Achilleas Anastasopoulos (joint work with Lihua Weng and Sandeep Pradhan) April A Framework for Heterogeneous Quality-of-Service Guarantees in Wireless Networks: A Communication-theoretic Approach

2 Outline Motivation Background: error exponents for single- user channels The concept of error exponent region (EER) Scalar Gaussian broadcast channel (SGBC) MIMO Fading broadcast channel Conclusions

3 Motivation: Scenario 1 User 1: FTP application -High data rate -High reliability User 2: Voice -Low data rate -Low reliability Base Station Solution: allocate more resources (e.g., time slots, or BW) to user 1

4 Motivation: Scenario 2 User 1: FTP application -High data rate -High reliability User 2: Telemetry data -Low data rate -High reliability Base Station Solution: trade data rate for reliability for user 2 (e.g., using higher power and/or channel coding)

5 Motivation: Scenario 3 User 1: FTP application -High data rate -High reliability User 2: Multi-media -High data rate -Low reliability Base Station Solution 1: trade reliability for data rate for user 2 (e.g., no channel coding) Solution 2: allocate more resources to user 1 (e.g., power, or BW to utilize in channel coding)

6 Comments/Questions An individual user can trade its own data rate for reliability (scenario 2, 3) There are several techniques (usually referred to as “unequal error protection”) that provide solutions through asymmetric resource allocation  What is the the best you can do for a given channel and given resources?  Can available reliability be treated as another resource (like power, or BW) that can be allocated to different users?  Can communication theory provide answers to these questions?  How do you do that in practice?

7 Basic result of this work As in single-user channels, there is a basic trade-off between data rate and reliability Multi-user channels provide an additional degree of freedom:  Users can trade reliabilities with each other (even for fixed data rates)  The above seems like an obvious statement… There is a way to formulate this problem as a communication theoretic problem and study its fundamental limits

8 Outline Motivation Background: error exponents for single- user channels The concept of error exponent region (EER) Scalar Gaussian broadcast channel (SGBC) MIMO Fading multi-user channels Conclusions

9 Error exponent: Single-user channel Channel capacity, C: highest possible transmission rate that results in arbitrarily low probability of codeword error with long codewords Error Exponent, E: rate of exponential decay of codeword error probability For a codeword of length N, the probability of codeword error behaves as where E(R) is the error exponent (as a function of the transmission rate R)  DMC (Gallager65; Shannon et al67)  AWGN (Shannon59; Gallager65)

10 Error exponent: Single-user channel Error exponent E(R) is an increasing function of the distance between R and C Only trade-off: increase E(R) by decreasing R, i.e, trade reliability for rate Upper bounds on P err  Lower bounds on E  simple  Random coding bound, expurgated bound Lower bounds on P err  Upper bounds on E  not that simple  Sphere packing bound, minimum distance bound, straight line bound

11 Error exponent: Multi-user channel Channel capacity region: all possible transmission rate vectors (R 1,R 2 ) for arbitrarily low system error probability System error probability: for correct transmission, all users have to be decoded correctly

12 Error exponent: Multi-user channel Error Exponent: rate of exponential decay of system error probability For a codeword of length N, the probability of system error behaves as where E(R 1,R 2 ) is the error exponent  Gaussian MAC (Gallager85; Guess&Varanasi00)  Wireless MIMO MAC at high SNR (Zheng&Tse03)

13 Error exponent: Multi-user channel (conclusions) We saw (scenario 1, 3) that different users might have different reliability requirements (e.g., FTP and multi-media) Based on a single probability of system error, a network can only be designed to satisfy the most stringent reliability requirement (equal QoS for all users), which might result in a suboptimum resource allocation Information/communication theory seems inadequate (so far) to address heterogeneous QoS requirements

14 Outline Motivation Background: error exponents for single- user channels The concept of error exponent region (EER) Scalar Gaussian broadcast channel (SGBC) MIMO Fading multi-user channels Conclusions

15 A straightforward extension Since a single system error probability is inadequate to characterize the requirements of multiple users, let us consider multiple error probabilities; one for each user Implication: multiple error exponents; one for each user

16 A straightforward extension We have trade-off between error exponents and rates (as in the single-user channel). Is there any other trade-off available for error exponents in a multi-user channel?

