Achilleas Anastasopoulos

1
A Framework for Heterogeneous Quality-of-Service
Guarantees in Wireless Networks A
Communication-theoretic Approach

• Achilleas Anastasopoulos
• (joint work with Lihua Weng and Sandeep Pradhan)
• April 30 2004

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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
Base Station

• User 1 FTP application
• -High data rate
• -High reliability

User 2 Voice -Low data rate -Low reliability

• Solution allocate more resources (e.g., time
  slots, or BW) to user 1

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Motivation Scenario 2
Base Station

• User 1 FTP application
• -High data rate
• -High reliability

User 2 Telemetry data -Low data rate -High
reliability

• Solution trade data rate for reliability for
  user 2 (e.g., using higher power and/or channel
  coding)

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Motivation Scenario 3
Base Station

• User 1 FTP application
• -High data rate
• -High reliability

User 2 Multi-media -High data rate -Low
reliability

• 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)

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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?

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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

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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

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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)

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Error exponent Single-user channel

• Upper bounds on Perr? Lower bounds on E ? simple
• Random coding bound, expurgated bound
• Lower bounds on Perr? Upper bounds on E ? not
  that simple
• Sphere packing bound, minimum distance bound,
  straight line bound

• 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

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Error exponent Multi-user channel

• Channel capacity region all possible
  transmission rate vectors (R1,R2) for arbitrarily
  low system error probability
• System error probability for correct
  transmission, all users have to be decoded
  correctly

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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(R1,R2) is the error exponent
• Gaussian MAC (Gallager85 GuessVaranasi00)
• Wireless MIMO MAC at high SNR (ZhengTse03)

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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

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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

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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

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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?

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The concept of EER

• Fix an operating point (R1,R2)

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The concept of EER

• Fix an operating point (R1,R2)
• Which point from the capacity boundary do we back
  off to reach A?

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The concept of EER

• Fix an operating point (R1,R2)
• Which point from the capacity boundary do we back
  off to reach A?
• B ? A E1 lt E2

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The concept of EER

• Fix an operating point (R1,R2)
• Which point from the capacity boundary do we back
  off to reach A?
• B ? A E1 lt E2
• D ? A E1 gt E2

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The concept of EER

• Fix an operating point (R1,R2)
• Which point from the capacity boundary do we back
  off to reach A?
• B ? A E1 lt E2
• D ? A E1 gt E2

• In addition to error exponent/rate trade-off,
  given a fixed (R1,R2), one can potentially
  trade-off E1 with E2

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The concept of EER Definition

• Definition The error exponent region (EER) 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 pair of (R1,R2)

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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

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SGBC definitions

• Scalar Gaussian Broadcast Channel
• Observe two messages joint encoder separate
  decoders
• This is a degraded broadcast channel (i.e., if
  s2gts1 then, Y2XN1N2Y1 N2, with E(N2)2
  s22-s12 )

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SGBC EER Inner Bound Time-sharing

• Achievable EER by time-sharing
• where E(R,SNR) is any of the error exponent
  lower bounds for a single-user AWGN channel

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SGBC EER Inner Bound Time-sharing
R1 R2 0.5 P/s12 P/s22 10

• Indeed, there is a trade-off for error exponents,
  given a fixed pair of rates for time-sharing

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SGBC EER Inner Bound Superposition

• Superposition encoding
• Generate two independent codebooks Ci, 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

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SGBC EER Inner Bound Superposition

• Decoding two options (at least)
• Individual ML decoding (optimal)
• Joint Maximum-Likelihood (ML) decoding

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SGBC EER Inner Bound Superposition

• Upper bound derivation for joint ML decoding
• Let us look at user 1
• Type 1 M1 is decoded erroneously, but M2 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)

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SGBC EER Inner Bound Superposition

• 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
  Et3(R,SNR1,SNR2) is a slightly more complicated
  expression (for type 3 errors)

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SGBC EER Inner Bound
R1 R2 0.5 P/s12 P/s22 10

• Observation although superposition achieves
  capacity (while time-sharing does not always
  achieve it), time sharing can help in expanding
  the EER. Why?

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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 Et3 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

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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

• We need an outer bound for the EER

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SGBC EER Outer Bound Single-user
Any broadcast channel
is always worse than two separate single-user
channels with same marginals
thus
and

• where Esu (R,SNR) is any error exponent upper
  bound for the AWGN channel

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SGBC EER Outer Bound Sato

• For any Q(Y1,Y2X) with the same marginals as
  P(Y1,Y2 X)

is always worse than
By choosing the worst-case Q(Y1,Y2 X)
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SGBC EER Outer Bound
R1 R2 0.5 SNR1 SNR2 10

• This is a proof that the true EER implies a
  trade-off between users reliabilities

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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

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Background Single-user channel

• MIMO fading single-user channel (ZhengTse03)
• Diversity and Multiplexing trade-off (high SNR)
• r multiplexing gain
• d diversity gain

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Background Single-user channel
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Multiplexing Gain Region (MGR)Diversity Gain
Region (DGR)

• MIMO fading multi-user channel
• Multiplexing Gain Region the set of all
  achievable multiplexing-gain vector (r1,,rK)
• Diversity Gain Region the set of all achievable
  diversity-gain vector (d1,,dK), given a
  multiplexing-gain vector.

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MIMO Fading Broadcast Channel

• MIMO Fading Broadcast Channel (MFBC) block
  fading
• X m x t channel input matrix
• Yi ni x t channel output matrix
• Zi ni x t noise matrix i.i.d. element CN(0,1)
• Hi ni x m fading matrix i.i.d. element
  CN(0,1)
• Assume Hi is known at receivers, but not at
  transmitter

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MFBC Multiplexing Gain Region

• Proposition For a MIMO fading broadcast channel,
  the multiplexing gain region is the same region
  achieved by time-sharing.

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MFBC DGR Inner Bound Time-Sharing

• Time-Sharing

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MFBC DGR Inner Bound Superposition

• Superposition X X1 X2
• X1 m x l matrix with i.i.d. element CN(0,1)
• X2 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

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MFBC DGR Inner Bound Superposition

• Superposition X X1 X2
• X1 m x t matrix with i.i.d. element CN(0,1)
• X2 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.

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Naïve Single-user Diversity Gain Region
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MFBC DGR Outer Bound

• MFBC DGR Outer Bound

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Diversity Gain Region Inner/Outer Bound
m n1 n2 4 t 120 r1 r2 0.5

• Observation For a symmetric MFBC, inner and
  outer bounds are tight at d1 d2
• Observation For a MFBC, either user 1 (or user
  2) can achieve his maximum (single-user)
  diversity gain if r1r2 lt 1

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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