Outline I. Discrete Alphabets II. AWGN Channels III. Network - PowerPoint PPT Presentation

Outline I. Discrete Alphabets II. AWGN Channels III. Network Applications

Gaussian Multiple-Access Channel Rate Region x 1 z w 1 E 1 R 1 < 1 � � 1 + P 1 2 log y w 1 ˆ N D w 2 ˆ � � R 2 < 1 1 + P 2 x 2 2 log w 2 E 2 N R 1 + R 2 < 1 � 1 + P 1 + P 2 � 2 log Power constraints P 1 , P 2 . Noise variance N . N Successive Cancellation � 1 � � � �� P 1 , 1 1 + P 2 R 2 2 log 1 + 2 log N + P 2 N Corner Point 1. Decode x 1 , treating x 2 as noise. 2. Subtract x 1 from y . 3. Decode x 2 . R 1

Lattice Achievability “Recipe” – Multiple-Access Corner Point Codebook Generation Tx 1 Select a nested lattice code: • Coarse lattice Λ = B Z n for shaping. • Fine lattice from q -ary linear code G for coding. Encoding Tx 2

Lattice Achievability “Recipe” – Multiple-Access Corner Point Codebook Generation Tx 1 Select a nested lattice code: • Coarse lattice Λ = B Z n for shaping. • Fine lattice from q -ary linear code G for coding. t 1 = [ B γ Gw 1 ] mod Λ Encoding • Map messages w 1 , w 2 to lattice Tx 2 points t 1 , t 2 . t 2 = [ B γ Gw 2 ] mod Λ

Lattice Achievability “Recipe” – Multiple-Access Corner Point Codebook Generation Tx 1 Select a nested lattice code: • Coarse lattice Λ = B Z n for shaping. • Fine lattice from q -ary linear code G for coding. t 1 = [ B γ Gw 1 ] mod Λ Encoding • Map messages w 1 , w 2 to lattice Tx 2 points t 1 , t 2 . • Choose independent dithers d 1 , d 2 uniformly over Voronoi region V . t 2 = [ B γ Gw 2 ] mod Λ

Lattice Achievability “Recipe” – Multiple-Access Corner Point Codebook Generation Tx 1 Select a nested lattice code: • Coarse lattice Λ = B Z n for shaping. • Fine lattice from q -ary linear code G for coding. t 1 = [ B γ Gw 1 ] mod Λ x 1 = [ t 1 + d 1 ] mod Λ Encoding • Map messages w 1 , w 2 to lattice Tx 2 points t 1 , t 2 . • Choose independent dithers d 1 , d 2 uniformly over Voronoi region V . • Add dithers to lattice points and t 2 = [ B γ Gw 2 ] mod Λ take mod Λ to get transmitted x 2 = [ t 1 + d 2 ] mod Λ signals x 1 , x 2 .

Lattice Achievability “Recipe” – Multiple-Access Corner Point Codebook Generation Tx 1 Select a nested lattice code: • Coarse lattice Λ = B Z n for shaping. • Fine lattice from q -ary linear code G for coding. t 1 = [ B γ Gw 1 ] mod Λ x 1 = [ t 1 + d 1 ] mod Λ Encoding • Map messages w 1 , w 2 to lattice Tx 2 points t 1 , t 2 . • Choose independent dithers d 1 , d 2 uniformly over Voronoi region V . • Add dithers to lattice points and t 2 = [ B γ Gw 2 ] mod Λ take mod Λ to get transmitted x 2 = [ t 1 + d 2 ] mod Λ signals x 1 , x 2 .

Lattice Achievability “Recipe” – Multiple-Access Corner Point Codebook Generation Tx 1 Select a nested lattice code: • Coarse lattice Λ = B Z n for shaping. • Fine lattice from q -ary linear code G for coding. t 1 = [ B γ Gw 1 ] mod Λ x 1 = [ t 1 + d 1 ] mod Λ Encoding • Map messages w 1 , w 2 to lattice Tx 2 points t 1 , t 2 . • Choose independent dithers d 1 , d 2 uniformly over Voronoi region V . • Add dithers to lattice points and t 2 = [ B γ Gw 2 ] mod Λ take mod Λ to get transmitted x 2 = [ t 1 + d 2 ] mod Λ signals x 1 , x 2 .

