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 & Department of Electrical and Computer Engineering nalysis Drexel University, Philadelphia, PA 19104 of etworks Conference on Information Sciences & Systems Princeton, NJ March 20 th , 2014 B. D. Boyle (Drexel MANL) DSQ CISS 2014 1 / 25
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
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 X (1) 1 , . . . , X ( M ) User 1 subband User Z (1) , . . . , Z ( M ) gain subband Z ( j ) = arg max n o X ( j ) 1 , X ( j ) 1 2 3 gain 2 Subband index 1 2 3 E NC S 1 Subband index Z = g ( X 1 , X 2 ) MS 1 X (1) 2 , . . . , X ( M ) D EC 2 User ˆ Z = f ( S 1 , S 2 ) subband S 2 E NC BS gain MS 2 1 2 3 Subband index • 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
Introduction Related Work CEO—Indirect Distributed Lossy Source Coding R X 1 S 1 E NC 1 � � ˆ D EC Z, D R X 2 S 2 E NC 2 R ( D ) D • S i ∈ { 1 , . . . , 2 nR i } • R achievable ( R , D ) pairs • Z = g ( X 1 , X 2 ), ˆ • R ( D ) smallest R s.t. Z = f ( S 1 , S 2 ) ( R , D ) ∈ R � � d ( Z , ˆ • D = E Z ) , R = R 1 + R 2 Cover & Thomas 2006 ([1]) and El Gamal & Kim 2011 ([2]) B. D. Boyle (Drexel MANL) DSQ CISS 2014 4 / 25
Introduction Related Work Distributed Functional SQ & Layered Architectures Zamir & Berger (1999) [3] Wagner et al. (2008) [5] • Lossy, continous sources • Lossy (MSE),Gaussian sources • Lattice quantizers w/ • Vector quantizers w/ SW coding Slepian-Wolf (SW) coding • Optimal for all rates • Optimal asymptotically in rate Misra et al. (2011) [6] Servetto (2005) [4] • Lossy (MSE), function of • Lossy, discrete sources sources • Scalar quantizer w/ SW coding • High rate regime • Optimal for all rates • 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
Introduction Summary of Contributions 1. Distortion optimal HomSQs 2. Entropy-constrained HomSQs S i x 1 ˆ x 2 ˆ x 3 ˆ x 4 ˆ Distortion 0.2 00 01 10 11 X 0.1 0.0 min X max X ` 1 ` 2 ` 3 0.0 0.2 0.4 0.6 0.8 2.0 Rate 1.0 0.0 0.0 0.2 0.4 0.6 0.8 ℓ 3. HetSQs superior to HomSQs 4. HetSQ can be close to for i.i.d. sources fundamental limit R 8 HomSQ 7 HetSQ Total Rate [bits] 6 R ( D ) 5 HomSQ 4 3 2 1 HetSQ 0 0.001 0.01 0.1 D Normalized Distortion B. D. Boyle (Drexel MANL) DSQ CISS 2014 6 / 25
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
Problem Model Problem Model & Notation X (1) 1 , . . . , X ( M ) User 1 subband User Z (1) , . . . , Z ( M ) gain subband Z ( j ) = arg max n X ( j ) 1 , X ( j ) o 1 2 3 gain 2 Subband index 1 2 3 S 1 E NC Subband index Z = g ( X 1 , X 2 ) MS 1 X (1) 2 , . . . , X ( M ) D EC 2 User ˆ Z = f ( S 1 , S 2 ) subband E NC S 2 BS gain MS 2 1 2 3 Subband index • ˆ • X i i.i.d. chan. capacity for MS i Z estimated subband allocation • Z optimal subband allocation • S i coded message from MS i • R i rate achieved by MS i Z = arg max { X i : i = 1 , 2 } • d distortion i d ( Z , ˆ Z ) = X Z − X ˆ Z We focus on two users w/ single subband B. D. Boyle (Drexel MANL) DSQ CISS 2014 8 / 25
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
Optimal Scalar Quantizer Design Scalar Quantizers S i x 1 ˆ x 2 ˆ x 3 ˆ x 4 ˆ • K -bin SQ parameterized by 00 01 10 11 • decision boundaries ℓ i X • reconstruction points ˆ x i min X max X ` 1 ` 2 ` 3 • Designed to meet distortion and/or rate constraints ℓ 0 � min X ℓ K � max X • Encoding: report the index of the bin containing X i X i MS i ’s channel capacity • Decoding: map bin index to X support set for r.v. X i reconstruction points S i MS i ’s message to BS B. D. Boyle (Drexel MANL) DSQ CISS 2014 10 / 25
