Amplify-and-Forward Schemes for Wireless Communications

2

Wireless Relay Network

s t

s t

Fixed channel

The network is the channel

“Tunable” channel

Problem: Design the optimal channel

3

Relay Networks: Advantages

Enhanced coverage

Increased throughput

Resilient communication

4

Wireless Relay Networks

NoiseInterferenceSynchronizationChannel Parameters

Challenge:Low complexity communication schemes for Wireless relay Networks

SourceReceiver

Three Candidates

A: DNC“Noisy” Network

Coding

B: PNCAmplify-and-forward

C: Quantize-map-and-forward

Alice Bob⁞

⁞

α

β

2mF

111 101 001 …… 110 101 011 ……A single link:

Overall network bit-error ~Ber(p)No more than pEmn 1s (Worst-case)

“Noisy” Network coding

n

“Noisy” Network coding: Bounds

Q. Wang, S. Jaggi, S.-Y. R. Li. Binary error correcting network codes. In Proc. ITW 2011.

TX(1)

2pEmn

TX(2) 2pEmn

TX(3)

2pEmn

)2(1 pHc

ERGV

For both coherent & incoherent NC

pEmnpEmn

pEmn

)(1Hamming pHc

ER

9

ii PEX 2

),0(~ 2NZ i

Amplify-and-Forward Relaying

10

s t

“Intersymbol Interference Channel with Colored Gaussian Noise”

Amplify-and-Forward in Wireless Networks

11

Lemma (Achievable rate for AF relay network):For an AF-relay network with M nodes, the rate achievable with a given amplification vector β is

Maximum Achievable rate:

Achievable Rate for AF Relay Networks

12

Part I: Approximating IAF(Ps)

Relay without Delay Approximation:

In some scenarios, almost optimal performance

Lower-bound within a constant gap from cutset upper-bound

Computing IAF(Ps) is ``hard’’

S. Agnihotri, S. Jaggi, M. Chen. Amplify-and-forward in wireless relay networks. In Proc. ITW 2011.

13

Layered Wireless Networks

s t

“No Intersymbol Interference, White Gaussian Noise”

14

AF Rate in Layered Networks

Previous Work– High SNR– Max. Transmit Power– Few layers

Our Work– Arbitrary SNR– Optimal Transmit Power– Any number of layers

Function of βli

15

Part II: Computing

can be computed layer-by-layer

Lemma (Computing Optimal β):

- maximize the sum rate to the next layer

- exponential reduction in the search space: NL L N

The optimal AF rates for

s t

Equal channel gains along all links between two adjacent layers

S. Agnihotri, S. Jaggi, M. Chen. Analog network coding in general SNR regime. To appear in ISIT 2012.

16

Part III: A Greedy Scheme

s t

- The optimal AF rate for the Diamond Network- First analytical characterization

For general layered networks: better rate approximation

The optimal AF rates fors t

Equal channel gains along all outgoing links from every node

S. Agnihotri, S. Jaggi, M. Chen. Analog network coding in general SNR regime: performance of a greedy scheme. To appear in NetCod 2012.

17

Part IV: Network SimplificationWhat fraction of the optimal rate can be maintained by using k out of N relays in each layer?

RN – Rk = 2L log(N/k)RN/Rk = (N/k)2L-1

Diamond Network:s t

RN – Rk = log(N/k) RN/Rk = N/k

s tECGAL Network:

S. Agnihotri, S. Jaggi, M. Chen. Analog network coding in general SNR regime: performance of network simplification. Submitted to ITW 2012.

18

Project Outcome So Far

New fundamental results for layered AF-networks: many firsts

New insights useful for:• characterization of the optimal rate in general AF networks• design of the optimal relay scheme for layered networks

Communication over a point-to-point channel

1X 2X 3X 1nX nX

);(max12

YXICEX

iii NXY

iX iY

iN

)1,0(~ NNi

P 1log2

1

PEX i 2, ,

iX is an integer

. . .= = = = =

5 42 3 19 90 1 0 0 00 0 0 1 00 1 0 0 11 0 0 0 00 1 1 1 01 0 1 1 1

. . .

. . .

. . .

. . .

. . .

= = = = =

5 42 3 19 9

and we take its binary representation

54321

6

122 ,6 Pm

Communication over a point-to-point channel

1X 2X 3X 1nX nX

);(max12

YXICEX

iii NXY

iX iY

iN

)1,0(~ NNi

P 1log2

1

PEX i 2, ,

iX is an integer

. . .

0 00 0 0 1 00 1 0 0 11 0 0 0 00 1 1 1 01 0 1 1 1

. . .

. . .

. . .

. . .

. . .

= = = = =

5 42 3 19 9

and we take its binary representation

iN0 1 0

1Y 2Y 3Y 1nY nY. . .

00 0 0 1 11 1 0 0 11 1 0 0 00 0 0 0 10 0 0 1 1

. . .

. . .

. . .

. . .

. . .

