The DoF of the MIMO Y-channel Anas Chaaban, Karlheinz Ochs, and - - PowerPoint PPT Presentation

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RUB Chair of Communication Systems The DoF of the MIMO Y-channel Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin Chair of Digital Communication Systems RUB, Bochum Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 1

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RUB Chair of Communication Systems

The DoF of the MIMO Y-channel

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin

Chair of Digital Communication Systems RUB, Bochum

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 1

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RUB Chair of Communication Systems

Outline

1 Introduction and previous work 2 MIMO Y-channel 3 Transmission Strategy

An Example Generalization

4 Upper Bounds 5 Conclusion

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 2

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The Y-channel

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 3

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RUB Chair of Communication Systems

The Y-channel

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 3

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The Y-channel

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 3

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RUB Chair of Communication Systems

The Y-channel

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 3

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RUB Chair of Communication Systems

The Y-channel

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 3

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RUB Chair of Communication Systems

The Y-channel

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 3

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RUB Chair of Communication Systems

The Y-channel

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 3

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RUB Chair of Communication Systems

The Y-channel

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 3

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RUB Chair of Communication Systems

1 Introduction and previous work 2 MIMO Y-channel 3 Transmission Strategy

An Example Generalization

4 Upper Bounds 5 Conclusion

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 4

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RUB Chair of Communication Systems

Two-way Communications

  • Bi-directional channel (Two-way channel) [Shannon 61]

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 5

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RUB Chair of Communication Systems

Two-way Communications

  • Bi-directional channel (Two-way channel) [Shannon 61]
  • Bi-directional relay channel (BRC) [Rankov et al. 05]

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 5

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RUB Chair of Communication Systems

Two-way Communications

  • Bi-directional channel (Two-way channel) [Shannon 61]
  • Bi-directional relay channel (BRC) [Rankov et al. 05]
  • Capacity of the BRC within 1/2 bit [G¨

und¨ uz et al. 08]

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 5

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RUB Chair of Communication Systems

Two-way Communications

  • Bi-directional channel (Two-way channel) [Shannon 61]
  • Bi-directional relay channel (BRC) [Rankov et al. 05]
  • Capacity of the BRC within 1/2 bit [G¨

und¨ uz et al. 08]

  • Deterministic BRC [Avestimehr et al. 09]

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 5

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Multi-way Communications

  • Approximate capacity of the multi-pair BRC [Sezgin et al. 09]

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 6

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Multi-way Communications

  • Approximate capacity of the multi-pair BRC [Sezgin et al. 09]
  • Approximate capacity of the multi-way relay channel (multicast) [G¨

und¨ uz et al. 09], [Ong et al. 10]

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 6

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RUB Chair of Communication Systems

Multi-way Communications

  • Approximate capacity of the multi-pair BRC [Sezgin et al. 09]
  • Approximate capacity of the multi-way relay channel (multicast) [G¨

und¨ uz et al. 09], [Ong et al. 10]

  • Capacity region of the Y-channel within 7/6 bit [Chaaban et al. 12]

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 6

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1 Introduction and previous work 2 MIMO Y-channel 3 Transmission Strategy

An Example Generalization

4 Upper Bounds 5 Conclusion

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 7

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The MIMO Y-channel

Uplink:

  • Tx signals xj(i) ∈ RMj, power P

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 8

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RUB Chair of Communication Systems

The MIMO Y-channel

Uplink:

  • Tx signals xj(i) ∈ RMj, power P
  • Uplink channels Hj ∈ RN×Mj,

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 8

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RUB Chair of Communication Systems

The MIMO Y-channel

Uplink:

  • Tx signals xj(i) ∈ RMj, power P
  • Uplink channels Hj ∈ RN×Mj,

Downlink:

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 8

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RUB Chair of Communication Systems

The MIMO Y-channel

Uplink:

  • Tx signals xj(i) ∈ RMj, power P
  • Uplink channels Hj ∈ RN×Mj,

Downlink:

  • Relay signal xr(i) ∈ RN, power P

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 8

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RUB Chair of Communication Systems

The MIMO Y-channel

Uplink:

  • Tx signals xj(i) ∈ RMj, power P
  • Uplink channels Hj ∈ RN×Mj,

Downlink:

  • Relay signal xr(i) ∈ RN, power P
  • Downlink channels Dj ∈ RMj×N,

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 8

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The MIMO Y-channel

Uplink:

  • Tx signals xj(i) ∈ RMj, power P
  • Uplink channels Hj ∈ RN×Mj,

Downlink:

  • Relay signal xr(i) ∈ RN, power P
  • Downlink channels Dj ∈ RMj×N,

DoF The DoF per message is defined as djk = limP →∞

Rjk

1 2 log(P ), and the sum DoF

is d = djk.

