cyclic communication and inseparability in mimo relay...

RUB Institute of Digital Communication Systems

Cyclic Communication and Inseparability in MIMORelay Channels

Aydin Sezginjoint work with Anas Chaaban

Institute of Digital Communication SystemsRUB, Bochum

Anas Chaaban, Aydin Sezgin Multi-way Communications 1

Outline

1 Motivation: From one-way to multi-way

2 The MIMO Y-channel: From single-antenna to multiple-antennas

3 Main result: From capacity to DoF

4 Insights and ingredientsChannel diagonalization: Separability of communicationstructureAlignment, Compute-and-forwardTransmission strategy(3-users)

5 Extensions and Conclusion

Outline

Towards 50B devices in 2020!

Increasing number of connected devices (IoT, M2M, etc.)

More sophisticated network topologies!

Machine-to-machinePoint↔multi-point

Car-to-carMulti-hop

Device-to-device

X-Channel

Interferencechannel

• Important factor: Relaying!

Question

Is communication mainly one-way?

Car-to-carMulti-hop

Device-to-device

X-Channel

Interferencechannel

Question

Car-to-carMulti-hop

Device-to-device

X-Channel

Interferencechannel

Question

Car-to-carMulti-hop

Device-to-device

X-Channel

Interferencechannel

Question

Car-to-carMulti-hop

Device-to-device

X-Channel

Interferencechannel

Question

Car-to-carMulti-hop

Device-to-device

X-Channel

Interferencechannel

Question

Car-to-carMulti-hop

Device-to-device

X-Channel

Interferencechannel

Question

Car-to-carMulti-hop

Device-to-device

X-Channel

Interferencechannel

Question

Multi-way Communications

• Part of our daily communication is uni-directional

• A large share is bi-directional (video conferencing e.g.)⇒ Two-way channel

• Distant nodes⇒ Two-way relay channel

• More than 2 nodes⇒ Multi-way relay channel (MRC).

Multi-way Relaying

• Multi-cast MRC: a message has multiple destinations

• Uni-cast MRC (Y-channel): a message has one destination

Multi-way Relaying

• Multi-cast MRC: a message has multiple destinations

• Uni-cast MRC (Y-channel): a message has one destination

Outline

MIMO Y-channel

· · ·M

· · ·N

· · ·M

· · · M

H1H1x1 H2H2

· · ·M

· · ·N

· · ·M

· · · M

D1D1y1 D2D2

Uplink:

• Tx signal: xi ∈ CM , power P

• Uplink channels: Hi ∈ CN×M

Downlink:

• Relay signal: xr ∈ CN , power P

• Downlink channels: Di ∈ CM×N

MIMO Y-channel

· · ·M

· · ·N

· · ·M

· · · M

H1H1x1 H2H2

· · ·M

· · ·N

· · ·M

· · · M

D1D1y1 D2D2

Uplink:

Downlink:

MIMO Y-channel

· · ·M

· · ·N

· · ·M

· · · M

H1H1x1 H2H2

· · ·M

· · ·N

· · ·M

· · · M

D1D1y1 D2D2

Uplink:

Downlink:

MIMO Y-channel

· · ·M

· · ·N

· · ·M

· · · M

H1H1x1 H2H2

· · ·M

· · ·N

· · ·M

· · · M

D1D1y1 D2D2

Uplink:

Downlink:

MIMO Y-channel

· · ·M

· · ·N

· · ·M

· · · M

H1H1x1 H2H2

· · ·M

· · ·N

· · ·M

· · · M

D1D1y1 D2D2

Uplink:

Downlink:

Capacity to Capacity Region

Single-user (MIMO P2P):

Tx RxH

Input covariance Q, tr(Q) ≤ P

Capacity: C = log |I+HQHH |

achievable rate

Multi-user (MIMO MAC):

Input covariance Qi, i = 1, 2,tr(Qi) ≤ P

Capacity region:

Ri ≤ log |I+HiQiHHi |,

R1 +R2 ≤ log |I+H1Q1HH1 +H2Q2H

achievable

rate region

Tx RxH

achievable rate

Capacity region:

achievable

rate region

Tx RxH

achievable rate

Capacity region:

achievable

rate region

Tx RxH

achievable rate

Capacity region:

achievable

rate region

Tx RxH

achievable rate

Capacity region:

achievable

rate region

Tx RxH

achievable rate

Capacity region:

achievable

rate region

Outline

Capacity to DoF

Tx RxH

Capacity:

C = log |I+HQHH |

Optimization: water-filling

• M Tx antennas, N Rx antennas

⇒ H ∈ CN×M ⇒ rank(H) = min{M,N}⇒ C ≈ min{M,N} log(P ) at high P

• DoF: d = limP→∞C

log(P )= rank(H) ⇒ d = min{M,N}

⇒ Capacity equivalent to that of d parallel SISO P2P channels!

