1. Amplify-and-Forward Schemes for Wireless Communications

2. Wireless Relay Network
Fixed channel s t
The network is the channel
“Tunable” channel t s
Problem: Design the optimal channel

3. Relay Networks: Advantages
Enhanced coverage
Increased throughput
Resilient communication

4. Wireless Relay Networks
Source
Receiver
Noise
Interference
Synchronization
Channel Parameters
Challenge:
Low complexity communication schemes for Wireless relay Networks

5. A: DNC “Noisy” Network Coding
Three Candidates
A: DNC “Noisy” Network Coding
B: PNC
Amplify-and-forward
C: Quantize-map-and-forward

6. “Noisy” Network codingα ⁞ ⁞
Alice β
Bob
A single link:
111
101
001 ……
110
101
011 ……
Overall network bit-error ~Ber(p)
No more than pEmn 1s (Worst-case)

7. “Noisy” Network coding: Bounds
TX(1)
2pEmn
TX(2)
TX(3)
For both coherent & incoherent NC
pEmn
Q. Wang, S. Jaggi, S.-Y. R. Li. Binary error correcting network codes. In Proc. ITW 2011.

8. Amplify-and-Forward Relaying

9. Amplify-and-Forward in Wireless Networks
“Intersymbol Interference Channel with Colored Gaussian Noise”

10. Achievable Rate for AF Relay Networks
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:

11. Part I: Approximating IAF(Ps)
Computing IAF(Ps) is ``hard’’
Relay without Delay Approximation:
In some scenarios, almost optimal performance
Lower-bound within a constant gap from cutset upper-bound
S. Agnihotri, S. Jaggi, M. Chen. Amplify-and-forward in wireless relay networks. In Proc. ITW 2011.

12. Layered Wireless Networks
“No Intersymbol Interference, White Gaussian Noise”

13. 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

14. 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.

15. Part III: A Greedy Scheme
- The optimal AF rate for the Diamond Network
- First analytical characterization
The optimal AF rates for s t
Equal channel gains along all outgoing links from every node
For general layered networks: better rate approximation
S. Agnihotri, S. Jaggi, M. Chen. Analog network coding in general SNR regime: performance of a greedy scheme. To appear in NetCod 2012.

16. Part IV: Network Simplification
What fraction of the optimal rate can be maintained by using k out of N relays in each layer?
Diamond Network: s t
RN – Rk = log(N/k)
RN/Rk = N/k s t
ECGAL Network:
RN – Rk = 2L log(N/k)
RN/Rk = (N/k)2L-1
S. Agnihotri, S. Jaggi, M. Chen. Analog network coding in general SNR regime: performance of network simplification. Submitted to ITW 2012.

17. 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

18. Communication over a point-to-point channel
is an integer
and we take its binary representation
. . . = = = = = 6 1 5 42 3 19 9
. . . 5 1
. . . 4 1 1
. . . 3 1 2 1 1
. . . 1
. . . 1 1 1 1 1 = = = = = 5 42 3 19 9

19. Communication over a point-to-point channel
is an integer
and we take its binary representation
. . .
. . . 6 1 6 1
. . .
. . . 5 1 5 1 1
. . .
. . . 4 1 1 4 1 1 1
. . .
. . . 3 1 3 1 1 2 1 1
. . . 1 2
. . . 1
. . .
. . . 1 1 1 1 1 1 1 1 = = = = =
Bit flips 5 42 3 19 9

20. Communication over a point-to-point channel
is an integer
and we take its binary representation
. . .
. . . 6 1 6 1
. . .
. . . 5 1 5 1 1
. . .
. . . 4 1 1 4 1 1 1
. . .
. . . 3 1 3 1 1 2 1 1
. . . 1 2
. . . 1
. . .
. . . 1 1 1 1 1 1 1 1 = = = = =
Bit flips 5 42 3 19 9

21. Communication over a point-to-point channel
is an integer
and we take its binary representation
Dependent bit flips
. . .
. . . 6 1 6 1 1
. . .
. . . 5 1 5 1 1
. . .
. . . 4 1 1 4 1 1 1
. . .
. . . 3 1 3 1 1 2 1 1
. . . 1 2
. . . 1
. . .
. . . 1 1 1 1 1 1 1 1 = = = = =
Bit flips 5 42 3 19 9

22. Communication over a point-to-point channel
is an integer
and we take its binary representation
Dependent bit flips
. . .
. . . 6 1 6 1 1
. . .
. . . 5 1 5 1 1
. . .
. . . 4 1 1 4 1 1 1
. . .
. . . 3 1 3 1 1
Very noisy
bit levels
Less noisy bit levels 2 1 1
. . . 1 2
. . . 1
. . .
. . . 1 1 1 1 1 1 1 1 = = = = =
Bit flips 5 42 3 19 9

23. Communication over a point-to-point channel
T. Dikaliotis, H. Yao, A. S. Avestimehr, S. Jaggi,
T. Ho. Low-Complexity Near-Optimal
Codes for Gaussian Relay Networks.
In SPCOM 2012.
is an integer
and we take its binary representation
. . .
Code to correct adversarial errors
. . . 6 1 6 1 1
. . .
. . . 5 1 5 1 1
. . .
. . . 4 1 1 4 1 1 1
. . .
. . . 3 1 3 1 1
. . .
Very noisy
bit levels
Less noisy bit levels 2 1 1 1 2
. . . 1
. . .
. . . 1 1 1 1 1 1 1 1 = = = = =
Bit flips 5 42 3 19 9

24. Publications
S. Agnihotri, S. Jaggi, M. Chen. Amplify-and-forward in wireless relay networks. In Proc. ITW 2011.
Q. Wang, S. Jaggi, S.-Y. R. Li. Binary error correcting network codes. In Proc. ITW 2011.
S. Agnihotri, S. Jaggi, M. Chen. Analog network coding in general SNR regime. To appear in ISIT 2012.
S. Agnihotri, S. Jaggi, M. Chen. Analog network coding in general SNR regime: performance of a greedy scheme. To appear in NetCod 2012.
S. Agnihotri, S. Jaggi, M. Chen. Analog network coding in general SNR regime: performance of network simplification. To appear in ITW 2012.
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.
S. Agnihotri, S. Jaggi, M. Chen. Analog network coding in general SNR regime. In preparation for submission to Trans. Info. Theory.

25. 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
Incorporating “simple” error-correction
“The capacity of relay channel”

26. Thank You