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 8

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MIMO Y-channel

If the MIMO Y-channel has M1 = M2 = M3 = M and N ≥ ⌈3M/2⌉, then the cut-set bound is achievable, i.e., d = 3M [Lee et al. 10].

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 9

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MIMO Y-channel

If the MIMO Y-channel has M1 = M2 = M3 = M and N ≥ ⌈3M/2⌉, then the cut-set bound is achievable, i.e., d = 3M [Lee et al. 10]. Question 1: What is the DoF of the MIMO Y-channel in general?

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 9

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MIMO Y-channel

If the MIMO Y-channel has M1 = M2 = M3 = M and N ≥ ⌈3M/2⌉, then the cut-set bound is achievable, i.e., d = 3M [Lee et al. 10]. Question 1: What is the DoF of the MIMO Y-channel in general? Question 2: Is the DoF characterized by the cut-set bounds?

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 9

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MIMO Y-channel

If the MIMO Y-channel has M1 = M2 = M3 = M and N ≥ ⌈3M/2⌉, then the cut-set bound is achievable, i.e., d = 3M [Lee et al. 10]. Question 1: What is the DoF of the MIMO Y-channel in general? Question 2: Is the DoF characterized by the cut-set bounds? Theorem The DoF of the MIMO Y-channel with M1 ≥ M2 ≥ M3 (wlog) is given by d = min{M1 + M2 + M3

  • Cut-set bound

, }.

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 9

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MIMO Y-channel

If the MIMO Y-channel has M1 = M2 = M3 = M and N ≥ ⌈3M/2⌉, then the cut-set bound is achievable, i.e., d = 3M [Lee et al. 10]. Question 1: What is the DoF of the MIMO Y-channel in general? Question 2: Is the DoF characterized by the cut-set bounds? Theorem The DoF of the MIMO Y-channel with M1 ≥ M2 ≥ M3 (wlog) is given by d = min{M1 + M2 + M3

  • Cut-set bound

, 2M2 + 2M3, 2N

  • New bounds

}.

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 9

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1 Introduction and previous work 2 MIMO Y-channel 3 Transmission Strategy

An Example Generalization

4 Upper Bounds 5 Conclusion

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 10

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Transmission Strategy

Transmission strategy is signal-space alignment for network-coding [Lee et al. 10] with asymmetric DoF allocation

  • The signals H1x1, H2x2, and H3x3 fill the entire space at the relay

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 11

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Transmission Strategy

Transmission strategy is signal-space alignment for network-coding [Lee et al. 10] with asymmetric DoF allocation

  • The signals H1x1, H2x2, and H3x3 fill the entire space at the relay
  • Relay zero-forces H2x2 using N13 and H3x3 using N12

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 11

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Transmission Strategy

Transmission strategy is signal-space alignment for network-coding [Lee et al. 10] with asymmetric DoF allocation

  • The signals H1x1, H2x2, and H3x3 fill the entire space at the relay
  • Relay zero-forces H2x2 using N13 and H3x3 using N12
  • Result: N12H1x1 + N12H2x2 (2D) and N13H1x1 + N13H3x3 (1D)

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 11

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RUB Chair of Communication Systems

Transmission Strategy

Transmission strategy is signal-space alignment for network-coding [Lee et al. 10] with asymmetric DoF allocation

  • The signals H1x1, H2x2, and H3x3 fill the entire space at the relay
  • Relay zero-forces H2x2 using N13 and H3x3 using N12
  • Result: N12H1x1 + N12H2x2 (2D) and N13H1x1 + N13H3x3 (1D)
  • Tx’s diagonalize their effective channels

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 11

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RUB Chair of Communication Systems

