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

R1 < 1 2 log

• 1 + P1

N

• R2 < 1

2 log

• 1 + P2

N

• R1 + R2 < 1

2 log

• 1 + P1 + P2

N

• w1

E1 x1 w2 E2 x2 z y D ˆ w1 ˆ w2 Power constraints P1, P2. Noise variance N.

Successive Cancellation

R2 R1 1 2 log

• 1 +

P1 N + P2

• , 1

2 log

• 1 + P2

N

• Corner Point
• 1. Decode x1, treating x2 as noise.
• 2. Subtract x1 from y.
• 3. Decode x2.

Lattice Achievability “Recipe” – Multiple-Access Corner Point

Codebook Generation Select a nested lattice code:

• Coarse lattice Λ = BZn for shaping.
• Fine lattice from q-ary linear code G

for coding. Encoding Tx 1 Tx 2

Lattice Achievability “Recipe” – Multiple-Access Corner Point

Codebook Generation Select a nested lattice code:

• Coarse lattice Λ = BZn for shaping.
• Fine lattice from q-ary linear code G

for coding. Encoding

• Map messages w1, w2 to lattice

points t1, t2. t1 = [BγGw1] mod Λ Tx 1 t2 = [BγGw2] mod Λ Tx 2

Lattice Achievability “Recipe” – Multiple-Access Corner Point

Codebook Generation Select a nested lattice code:

• Coarse lattice Λ = BZn for shaping.
• Fine lattice from q-ary linear code G

for coding. Encoding

• Map messages w1, w2 to lattice

points t1, t2.

• Choose independent dithers d1, d2

uniformly over Voronoi region V. t1 = [BγGw1] mod Λ Tx 1 t2 = [BγGw2] mod Λ Tx 2

Lattice Achievability “Recipe” – Multiple-Access Corner Point

Codebook Generation Select a nested lattice code:

• Coarse lattice Λ = BZn for shaping.
• Fine lattice from q-ary linear code G

for coding. Encoding

• Map messages w1, w2 to lattice

points t1, t2.

• Choose independent dithers d1, d2

uniformly over Voronoi region V.

• Add dithers to lattice points and

take mod Λ to get transmitted signals x1, x2. t1 = [BγGw1] mod Λ x1 = [t1 + d1] mod Λ Tx 1 t2 = [BγGw2] mod Λ x2 = [t1 + d2] mod Λ Tx 2

Lattice Achievability “Recipe” – Multiple-Access Corner Point

Codebook Generation Select a nested lattice code:

• Coarse lattice Λ = BZn for shaping.
• Fine lattice from q-ary linear code G

for coding. Encoding

• Map messages w1, w2 to lattice

points t1, t2.

• Choose independent dithers d1, d2

uniformly over Voronoi region V.

• Add dithers to lattice points and

take mod Λ to get transmitted signals x1, x2. t1 = [BγGw1] mod Λ x1 = [t1 + d1] mod Λ Tx 1 t2 = [BγGw2] mod Λ x2 = [t1 + d2] mod Λ Tx 2

Lattice Achievability “Recipe” – Multiple-Access Corner Point

Codebook Generation Select a nested lattice code:

• Coarse lattice Λ = BZn for shaping.
• Fine lattice from q-ary linear code G

for coding. Encoding

• Map messages w1, w2 to lattice

points t1, t2.

• Choose independent dithers d1, d2

uniformly over Voronoi region V.

• Add dithers to lattice points and

take mod Λ to get transmitted signals x1, x2. t1 = [BγGw1] mod Λ x1 = [t1 + d1] mod Λ Tx 1 t2 = [BγGw2] mod Λ x2 = [t1 + d2] mod Λ Tx 2

Lattice Achievability “Recipe” – Multiple-Access Corner Point

Receiver observes y = x1 + x2 + z. Decoding Rx

Lattice Achievability “Recipe” – Multiple-Access Corner Point

Receiver observes y = x1 + x2 + z. Decoding Rx

Lattice Achievability “Recipe” – Multiple-Access Corner Point

Receiver observes y = x1 + x2 + z. Decoding

• Scale by α.

