Gaussian 1-2-1 Networks with Imperfect Beamforming Yahya H. Ezzeldin - - PowerPoint PPT Presentation

Gaussian 1-2-1 Networks with Imperfect Beamforming

Yahya H. Ezzeldin‡, Martina Cardone†, Christina Fragouli‡ and Giuseppe Caire★

‡University of California Los Angeles †University of Minnesota Twin Cities ★Technische Universität Berlin

2020 IEEE International Symposium

• n Information Theory

Supported by NSF Awards 1514531, 1824568 and UC-NL grant LFR-18-548554

+ Abundant spectrum resource

• Severe propagation loss and blockage at high frequencies

(using omnidirectional communication)

[www.rcrwireless.com]

mmWave Communication

1

Transmitter Antenna Array Receiver Antenna Array Steerable high-gain directional antenna arrays Multi-hop communication

mmWave Communication

2

mmWave multi-hop network

Goal: What is the maximum unicast traffic rate that we can send between any two nodes in the network ?

D S

3

mmWave studies

Rate coverage and interference-noise ratio ▪

Andrew Thornburg et al., "Performance analysis of outdoor mmWave ad hoc networks." IEEE Transactions on Signal Processing (2016)

▪

James C. Martin, et al., "Receiver Adaptive Beamforming and Interference of Indoor Environments in mmWave." PIMRC (2018)

Potential connectivity through multi-hop ▪

Xingqin Lin et al., "Connectivity of Millimeter Wave Networks With Multi-Hop Relaying“, IEEE Wireless Communications Letters (2015)

What is the potential unicast capacity if all intermediate network nodes are used to relay information ? (with mmWave transmission constraints) 4

To model abstract aspects enabling mmWave communication: ➢ mmWave radios to use phased antenna arrays to focus/receive power along very narrow beams. ➢ Efficient communication possible when beams are aligned between two nodes.

Gaussian 1-2-1 network model [Ezzeldin et al. ISIT 2018]

5

To model abstract aspects enabling mmWave communication: ➢ mmWave radios to use phased antenna arrays to focus/receive power along very narrow beams. ➢ Efficient communication possible when beams are aligned between two nodes. ➢ Beam steering/alignment need to be optimized for maximizing data rate.

Gaussian 1-2-1 network model [Ezzeldin et al. ISIT 2018]

5

Ideal beamforming

[Ezzeldin et al. ISIT 2018]

6

Ideal 1-2-1 network model

Ideal beamforming

[Ezzeldin et al. ISIT 2018]

6

Imperfect beamforming

(this work) Ideal 1-2-1 network model

Ideal beamforming

[Ezzeldin et al. ISIT 2018]

6

?

Ideal 1-2-1 network model Imperfect 1-2-1 network model

Ideal beamforming

[Ezzeldin et al. ISIT 2018]

Imperfect beamforming

(this work)

6

?

Ideal 1-2-1 network model Imperfect 1-2-1 network model

Ideal beamforming

[Ezzeldin et al. ISIT 2018]

Imperfect beamforming

(this work)

Main Question

How can we properly incorporate side-lobe leakage in

• ur abstract modeling of the network ?

6

Nodes At any time:

• Each node can point its transmitting beam to at most one node.
• Each node can point its receiving beam to at most one node.
• In full-duplex, both beams can be simultaneously active.
• A link a → b is active only if node a points its Tx beam towards

node b and node b points its Rx beam towards node a.

Gaussian full-duplex 1-2-1 network model [Ezzeldin et al. ISIT 2018]

7

Nodes At any time:

• Each node can point its transmitting beam to at most one node.
• Each node can point its receiving beam to at most one node.
• In full-duplex, both beams can be simultaneously active.
• A link a → b is active only if node a points its Tx beam towards

node b and node b points its Rx beam towards node a.

Gaussian full-duplex 1-2-1 network model [Ezzeldin et al. ISIT 2018]

7

Gaussian full-duplex 1-2-1 network model [Ezzeldin et al. ISIT 2018]

Nodes At any time:

• Each node can point its transmitting beam to at most one node.
• Each node can point its receiving beam to at most one node.
• In full-duplex, both beams can be simultaneously active.
• A link a → b is active only if node a points its Tx beam towards

node b and node b points its Rx beam towards node a. Topology An edge exists between nodes a and b only if the link can be established by beam alignment (no blockage)

7

D S D S

Full-Duplex 1-2-1 network states Full-Duplex wireless network a single state

Gaussian full-duplex 1-2-1 network model [Ezzeldin et al. ISIT 2018]

Network states

At any time, the network has a particular state based on beam orientations of the N nodes. 8

D S Network schedule

Fraction of time each state is active

D S D S

state s = 1 state s = 2 state s = 3

Gaussian full-duplex 1-2-1 network model [Ezzeldin et al. ISIT 2018]

9

The capacity of a Gaussian 1-2-1 network with N nodes can be approximated to within a constant gap that depends only on the network size N. For the full-duplex 1-2-1 network, the approximate capacity and an optimal schedule that achieves it can be computed in time.

potential states !