17 The concept of EER Fix an operating point (R 1,R 2 )

18 The concept of EER Fix an operating point (R 1,R 2 ) Which point from the capacity boundary do we back off to reach A?

19 The concept of EER Fix an operating point (R 1,R 2 ) Which point from the capacity boundary do we back off to reach A? B  A : E 1 < E 2

20 The concept of EER Fix an operating point (R 1,R 2 ) Which point from the capacity boundary do we back off to reach A? B  A : E 1 < E 2 D  A : E 1 > E 2

21 The concept of EER Fix an operating point (R 1,R 2 ) Which point from the capacity boundary do we back off to reach A? B  A : E 1 < E 2 D  A : E 1 > E 2 In addition to error exponent/rate trade-off, given a fixed (R 1,R 2 ), one can potentially trade-off E 1 with E 2

22 The concept of EER: Definition Definition: The error exponent region (EER) 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 pair of (R 1,R 2 ) Possible shape for EER

23 Outline Motivation Background: error exponents for single- user channels The concept of error exponent region (EER) Scalar Gaussian broadcast channel (SGBC) MIMO Fading multi-user channels Conclusions

24 Scalar Gaussian Broadcast Channel Observe: two messages; joint encoder; separate decoders This is a degraded broadcast channel (i.e., if     then, Y 2 =X+N 1 +N’ 2 =Y 1 + N’ 2, with E{(N’ 2 ) 2 }=        ) SGBC definitions

25 Achievable EER by time-sharing: where E(R,SNR) is any of the error exponent lower bounds for a single-user AWGN channel SGBC EER Inner Bound: Time-sharing

26 SGBC EER Inner Bound: Time-sharing Indeed, there is a trade-off for error exponents, given a fixed pair of rates for time-sharing R 1 = R 2 =0.5 P/  1 2 = P/  2 2 =10

27 SGBC EER Inner Bound: Superposition Superposition encoding:  Generate two independent codebooks C i, each of size and power  Select a codeword from each codebook based on the individual messages and transmit their sum  Note: this is a capacity-achieving strategy for any degraded broadcast channel

28 SGBC EER Inner Bound: Superposition Decoding: two options (at least)  Individual ML decoding (optimal)  Joint Maximum-Likelihood (ML) decoding

29 Upper bound derivation for joint ML decoding  Let us look at user 1:  Type 1: M 1 is decoded erroneously, but M 2 is decoded correctly  same as if only user 1 was present in the channel  Type 3: both messages are decoded erroneously (similar bound as in Gallager85 for MAC channels) SGBC EER Inner Bound: Superposition

30 Superposition Inner Bound with joint ML decoding where E(R,SNR) is any of the error exponent lower bounds for a single-user AWGN channel, and E t3 (R,SNR 1,SNR 2 ) is a slightly more complicated expression (for type 3 errors) SGBC EER Inner Bound: Superposition

31 SGBC EER Inner Bound Observation: although superposition achieves capacity (while time-sharing does not always achieve it), time sharing can help in expanding the EER. Why? R 1 = R 2 =0.5 P/  1 2 = P/  2 2 =10

32 Time-Sharing vs. Superposition Three possible reasons:  The superposition EER is derived based on joint ML decoding, but the optimum decoder is individual ML decoding  Joint ML decoding might be still a good strategy, but E t3 is a loose bound  Time-Sharing can sometimes indeed expand the EER obtained by superposition: when we need very high reliability for one user, it might be better to separate the users

33 SGBC EER Inner Bound: Summary We can keep expanding the inner bound by finding better and better strategies It is not clear yet that the exact EER implies a trade-off between users’ reliabilities Possible true EER We need an outer bound for the EER

