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Distributed Scalar Quantizers for Subband Allocation John MacLaren Walsh Bradford D. Boyle Steven Weber jwalsh@coe.drexel.edu bradford@drexel.edu sweber@coe.drexel.edu odeling Modeling & Analysis of Networks Laboratory &

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SLIDE 1

Distributed Scalar Quantizers for Subband Allocation

Bradford D. Boyle

bradford@drexel.edu

John MacLaren Walsh

jwalsh@coe.drexel.edu

Steven Weber

sweber@coe.drexel.edu

Modeling & Analysis of Networks Laboratory Department of Electrical and Computer Engineering Drexel University, Philadelphia, PA 19104

  • deling

&

nalysis

  • f

etworks

Conference on Information Sciences & Systems Princeton, NJ March 20th, 2014

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 1 / 25

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SLIDE 2

Introduction

Outline

1 Introduction 2 Problem Model 3 Optimal Scalar Quantizer Design

Homogeneous Scalar Quantizers Heterogeneous Scalar Quantizers

4 Results 5 Conclusions

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 2 / 25

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SLIDE 3

Introduction

Motivation

  • Subbands in an OFDMA system must be assigned to a unique MS
  • AMC: BS wants MS w/ best channel & the gain on that channel
  • Rateless codes (e.g., ARQ): BS wants MS w/ best channel

MS 1 MS 2 BS ENC ENC DEC Subband index User subband gain 1 2 3 Subband index User subband gain 1 2 3 X(1)

1 , . . . , X(M) 1

X(1)

2 , . . . , X(M) 2

S1 Subband index User subband gain 1 2 3 Z(1), . . . , Z(M) Z(j) = arg max n X(j)

1 , X(j) 2

  • Z = g(X1, X2)

ˆ Z = f(S1, S2) S2

  • BS does not need to reproduce MS local state
  • Trade-off in feedback overhead & system efficiency
  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 3 / 25

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SLIDE 4

Introduction

Related Work

CEO—Indirect Distributed Lossy Source Coding

ENC 1 ENC 2 DEC X1 X2 S1 S2

  • ˆ

Z, D

  • D

R R(D) R

  • Si ∈ {1, . . . , 2nRi}
  • Z = g(X1, X2), ˆ

Z = f (S1, S2)

  • D = E
  • d(Z, ˆ

Z)

  • , R = R1 + R2
  • R achievable (R, D) pairs
  • R(D) smallest R s.t.

(R, D) ∈ R

Cover & Thomas 2006 ([1]) and El Gamal & Kim 2011 ([2])

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 4 / 25

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SLIDE 5

Introduction

Related Work

Distributed Functional SQ & Layered Architectures

Zamir & Berger (1999) [3]

  • Lossy, continous sources
  • Lattice quantizers w/

Slepian-Wolf (SW) coding

  • Optimal asymptotically in rate

Servetto (2005) [4]

  • Lossy, discrete sources
  • Scalar quantizer w/ SW coding
  • Optimal for all rates

Wagner et al. (2008) [5]

  • Lossy (MSE),Gaussian sources
  • Vector quantizers w/ SW coding
  • Optimal for all rates

Misra et al. (2011) [6]

  • Lossy (MSE), function of

sources

  • High rate regime
  • Optimal asymptotically in rate

“Layered” Achievable Scheme Scalar Quantizers at each user followed by entropy coding

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 5 / 25

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SLIDE 6

Introduction

Summary of Contributions

  • 1. Distortion optimal HomSQs

X 00 01 10 11 `1 `2 `3 ˆ x1 ˆ x2 ˆ x3 ˆ x4 min X max X Si

  • 2. Entropy-constrained HomSQs

0.0 0.1 0.2 0.0 0.2 0.4 0.6 0.8 Distortion 0.0 1.0 2.0 0.0 0.2 0.4 0.6 0.8 Rate ℓ

