Compute-and-Forward: Harnessing Interference with ... · Pros and Cons Decode-and-Forward: • Pro:...

ISIT 2008: Compute-and-Forward 1 / 23

Compute-and-Forward: Harnessing Interference with

Structured Codes

Bobak Nazer and Michael Gastpar

Department of Electrical Engineering and Computer SciencesUniversity of California, Berkeley

July 8, 2008

ISIT 2008

UC Berkeley Wireless Foundations Nazer and Gastpar

ISIT 2008: Compute-and-Forward > Motivation 2 / 23

Wireless Relay Network

E1

E2

• Encoders want to send messages




E1

E2

D1

D2

D3

• Encoders want to send messages to decoders




E1

E2

R1

R2

R3

R4

D1

D2

D3


• Assisted by relays




E1

E2

R1

R2

R3

R4

D1

D2

D3


• Assisted by relays

• Wireless channel: interference and noise



Basic Relaying Strategies

Relaying schemes are usually some mixture of:

• Decode-and-Forward: Relays reliably recover transmittedcodewords.(Cover-El Gamal ’79, Kramer-Gastpar-Gupta ’05,

Sanderovich-Somekh-Poor-Shamai ’08, . . . )







• Compress-and-Forward: Relays quantize their noisy observationsand send them towards the destination.(Cover-El Gamal ’79, Kramer-Gastpar-Gupta ’05, Cover-Kim ’07,

Aleksic-Razaghi-Yu ’07, Sanderovich-Somekh-Poor-Shamai ’08, . . . )







• Compress-and-Forward: Relays quantize their noisy observationsand send them towards the destination.(Cover-El Gamal ’79, Kramer-Gastpar-Gupta ’05, Cover-Kim ’07,

Aleksic-Razaghi-Yu ’07, Sanderovich-Somekh-Poor-Shamai ’08, . . . )

• Amplify-and-Forward: Relays retransmit their noisy observations.

(Schein-Gallager ’00, Gastpar-Vetterli ’05, Borade-Zheng-Gallager ’06,

El Gamal-Hassanpour-Mammen ’07, . . . )



Pros and Cons

Decode-and-Forward:

• Pro: Relays remove noise by decoding codewords.



Pros and Cons

Decode-and-Forward:


• Con: If there is more than one transmitter, relays areinterference-limited.

Compress-and-Forward and Amplify-and-Forward:



Pros and Cons

Decode-and-Forward:




• Pro: Interference left for receiver which can treat network as onebig MIMO channel.



Pros and Cons

Decode-and-Forward:





• Con: Relays do not decode so noise builds along the way to thereceiver.



Pros and Cons

Decode-and-Forward:





• Con: Relays do not decode so noise builds along the way to thereceiver.

Can a strategy handle both interference and noise?



Something in Between

Tx1x1

Tx2x2

Channel Rx x1 + x2

• Compute-and-Forward: Relays reliably recover linear functions oftransmitted codewords.

• Nazer-Gastpar IT Trans. Oct. ’07 : Can exploit channels to reliablycompute functions using structured codes (computation coding).

• Example: Multiple-access channel is a noisy sum. Decode just thesum of codewords.



Motivating Example: Sum-Difference Relay Channel

m1 E1

X1

m2 E2

X2

H

Z1

Y1

Z2

Y2

R1

R0

R2

R0

Dm1

m2




m1 E1

X1

m2 E2

X2

[

1 11 −1

]

Z1

Y1

Z2

Y2

R1

R0

R2

R0

Dm1

m2

• Y1 = X1 + X2 + Z1 and Y2 = X1 − X2 + Z2




m1 E1

X1

m2 E2

X2

[

1 11 −1

]

Z1

Y1

Z2

Y2

R1

X1 + X2

R0

R2

X1 − X2

R0

Dm1

m2

• Y1 = X1 + X2 + Z1 and Y2 = X1 − X2 + Z2

• Compute-and-Forward: Relay 1 decodes the sum. Relay 2 decodesthe difference.



“Structural Gain”

• Compute-and-Forwardnearly reaches upperbound

0 2 4 6 8 100

0.5

1

1.5

2

SNR

Rat

e

Upper Bound

Compute−Forward

Compress−Forward

Decode−Forward



Overview

Main ideas for this talk:

=⇒ Reducing a relay network into a system of (reliable) linearequations.