Previous Results [Ezzeldin et al. ISIT 2018]

Efficient Scheduling Capacity approximation Guarantees on simplified operation

• An optimal schedule activates at most 𝑂2 + 1 states in full-duplex.
• At most 2N+2 paths need to be active for approximate capacity in Gaussian

full-duplex 1-2-1 networks (out of potentially exponential).

10

?

Ideal 1-2-1 network model Imperfect 1-2-1 network model

Ideal beamforming

[Ezzeldin et al. ISIT 2018]

Imperfect beamforming

(this work)

11

Gaussian Imperfect 1-2-1 network model

Topology An edge exists between nodes a and b only if the communication can be established by beamforming (no blockage) Nodes At any time:

• Each node can point its main TX lobe to at most one node.
• Each node can point its main RX lobe to at most one node.
• In full-duplex, both beams can be simultaneously active.
• If Tx lobe an Rx lobe are aligned then channel coefficient

a → b is enhanced by a gain of .

• Otherwise, channel coefficient a → b is attenuated by a factor
• f .

12

Gaussian Imperfect 1-2-1 network model

Topology An edge exists between nodes a and b only if the communication can be established by beamforming (no blockage) Nodes At any time:

• Each node can point its main TX lobe to at most one node.
• Each node can point its main RX lobe to at most one node.
• In full-duplex, both beams can be simultaneously active.
• If Tx lobe an Rx lobe are aligned then channel coefficient

a → b is enhanced by a gain of .

• Otherwise, channel coefficient a → b is attenuated by a factor
• f .

12

Ideal 1-2-1 network model Imperfect 1-2-1 network model

Ideal beamforming

[Ezzeldin et al. ISIT 2018]

Imperfect beamforming

(this work)

13

Ideal vs Imperfect 1-2-1 network model

Constant gap capacity approximation

(ISIT 2018)

14

Ideal vs Imperfect 1-2-1 network model

Constant gap capacity approximation

(this work)

Constant gap capacity approximation

(ISIT 2018)

14

Ideal vs Imperfect 1-2-1 network model

Efficient polynomial-time scheduling

?

Constant gap capacity approximation

(ISIT 2018)

Constant gap capacity approximation

(this work)

14

Ideal vs Imperfect 1-2-1 network model

Efficient polynomial-time scheduling

?

Guarantees on operating only a fraction of the network paths

?

Constant gap capacity approximation

(ISIT 2018)

Constant gap capacity approximation

(this work)

14

Ideal vs Imperfect 1-2-1 network model

Efficient polynomial-time scheduling

?

Guarantees on operating only a fraction of the network paths

?

?

Constant gap capacity approximation

(ISIT 2018)

Constant gap capacity approximation

(this work)

14

Ideal beamforming Imperfect beamforming

Ideal 1-2-1 network model Imperfect 1-2-1 network model

Question

For what values of the tuple ( , ) is the ideal 1-2-1 network model a good approximation of the imperfect 1-2-1 network model ?

15

Ideal beamforming Imperfect beamforming

Ideal 1-2-1 network model Imperfect 1-2-1 network model

Question

For what values of the tuple ( , ) is the ideal 1-2-1 network model a good approximation of the imperfect 1-2-1 network model ? (sufficient conditions)

15

Theorem: Consider an N-relay Gaussian Imperfect 1-2-1 network with channel coefficients given .

Main Result : From imperfect to ideal 1-2-1 networks

16

Theorem: Consider an N-relay Gaussian Imperfect 1-2-1 network with channel coefficients given . Let approximate capacity of the network for beamforming parameters ,

Main Result : From imperfect to ideal 1-2-1 networks

16

Theorem: Consider an N-relay Gaussian Imperfect 1-2-1 network with channel coefficients given . Let approximate capacity of the network for beamforming parameters , and be the maximum degree of the graph representing the network topology.

Main Result : From imperfect to ideal 1-2-1 networks

16

Theorem: Consider an N-relay Gaussian Imperfect 1-2-1 network with channel coefficients given . Let approximate capacity of the network for beamforming parameters , and be the maximum degree of the graph representing the network topology. If the beamforming parameters satisfy that

Main Result : From imperfect to ideal 1-2-1 networks

16

Theorem: Consider an N-relay Gaussian Imperfect 1-2-1 network with channel coefficients given . Let approximate capacity of the network for beamforming parameters , and be the maximum degree of the graph representing the network topology. If the beamforming parameters satisfy that then

Main Result : From imperfect to ideal 1-2-1 networks

16

Theorem: Consider an N-relay Gaussian Imperfect 1-2-1 network with channel coefficients given . Let approximate capacity of the network for beamforming parameters , and be the maximum degree of the graph representing the network topology. If the beamforming parameters satisfy that then independent of the operating SNR P.

Main Result : From imperfect to ideal 1-2-1 networks

16

Theorem: Consider an N-relay Gaussian Imperfect 1-2-1 network with channel coefficients given . Let approximate capacity of the network for beamforming parameters , and be the maximum degree of the graph representing the network topology. If the beamforming parameters satisfy that then independent of the operating SNR P.