Rx

Lattice Achievability “Recipe” – Multiple-Access Corner Point

Receiver observes y = x1 + x2 + z. Decoding

• Scale by α.
• Subtract dither d1.

Rx

Lattice Achievability “Recipe” – Multiple-Access Corner Point

Receiver observes y = x1 + x2 + z. Decoding

• Scale by α.
• Subtract dither d1.
• Take mod Λ.

Rx

Lattice Achievability “Recipe” – Multiple-Access Corner Point

Receiver observes y = x1 + x2 + z. Decoding

• Scale by α.
• Subtract dither d1.
• Take mod Λ.
• Decode to nearest codeword.

[αy − d1] mod Λ = [α(x1 + x2 + z) − d1] mod Λ = [x1 − d1 + αz + αx2 − (1 − α)x1] mod Λ =

• [t1 + d1] mod Λ − d1 + αz + αx2 − (1 − α)x1
• mod Λ

= [t1 + αz + αx2 − (1 − α)x1] Effective Noise Rx

Lattice Achievability “Recipe” – Multiple-Access Corner Point

• Effective noise after scaling is NEFFEC = α2(N + P2) + (1 − α)2P1.
• Minimized by setting α to be the MMSE coefficient:

αMMSE = P1 N + P1 + P2

• Plugging in, we get

NEFFEC = (N + P2)P1 N + P1 + P2

• Resulting rate is

R = 1 2 log

• P1

NEFFEC

• = 1

2 log

• 1 +

P1 N + P2

• To obtain different rates for x1 and x2, use nested linear codes G1

and G2 inside Voronoi region V.

AWGN Two-Way Relay Channel – Symmetric Rates

w1

Has Wants w2

w1

Has Wants

w2

Relay

AWGN Two-Way Relay Channel – Symmetric Rates

zMAC yMAC Relay xBC z2 z1 User 1 x1 w1 ˆ w2 User 2 x2 w2 ˆ w1

• Equal power constraints P.
• Equal noise variances N.
• Equal rates R.

AWGN Two-Way Relay Channel – Symmetric Rates

zMAC yMAC Relay xBC z2 z1 User 1 x1 w1 ˆ w2 User 2 x2 w2 ˆ w1

• Equal power constraints P.
• Equal noise variances N.
• Equal rates R.
• Upper Bound:

R ≤ 1 2 log

• 1 + P

N

• Decode-and-Forward: Relay decodes w1, w2 and transmits w1 ⊕ w2.

R = 1 4 log

• 1 + 2P

N

• Compress-and-Forward: Relay transmits quantized y.

R = 1 2 log

• 1 + P

N P 3P + N

AWGN Two-Way Relay Channel – Symmetric Rates

5 10 15 20 0.5 1 1.5 2 2.5 3 3.5 SNR in dB Rate per User Upper Bound Compress Decode

Decoding the Sum of Lattice Codewords

Encoders use the same nested lattice codebook. Transmit lattice codewords: x1 = t1 x2 = t2 t1 E1 x1 t2 E2 x2 z y D ˆ v v = [t1 + t2] mod Λ Decoder recovers modulo sum. [y] mod Λ = [x1 + x2 + z] mod Λ = [t1 + t2 + z] mod Λ =

• [t1 + t2] mod Λ + z
• mod Λ

Distributive Law = [v + z] mod Λ R = 1 2 log P N

Decoding the Sum of Lattice Codewords – MMSE Scaling

Encoders use the same nested lattice codebook. Transmit dithered codewords: x1 = [t1 + d1] mod Λ x2 = [t2 + d2] mod Λ t1 E1 x1 t2 E2 x2 z y D ˆ v v = [t1 + t2] mod Λ Decoder scales by α, removes dithers, recovers modulo sum. [αy − d1 − d2] mod Λ = [α(x1 + x2 + z) − d1 − d2] mod Λ = [x1 + x2 − (1 − α)(x1 + x2) + αz − d1 − d2] mod Λ =