0 1 0

Bit flips

54321

654321

6

122 ,6 Pm

0

Communication over a point-to-point channel

1X 2X 3X 1nX nX

);(max12

YXICEX

iii NXY

iX iY

iN

)1,0(~ NNi

P 1log2

1

PEX i 2, ,

iX is an integer

. . .

0 00 0 0 1 00 1 0 0 11 0 0 0 00 1 1 1 01 0 1 1 1

. . .

. . .

. . .

. . .

. . .

= = = = =

5 42 3 19 9

and we take its binary representation

0 1 054321

6iN

1Y 2Y 3Y 1nY nY

00 0 0 1 11 1 0 0 11 1 0 0 00 0 0 0 10 0 0 1 1

. . .

. . .

. . .

. . .

. . .

0 1 0

Bit flips

54321

6

. . .

122 ,6 Pm

0

Communication over a point-to-point channel

1X 2X 3X 1nX nX

);(max12

YXICEX

iii NXY

iX iY

iN

)1,0(~ NNi

P 1log2

1

PEX i 2, ,

iX is an integer

. . .

0 00 0 0 1 00 1 0 0 11 0 0 0 00 1 1 1 01 0 1 1 1

. . .

. . .

. . .

. . .

. . .

= = = = =

5 42 3 19 9

and we take its binary representation

iN0 1 0

54321

61Y 2Y 3Y 1nY nY

1 00 0 0 1 11 1 0 0 11 1 0 0 00 0 0 0 10 0 0 1 1

. . .

. . .

. . .

. . .

. . .

0 1 0

Bit flips

54321

6

Dependent bit flips

. . .

122 ,6 Pm

Communication over a point-to-point channel

1X 2X 3X 1nX nX

);(max12

YXICEX

iii NXY

iX iY

iN

)1,0(~ NNi

P 1log2

1

PEX i 2, ,

iX is an integer

. . .

0 00 0 0 1 00 1 0 0 11 0 0 0 00 1 1 1 01 0 1 1 1

. . .

. . .

. . .

. . .

. . .

= = = = =

5 42 3 19 9

and we take its binary representation

iN0 1 0

54321

61Y 2Y 3Y 1nY nY

1 00 0 0 1 11 1 0 0 11 1 0 0 00 0 0 0 10 0 0 1 1

. . .

. . .

. . .

. . .

. . .

0 1 0

Bit flips

54321

6

Dependent bit flips

. . .

Less noisy bit levels

Very noisy bit levels

122 ,6 Pm

Communication over a point-to-point channel

1X 2X 3X 1nX nX

);(max12

YXICEX

iii NXY

iX iY

iN

)1,0(~ NNi

P 1log2

1

PEX i 2, ,

iX is an integer

. . .

0 00 0 0 1 00 1 0 0 11 0 0 0 00 1 1 1 01 0 1 1 1

. . .

. . .

. . .

. . .

. . .

= = = = =

5 42 3 19 9

and we take its binary representation

iN0 1 0

54321

61Y 2Y 3Y 1nY nY

1 00 0 0 1 11 1 0 0 11 1 0 0 00 0 0 0 10 0 0 1 1

. . .

. . .

. . .

. . .

. . .

0 1 0

Bit flips

54321

6

. . .

Less noisy bit levels

Very noisy bit levels

Code to correct adversarial errors

122 ,6 Pm

T. Dikaliotis, H. Yao, A. S. Avestimehr, S. Jaggi, T. Ho. Low-Complexity Near-Optimal Codes for Gaussian Relay Networks. In SPCOM 2012.

25

Publications1. S. Agnihotri, S. Jaggi, M. Chen. Amplify-and-forward in wireless relay

networks. In Proc. ITW 2011.2. Q. Wang, S. Jaggi, S.-Y. R. Li. Binary error correcting network codes. In

Proc. ITW 2011.3. S. Agnihotri, S. Jaggi, M. Chen. Analog network coding in general SNR

regime. To appear in ISIT 2012.4. S. Agnihotri, S. Jaggi, M. Chen. Analog network coding in general SNR

regime: performance of a greedy scheme. To appear in NetCod 2012.5. S. Agnihotri, S. Jaggi, M. Chen. Analog network coding in general SNR

regime: performance of network simplification. To appear in ITW 2012.6. T. Dikaliotis, H. Yao, A. S. Avestimehr, S. Jaggi, T. Ho. Low-Complexity

Near-Optimal Codes for Gaussian Relay Networks. To appear in SPCOM 2012.

7. S. Agnihotri, S. Jaggi, M. Chen. Analog network coding in general SNR regime. In preparation for submission to Trans. Info. Theory.

26

Current and Future Work

Optimal and efficient relay schemes for layered networks

Distributed relay schemes

“Back to general AF networks”- the optimal rate, distributed schemes

General wireless relay networks- resource-performance tradeoff- optimal relay scheme, capacity

“The capacity of relay channel”

Incorporating “simple” error-correction

27

Thank You