Overview



=⇒ Using a new coding technique, Compute-and-Forward, thatlets relays decode linear functions of codewords.



Overview



=⇒ Using a new coding technique, Compute-and-Forward, thatlets relays decode linear functions of codewords.

=⇒ Based on (algebraically) structured codes (i.e., lattices)that can exploit the structure of the interference andprotect against noise.


ISIT 2008: Compute-and-Forward > Problem Statement 9 / 23

Problem Statement

m1 E1

X1

m2 E2

X2

.

.

.

mM EM

XM

H

Z1

Y1

Z2

Y2

ZM

YM

R1

R0

R2

R0

.

.

.

RM

R0

D

m1

m2.

.

.

mM

• M encoders, M relays, 1 decoder• Usual power constraints: 1

n

∑ni=1

(Xk[i])2 ≤ SNR

• i.i.d. Gaussian noise Zk ∼ N (0, 1)• Each relay observes a noisy linear combination:

Yk[i] =

M∑

j=1

hjkXj [i] + Zk[i]



Oblivious Transmitters

m1 E1

X1

m2 E2

X2

.

.

.

mM EM

XM

H

Z1

Y1

Z2

Y2

ZM

YM

R1

R0

R2

R0

.

.

.

RM

R0

D

m1

m2.

.

.

mM

• Fixed channel matrix H

• Encoders: No channel knowledge

• Relays: Each knows its own coefficients hk = [h1k h2k · · · hMk]

• Decoder: Knows entire channel matrix H



Metrics of Interest

• Rate per User: maximize symmetric rate, R

mj ∈ {1, 2, . . . , 2nR} j = 1, 2, . . . ,M

Pr(mj 6= mj) → 0



Metrics of Interest

• Rate per User: maximize symmetric rate, R

mj ∈ {1, 2, . . . , 2nR} j = 1, 2, . . . ,M

Pr(mj 6= mj) → 0

• Degrees of Freedom: high SNR analysis

d = limSNR→∞

R1

2log (1 + SNR)


ISIT 2008: Compute-and-Forward > Encoding Strategy 12 / 23

Compute-and-Forward with Lattices

First, pick a good lattice Λ (using Erez-Litsyn-Zamir ’05):





1 Each encoder transmits a point from the lattice Λ.






2 Each relay decodes a linear function with integer coefficients, Uk,of the transmitted codewords. Integer coefficients approximatechannel coefficients.

Yk =∑

j=1

hjkXj + Zk

Uk =∑

j=1

ajkXj where ajk ∈ Z






2 Each relay decodes a linear function with integer coefficients, Uk,of the transmitted codewords. Integer coefficients approximatechannel coefficients.

Yk =∑

j=1

hjkXj + Zk

Uk =∑

j=1

ajkXj where ajk ∈ Z

3 Decoder collects equations of codewords. If integer equations arefull rank, decoding is successful.



Relays Decode Linear Functions

m1 E1

X1

m2 E2

X2

...

mM EM

XM

H

Z1

Y1

Z2

Y2

ZM

YM

R1

∑

j aj1Xj

R0

R2

∑

j aj2Xj

R0

...

RM

∑

j ajMXj

R0

D

m1

m2...

mM



Random Coding vs. Lattice Coding




• Sum of codewords is not acodeword.

• Must decode individualmessages.




• Sum of codewords is not acodeword.

• Must decode individualmessages.

• Sum of codewords is acodeword.

• Can decode linearfunctions of messages.


ISIT 2008: Compute-and-Forward > Main Results 15 / 23

Achievable Rates

• Channel to relay k given by Yk = hTk X + Zk.

• Relay k decodes integer equation Uk = aTk X

Theorem

The following symmetric rate is achievable for our relay network:

R = mink

maxak∈ZM

1

2log

(

SNR

1 + SNR‖hk − ak‖2

)

• ‖hk − ak‖2 is a mismatch penalty (or approximation error).

• Actually, we can do better!



Noise-Approximation Tradeoff

• Channel to relay k given by Yk = hTk X + Zk.

• hk may not be close to integer vector (large approximation error‖hk − ak‖

2)

• Idea: Relay scales observed channel output by λ ∈ R beforedecoding:

λYk = λhTk X + λZk

Yk = hTk X + Zk

• New approximation error minak‖λhk − ak‖ may be smaller than

original minak‖hk − ak‖.