• [t1 + t2] mod Λ − (1 − α)(x1 + x2) + αz
• mod Λ

= [v − (1 − α)(x1 + x2) + αz] mod Λ Effective Noise NEFFEC = (1 − α)22P + α2N

Decoding the Sum of Lattice Codewords – MMSE Scaling

• Effective noise after scaling is NEFFEC = (1 − α)22P + α2N.
• Minimized by setting α to be the MMSE coefficient:

αMMSE = 2P N + 2P

• Plugging in, we get

NEFFEC = 2NP N + 2P

• Resulting rate is

R = 1 2 log

• P

NEFFEC

• = 1

2 log 1 2 + P 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

t1 = φ(w1) = [BγGw1] mod Λ t2 = φ(w2) = [BγGw2] mod Λ

• Mapping between finite field messages and lattice codewords

preserves linearity: φ−1 [t1 + t2] mod Λ

• = w1 ⊕ w2
• 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: t1 = φ(w1) t2 = φ(w2) Transmit dithered codewords: x1 = [t1 + d1] mod Λ x2 = [t2 + d2] mod Λ w1 E1 x1 w2 E2 x2 z y D ˆ u u = w1 ⊕ w2

• If decoder can recover [t1 + t2] mod Λ, it also can get the sum of

the messages w1 ⊕ w2 = φ−1 [t1 + t2] mod Λ

• .
• Achievable rate R = 1

2 log 1 2 + P N

• .

AWGN Two-Way Relay Channel – Symmetric Rates

w1

Has Wants w2

w1

Has Wants

w2

Relay

• Equal power constraints P.
• Equal noise variances N.
• Equal rates R.
• Upper Bound:

R ≤ 1 2 log

• 1 + P

N

• Compute-and-Forward: Relay decodes w1 ⊕ w2 and retransmits.

R = 1 2 log 1 2 + P N

• Wilson-Narayanan-Pfister-Sprintson ’10: Applies nested lattice codes

to the two-way relay channel.

AWGN Two-Way Relay Channel – Symmetric Rates

zMAC yMAC Relay xBC z2 z1 User 1 x1 w1 ˆ w2 User 2 x2 w2 ˆ w1

• Equal power constraints P.
• Equal noise variances N.
• Equal rates R.
• Upper Bound:

R ≤ 1 2 log

• 1 + P

N

• Compute-and-Forward: Relay decodes w1 ⊕ w2 and retransmits.

R = 1 2 log 1 2 + P N

• Wilson-Narayanan-Pfister-Sprintson ’10: Applies nested lattice codes

to the two-way relay channel.

AWGN Two-Way Relay Channel – Symmetric Rates

5 10 15 20 0.5 1 1.5 2 2.5 3 3.5 SNR in dB Rate per User Upper Bound Compute Compress Decode

Compute-and-Forward Illustration

w2 w1 x1 x2 z y w1 ⊕ w2

Compute-and-Forward Illustration

w2 w1 x1 x2 z y w1 ⊕ w2

Random i.i.d. codes are not good for computation

2nR codewords each. 2n2R possible sums of codewords.

Random i.i.d. codes are not good for computation

2nR codewords each. 2n2R possible sums of codewords. x1 x2 z y

Random i.i.d. codes are not good for computation

2nR codewords each. 2n2R possible sums of codewords. x1 x2 z y

Random i.i.d. codes are not good for computation

2nR codewords each. 2n2R possible sums of codewords. x1 x2 z y

Multiple-Access Networks

w Z2 Z1 Z3 ˆ w ˆ w ˆ w

• Multicast

demands

• Multi-access

interference

• No broadcast

constraints

• Compute-and-forward is well-suited for multicasting over

multiple-access networks.

• Equal transmitter powers: Nazer-Gastpar ’07.

