: highest data rate for arbitrarily low probability of codeword error with long codewords • 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 (Elias 55; Fano 61; Gallager 65; Shannon 67) - AWGN (Shannon 59; Gallager 65) 5

Error Exponent • We have a tradeoff between error exponent and rate 6

Error Exponent • We have a tradeoff between error exponent and rate 6

Capacity: Multi-User Channel • • Channel capacity region: all possible transmission rate vectors (R

Capacity: Multi-User Channel • • Channel capacity region: all possible transmission rate vectors (R 1, R 2) for arbitrarily low probability of system error with long codewords Probability of system error: any user’s codeword is decoded in error 7

Error Exponent: Multi-User Channel • Error Exponent: rate of exponential decay of the smallest

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) 8

Single Error Exponent: Drawback • Multi-user channel – single error exponent - Different applications

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. 9

Outline • Background: Error Exponent • Error Exponent Region (EER) • Gaussian Broadcast Channel

Outline • Background: Error Exponent • Error Exponent Region (EER) • Gaussian Broadcast Channel (GBC) • Conjectured GBC EER Outer Bound • Conclusion 10

Multiple Error Exponents: Tradeoff 1 • We have tradeoff between error exponents (E 1,

Multiple Error Exponents: Tradeoff 1 • We have tradeoff between error exponents (E 1, E 2) and rates (R 1, R 2) as in the single-user channel. 11

Multiple Error Exponents: Tradeoff 2 • • • Fix an operating point (R 1,

Multiple Error Exponents: Tradeoff 2 • • • Fix an operating point (R 1, R 2), which point from the capacity boundary can we back off to reach A? B A : E 1 < E 2 D A : E 1 > E 2 Given a fixed (R 1, R 2), one can potentially tradeoff E 1 with E 2 12

Error Exponent Region (EER) • • Definition: Given (R 1, R 2), error exponent

Error Exponent Region (EER) • • Definition: Given (R 1, R 2), error exponent region is the set of all achievable error exponent pairs (E 1, E 2) Careful!!! - Channel capacity region: one for a given channel - EER: numerous, i. e. , one for each rate pair (R 1, R 2) 13

Outline • Background: Error Exponent • Error Exponent Region (EER) • Gaussian Broadcast Channel

Outline • Background: Error Exponent • Error Exponent Region (EER) • Gaussian Broadcast Channel (GBC) - EER Inner Bound • Single-Code Encoding • Superposition Encoding - EER Outer Bound • Conjectured GBC EER Outer Bound • Conclusion 14

Gaussian Broadcast Channel 15

Gaussian Broadcast Channel 15

Single-Code Encoding CB = {Ck | k=(i-1)*M 2+j; i = 1, … , M

Single-Code Encoding CB = {Ck | k=(i-1)*M 2+j; i = 1, … , M 1; j = 1, … , M 2} 16

Superposition Encoding 17

Superposition Encoding 17

Individual and Joint ML Decoding • Individual ML Decoding (optimal) • Joint ML Decoding

Individual and Joint ML Decoding • Individual ML Decoding (optimal) • Joint ML Decoding - Type 1 error: one user’s own message decoded erroneously, but the other user’s message decoded correctly 18

Joint ML Decoding (cont. ) • Joint ML Decoding - Type 3 error: both

Joint ML Decoding (cont. ) • Joint ML Decoding - Type 3 error: both users’ messages are decoded erroneously • Achievable Error Exponents 19

Naïve Single-User Decoding • Naïve Single-user decoding: Decode one user’s signal by regarding the

Naïve Single-User Decoding • Naïve Single-user decoding: Decode one user’s signal by regarding the other user’s signal as noise 20

Special Case 1: Uniform Superposition 21

Special Case 1: Uniform Superposition 21

Special Case 2: On-Off Superposition (Time-Sharing) 22

Special Case 2: On-Off Superposition (Time-Sharing) 22

EER Inner Bound R 1 = 1 R 2 = 0. 1 SNR 1

EER Inner Bound R 1 = 1 R 2 = 0. 1 SNR 1 = 10 SNR 2 = 5 23

EER Inner Bound R 1 = 0. 5 R 2 = 0. 5 SNR

EER Inner Bound R 1 = 0. 5 R 2 = 0. 5 SNR 1 = 10 SNR 2 = 10 24

Superposition vs. Uniform 25

Superposition vs. Uniform 25

Superposition vs. Uniform (cont. ) 26

Superposition vs. Uniform (cont. ) 26

Joint ML vs. Naïve Single-User 27

Joint ML vs. Naïve Single-User 27

Outline • Background: Error Exponent • Error Exponent Region (EER) • Gaussian Broadcast Channel

Outline • Background: Error Exponent • Error Exponent Region (EER) • Gaussian Broadcast Channel (GBC) - EER Inner Bound - EER Outer Bound • Single-User Outer Bound • Sato Outer Bound • Conjectured GBC EER Outer Bound • Conclusion 28

EER Outer Bound: Single-User 29

EER Outer Bound: Single-User 29

EER Outer Bound: Sato 30

EER Outer Bound: Sato 30

EER Inner & Outer Bounds valid R 1 = R 2 =0. 5 impossible

EER Inner & Outer Bounds valid R 1 = R 2 =0. 5 impossible SNR 1 = SNR 2 =10 • This is a proof that the true EER implies a tradeoff between users’ reliabilities 31

Outline • Background: Error Exponent • Error Exponent Region (EER) • Gaussian Broadcast Channel

Outline • Background: Error Exponent • Error Exponent Region (EER) • Gaussian Broadcast Channel (GBC) • Conjectured GBC EER Outer Bound • Conclusion 32

Review: GBC EER Outer Bound • Each outer bound is based on single-user error

Review: GBC EER Outer Bound • Each outer bound is based on single-user error exponent upper bounds. The right hand side of the inequalities depends only on R 1 and R 2 33

Gaussian Single-User Channel (GSC) with Two Messages 34

Gaussian Single-User Channel (GSC) with Two Messages 34

Background: Minimum Distance Bound 35

Background: Minimum Distance Bound 35

GSC EER Outer Bound - Partition 36

GSC EER Outer Bound - Partition 36

Union of Circles 37

Union of Circles 37

Union of Circles • • C = {C 1, C 2, …, CM} A(C,

Union of Circles • • C = {C 1, C 2, …, CM} A(C, r): area of the union of the circles with radius r 38

Minimum-Area Code 1. What is the maximum of dmin(C) under the constraint A(C, 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’? 39

Intuition: Surface Cap 40

Intuition: Surface Cap 40

Conjectured Solution 41

Conjectured Solution 41

Conjectured GSC EER Outer Bound What is the maximum of dmin(C) under the constraint

Conjectured GSC EER Outer Bound What is the maximum of dmin(C) under the constraint A(C, r) is at most A’? 42

Conjectured GBC EER Outer Bound R 1 = 0. 5 R 2 = 2.

Conjectured GBC EER Outer Bound R 1 = 0. 5 R 2 = 2. 4 SNR 1 = 100 SNR 2 = 1000 43

Conclusion • EER for Multi-User Channel • - The set of achievable error exponent

Conclusion • EER for Multi-User Channel • - The set of achievable error exponent pair (E 1, E 2) Gaussian Broadcast Channel - EER inner bound : single-code, superposition - EER outer bound : single-user, Sato Conjectured GBC EER Outer Bound • • Gaussian Multiple Access Channel - EER is known for some operating points • MIMO Fading Broadcast Channel MIMO Fading Multiple Access Channel - Diversity Gain Region 44