Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
An Introduction to Physical Layer Network Coding: Lattice Codes as Groups
September 11, 2015 2015 ソサイエティ大会
Sendai, Miyagi, Japan
Brian M. Kurkoski Japan Advanced Institute of Science and Technology
Cooperative Wireless NetworksCooperative Wireless Networks
Wireless networks must deal with interference & noise
Cooperative Wireless NetworksCooperative Wireless Networks
Wireless networks must deal with interference & noise
OverviewOverview
3
1 Motivation for physical-layer network coding 1A Network Coding
1B Physical Layer Network Coding
2 Nested Lattice Codes 2A Quotient Groups
2B Lattice Quotient Groups
2C Nested Lattice Codes
3 Encoding and Isomorphisms in Nested Lattice Codes 3A Self-similar Voronoi codes
3B Non-Self-similar Vornoi Codes
Routing vs. Network Coding
Capacity: max rate from source to destination
Routing
• Capacity = 3/2
4
relay
1A
5
relay
Routing vs. Network Coding
Capacity: max rate from source to destination
Routing
• Capacity = 3/2
Network Coding
• Internal nodes perform linear operations
• Capacity = 2
Forwarding combinations of messages can increase capacity
5
relay
Routing vs. Network Coding
Capacity: max rate from source to destination
Routing
• Capacity = 3/2
Network Coding
• Internal nodes perform linear operations
• Capacity = 2
Forwarding combinations of messages can increase capacity
matrix form…
2 received messages and 2 desired messages:
Matrix Form Recovery of Messages
6
w 1 w 1 w 2
relay
u 1,u 2 received messages desired messages
Destination
w, u, q in a field. Allow relay to multiply by q
If Q has rank L, then all messages w recoverable How to design Q ? • Algorithmic approach (Jaggi et al.) Success if field size p > number of destinations • Random approach (Kotter and Medard. Ho et al.) Probability of valid solutions increases with p
Source Generalized Network Coding
7
Action of One Row A “Relay”
8
Source has w 1, w 2, …
Destination has u 1, u 2, …
received messages desired messages
Action of One Row A “Relay”
9
q1w1⨁ … ⨁qLwL
q1w1
relay
qLwL
q2w2...
received messages desired messages
Action of One Row A “Relay”
9
q1w1⨁ … ⨁qLwL
q1w1
relay
qLwL
q2w2...
received messages desired messages
What if the relay is wireless…?
PLNC = Physical Layer Network Coding
User 1
User M
User 2
Relay
Wireless Multiple-Access Channel
x1
x2
xM
y = x1 + … + xM
+ noise
10
xrelay
Addition occurs over the air
1B
Network Coding vs. PLNC
11
q 1w 1 ⨁ q 2w 2
Network Coding
Physical Layer Network Coding
x 1
x 2
h 1x 1 + h 2x 2
relay
relay
q 1w 1
q 2w 2
Network Coding:relay adds incoming messages
PLNC:addition over the air fading plays a role combat noise
PLNC with Error-Correction
12
w 1
w 2 z
x 1
x 2
Perform error-correction coding on vectors:
x i = Enc(w i) Relay performs two functions:
x 1 + x 2 = Decoder(y)
w 1 ⨁ w 2 = Enc–1(x 1 + x 2)
w 1 ⨁ w 2yrelay
PLNC with Error-Correction
13
w 1
w 2
relay
z
x 1
x 2w 1 ⨁ w 2y
Powerful idea: • Relay only eliminates noise • Relay does not need to separate inference • Converted a noisy network into a noiseless network
We Need A Code to Perform PLNC
Code must correct errors, for noisy wireless channels
• Code must satisfy a power constraint.
Code must form a group over addition
• so addition over the channel makes sense.
Code must have a group isomorphism: Enc(w 1 ⨁ w )d2) = x 1 + x 2,
• so network coding can be performed
These properties are satisfied by nested lattice codes.
14
Quotient Groups2A
15
group subgroup
Definition of a Coset
16
Quotient Groups
Example
17
Coset leaders: {0, 1, 2, 3} Coset leaders: {–2, –1, 0, 1}
Coset Leader (Coset Representative)
9
A coset leader is a single representative element from each coset.