Unequal transmitter powers: Nam-Chung-Lee ’09.

Exercise: Sum-Difference Network

x1 x2 z1 y1 z2 y2 x1 x2 Find the achievable rate for compute-and-forward. What is the requirement on the backhaul rates?

Outline

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

Dirty Paper Coding

s is interference known noncausally to the encoder. Assume s i.i.d. Gaussian, very large variance PS.

Erez-Shamai-Zamir ’05:

Encoder subtracts αs, dithers, and takes mod Λ. x = [t − αs + d] mod Λ w E x s z y D ˆ w Decoder scales by α, removes dither, takes mod Λ, and recovers t. Interference is cancelled.

[αy − d] mod Λ = [x + αs − d + z − (1 − α)x] mod Λ =

• [t − αs + d] mod Λ + αs − d + z − (1 − α)x
• mod Λ

=

• t + z − (1 − α)x
• mod Λ

Dirty Paper Coding

s is interference known noncausally to the encoder. Assume s i.i.d. Gaussian, very large variance PS.

Erez-Shamai-Zamir ’05:

Encoder subtracts αs, dithers, and takes mod Λ. x = [t − αs + d] mod Λ w E x s z y D ˆ w Decoder scales by α, removes dither, takes mod Λ, and recovers t. Interference is cancelled.

[αy − d] mod Λ = [x + αs − d + z − (1 − α)x] mod Λ =

• [t − αs + d] mod Λ + αs − d + z − (1 − α)x
• mod Λ

=

• t + z − (1 − α)x
• mod Λ

Dirty Paper Coding

s is interference known noncausally to the encoder. Assume s i.i.d. Gaussian, very large variance PS.

Erez-Shamai-Zamir ’05:

Encoder subtracts αs, dithers, and takes mod Λ. x = [t − αs + d] mod Λ w E x s z y D ˆ w Decoder scales by α, removes dither, takes mod Λ, and recovers t. Interference is cancelled.

[αy − d] mod Λ = [x + αs − d + z − (1 − α)x] mod Λ =

• [t − αs + d] mod Λ + αs − d + z − (1 − α)x
• mod Λ

=

• t + z − (1 − α)x
• mod Λ

Dirty Paper Coding

s is interference known noncausally to the encoder. Assume s i.i.d. Gaussian, very large variance PS.

Erez-Shamai-Zamir ’05:

Encoder subtracts αs, dithers, and takes mod Λ. x = [t − αs + d] mod Λ w E x s z y D ˆ w Decoder scales by α, removes dither, takes mod Λ, and recovers t. Interference is cancelled.

[αy − d] mod Λ = [x + αs − d + z − (1 − α)x] mod Λ =

• [t − αs + d] mod Λ + αs − d + z − (1 − α)x
• mod Λ

=

• t + z − (1 − α)x
• mod Λ

Dirty Gaussian Multiple-Access Channel

w1 E1 x1 s1 w2 E2 x2 s2 z y D ˆ w1, ˆ w2

Philosof-Zamir-Erez-Khisti ’11:

• Encoder 1 knows interference s1.
• Encoder 2 knows interference s2.
• Need to cancel out interference in a distributed fashion.
• Assume i.i.d. Gaussian interference with very large variance PS.

Random i.i.d. methods yield rate that goes to 0 as PS goes to infinity.

Exercise: Dirty Gaussian Multiple-Access Channel

Subtract (part of) the interference signals ahead of time: x1 = [t1 − αs1 + d1] mod Λ x2 = [t2 − αs2 + d2] mod Λ

Exercise: Dirty Gaussian Multiple-Access Channel

Subtract (part of) the interference signals ahead of time: x1 = [t1 − αs1 + d1] mod Λ x2 = [t2 − αs2 + d2] mod Λ Decoder removes dithers: [αy − d1 − d2] mod Λ = [α(x1 + x2 + s1 + s2 + z) − d1 − d2] mod Λ = [x1 + x2 + α(s1 + s2) − (1 − α)(x1 + x2) + αz) − d1 − d2] mod Λ =

