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

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

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)

Error Exponent 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

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)

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

Multiple Error Exponents: Tradeoff 1 We have tradeoff between error exponents (E1,E2) and rates (R1,R2) as in the single-user channel.

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?

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)

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

Gaussian Broadcast Channel

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

Superposition Encoding

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

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

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

Special Case 1: Uniform Superposition

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

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

EER Inner Bound R1 = 0.5 R2 = 0.5 SNR1 = 10 SNR2 = 10

Superposition vs. Uniform

Superposition vs. Uniform (cont.)

Joint ML vs. Naïve Single-User

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

EER Outer Bound: Single-User

EER Outer Bound: Sato

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

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

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

Gaussian Single-User Channel (GSC) with Two Messages

Background: Minimum Distance Bound

GSC EER Outer Bound - Partition

Union of Circles

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

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

Conjectured GBC EER Outer Bound SNR1 = 100 SNR2 = 1000

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