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
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 S i x 1 ˆ x 2 ˆ x 3 ˆ ˆ x 4 • Distortion is a function of ℓ 00 01 10 11 X • Select ℓ to minimize D ( ℓ ) min X max X ` 1 ` 2 ` 3 Theorem If ℓ is an optimal HomSQ then there exists µ ≥ 0 such that � ℓ k +1 f X ( ℓ k ) ( ℓ k − x ) f X ( x ) d x − µ k + µ k +1 = 0 ℓ k − 1 µ k ( ℓ k − 1 − ℓ k ) = 0 . B. D. Boyle (Drexel MANL) DSQ CISS 2014 12 / 25
Optimal Scalar Quantizer Design Homogeneous Scalar Quantizers Entropy Constrained Scalar Quantizers Distortion 0.2 • Rate is also a function of ℓ 0.1 • Select ℓ to minimize D ( ℓ ) w/ 0.0 0.0 0.2 0.4 0.6 0.8 an upper-limit R 0 on rate 2.0 Rate 1.0 p k � P ( S i = k ) = F X ( ℓ k ) − F X ( ℓ k − 1 ) 0.0 0.0 0.2 0.4 0.6 0.8 R HomSQ ( ℓ ) = H ( S 1 )+ H ( S 2 ) = 2 H ( s ) ℓ Problem is non-convex Theorem If ℓ is an optimal ECSQ, then there exists µ ≥ 0 and µ R ≥ 0 such that �� ℓ k +1 � � � p k +1 f X ( ℓ k ) ( ℓ k − x ) f X ( x ) d x + 2 µ R log 2 − µ k + µ k +1 = 0 p k ℓ k − 1 µ k ( ℓ k − 1 − ℓ k ) = 0 and µ R ( R HomSQ ( ℓ ) − R 0 ) = 0 . B. D. Boyle (Drexel MANL) DSQ CISS 2014 13 / 25
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
Optimal Scalar Quantizer Design Heterogeneous Scalar Quantizers From HomSQs to HetSQs Some Insights HomSQ 1 HomSQ 2 Deterministic HomSQ X 2 X 2 X 2 2 2 2 ? 2 2 2 1 2 ? 2 2 ? 1 2 2 1 1 2 ? 1 1 2 1 1 1 ? 1 ? 1 1 1 1 1 1 1 X 1 X 1 X 1 • arg max not symmetric • Obvious for S 1 � = S 2 • Distortion is • Flip a coin for S 1 = S 2 • Same distortion for a fixed • Distortion only along diagonal mapping along diagonal B. D. Boyle (Drexel MANL) DSQ CISS 2014 15 / 25
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 Mapping Two X 2 X 2 2 2 2 2 2 2 2 1 2 2 2 1 2 2 1 1 2 1 1 1 2 2 1 1 1 1 1 1 2 1 1 1 X 1 X 1 R (1) HetSQ ≤ R (1) R (2) HetSQ ≤ R (2) HomSQ 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
Optimal Scalar Quantizer Design Heterogeneous Scalar Quantizers Staggered HetSQ Staggered Mapping Theorem X 2 For a HetSQ, if there exists an quantization interval for a user that 2 2 2 2 is completely contained in the quantization interval for another 2 2 1 1 user, then the quantizer is not 2 2 1 1 optimal. 1 1 1 1 Design of HetSQ X 1 1: Select the total # of bins K 2: Design optimal HomSQ R (1) HetSQ ≤ R (1) boundaries ℓ HomSQ HomSQ 3: Assign ℓ ( k ) R (2) HetSQ ≤ R (2) HomSQ to MS1 if k odd; HomSQ else, MS2 B. D. Boyle (Drexel MANL) DSQ CISS 2014 17 / 25
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
Results Example 1 Uniform ( a , b ) Channel Capacity For k = 1 , . . . , K − 1, the optimal quantizer is given as k = aK + ( b − a ) k ℓ ∗ , µ ∗ k = 0 K Uniform(0 , 1) & K = 1 , . . . , 6 HetSQ— No free lunch ; consider 5 HomSQ • K = 3: 42 . 1% fewer bits EC HomSQ 4 Total Rate [bits] HetSQ • MS2 scheduled w.p. 0 . 556 3 • R (2) HetSQ = R (1) HetSQ 2 • K = 4, 37 . 5% fewer bits 1 • MS2 scheduled w.p. 0 . 500 • R (2) HetSQ = 0 . 67 R (1) 0 HetSQ 0.01 0.1 Distortion [bits] B. D. Boyle (Drexel MANL) DSQ CISS 2014 19 / 25
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