• t1 + t2 + (1 − α)(x1 + x2) + αz
• mod Λ

Exercise: Dirty Gaussian Multiple-Access Channel

Subtract (part of) the interference signals ahead of time: x1 = [t1 − αs1 + d1] mod Λ x2 = [t2 − αs2 + d2] mod Λ Decoder removes dithers: [αy − d1 − d2] mod Λ = [α(x1 + x2 + s1 + s2 + z) − d1 − d2] mod Λ = [x1 + x2 + α(s1 + s2) − (1 − α)(x1 + x2) + αz) − d1 − d2] mod Λ =

• t1 + t2 + (1 − α)(x1 + x2) + αz
• mod Λ

Select α = 2P/(2P + N) to obtain R1 + R2 ≤

• 1

2 log 1 2 + P N +

Computation over Fading Channels

Transmitters do not know channel realization. Encoders use the same nested lattice codebook. Transmit dithered codewords: xℓ = [tℓ + dℓ] mod Λ t1 E1 x1 h1 t2 E2 x2 h2 tK EK xK hK . . . z y D ˆ v

v =

• K
• ℓ=1

aℓtℓ

• mod Λ
• Decoder removes dithers and recovers integer combination

v = K

• ℓ=1

aℓtℓ

• mod Λ
• Receiver can use its knowledge of the channel gains to match the

equation coefficients aℓ to the channel coefficients hℓ.

Distributive Law

• Distributive Law also holds for integer combinations. Let a, b ∈ Z.
• a[x1] mod Λ + b[x2] mod Λ
• mod Λ

=

• a
• x1 − QΛ(x1)
• + b
• x2 − QΛ(x2)
• mod Λ

=

• ax1 + bx2 − aQΛ(x1) − bQΛ(x2)
• mod Λ

= [ax1 + bx2] mod Λ

• Last step follows since since aQΛ(x1) and bQΛ(x2) are elements of

the lattice Λ.

Computation over Fading Channels

• Transmit dithered codewords xℓ = [tℓ + dℓ] mod Λ
• Decoder removes dithers and recovers integer combination
• y −

K

• ℓ=1

aℓdℓ

• mod Λ

= K

• ℓ=1

hℓxℓ + z −

K

• ℓ=1

aℓdℓ

• mod Λ

= K

• ℓ=1

aℓ(xℓ − dℓ) +

K

• ℓ=1

(hℓ − aℓ)xℓ + z

• mod Λ

= K

• ℓ=1

aℓtℓ

• mod Λ +

K

• ℓ=1

(hℓ − aℓ)xℓ + z

• mod Λ

Distributive Law Effective Noise

Computation over Fading Channels – Effective Noise

• Effective noise due to mismatch between channel coefficients

h = [h1 · · · hK]T and equation coefficients a = [a1 · · · aK]T . NEFFEC = N + Ph − a2 R = 1 2 log

• P

N + Ph − a2

Computation over Fading Channels – Effective Noise

• Effective noise due to mismatch between channel coefficients

h = [h1 · · · hK]T and equation coefficients a = [a1 · · · aK]T . NEFFEC = N + Ph − a2 R = 1 2 log

• P

N + Ph − a2

• Can do better with MMSE scaling.

NEFFEC = α2N + Pαh − a2 R = max

α

1 2 log

• P

α2N + Pαh − a2

• = 1

2 log

• N + Ph2

Na2 + P(h2a2 − (hT a)2)

• See Nazer-Gastpar ’11 for more details.

Computation over Fading Channels – Special Cases

• The rate expression simplifies in some special cases.

R = 1 2 log

• N + Ph2

Na2 + P(h2a2 − (hT a)2)

• Integer channels: h = a.

R = 1 2 log

• 1

a2 + P N

• Recovering a single message: Set a = δm, the mth unit vector.