  • 3. HetSQs superior to HomSQs

for i.i.d. sources

D R HetSQ HomSQ

  • 4. HetSQ can be close to

fundamental limit

1 2 3 4 5 6 7 8 0.001 0.01 0.1 Total Rate [bits] Normalized Distortion HomSQ HetSQ R(D)

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 6 / 25

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SLIDE 7

Problem Model

Outline

1 Introduction 2 Problem Model 3 Optimal Scalar Quantizer Design

Homogeneous Scalar Quantizers Heterogeneous Scalar Quantizers

4 Results 5 Conclusions

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 7 / 25

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SLIDE 8

Problem Model

Problem Model & Notation

MS 1 MS 2 BS ENC ENC DEC Subband index User subband gain 1 2 3 Subband index User subband gain 1 2 3 X(1)

1 , . . . , X(M) 1

X(1)

2 , . . . , X(M) 2

S1 Subband index User subband gain 1 2 3 Z(1), . . . , Z(M) Z(j) = arg max n X(j)

1 , X(j) 2

  • Z = g(X1, X2)

ˆ Z = f(S1, S2) S2

  • Xi i.i.d. chan. capacity for MS i
  • Z optimal subband allocation

Z = arg max

i

{Xi : i = 1, 2}

  • ˆ

Z estimated subband allocation

  • Si coded message from MS i
  • Ri rate achieved by MS i
  • d distortion

d(Z, ˆ Z) = XZ − Xˆ

Z

We focus on two users w/ single subband

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 8 / 25

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SLIDE 9

Optimal Scalar Quantizer Design

Outline

1 Introduction 2 Problem Model 3 Optimal Scalar Quantizer Design

Homogeneous Scalar Quantizers Heterogeneous Scalar Quantizers

4 Results 5 Conclusions

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 9 / 25

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SLIDE 10

Optimal Scalar Quantizer Design

Scalar Quantizers

  • K-bin SQ parameterized by
  • decision boundaries ℓi
  • reconstruction points ˆ

xi

  • Designed to meet distortion

and/or rate constraints

  • Encoding: report the index of

the bin containing Xi

  • Decoding: map bin index to

reconstruction points

X 00 01 10 11 `1 `2 `3 ˆ x1 ˆ x2 ˆ x3 ˆ x4 min X max X Si

ℓ0 min X ℓK max X Xi MSi’s channel capacity X support set for r.v. Xi Si MSi’s message to BS

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 10 / 25

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SLIDE 11

Optimal Scalar Quantizer Design Homogeneous Scalar Quantizers

Outline

1 Introduction 2 Problem Model 3 Optimal Scalar Quantizer Design

Homogeneous Scalar Quantizers Heterogeneous Scalar Quantizers

4 Results 5 Conclusions

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 11 / 25

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SLIDE 12

Optimal Scalar Quantizer Design Homogeneous Scalar Quantizers

Minimum Distortion Scalar Quantizers

Obs: Not reproducing local state ⇒ reconstruction points not needed HomSQ: Both users have identical quantizer decision boundaries

X 00 01 10 11 `1 `2 `3 ˆ x1 ˆ x2 ˆ x3 ˆ x4 min X max X Si

  • Distortion is a function of ℓ
  • Select ℓ to minimize D(ℓ)

Theorem If ℓ is an optimal HomSQ then there exists µ ≥ 0 such that fX(ℓk) ℓk+1

ℓk−1

(ℓk − x)fX(x) dx − µk + µk+1 = 0 µk(ℓk−1 − ℓk) = 0.

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 12 / 25

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SLIDE 13

Optimal Scalar Quantizer Design Homogeneous Scalar Quantizers

Entropy Constrained Scalar Quantizers

  • Rate is also a function of ℓ
  • Select ℓ to minimize D(ℓ) w/

an upper-limit R0 on rate pk P(Si = k) = FX(ℓk)−FX(ℓk−1) RHomSQ(ℓ) = H(S1)+H(S2) = 2H(s)

0.0 0.1 0.2 0.0 0.2 0.4 0.6 0.8 Distortion 0.0 1.0 2.0 0.0 0.2 0.4 0.6 0.8 Rate ℓ

Problem is non-convex Theorem If ℓ is an optimal ECSQ, then there exists µ ≥ 0 and µR ≥ 0 such that fX(ℓk) ℓk+1

ℓk−1

(ℓk − x)fX(x) dx + 2µR log2 pk+1 pk

  • − µk + µk+1 = 0

µk(ℓk−1 − ℓk) = 0 and µR(RHomSQ(ℓ) − R0) = 0.