Transmission Strategy

Transmission strategy is signal-space alignment for network-coding [Lee et al. 10] with asymmetric DoF allocation

  • The signals H1x1, H2x2, and H3x3 fill the entire space at the relay
  • Relay zero-forces H2x2 using N13 and H3x3 using N12
  • Result: N12H1x1 + N12H2x2 (2D) and N13H1x1 + N13H3x3 (1D)
  • Tx’s diagonalize their effective channels ⇒ desired channel structure
  • Relay obtains net-coded signals: u12 + u21, u′

12 + u′ 21, and u13 + u31 Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 11

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Transmission strategy

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 12

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Transmission strategy

  • Relay uses a similar beam-forming strategy to deliver the net-coded

signals to their desired destinations

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 12

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Transmission strategy

  • Relay uses a similar beam-forming strategy to deliver the net-coded

signals to their desired destinations

  • User 1 gets u12 + u21, u′

12 + u′ 21, and u13 + u31, and extracts u21, u′ 21,

and u31

  • User 2 gets u12 + u21 and u′

12 + u′ 21 and extracts u12 and u′ 12

  • User 3 gets u13 + u31 and extracts u13

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 12

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Transmission strategy

  • Relay uses a similar beam-forming strategy to deliver the net-coded

signals to their desired destinations

  • User 1 gets u12 + u21, u′

12 + u′ 21, and u13 + u31, and extracts u21, u′ 21,

and u31

  • User 2 gets u12 + u21 and u′

12 + u′ 21 and extracts u12 and u′ 12

  • User 3 gets u13 + u31 and extracts u13
  • 6 symbols delivered successfully ⇒ 6 DoF (optimal)

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 12

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1 Introduction and previous work 2 MIMO Y-channel 3 Transmission Strategy

An Example Generalization

4 Upper Bounds 5 Conclusion

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 13

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General Transmission Strategy

Case 1: d = min{2M2 + 2M3, M1 + M2 + M3, 2N} = 2M2 + 2M3

  • Use only M2 + M3 antennas at the relay

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 14

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General Transmission Strategy

Case 1: d = min{2M2 + 2M3, M1 + M2 + M3, 2N} = 2M2 + 2M3

  • Use only M2 + M3 antennas at the relay
  • Users send u12 and u21 (M2-dim), and u13 and u31 (M3-dim)

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 14

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General Transmission Strategy

Case 1: d = min{2M2 + 2M3, M1 + M2 + M3, 2N} = 2M2 + 2M3

  • Use only M2 + M3 antennas at the relay
  • Users send u12 and u21 (M2-dim), and u13 and u31 (M3-dim)
  • Align u12 and u21, and align u13 and u31 @ relay

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 14

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General Transmission Strategy

Case 1: d = min{2M2 + 2M3, M1 + M2 + M3, 2N} = 2M2 + 2M3

  • Use only M2 + M3 antennas at the relay
  • Users send u12 and u21 (M2-dim), and u13 and u31 (M3-dim)
  • Align u12 and u21, and align u13 and u31 @ relay
  • Relay decodes L(u12, u21) and L(u13, u31)

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 14

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General Transmission Strategy

Case 1: d = min{2M2 + 2M3, M1 + M2 + M3, 2N} = 2M2 + 2M3

  • Use only M2 + M3 antennas at the relay
  • Users send u12 and u21 (M2-dim), and u13 and u31 (M3-dim)
  • Align u12 and u21, and align u13 and u31 @ relay
  • Relay decodes L(u12, u21) and L(u13, u31)
  • Beam-form L(u12, u21) and L(u13, u31) orthogonal to users 3 and 2,

respectively

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 14

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General Transmission Strategy

Case 1: d = min{2M2 + 2M3, M1 + M2 + M3, 2N} = 2M2 + 2M3

  • Use only M2 + M3 antennas at the relay
  • Users send u12 and u21 (M2-dim), and u13 and u31 (M3-dim)
  • Align u12 and u21, and align u13 and u31 @ relay
  • Relay decodes L(u12, u21) and L(u13, u31)
  • Beam-form L(u12, u21) and L(u13, u31) orthogonal to users 3 and 2,

respectively

  • Each user decodes the desired linear combinations, and extracts the

desired signals ⇒ 2M2 + 2M3 DoF

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 14

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General Transmission Strategy