R = 1 2 log

• 1 +

h2

mP

N + P

ℓ=m h2 ℓ

Finite Field Computation over Fading Channels

Transmitters do not know channel realization. Encoders use the same nested lattice codebook. Transmit dithered codewords: xℓ = [tℓ + dℓ] mod Λ w1 E1 x1 h1 w2 E2 x2 h2 wK EK xK hK . . . z y D ˆ u u =

K

• ℓ=1

aℓwℓ

• Recall that mapping tℓ = φ(wℓ) between messages and lattice

points preserves linearity. φ−1 K

• ℓ=1

aℓtℓ

• mod Λ
• =

K

• ℓ=1

aℓwℓ

• mod q =

K

• ℓ=1

aℓwℓ

• Digital interface that fits well with network coding.

Computation Coding

All users pick the same nested lattice code:

Computation Coding

Choose messages over field wℓ ∈ Fk

q:

w2 w1

Computation Coding

Map wℓ to lattice point tℓ = φ(wℓ): w2 w1

Computation Coding

Transmit lattice points over the channel: w2 w1 x1 h1 x2 h2 z y h = [ 1.4 2.1 ] a = [ 2 3 ]

Computation Coding

Transmit lattice points over the channel: w2 w1 x1 h1 x2 h2 z y h = [ 1.4 2.1 ] a = [ 2 3 ]

Computation Coding

Lattice codewords are scaled by channel coefficients: w2 w1 x1 h1 x2 h2 z y h = [ 1.4 2.1 ] a = [ 2 3 ]

Computation Coding

Scaled codewords added together plus noise: w2 w1 x1 h1 x2 h2 z y h = [ 1.4 2.1 ] a = [ 2 3 ]

Computation Coding

Scaled codewords added together plus noise: w2 w1 x1 h1 x2 h2 z y h = [ 1.4 2.1 ] a = [ 2 3 ]

Computation Coding

Extra noise penalty for non-integer channel coefficients: w2 w1 x1 h1 x2 h2 z y h = [ 1.4 2.1 ] a = [ 2 3 ] Effective noise: N + Ph − a2

Computation Coding

Scale output by α to reduce non-integer noise penalty: w2 w1 x1 h1 x2 h2 z y αh = [ α1.4 α2.1 ] a = [ 2 3 ] Effective noise: α2N + Pαh − a2

Computation Coding

Scale output by α to reduce non-integer noise penalty: w2 w1 x1 h1 x2 h2 z y αh = [ α1.4 α2.1 ] a = [ 2 3 ] Effective noise: α2N + Pαh − a2

Computation Coding

Decode to closest lattice point: w2 w1 x1 h1 x2 h2 z y αh = [ α1.4 α2.1 ] a = [ 2 3 ] Effective noise: α2N + Pαh − a2

Computation Coding

Compute sum of lattice points modulo the coarse lattice: w2 w1 x1 h1 x2 h2 z y αh = [ α1.4 α2.1 ] a = [ 2 3 ] Effective noise: α2N + Pαh − a2

Computation Coding

Map back to equation of message symbols over the field: w2 w1 x1 h1 x2 h2 z y αh = [ α1.4 α2.1 ] a = [ 2 3 ] Effective noise: α2N + Pαh − a2

K

• ℓ=1

aℓwℓ

Computation over Fading Channels – Multiple Receivers

w1 E1 x1 w2 E2 x2 . . . wK EK xK

H

z1 y1 z2 y2 zK yK D1 ˆ u1 D2 ˆ u2 . . . DK ˆ uK

• Equal rates R. No channel state information (CSI) at transmitters.
• Receivers use their CSI to select coefficients, decode linear equation

uk =

K

• ℓ=1

akℓwℓ

• Reliable decoding possible if

R < min

k:akℓ=0

1 2 log

• N + Phk2

Nak2 + P(hk2ak2 − (hT

k ak)2)