Tx RxTx RxTx Rx (more on that later)

Capacity to DoF

Tx RxH

Capacity:

C = log |I+HQHH |

Capacity to DoF

Tx RxH

Capacity:

C = log |I+HQHH |

Capacity to DoF

Tx RxH

Capacity:

C = log |I+HQHH |

Capacity to DoF

Tx RxH

Capacity:

C = log |I+HQHH |

⇒ H ∈ CN×M ⇒ rank(H) = min{M,N}

⇒ C ≈ min{M,N} log(P ) at high P

Capacity to DoF

Tx RxH

Capacity:

C = log |I+HQHH |

Capacity to DoF

Tx RxH

Capacity:

C = log |I+HQHH |

Capacity to DoF

Tx RxH

Capacity:

C = log |I+HQHH |

Capacity to DoF

Capacity region:

Ri ≤ log |I+HiQiHHi |

Optimization: Iterative water-filling

• Mi Tx antennas, N Rx antennas

⇒ Hi ∈ CN×Mi ⇒ rank(Hi) = min{Mi, N}• CΣ ≈ min{M1 +M2, N} log(P ) at high P

• DoF: di = limP→∞Ri

log(P )

⇒ DoF region: di ≤ rank(Hi), d1 + d2 ≤ rank([H1, H2])

⇒ d1 + d2 = min{M1 +M2, N}⇒ Sum-capacity equivalent to that of d1 + d2 parallel SISO P2P channels!

Capacity to DoF

Capacity region:

log(P )

Capacity to DoF

Capacity region:

log(P )

Capacity to DoF

Capacity region:

log(P )

Capacity to DoF

Capacity region:

⇒ Hi ∈ CN×Mi ⇒ rank(Hi) = min{Mi, N}

• CΣ ≈ min{M1 +M2, N} log(P ) at high P

log(P )

Capacity to DoF

Capacity region:

log(P )

Capacity to DoF

Capacity region:

log(P )

Capacity to DoF

Capacity region:

log(P )

Capacity to DoF

Capacity region:

log(P )

⇒ d1 + d2 = min{M1 +M2, N}

⇒ Sum-capacity equivalent to that of d1 + d2 parallel SISO P2P channels!

Capacity to DoF

Capacity region:

log(P )

Back to the Y-channel

Definition

Rij : Rate of signal from i to j

dij : DoF defined as limP→∞Rij

log(P )

dΣ: sum-DoF =∑

D: Set of simultaneously achievableDoF’s dij

+ dΣ is an overall performance metric for a network

− dΣ doesn’t provide insights on the trade-off between individual DoF’s

Find the DoF region of the MIMO Y-channel.

Definition

log(P )

dΣ: sum-DoF =∑

Definition

log(P )

dΣ: sum-DoF =∑

Definition

log(P )

dΣ: sum-DoF =∑

Definition

log(P )

dΣ: sum-DoF =∑

d12dΣ

Definition

log(P )

dΣ: sum-DoF =∑

d12dΣ

Definition

log(P )

dΣ: sum-DoF =∑

d12dΣ

Definition

log(P )

dΣ: sum-DoF =∑

d12dΣ

Definition

log(P )

dΣ: sum-DoF =∑

d12dΣ

Main result

The DoF region of a 3-user MIMO Y-channel with N ≤M is described by

d12 + d13 + d23 ≤ N

d12 + d13 + d32 ≤ N

......

... ≤...

Theorem (DoF region)

DoF region for N ≤M described by

dp1p2 + dp1p3 + dp2p3 ≤ N, ∀p

where p is a permutation of (1, 2, 3) and pi is its i-th component.

Main result

d12 + d13 + d23 ≤ N

d12 + d13 + d32 ≤ N

......

... ≤...

dp1p2 + dp1p3 + dp2p3 ≤ N, ∀p

Main result

d12 + d13 + d23 ≤ N

d12 + d13 + d32 ≤ N

......

... ≤...

dp1p2 + dp1p3 + dp2p3 ≤ N, ∀p

Main result

d12 + d13 + d23 ≤ N

d12 + d13 + d32 ≤ N

......

