Interference Management for Multi-hop Networks

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The document summarizes research on interference management techniques for multi-hop networks. It describes a two-user two-hop network model and discusses degrees of freedom (DoF) as a way to characterize the network capacity at high signal-to-noise ratios. An aligned interference neutralization scheme is presented that achieves the maximum DoF of 2 for this network using signal alignment across multiple channel extensions. Variations of this scheme for untrusted relays, confidential messages, and imperfect channel state information are also summarized.

Engineering

Interference Management for Multi-
hop Networks
Mohamed Seif, PhD Student

Network Model
• Two user two hop network

• Single antenna nodes

• Time varying channels

Degrees of Freedom Notion
• Rigorous approximation to the network capacity:
Csum = DoF log(P) + o(log(P))
• Proposed by D. N. Tse et.al. in IT 2003
• DoF: represents the number of interference-free signaling
in the network
• We are interested in the high SNR regime (neglect noise)

• What is the maximum DoF for this network?
DoF  2 (Cut set bound)
Goal: to achieve this upper bound

Two User Interference Channel Two User X Channel
W1
W2
ˆW1
ˆW2
W11
W21
ˆW11
ˆW21
ˆW12
W12
W22 ˆW22
DoF = 1 DoF =
4
3
Is there better scheme?

Channel extension
2
6
6
6
4
Y (1)
Y (2)
...
Y (L)
3
7
7
7
5
=
2
6
6
6
4
H(1)
H(2)
...
H(L)
3
7
7
7
5
2
6
6
6
4
X(1)
X(2)
...
X(L)
3
7
7
7
5
+
2
6
6
6
4
Z(1)
Z(2)
...
Z(L)
3
7
7
7
5
• Channel extension is coding across time
• Useful when there is no enough spatial dimension for transmission
• Can be viewed as MIMO channel with L ⇥ L dimensions
• Consider a transmitter and receiver with single antennas
• Causal system (why?)

Achievable Scheme: Aligned
Interference Neutralization Scheme

a1
a2
b1
a1
a1 + b1
a2 + b1
a1
a2 a1
b1
a2 2a2 + b1
Example: L = 2
DoF =
3
2
>
4
3

a1
a2
a3
a4
a5
a6
b1
b2
b3
b4
b5
a1
a1 + b1
a2 + b1
a6 + b5
a2 + b2
a5 + b5
a6
a5 + b4
a3 + b2
a4 + b3
a3 + b3
a4 + b4
a1
a2 a1
a3 a2
a4 a3
a5 a4
a6 a5
b1
b2 b1
b3 b2
b4 b3
b5 b4
2a6 + b5
L
L 1

DoF Calculation
• Generally, user 1 sends L symbols while user 2 sends
symbols.
• Then, in total
L 1
DoF =
2L 1
L
! 2

Design of Precoders
• Alignment Conditions for the ﬁrst hop:
• Design of precoders at relays will be designed with
the same manner
F11v1,2 = F12v2
F21v1,1 = F22v2
L = 2

v1,i+1 = (F 1
11 F12F 1
22 F21)i
v1,1
v2,i = (F 1
22 F21F 1
11 F12)i 1
F 1
22 F21v1,1
8i 2 {1, 2, . . . , L 1}
Question: Is these vectors linearly independent?
Generally, for L channel extensions:

Untrusted Relays and Conﬁdential
messages

• Proof Sketch:
n(R1 + R2) = I(W1, W2; Xn
R1
) + I(W1, W2; Y n
D2
|Xn
R1
) + H(W1, W2|Xn
R1
, Y n
D2
)
where,
(Secrecy constraint)
I(W1, W2; Y n
D2
|Xn
R1
)  n log(P) (By construction)
H(W1, W2|Xn
R1
, Y n
D2
)  no(log(P)) + n✏n
d1 + d2  1
[M. Seif, R. Tandon, Ming Li - 2017]
I(W1, W2; Xn
R1
)  n✏n
(Refer to our paper)

[M. Seif, R. Tandon, Ming Li - 2017]
MIMO Case

Secure DoF Region
Proof: Please refer to our paper

Imperfect CSIT
• Possible sources of error in the CSI feedback process:
Channel estimation
Quantization
Feedback delay

. . .
Tx
Rx1
Rx2
a1
a2
b1
b2
L1(a1, a2)
I1(a1, a2)
I1(b1, b2)
L1(b1, b2)
I1(a1, a2) + I1(b1, b2)
h1(t)
h2(t)
t = 1t = 2t = 3
t = 1 t = 2
I1(a1, a2) + I1(b1, b2)
t = 3
DoF 
4
3
Feedback delay
Transmitter knows the channel coeﬃcients after one time slot
[Maddah Ali, D.N.Tse - 2010]

• For the multi-hop network with delayed CSIT
DoF 
4
3
• For the multi-hop network with alternating CSIT

DoF 
4
3
[A.Wagdy, M. Seif, et.al. - 2017]
[Varanasi, Vaze - 2011]
These transmission schemes inspired by MAT scheme

Non-layered Interference
Network [Jafar et.al. - 2011]
Still 2 DoF achievable? Yes, by AIN schemes
Except direct link between transmitter and non-intended receiver,
DoF collapse to one

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