Case Study – Hadamard Relay Network

w1 E1 x1 w2 E2 x2 . . . wK EK xK

H

z1 y1 z2 y2 zK yK R1 x1R R2 x2R . . . RK xKR z1R y1R z2R y2R zKR yKR D ˆ w1 ˆ w2 . . . ˆ wK

• Equal rates R. H is a Hadamard matrix, HHT = KI

Upper Bound Compute-and-Forward 1 2 log

• 1 + P

N

• 1

2 log 1 K + P N

• Compress-and-Forward

Decode-and-Forward 1 2 log

• 1 + P

N P N + KP

• 1

2K log

• 1 + KP

N

Case Study – Hadamard Relay Network

w1 E1 x1 w2 E2 x2 . . . wK EK xK

1 1 · · · 1 1 1 · · · −1 . . . . . . ... . . . 1 −1 · · · −1

z1 y1 z2 y2 zK yK R1 x1R R2 x2R . . . RK xKR z1R y1R z2R y2R zKR yKR D ˆ w1 ˆ w2 . . . ˆ wK

• Equal rates R. H is a Hadamard matrix, HHT = KI

Upper Bound Compute-and-Forward 1 2 log

• 1 + P

N

• 1

2 log 1 K + P N

• Compress-and-Forward

Decode-and-Forward 1 2 log

• 1 + P

N P N + KP

• 1

2K log

• 1 + KP

N

Computation over Fading Channels – No CSIT

w1 E1 x1 h1 w2 E2 x2 h2 w3 E3 x3 h3 z y D ˆ u u =

K

• ℓ=1

aℓwℓ

• Three transmitters that

do not know the fading coefficients.

• Average rate plotted for

i.i.d. Gaussian fading. Relay either decodes some linear function of messages

• r an individual message.

5 10 15 20 25 0.5 1 1.5 2 2.5 Transmitter Power in dB Averate Rate in bits per channel use Decode an Equation Decode a Message Interference as Noise

Computation over Fading Channels – No CSIT

• Receiver observes y = x1 + hx2 + z.
• Recovers aw1 ⊕ bw2 for a, b = 0.

10dB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Channel coefficient h Message rate R

Upper Bound Compute Decode Both

Computation over Fading Channels – No CSIT

• Receiver observes y = x1 + hx2 + z.
• Recovers aw1 ⊕ bw2 for a, b = 0.

20dB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3 3.5

Channel coefficient h Message rate R

Upper Bound Compute Decode Both

Computation over Fading Channels – No CSIT

• Receiver observes y = x1 + hx2 + z.
• Recovers aw1 ⊕ bw2 for a, b = 0.

30dB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Channel coefficient h Message rate R

Upper Bound Compute Decode Both

Computation over Fading Channels – No CSIT

• Receiver observes y = x1 + hx2 + z.
• Recovers aw1 ⊕ bw2 for a, b = 0.

40dB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7

Channel coefficient h Message rate R

Upper Bound Compute Decode Both

Computation over Fading Channels – No CSIT

• Receiver observes y = x1 + hx2 + z.
• Recovers aw1 ⊕ bw2 for a, b = 0.

50dB

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9

Channel coefficient h Message rate R

Upper Bound Compute Decode Both

Diophantine Approximation

• Choose equation coefficients to maximize rate:

RCOMP = max

a∈ZK max α

1 2 log

• P

α2N + Pαh − a2

• Equivalently min

a∈ZK min α α2N + Pαh − a2.

• Closely connected to Diophantine approximation, i.e. approximating

irrationals with rationals.

Computation over Fading Channels – Multiple Receivers

w1 E1 x1 w2 E2 x2 . . . wK EK xK

H

z1 y1 z2 y2 zK yK D1 ˆ u1 D2 ˆ u2 . . . DK ˆ uK

• Equal rates R. No channel state information (CSI) at transmitters.
• Receivers use their CSI to select coefficients, decode linear equation

uk =

K

• ℓ=1

akℓwℓ

• Reliable decoding possible if

R < min

k:akℓ=0

1 2 log

• N + Phk2

Nak2 + P(hk2ak2 − (hT

k ak)2)

Diophantine Approximation

• For each receiver, choose equation coefficients to maximize rate:

RCOMP = max

a∈ZK max α

1 2 log

• P

α2N + Pαh − a2

• Equivalently min

a∈ZK min α α2N + Pαh − a2.