19
g1
g2
−1 0 1 2 3
−1
0
1
2
0
g1 g2
Lattice: Linear code over real numbers2BA2 or “hex” lattice
Quotient Groups Based on Lattices
20
Cosets form a group under addition
Nested Lattice Codes2C
26
Voronoi region
Fundamental Regions
parallelotope
Voronoi region (hyper-) rectangle
Fundamental Regions
28
Selecting the Coset Leaders
29
Nested Lattice Codes Form a Group
Various Nested Lattice Codes
3 Different Voronoi regions — All codes form Groups
Various Nested Lattice Codes
Various Nested Lattice Codes
(hyper-) rectangle
parallelotope Important for theory, not very practical
Voronoi region Best transmit power, not always easy to implement
Rectangle Easier to implement, no shaping gain
Voronoi is Best for AWGN Channel
Voronoi regions are sphere-like in high dimension. A sphere satisfies the AWGN power constraint
Encoding and Isomorphism3A
33
Encoding: mapping information to codewords
Indexing: mapping codewords to information
Isomorphism between information (ring) and codewords (group)
Quantization and Modulo
34
0
Q(y)
y
Quantization and Modulo
34
0
Q(y)
y
Quantization has exponential complexity in general
35
0
Quantization and Modulo
Voronoi region at origin
36
0
x
Q(x)
Quantization and Modulo
36
0
x
Q(x)x – Q(x)
Quantization and Modulo
36
0
x
Q(x)x – Q(x)
Quantization and Modulo
37
relay
Recall the multiple-access scenario
Real Addition with Lattice Codes
Real Addition with Lattice Codes
38
c1
c2
c1 + c2
Real Addition with Lattice Codes
38
c1
c2
c1 + c2
Encoding and Indexing
Encodingb x
Indexingbx
[0, 0]
[0, 1]
[0, 2]
[0, 3]
[1, 0]
[1, 1]
[1, 2]
[1, 3]
[2, 0]
[2, 1]
[2, 2]
[2, 3]
[3, 0]
[3, 1]
[3, 2]
[3, 3]
Encoding Parallelotope
g1g2
[0, 0]
[0, 1]
[0, 2]
[0, 3]
[1, 0]
[1, 1]
[1, 2]
[1, 3]
[2, 0]
[2, 1]
[2, 2]
[2, 3]
[3, 0]
[3, 1]
[3, 2]
[3, 3]Encoding the Voronoi Region
Encoding the Voronoi Region
[0, 0] [0, 0]
[0, 1] [0, 1]
[0, 2] [0, 2]
[0, 3]
[0, 3]
[1, 0] [1, 0]
[1, 1] [1, 1]
[1, 2]
[1, 2]
[1, 3]
[1, 3]
[2, 0] [2, 0]
[2, 1]
[2, 1]
[2, 2]
[2, 2]
[2, 3]
[2, 3]
[3, 0]
[3, 0]
[3, 1]
[3, 1]
[3, 2]
[3, 2]
[3, 3]
[3, 3]
Group Isomorphism for PLNC
18
Group Isomorphism for PLNC
Even if the decoder cannot recover the original data b1, b218
Decoder
User 1
Group Isomorphism for PLNC
b1
Simple multiple access channel
User 2b2
18
index
Decoder
User 1
Group Isomorphism for PLNC
Even if the decoder cannot recover the original data b1, b2
b1 x1
x2Assuming successful decoding: Decoder produces x1 + x2 (not x1, x2 individually) Indexing produces b1 ⊕ b2 (not b1, b2 individually) Highly suitable for network coding
Simple multiple access channel
User 2b2
18
index
/34Brian Kurkoski, JAIST
Nested lattice codes with non-self-similar lattices • High dimension lattices (LDLC, etc.): excellent coding gain, computationally hard to perform shaping,
• Low dimension lattices (E8, Barnes-Wall): Good shaping gain with efficient algorithms, not very good coding gain.
And now for something new…
45
3B
/34Brian Kurkoski, JAIST
Proposed method. Construct a quotient group:
Nested lattice codes with non-self-similar lattices
46
High-dimension lattice: n = 1,000 to 105
E8, Barnes-Wall, etc. lattice n = 8, 16
/34Brian Kurkoski, JAIST
Proposed method. Construct a quotient group:
Nested lattice codes with non-self-similar lattices
47
High-dimension lattice: n = 1,000 to 105
E8, Barnes-Wall, etc. lattice n = 8, 16
/34Brian Kurkoski, JAIST
Sufficient Conditions to form a Group
48
/34Brian Kurkoski, JAIST
-8 -6 -4 -2 0 2 4 6 8-5
0
5
10
49
Indexing Non-Nested Lattice Codes
/34Brian Kurkoski, JAIST
-8 -6 -4 -2 0 2 4 6 8-5
0
5
10
Indexing Non-Nested Lattice Codes
50
/34Brian Kurkoski, JAIST
-8 -6 -4 -2 0 2 4 6 8-5
0
5
10
Indexing Non-Nested Lattice Codes
51
/34Brian Kurkoski, JAIST
-8 -6 -4 -2 0 2 4 6 8-5
0
5
10
Indexing Non-Nested Lattice Codes
52
What about a change of basis?
/34Brian Kurkoski, JAIST
Finding a Basis Suitable for Encoding
53
/34Brian Kurkoski, JAIST
Finding a Basis Suitable for Encoding
53
/34Brian Kurkoski, JAIST
Finding a Basis Suitable for Encoding
53
linearly dependent
/34Brian Kurkoski, JAIST 54-8 -6 -4 -2 0 2 4 6 8
-5
0
5
10
Indexing Non-Nested Lattice Codes
Using a Suitable Basis
Summary – Physical Layer Network Coding
PLNC:
• Technique for cooperative wireless networks
• Exploit network coding to increase capacity
• Lattices: real codes to correct errors, shaping gain
• Remove noise first, and interference later
•Compute-and-Forward relaying also deals with fading
55
John Conway and Neil Sloane, Sphere Packings, Lattices and Groups, Springer, Third Edition, 1999
Recommended Reading
G. David Forney, Lecture notes for Principles of Digital Communications II Course at MIT http://dspace.mit.edu/
Ram Zamir, Lattice Coding for Signals and Networks, Cambridge Univ Press, September 2014
Bobak Nazer and Michael Gastpar, “Reliable Physical Layer Network Coding,” Proceedings of the IEEE, March 2011