Main Result : From imperfect to ideal 1-2-1 networks

16

Theorem: Consider an N-relay Gaussian Imperfect 1-2-1 network with channel coefficients given . Let approximate capacity of the network for beamforming parameters , and be the maximum degree of the graph representing the network topology. If the beamforming parameters satisfy that then independent of the operating SNR P.

Main Result : From imperfect to ideal 1-2-1 networks

16

Ideal vs Imperfect 1-2-1 network model

Efficient polynomial-time scheduling Guarantees on operating only a fraction of the network paths

Under condition in previous theorem

Efficient polynomial-time scheduling Guarantees on operating only a fraction of the network paths With a constant gap Constant gap capacity approximation

(ISIT 2018)

Constant gap capacity approximation

(this work)

17

Main Result : From imperfect to ideal 1-2-1 networks (Proof Sketch)

Imperfect 1-2-1 approximate capacity

18

Main Result : From imperfect to ideal 1-2-1 networks (Proof Sketch)

Imperfect 1-2-1 approximate capacity diagonal matrix

*

18

Main Result : From imperfect to ideal 1-2-1 networks (Proof Sketch)

Imperfect 1-2-1 approximate capacity diagonal matrix

(*) Show that

Upper bound: Direct consequence of Hadamard-Fischer inequality; Lower bound: Using a result by [Ostrowski 1952] that lower bounds the product of eigenvalues of a diagonally dominant matrix by the product of its diagonal terms.

*

18

Main Result : From imperfect to ideal 1-2-1 networks (Proof Sketch)

Imperfect 1-2-1 approximate capacity diagonal matrix

(*) Show that

Upper bound: Direct consequence of Hadamard-Fischer inequality; Lower bound: Using a result by [Ostrowski 1952] that lower bounds the product of eigenvalues of a diagonally dominant matrix by the product of its diagonal terms.

A matrix is diagonally dominant if, we have that i.e., the diagonal term on each row is stronger than the sum of all off-diagonal terms of that row.

*

18

Main Result : From imperfect to ideal 1-2-1 networks (Proof Sketch)

Imperfect 1-2-1 approximate capacity diagonal matrix

*

(*) Show that

Upper bound: Direct consequence of Hadamard-Fischer inequality; Lower bound: Using a result by [Ostrowski 1952] that lower bounds the product of eigenvalues of a diagonally dominant matrix by the product of its diagonal terms. If in the theorem, then is diagonally dominant for all .

18

Main Result : From imperfect to ideal 1-2-1 networks (Proof Sketch)

Imperfect 1-2-1 approximate capacity Ideal 1-2-1 approximate capacity diagonal matrix

*

(*) Show that

Upper bound: Direct consequence of Hadamard-Fischer inequality; Lower bound: Using a result by [Ostrowski 1952] that lower bounds the product of eigenvalues of a diagonally dominant matrix by the product of its diagonal terms. If in the theorem, then is diagonally dominant for all .

18

Main Result : Simple achievable schemes

Ideal 1-2-1 network model: Approximate capacity achieved by routing. Imperfect 1-2-1 network model: Approximate capacity achieved by physical layer cooperation [Avestimehr et al. 2011], [Lim et al. 2011].

How much rate can simple schemes based on routing achieve in the Imperfect 1-2-1 model ?

19

Main Result : Treating Sidelobes as Noise (TSN)

20

Main Result : Treating Sidelobes as Noise (TSN)

Theorem: Consider an N-relay Gaussian Imperfect 1-2-1 network with channel coefficients given . Let approximate capacity of the network for beamforming parameters , and be the maximum degree of the graph representing the network topology. Let be the rate achieved by Treating Side-lobes as Noise. Then, we have where . 20

Main Result : Treating Sidelobes as Noise (TSN)

Theorem: Consider an N-relay Gaussian Imperfect 1-2-1 network with channel coefficients given . Let approximate capacity of the network for beamforming parameters , and be the maximum degree of the graph representing the network topology. Let be the rate achieved by Treating Side-lobes as Noise. Then, we have where .

A typical vehicle platooning scenario

20

Main Result : Treating Sidelobes as Noise (TSN)

is sufficiently small for to dominate the gap

Theorem: Consider an N-relay Gaussian Imperfect 1-2-1 network with channel coefficients given . Let approximate capacity of the network for beamforming parameters , and be the maximum degree of the graph representing the network topology. Let be the rate achieved by Treating Side-lobes as Noise. Then, we have where .

A typical vehicle platooning scenario

20

Summary and Takeaways

• Characterization of approximate of the capacity of Gaussian full-duplex 1-2-1 networks

with imperfect beamforming.

• Finding sufficient conditions for the imperfect 1-2-1 model to be approximated by the

ideal 1-2-1 model that depend on:

➢ The size of the network ➢ Ratio between channel coefficients in the network

• Characterizing the gap between the rate achieved by the treating sidelobe receptions

as noise and the approximate capacity of the ideal 1-2-1 network model. 21

Thank you Questions ?

Email : yezzeldin@g.ucla.edu