... ≤...

dp1p2 + dp1p3 + dp2p3 ≤ N, ∀p

Outline

Insight from the upper bounds

Message-flow-graph for upperbounds:

dp1p2 + dp1p3 + dp2p3 ≤ N

Message-flow-graph ford = (2, 0, 1, 1, 1, 0) ∈ D:

Message-flow-graph for anyd ∈ D:

p1 p2 p3

dp1p2 dp2p3

No cycles!

1 2 32

2-cycles and 3-cycles!

1 2 3d12

Optimal strategy should be able to ‘resolve’ cycles!

p1 p2 p3

dp1p2 dp2p3

No cycles!

1 2 32

1 2 3d12

p1 p2 p3

dp1p2 dp2p3

No cycles!

1 2 32

1 2 3d12

p1 p2 p3

dp1p2 dp2p3

No cycles!

1 2 32

1 2 3d12

p1 p2 p3

dp1p2 dp2p3

dp1p3No cycles!

1 2 32

1 2 3d12

p1 p2 p3

dp1p2 dp2p3

dp1p3No cycles!

1 2 32

1 2 3d12

Optimal strategy should be able to ‘resolve’ cycles!Anas Chaaban, Aydin Sezgin Multi-way Communications 17

Outline

Uplink-downlink Separability

Node 1 Node 2Channel

Two-way channels are in general not separable

⇒ uplink and downlink have to be considered jointly

⇒ xi and yi are dependent

Node 1 Node 2

Tx 2Rx 1

Basic Gaussian two-way channel is separable

⇒ uplink and downlink can be considered separately

⇒ xi and yi are independent

Node 1 Node 2

Tx 2Rx 1

Node 1 Node 2

Tx 2Rx 1

Node 1 Node 2+

Tx 2Rx 1

Node 1 Node 2

Tx 2Rx 1

Rx 2x1

Node 1 Node 2

Tx 2Rx 1

Rx 2x1

Channel Diagonalization

Definition (Channel diagonlization)

Transform an arbitrary MIMO channel matrix H to a diagonal matrix.

MIMO M ×N P2P channel can be diagonalized by zero-forcing (ZF)

M ≥ N : ZF pre-coding:

• Pseudo-inverse:H† = HH [HHH ]−1

• H†H = I

x yH†

N parallel SISO P2P channels!

M ≤ N : ZF post-coding:

• Pseudo-inverse:H‡ = [HHH]−1HH

• HH‡ = I

x yH‡

yyH‡

M parallel SISO P2P channels!

DoF achievable by treating each sub-channel separately ⇒ Separability!MAC and BC are also separable

• H†H = I

H†uu

N parallel SISO P2P channels!M ≤ N : ZF post-coding:

• HH‡ = I

x yH‡

yyH‡

• H†H = I

• HH‡ = I

x yH‡

yyH‡

• H†H = I

• HH‡ = I

x yH‡

yyH‡

• H†H = I

N parallel SISO P2P channels!

M ≤ N : ZF post-coding:

• HH‡ = I

x yH‡

yyH‡

• H†H = I

• HH‡ = I

H‡yy

• H†H = I

• HH‡ = I

• H†H = I

• HH‡ = I

• H†H = I

• HH‡ = I

• H†H = I

• HH‡ = I

DoF achievable by treating each sub-channel separately ⇒ Separability!

MAC and BC are also separable

• H†H = I

• HH‡ = I

Outline

Signal Alignment

Definition (Signal alignment)

Placing two signals x1 and x2 in signal space so that span(x1) = span(x2).

Two signals can be aligned bypre-coding:x1 = V1u1

x2 = V2u2

• V1, V2 arbitrary

• H1V1 = H2V2

Node 1

Node 2

Node 3

H1x1H2x2

Signal Alignment

x2 = V2u2

• H1V1 = H2V2

Node 1

Node 2

Node 3

H1x1H2x2

Signal Alignment

x2 = V2u2

• H1V1 = H2V2

Node 1

Node 2

Node 3

H1x1H2x2

Signal Alignment

x2 = V2u2

• H1V1 = H2V2

Node 1

Node 2

Node 3

H1x1H2x2

Signal Alignment

x2 = V2u2

• H1V1 = H2V2

Node 1

Node 2

Node 3

H1x1H2x2

Signal Alignment

x2 = V2u2

• H1V1 = H2V2

Node 1

Node 2

Node 3

H1x1H2x2

Compute-forward

Definition (Compute-forward (CF))

Computing and forwarding a linear combination of two signals a1x1 + a2x2.

CF can be accomplished byusing lattice codes.

• Property: u1 and u2

lattice codes ⇒ u1 + u2

lattice code!

