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

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. 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 (Elias55;Fano61;Gallager65; Shannon67)
AWGN (Shannon 59; Gallager 65)

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 (R1,R2) 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 (E1,E2) and rates (R1,R2) as in the single-user channel.

12. 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
D  A : E1 > E2
Given a fixed (R1,R2), one can potentially tradeoff E1 with E2
Fix an operating point (R1,R2), which point from the capacity boundary can we back off to reach A?
B  A : E1 < E2
D  A : E1 > E2
Fix an operating point (R1,R2), which point from the capacity boundary can we back off to reach A?
B  A : E1 < E2
Fix an operating point (R1,R2), which point from the capacity boundary can we back off to reach A?

13. Error Exponent Region (EER)
Definition: Given (R1,R2), error exponent region is the set of all achievable error exponent pairs (E1,E2)
Careful!!!
Channel capacity region: one for a given channel
EER: numerous, i.e., one for each rate pair (R1,R2)

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

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.)
Type 3 error: both users’ messages are decoded erroneously
Achievable Error Exponents

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

21. Special Case 1: Uniform Superposition

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

23. EER Inner Bound
R1 = 1
R2 = 0.1
SNR1 = 10
SNR2 = 5

24. EER Inner Bound
R1 = 0.5
R2 = 0.5
SNR1 = 10
SNR2 = 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
valid
impossible
R1 = R2 =0.5
SNR1 = SNR2 =10
This is a proof that the true EER implies a tradeoff between users’ reliabilities

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

33. 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 R1 and R2

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 = {C1, C2, …, CM}
A(C,r): area of the union of the circles with radius r

39. 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’?

40. Intuition: Surface Cap

41. Conjectured Solution

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

43. Conjectured GBC EER Outer Bound
SNR1 = 100
SNR2 = 1000

44. Conclusion EER for Multi-User Channel Gaussian Broadcast 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
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