Case 2: d = min{2M2 + 2M3, M1 + M2 + M3, 2N} = M1 + M2 + M3

  • Similar to [Lee et al. 10] but with an asymmetric DoF allocation

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 15

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General Transmission Strategy

Case 2: d = min{2M2 + 2M3, M1 + M2 + M3, 2N} = M1 + M2 + M3

  • Similar to [Lee et al. 10] but with an asymmetric DoF allocation
  • Use only M1+M2+M3

2

antennas at the relay

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 15

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General Transmission Strategy

Case 2: d = min{2M2 + 2M3, M1 + M2 + M3, 2N} = M1 + M2 + M3

  • Similar to [Lee et al. 10] but with an asymmetric DoF allocation
  • Use only M1+M2+M3

2

antennas at the relay

  • Align u12 and u21 in a d12-dim subspace

(d12 = dim(span(H1) ∩ span(H2)) = M1+M2−M3

2

)

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 15

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General Transmission Strategy

Case 2: d = min{2M2 + 2M3, M1 + M2 + M3, 2N} = M1 + M2 + M3

  • Similar to [Lee et al. 10] but with an asymmetric DoF allocation
  • Use only M1+M2+M3

2

antennas at the relay

  • Align u12 and u21 in a d12-dim subspace

(d12 = dim(span(H1) ∩ span(H2)) = M1+M2−M3

2

)

  • similarly, align u13 and u31 in d13 = M1+M3−M2

2

dimensions, and align u23 and u32 in d23 = M2+M3−M1

2

dimensions

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 15

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General Transmission Strategy

Case 2: d = min{2M2 + 2M3, M1 + M2 + M3, 2N} = M1 + M2 + M3

  • Similar to [Lee et al. 10] but with an asymmetric DoF allocation
  • Use only M1+M2+M3

2

antennas at the relay

  • Align u12 and u21 in a d12-dim subspace

(d12 = dim(span(H1) ∩ span(H2)) = M1+M2−M3

2

)

  • similarly, align u13 and u31 in d13 = M1+M3−M2

2

dimensions, and align u23 and u32 in d23 = M2+M3−M1

2

dimensions

  • Achieve d = 2d12 + 2d13 + 2d23 = M1 + M2 + M3 DoF

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 15

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General Transmission Strategy

Case 3: d = min{2M2 + 2M3, M1 + M2 + M3, 2N} = 2N

  • Reduce the number of antennas at the users so that

N = min{2M2 + 2M3, M1 + M2 + M3}

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 16

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General Transmission Strategy

Case 3: d = min{2M2 + 2M3, M1 + M2 + M3, 2N} = 2N

  • Reduce the number of antennas at the users so that

N = min{2M2 + 2M3, M1 + M2 + M3}

  • Use same scheme as case 1 and case 2

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 16

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1 Introduction and previous work 2 MIMO Y-channel 3 Transmission Strategy

An Example Generalization

4 Upper Bounds 5 Conclusion

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 17

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Upper Bounds: Cut-set

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 18

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Upper Bounds: Cut-set

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Upper Bounds: Cut-set

  • ⇒ Cut-set bounds: djk + djl ≤ min{N, Mj, Mk + Ml}

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 18

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Upper Bounds: Cut-set

  • ⇒ Cut-set bounds: djk + djl ≤ min{N, Mj, Mk + Ml}
  • ⇒ d ≤ M1 + M2 + M3

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 18

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Upper Bounds: Genie-aided

  • User 2 decodes (m12, m32) from (y2, m21, m23),

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 19

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Upper Bounds: Genie-aided

  • User 2 decodes (m12, m32) from (y2, m21, m23),
  • give (y3, m31) to user 2 as side information

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 19

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Upper Bounds: Genie-aided

  • User 2 decodes (m12, m32) from (y2, m21, m23),
  • give (y3, m31) to user 2 as side information
  • now user 2 knows (y3, m31, m32) and can decode m13

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 19

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Upper Bounds: Genie-aided

  • User 2 decodes (m12, m32) from (y2, m21, m23),
  • give (y3, m31) to user 2 as side information
  • now user 2 knows (y3, m31, m32) and can decode m13
  • ⇒ R12 + R32 + R13 ≤ I(m12, m32, m13; y2, y3, m21, m23, m31)