Lattice Achievability “Recipe” – Multiple-Access Corner Point Receiver observes y = x 1 + x 2 + z . Decoding Rx

Lattice Achievability “Recipe” – Multiple-Access Corner Point Receiver observes y = x 1 + x 2 + z . Decoding Rx

Lattice Achievability “Recipe” – Multiple-Access Corner Point Receiver observes y = x 1 + x 2 + z . Decoding Rx • Scale by α .

Lattice Achievability “Recipe” – Multiple-Access Corner Point Receiver observes y = x 1 + x 2 + z . Decoding Rx • Scale by α . • Subtract dither d 1 .

Lattice Achievability “Recipe” – Multiple-Access Corner Point Receiver observes y = x 1 + x 2 + z . Decoding Rx • Scale by α . • Subtract dither d 1 . • Take mod Λ .

Lattice Achievability “Recipe” – Multiple-Access Corner Point Receiver observes y = x 1 + x 2 + z . Decoding Rx • Scale by α . • Subtract dither d 1 . • Take mod Λ . • Decode to nearest codeword. [ α y − d 1 ] mod Λ = [ α ( x 1 + x 2 + z ) − d 1 ] mod Λ = [ x 1 − d 1 + α z + α x 2 − (1 − α ) x 1 ] mod Λ � � = [ t 1 + d 1 ] mod Λ − d 1 + α z + α x 2 − (1 − α ) x 1 mod Λ = [ t 1 + α z + α x 2 − (1 − α ) x 1 ] Effective Noise

Lattice Achievability “Recipe” – Multiple-Access Corner Point • Effective noise after scaling is N EFFEC = α 2 ( N + P 2 ) + (1 − α ) 2 P 1 . • Minimized by setting α to be the MMSE coefficient: P 1 α MMSE = N + P 1 + P 2 • Plugging in, we get N EFFEC = ( N + P 2 ) P 1 N + P 1 + P 2 • Resulting rate is R = 1 � P 1 � = 1 � P 1 � 2 log 2 log 1 + N EFFEC N + P 2 • To obtain different rates for x 1 and x 2 , use nested linear codes G 1 and G 2 inside Voronoi region V .

AWGN Two-Way Relay Channel – Symmetric Rates Has w 1 Relay Has w 2 Wants w 2 Wants w 1

AWGN Two-Way Relay Channel – Symmetric Rates z MAC x 1 x 2 w 1 w 2 y MAC • Equal power constraints P . User 1 User 2 • Equal noise variances N . Relay • Equal rates R . x BC w 2 ˆ w 1 ˆ z 1 z 2

AWGN Two-Way Relay Channel – Symmetric Rates z MAC x 1 x 2 w 1 w 2 y MAC • Equal power constraints P . User 1 User 2 • Equal noise variances N . Relay • Equal rates R . x BC w 2 ˆ w 1 ˆ z 1 z 2 • Upper Bound: R ≤ 1 � 1 + P � 2 log N • Decode-and-Forward: Relay decodes w 1 , w 2 and transmits w 1 ⊕ w 2 . R = 1 � 1 + 2 P � 4 log N • Compress-and-Forward: Relay transmits quantized y . R = 1 � 1 + P P � 2 log N 3 P + N

AWGN Two-Way Relay Channel – Symmetric Rates 3.5 Upper Bound 3 Compress 2.5 Decode Rate per User 2 1.5 1 0.5 0 0 5 10 15 20 SNR in dB

Decoding the Sum of Lattice Codewords Encoders use the same nested x 1 z lattice codebook. t 1 E 1 y Transmit lattice codewords: D v ˆ x 1 = t 1 x 2 t 2 E 2 v = [ t 1 + t 2 ] mod Λ x 2 = t 2 Decoder recovers modulo sum. [ y ] mod Λ = [ x 1 + x 2 + z ] mod Λ = [ t 1 + t 2 + z ] mod Λ � � = [ t 1 + t 2 ] mod Λ + z mod Λ Distributive Law = [ v + z ] mod Λ � P R = 1 � 2 log N