34 SGBC EER Outer Bound: Single-user where E su (R,SNR) is any error exponent upper bound for the AWGN channel is always worse than two separate single-user channels with same marginals Any broadcast channel thusand

35 SGBC EER Outer Bound: Sato For any Q(Y 1,Y 2 |X) with the same marginals as P(Y 1,Y 2 |X) is always worse than By choosing the worst-case Q(Y 1,Y 2 |X)

36 SGBC EER Outer Bound R 1 = R 2 =0.5 SNR 1 = SNR 2 =10 This is a proof that the true EER implies a trade-off between users’ reliabilities impossible valid

37 Outline Motivation Background: error exponents for single- user channels The concept of error exponent region (EER) Scalar Gaussian broadcast channel (SGBC) MIMO Fading multi-user channels Conclusions

38 Background: Single-user channel MIMO Fading Single-user Channel (Tse, 2003) : block fading  X: m x t channel input matrix  Y: n x t channel output matrix  Z: n x t noise matrix; i.i.d. with CN(0,1)  H: n x m fading matrix; i.i.d. with CN(0,1) Assume H is known at receiver, but not at transmitter

39 Background: Single-user channel MIMO fading single-user channel (Zheng&Tse03)  Diversity and Multiplexing trade-off (high SNR) r: multiplexing gain d: diversity gain

40 Background: Single-user channel

41 Multiplexing Gain Region (MGR) Diversity Gain Region (DGR) MIMO fading multi-user channel  Multiplexing Gain Region: the set of all achievable multiplexing-gain vector (r 1,…,r K )  Diversity Gain Region: the set of all achievable diversity-gain vector (d 1,…,d K ), given a multiplexing-gain vector.

42 MIMO Fading Broadcast Channel MIMO Fading Broadcast Channel (MFBC) : block fading  X : m x t channel input matrix  Y i : n i x t channel output matrix  Z i : n i x t noise matrix; i.i.d. element CN(0,1)  H i : n i x m fading matrix; i.i.d. element CN(0,1) Assume H i is known at receivers, but not at transmitter

43 MFBC Multiplexing Gain Region Proposition: For a MIMO fading broadcast channel, the multiplexing gain region is the same region achieved by time-sharing.

44 MFBC DGR Inner Bound: Time-Sharing Time-Sharing

45 MFBC DGR Inner Bound: Superposition Superposition: X = X 1 + X 2 X 1 : m x l matrix with i.i.d. element CN(0,1) X 2 : m x l matrix with i.i.d. element CN(0,SNR -(1-p) )  Joint Maximum-Likelihood (ML) decoding Note : The role of user 1 and user 2 can be exchanged

46 MFBC DGR Inner Bound: Superposition Superposition: X = X 1 + X 2 X 1 : m x t matrix with i.i.d. element CN(0,1) X 2 : m x t matrix with i.i.d. element CN(0,SNR -(1-p) )  Joint ML and naïve single-user decoding Note : The role of user 1 and user 2 can be exchanged.

47 Naïve Single-user Diversity Gain Region

48 MFBC DGR Outer Bound

49 Diversity Gain Region Inner/Outer Bound Observation: For a symmetric MFBC, inner and outer bounds are tight at d 1 = d 2 Observation: For a MFBC, either user 1 (or user 2) can achieve his maximum (single-user) diversity gain if r 1 +r 2 < 1 m = n 1 = n 2 =4 t = 120 r 1 = r 2 = 0.5

50 Conclusions  The concept of error exponent region for multi-user channels was presented  Inner (time-sharing/superposition) and outer (single- user/Sato) bounds were derived for the SGBC EER  Implication: Users can trade reliability between each other even for a fixed set of transmission rates Ongoing Work  Tighten EER inner/outer bounds for SGBC  EER for Gaussian multiple-access channels  Diversity/multiplexing trade-off region for wireless MIMO BC/MAC  Practical schemes that achieve EER