• Noise variance goes from 1 to λ2.




λh

λ = 1

h = (1, 5

4), λh = (1, 5

4)

Noise Variance = 1

Approximation Error =(

1

4

)2= 1

16




λh

λ = 2

h = (1, 5

4), λh = (2, 10

4)

Noise Variance = 4


1

2

)2= 1

4




λh

λ = 3

h = (1, 5

4), λh = (3, 15

4)

Noise Variance = 9


1

4

)2= 1

16




λh

λ = 4

h = (1, 5

4), λh = (4, 5)

Noise Variance = 16

Approximation Error = 0



Achievable Rates

Theorem


R = mink

maxak∈ZM

maxλk∈R

1

2log

(

SNR

λ2

k + SNR‖λkhk − ak‖2

)



Achievable Rates

Theorem


R = mink

maxak∈ZM

maxλk∈R

1

2log

(

SNR

λ2


)

= mink

maxak∈ZM

1

2log

(

1

‖a‖2 − λMMSE < a,h >

)

• The optimal choice of λ is always given by the MMSE coefficient:

λMMSE =SNR < h,a >

1 + SNR‖h‖2



Achievable Rates

Theorem


R = mink

maxak∈ZM

maxλk∈R

1

2log

(

SNR

λ2


)

= mink

maxak∈ZM

1

2log

(

1

‖a‖2 − λMMSE < a,h >

)

• The optimal choice of λ is always given by the MMSE coefficient:

λMMSE =SNR < h,a >

1 + SNR‖h‖2

• Optimal λ might be smaller than 1. (Example: h = [10 20 10])



Multiple-Access is a Special Case

• For any channel hk, if ak = [1 0 0 · · · 0], (meaning try to decodejust x1) then:

R =1

2log

(

1 +|h1k|

2SNR

1 + SNR∑M

j=2|hjk|2

)

• This is just the corner point of the MAC rate region.

• Multiple-access emerges naturally by simply trying to decode theindividual messages.



Degrees of Freedom

• If all the channel coefficients, hjk, are rational then all we need todo is pick λ large enough to make all the coefficients integers.Then we get to the full degrees of freedom:

dCPF = M, H is rational

• Otherwise, using Dirichlet’s Theorem we can lower bound theperformance for general hjk:

dCPF ≥ 1

• One can show that for Decode-and-Forward the sum degrees offreedom is dDF = 1.

• Compress-and-Forward can get all the way to the optimumdCF = M .



Example: Complex Phase Fading

• Complex uniform phasefading, unknown attransmitters.

• Relays have fixed rate tothe decoder (R0 = 4).

• Similar results for Rayleighfading.


ISIT 2008: Compute-and-Forward > Conclusions 22 / 23

Related Work

The key to our achievable scheme is the use of structured randomcodes. These are useful for proving new capacity results:

• Parallel Relay Channels (Kochman-Khina-Erez-Zamir ’08)

• Computation over MACs (Nazer-Gastpar ’05, ’07)

• Wireless Network Coding (Nazer-Gastpar ’06, ’07, ’08)

• Distributed Source Coding (Korner-Marton ’79, Krithivasan-Pradhan ’07,

’08, Tavildar-Viswanath-Wagner ’08)

• Two-Way Relay Channel (Wilson-Narayanan-Pfister-Sprintson ’07, ’08,

Nam-Chung-Lee ’08)

• Dirty Multiple-Access Channel (Philosof-Khisti-Erez-Zamir ’07)

• Interference Channels (Bresler-Parekh-Tse ’07, Jafar-Vishwanath ’08)

Interested? See our paper, The Case for Structured Random Codes in

Network Capacity Theorems in ETT July ’08 for more . . .


ISIT 2008: Compute-and-Forward > Conclusions 23 / 23

Conclusions

• Relays can remove noise without fully decoding the transmittedcodewords by decoding linear equations of codewords.

• Enabled by structured random codes (specifically lattice codes).

• Allows us to think of networks as systems of linear equations (verysimilar to network coding).

• Performs better than classical strategies in some regimes. Morework needed to characterize compute-and-forward performance ingeneral.


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