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

2 Motivation: Downlink Communication

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.

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

5 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

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

7 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

8 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)

9 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.

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

11 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.

12 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 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 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

13 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 )

14 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

15 Gaussian Broadcast Channel

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

17 Superposition Encoding

18 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

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

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

21 Special Case 1: Uniform Superposition

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

23 EER Inner Bound R 1 = 1 R 2 = 0.1 SNR 1 = 10 SNR 2 = 5

24 EER Inner Bound R 1 = 0.5 R 2 = 0.5 SNR 1 = 10 SNR 2 = 10

25 Superposition vs. Uniform

26 Superposition vs. Uniform (cont.)

27 Joint ML vs. Naïve Single-User

28 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

29 EER Outer Bound: Single-User

30 EER Outer Bound: Sato

31 EER Inner & Outer Bounds R 1 = R 2 =0.5 SNR 1 = SNR 2 =10 This is a proof that the true EER implies a tradeoff between users’ reliabilities impossible valid

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

33 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 Review: GBC EER Outer Bound

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

35 Background: Minimum Distance Bound

36 GSC EER Outer Bound - Partition

37 Union of Circles

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

39 Minimum-Area Code 1. What is the maximum of d min (C) under the constraint A(C,r) is at most A’? 2. What is the minimum of A(C,r) under the constraint d min (C) is at least d’?

40 Intuition: Surface Cap

41 Conjectured Solution

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

43 Conjectured GBC EER Outer Bound R 1 = 0.5 R 2 = 2.4 SNR 1 = 100 SNR 2 = 1000

44 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