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 13 / 25

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SLIDE 14

Optimal Scalar Quantizer Design Heterogeneous Scalar Quantizers

Outline

1 Introduction 2 Problem Model 3 Optimal Scalar Quantizer Design

Homogeneous Scalar Quantizers Heterogeneous Scalar Quantizers

4 Results 5 Conclusions

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 14 / 25

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SLIDE 15

Optimal Scalar Quantizer Design Heterogeneous Scalar Quantizers

From HomSQs to HetSQs

Some Insights

HomSQ 1

X1 X2

? ? 2 1 HomSQ 2

X1 X2 2 1 ? ? ? ? 2 2 2 2 2 1 1 1 1 1

Deterministic HomSQ

X1 X2 2 1 2 2 2 2 2 1 1 1 1 1 1 1 1 1

  • Obvious for S1 = S2
  • Flip a coin for S1 = S2
  • Distortion only along diagonal
  • arg max not symmetric
  • Distortion is
  • Same distortion for a fixed

mapping along diagonal

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 15 / 25

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SLIDE 16

Optimal Scalar Quantizer Design Heterogeneous Scalar Quantizers

From HomSQ to HetSQ

Rate Reduction

Obs: All mappings have the same distortion; some have better total rate Mapping One

X1 X2 2 1 2 2 2 2 2 1 1 1 1 1 1 2 1 2

R(1)

HetSQ ≤ R(1) HomSQ

Mapping Two

X1 X2 2 1 2 2 2 2 2 1 1 1 1 1 1 1 2 2

R(2)

HetSQ ≤ R(2) HomSQ

Follows from subadditivity of t log t Theorem For an optimal HomSQ ℓ∗ that achieves a distortion D(ℓ∗), there exists a HetSQ that achieves the same distortion but at a lower rate.

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 16 / 25

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SLIDE 17

Optimal Scalar Quantizer Design Heterogeneous Scalar Quantizers

Staggered HetSQ

Staggered Mapping

X1 X2 2 1 2 2 2 2 2 1 1 1 1 1 1 2 1 2

R(1)

HetSQ ≤ R(1) HomSQ

R(2)

HetSQ ≤ R(2) HomSQ

Theorem For a HetSQ, if there exists an quantization interval for a user that is completely contained in the quantization interval for another user, then the quantizer is not

  • ptimal.

Design of HetSQ

1: Select the total # of bins K 2: Design optimal HomSQ

boundaries ℓHomSQ

3: Assign ℓ(k)

HomSQ to MS1 if k odd;

else, MS2

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 17 / 25

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SLIDE 18

Results

Outline

1 Introduction 2 Problem Model 3 Optimal Scalar Quantizer Design

Homogeneous Scalar Quantizers Heterogeneous Scalar Quantizers

4 Results 5 Conclusions

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 18 / 25

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SLIDE 19

Results

Example 1

Uniform(a, b) Channel Capacity

For k = 1, . . . , K − 1, the optimal quantizer is given as ℓ∗

k = aK + (b − a)k

K , µ∗

k = 0

HetSQ—No free lunch; consider

  • K = 3: 42.1% fewer bits
  • MS2 scheduled w.p. 0.556
  • R(2)

HetSQ = R(1) HetSQ

  • K = 4, 37.5% fewer bits
  • MS2 scheduled w.p. 0.500
  • R(2)