• Closely connected to Diophantine approximation, i.e. approximating

irrationals with rationals.

• When requiring the K linear combinations decoded by the K

receivers to be linearly independent, Niesen-Whiting ’11 shows that DoF = lim

P →∞

RCOMP

1 2 log(1 + P) ≤ 2

• Also shows that by combining compute-and-forward with

interference alignment can get DoF to K (but this requires channel state information at the transmitters).

Static Linear Pre-processing: Integer-forcing Receiver

w1 E1 x1 w2 E2 x2 . . . wK EK xK

H

z1 y1 z2 y2 zK yK D1 ˆ u1 D2 ˆ u2 . . . DK ˆ uK

Static Linear Pre-processing: Integer-forcing Receiver

w1 E1 x1 w2 E2 x2 . . . wK EK xK

H

z1 y1 z2 y2 zK yK

B

D1 ˆ u1 D2 ˆ u2 . . . DK ˆ uK

Integer-forcing: Achievable rate

Theorem

Consider the MIMO channel with channel matrix H ∈ R2N×2M. Under the integer-forcing architecture, the following rate is achievable: R < min

m 2MR(H, am, bm)

R(H, am, bm) = 1 2 log+

• SNR

bm2 + SNRHT bm − am2

• for any full-rank integer matrix A ∈ Z2M×2M and any matrix

B ∈ R2M×2N.

Integer-forcing: Diophantine Approximation Problem

Example: H = 0.7 1.3 0.8 1.5

• .

We find the eigenvalues and eigenvectors to be λ1 ≈ 2.1954 v1 ≈ (−0.6561, −0.7547)T λ2 ≈ 0.0046 v2 ≈ (−0.8818, 0.4717)T

Integer-forcing: Diophantine Approximation Problem

Now, there is more wiggle room in the diophantine approximation...

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

1 λMAX vMAX 1 λMIN vMIN

a1 a2

b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b

1 λMAX vMAX 1 λMIN vMIN

a2 a1

Integer-forcing: Diophantine Approximation Problem

10 20 30 40 50 1 2 3 4 5 6 SNR (dB) Achievable Rate (bits per real symbol) Joint ML Integer MMSE−SIC MMSE Decorrelator

High-SNR (Ex: DMT for 4 × 4 with Rayleigh fading)

1 2 3 4 0.5 1 1.5 2 2.5 3 3.5 4 diversity gain d multiplexing gain r Joint ML Integer V−BLAST III V−BLAST II V−BLAST I Decorrelator

Concluding Remarks

• Algebraic structure appears to play a role in Network Information

Theory (good news or bad news?)

• In particular, codes with algebraic structure lead to the highest

known achievable rates for some communication scenarios of great interest.

• We have only considered linear/lattice codes in the

“compute-and-forward” channel coding perspective.

• Similar insights apply to source coding and to joint source-channel

coding.

• Another (though related) form of algebraic structure appears in

interference alignment.

Resources

For an extended list of references (as well as proofs), see

• B. Nazer and M. Gastpar, “Computation over multiple-access

channels,” IEEE Transactions on Information Theory, vol. 53,

• no. 10, pp. 3498–3516, Oct. 2007.
• B. Nazer and M. Gastpar, “Reliable physical layer network coding,”

Proceedings of the IEEE, vol. 99, no. 3, pp. 438–460, March 2011.

• B. Nazer and M. Gastpar, “Compute-and-forward: Harnessing

interference through structured codes,” IEEE Transactions on Information Theory, vol. 57, no. 10, pp. 6463-6486, Oct. 2011.