Decoding the Sum of Lattice Codewords – MMSE Scaling Encoders use the same nested x 1 z lattice codebook. t 1 E 1 Transmit dithered codewords: y D v ˆ x 1 = [ t 1 + d 1 ] mod Λ x 2 t 2 E 2 v = [ t 1 + t 2 ] mod Λ x 2 = [ t 2 + d 2 ] mod Λ Decoder scales by α , removes dithers, recovers modulo sum. [ α y − d 1 − d 2 ] mod Λ = [ α ( x 1 + x 2 + z ) − d 1 − d 2 ] mod Λ = [ x 1 + x 2 − (1 − α )( x 1 + x 2 ) + α z − d 1 − d 2 ] mod Λ � � = [ t 1 + t 2 ] mod Λ − (1 − α )( x 1 + x 2 ) + α z mod Λ = [ v − (1 − α )( x 1 + x 2 ) + α z ] mod Λ N EFFEC = (1 − α ) 2 2 P + α 2 N Effective Noise

Decoding the Sum of Lattice Codewords – MMSE Scaling • Effective noise after scaling is N EFFEC = (1 − α ) 2 2 P + α 2 N . • Minimized by setting α to be the MMSE coefficient: 2 P α MMSE = N + 2 P • Plugging in, we get 2 NP N EFFEC = N + 2 P • Resulting rate is R = 1 � P � = 1 � 1 2 + P � 2 log 2 log N EFFEC N • Getting the full “one plus” term is an open challenge. Does not seem possible with nested lattices.

From Messages to Lattice Points and Back • Map messages to lattice points t 1 = φ ( w 1 ) = [ B γ Gw 1 ] mod Λ t 2 = φ ( w 2 ) = [ B γ Gw 2 ] mod Λ • Mapping between finite field messages and lattice codewords preserves linearity: φ − 1 � � [ t 1 + t 2 ] mod Λ = w 1 ⊕ w 2 • This means that after decoding a mod Λ equation of lattice points we can immediately recover the finite field equation of the messages. See Nazer-Gastpar ’11 for more details.

Finite Field Computation over a Gaussian MAC Map messages to lattice points: t 1 = φ ( w 1 ) x 1 z w 1 E 1 t 2 = φ ( w 2 ) y u ˆ D Transmit dithered codewords: x 2 w 2 u = w 1 ⊕ w 2 E 2 x 1 = [ t 1 + d 1 ] mod Λ x 2 = [ t 2 + d 2 ] mod Λ • If decoder can recover [ t 1 + t 2 ] mod Λ , it also can get the sum of the messages w 1 ⊕ w 2 = φ − 1 � � [ t 1 + t 2 ] mod Λ . � 1 � • Achievable rate R = 1 2 + P 2 log . N

AWGN Two-Way Relay Channel – Symmetric Rates • Equal power constraints P . Has • Equal noise variances N . Relay w 1 Has w 2 • Equal rates R . Wants w 2 Wants w 1 • Upper Bound: � � R ≤ 1 1 + P 2 log N • Compute-and-Forward: Relay decodes w 1 ⊕ w 2 and retransmits. R = 1 � 1 2 + P � 2 log N • Wilson-Narayanan-Pfister-Sprintson ’10: Applies nested lattice codes to the two-way relay channel.

AWGN Two-Way Relay Channel – Symmetric Rates z MAC x 1 x 2 w 1 w 2 • Equal power constraints P . y MAC User 1 User 2 • Equal noise variances N . Relay • Equal rates R . x BC w 2 ˆ w 1 ˆ z 1 z 2 • Upper Bound: � � R ≤ 1 1 + P 2 log N • Compute-and-Forward: Relay decodes w 1 ⊕ w 2 and retransmits. R = 1 � 1 2 + P � 2 log N • Wilson-Narayanan-Pfister-Sprintson ’10: Applies nested lattice codes to the two-way relay channel.

AWGN Two-Way Relay Channel – Symmetric Rates 3.5 Upper Bound 3 Compute Compress 2.5 Decode Rate per User 2 1.5 1 0.5 0 0 5 10 15 20 SNR in dB

Compute-and-Forward Illustration w 1 x 1 z y w 1 ⊕ w 2 x 2 w 2

Compute-and-Forward Illustration w 1 x 1 z y w 1 ⊕ w 2 x 2 w 2