HetSQ = 0.67R(1) HetSQ

Uniform(0, 1) & K = 1, . . . , 6

1 2 3 4 5 0.01 0.1 Total Rate [bits] Distortion [bits] HomSQ EC HomSQ HetSQ

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 19 / 25

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SLIDE 20

Results

Example 2

Exponential (Exp(λ)) Channel Capacity

Let wk = λℓk; then the optimal quantizer is given as w∗

k = −e−w∗

k−1(1 + w∗

k−1) + e−w∗

k+1(1 + w∗

k+1)

(e−w∗

k+1 − e−w∗ k−1)

HetSQ—No free lunch; consider

  • K = 3: 43.3% fewer bits
  • MS2 scheduled w.p. 0.560
  • R(2)

HetSQ ≈ 0.73R(1) HetSQ

  • K = 4: 38.9% fewer bits
  • MS2 scheduled w.p. 0.534
  • R(2)

HetSQ ≈ 0.67R(1) HetSQ

Exp(2) & K = 1, . . . , 6

1 2 3 4 5 0.01 0.1 Total Rate [bits] Distortion [bits] HomSQ EC HomSQ HetSQ

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 20 / 25

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SLIDE 21

Results

Example 3: LTE CQI

Discrete Uniform Channel Capacity

Q: Can we compare HomSQ and HetSQ to a fundamental limit? A: R-D function computed from Berger-Tung bound Berger-Tung inner & outer bounds coincide for independent sources

  • 16 CQI levels in LTE [7]
  • D = 0 for R1 + R2 ≤ 8 bits
  • HetSQ within 0.124 bits
  • HomSQ within 1.622 bits

SQ w/ SW coding is optimal for recovery of discrete sources (Servetto 2005) [4]

1 2 3 4 5 6 7 8 0.001 0.01 0.1 Total Rate [bits] Normalized Distortion HomSQ HetSQ R(D)

Produced w/ help from Gwanmo Ku & Jie Ren

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 21 / 25

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SLIDE 22

Conclusions

Outline

1 Introduction 2 Problem Model 3 Optimal Scalar Quantizer Design

Homogeneous Scalar Quantizers Heterogeneous Scalar Quantizers

4 Results 5 Conclusions

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 22 / 25

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SLIDE 23

Conclusions

Review of Contributions

1 Distortion optimal HomSQs 2 Entropy-constrained distortion optimal HomSQs 3 Simple HetSQs achieve same distortion w/ lower rate as best HomSQs 4 HetSQ for discrete uniform distribution is close to fundamental limit

Future Work

  • Fundamental limit for continous sources
  • Generalize 2-user to N-user
  • Consider VQs for multiple subbands
  • Investigate low-rate performance as N → ∞
  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 23 / 25

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SLIDE 24

Conclusions

Acknowledgments

Supported by the AFOSR under agreement number FA9550-12-1-0086

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 24 / 25

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SLIDE 25

Conclusions

References

  • T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed.

Wiley Interscience, 2006.

  • A. E. Gamal and Y.-H. Kim, Network Information Theory.

Cambridge University Press, 2011.

  • R. Zamir and T. Berger, “Multiterminal source coding with high resolution,” IEEE Trans.
  • Inf. Theory, vol. 45, no. 1, pp. 106–117, 1999.
  • S. D. Servetto, “Achievable rates for multiterminal source coding with scalar quantizers,”

in Conf. Rec. of the 39th Asilomar Conf. Signals, Systems and Computers, 2005, pp. 1762–1766.

  • A. B. Wagner, S. Tavildar, and P. Viswanath, “Rate region of the quadratic gaussian

two-encoder source-coding problem,” IEEE Trans. Inf. Theory, vol. 54, no. 5, pp. 1938–1961, 2008.

  • V. Misra, V. K. Goyal, and L. R. Varshney, “Distributed scalar quantization for computing:

High-resolution analysis and extensions,” IEEE Trans. Inf. Theory, vol. 57, no. 8, pp. 5298–5325, 2011. Evolved Universal Terrestrial Radio Access (E-UTRA); Physical Layer Procedures, 3GPP Technical Specification TS 36.213, Rev. 8.8.0, Sep 2009.

  • B. D. Boyle (Drexel MANL)

DSQ CISS 2014 25 / 25