• x1 = V1u1,

x2 = V2u2,

H1V1 = H2V2 = [1, 1]T

(e.g.)

• receivey3 = (u1 +u2)[1, 1]T +n

• compute u1 + u2 from[1, 1]y3 = 2(u1+u2)+n′

0 1 2−1−2

u1u2 u1 + u2xx x

Node 1

Node 2

Node 3

Compute-forward

lattice code!

• x1 = V1u1,

x2 = V2u2,

H1V1 = H2V2 = [1, 1]T

(e.g.)

• receivey3 = (u1 +u2)[1, 1]T +n

0 1 2−1−2

u1u2 u1 + u2xx x

Node 1

Node 2

Node 3

Compute-forward

lattice code!

• x1 = V1u1,

x2 = V2u2,

H1V1 = H2V2 = [1, 1]T

(e.g.)

• receivey3 = (u1 +u2)[1, 1]T +n

0 1 2−1−2u1u2 u1 + u2xx x

Node 1

Node 2

Node 3

Compute-forward

lattice code!

• x1 = V1u1,

x2 = V2u2,

H1V1 = H2V2 = [1, 1]T

(e.g.)

• receivey3 = (u1 +u2)[1, 1]T +n

0 1 2−1−2u1u2 u1 + u2xx x

Node 1

Node 2

Node 3

Compute-forward

lattice code!

• x1 = V1u1,

x2 = V2u2,

H1V1 = H2V2 = [1, 1]T

(e.g.)

• receivey3 = (u1 +u2)[1, 1]T +n

0 1 2−1−2u1u2 u1 + u2xx x

Node 1

Node 2

Node 3

Compute-forward

lattice code!

• x1 = V1u1,

x2 = V2u2,

H1V1 = H2V2 = [1, 1]T

(e.g.)

• receivey3 = (u1 +u2)[1, 1]T +n

0 1 2−1−2u1u2 u1 + u2xx x

Node 1

Node 2

Node 3

Outline

Overview

Achievability of D is proved using:

Channel diagonalization:

MIMOY-channel

→N SISO

Y-channels(sub-channels)

Information exchange:

• Bi-directional: signal-alignment/compute-forward

• Cyclic: signal-alignment/compute-forward

• Uni-directional: decode-forward

Resource allocation: distribute sub-channels over users

Overview

MIMOY-channel

→N SISO

Overview

MIMOY-channel

→N SISO

Overview

MIMOY-channel

→N SISO

Overview

MIMOY-channel

→N SISO

Overview

MIMOY-channel

→N SISO

Overview

MIMOY-channel

→N SISO

Overview

MIMOY-channel

→N SISO

• a MIMO Y-channelwith M = N = 3

• actually looks likethis!

• Pre- and post-codeusing theMoore-Penrosepseudo inverse

• ChannelDiagonalization ⇒ Nsub-channels

User 1

User 2

Relay User 3

Uplink

User 1

User 2

User 3

Downlink

User 1

User 2

Relay User 3

Uplink

User 1

User 2

User 3

Downlink

User 1

User 2

Relay User 3

Uplink

User 1

User 2

User 3

Downlink

User 1

User 2

Relay User 3

Uplink

User 1

User 2

User 3

Downlink

Information transfer

Bi-directional:

• signal-alignment

• compute-forward

• exchanges 2 symbols

• requires 1 sub-channel(up- and down-link)

• efficiency 2DoF/dimension

User 1

User 2

Relay User 3

Uplink

u12+u21

User 1

User 2

Relay User 3

Downlink

u12+u21

Cyclic:

• signal-alignment

• compute-forward

• requires 2 sub-channels(up- and down-link)

• efficiency 3/2DoF/dimension

User 1

User 2

Relay User 3

Uplink

u12+u23

u31+u23

User 1

User 2

Relay User 3

Downlink

u12+u23

u31+u23

Uni-directional:

• decode-forward

• requires 1 sub-channel(up- and down-link)

• efficiency 1DoF/dimension

User 1

User 2

Relay User 3

Uplink

User 1

User 2

Relay User 3

Downlink

Example

DoF tuple d = (d12, d13, d21, d23, d31, d32) = (2, 0, 1, 1, 1, 0), Y-channel with3 = N ≤M

Uni-directional only:

• 5 sub-channels > N !

Bi-directional 2 symbols 1 sub-channelCyclic 3 symbols 2 sub-channels

Uni-directional 1 symbol 1 sub-channel

Bi-directional + uni-directional:

• bi-directional achieves db12 = db21 = 1 over 1 sub-channel