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 19

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Upper Bounds: Genie-aided

  • User 2 decodes (m12, m32) from (y2, m21, m23),
  • give (y3, m31) to user 2 as side information
  • now user 2 knows (y3, m31, m32) and can decode m13
  • ⇒ R12 + R32 + R13 ≤ I(m12, m32, m13; y2, y3, m21, m23, m31)
  • ⇒ d12 + d32 + d13 ≤ min{N, M2 + M3} (bound 1)

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 19

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Upper Bounds: Genie-aided

  • User 1 decodes (m21, m31) from (y1, m12, m13),

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Upper Bounds: Genie-aided

  • User 1 decodes (m21, m31) from (y1, m12, m13),
  • give (yr, m32) to user 1 as side information

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 20

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RUB Chair of Communication Systems

Upper Bounds: Genie-aided

  • User 1 decodes (m21, m31) from (y1, m12, m13),
  • give (yr, m32) to user 1 as side information
  • now user 1 knows (yr, m31, m32) which is a better observation than

(y3, m31, m32)

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 20

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RUB Chair of Communication Systems

Upper Bounds: Genie-aided

  • User 1 decodes (m21, m31) from (y1, m12, m13),
  • give (yr, m32) to user 1 as side information
  • now user 1 knows (yr, m31, m32) which is a better observation than

(y3, m31, m32)

  • then, user 1 can decode m23,

⇒ R21 + R31 + R23 ≤ I(m21, m31, m23; y1, yr, m12, m13, m32)

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 20

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RUB Chair of Communication Systems

Upper Bounds: Genie-aided

  • User 1 decodes (m21, m31) from (y1, m12, m13),
  • give (yr, m32) to user 1 as side information
  • now user 1 knows (yr, m31, m32) which is a better observation than

(y3, m31, m32)

  • then, user 1 can decode m23,

⇒ R21 + R31 + R23 ≤ I(m21, m31, m23; y1, yr, m12, m13, m32)

  • ⇒ d21 + d31 + d23 ≤ min{N, M2 + M3} (bound 2)

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 20

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RUB Chair of Communication Systems

Upper Bounds: Genie-aided

  • User 1 decodes (m21, m31) from (y1, m12, m13),
  • give (yr, m32) to user 1 as side information
  • now user 1 knows (yr, m31, m32) which is a better observation than

(y3, m31, m32)

  • then, user 1 can decode m23,

⇒ R21 + R31 + R23 ≤ I(m21, m31, m23; y1, yr, m12, m13, m32)

  • ⇒ d21 + d31 + d23 ≤ min{N, M2 + M3} (bound 2)

Adding bound 1 and bound 2 ⇒ d ≤ min{2N, 2M2 + 2M3}

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 20

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1 Introduction and previous work 2 MIMO Y-channel 3 Transmission Strategy

An Example Generalization

4 Upper Bounds 5 Conclusion

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 21

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RUB Chair of Communication Systems

Conclusion

  • We derived new upper bounds for the MIMO Y-channel

Thank

u!

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 22

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RUB Chair of Communication Systems

Conclusion

  • We derived new upper bounds for the MIMO Y-channel
  • We generalized the scheme of signal-space alignment for net-coding to

the MIMO Y-channel with asymmetric DoF allocation

Thank

u!

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 22

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RUB Chair of Communication Systems

Conclusion

  • We derived new upper bounds for the MIMO Y-channel
  • We generalized the scheme of signal-space alignment for net-coding to

the MIMO Y-channel with asymmetric DoF allocation

  • As a result, we characterized the DoF of the MIMO Y-channel with

arbitrary number of antennas

Thank

u!

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 22

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RUB Chair of Communication Systems

Conclusion

  • We derived new upper bounds for the MIMO Y-channel
  • We generalized the scheme of signal-space alignment for net-coding to

the MIMO Y-channel with asymmetric DoF allocation

  • As a result, we characterized the DoF of the MIMO Y-channel with

arbitrary number of antennas Future work:

  • Extension to networks with more users

Thank

u!

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 22

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RUB Chair of Communication Systems

Conclusion

  • We derived new upper bounds for the MIMO Y-channel
  • We generalized the scheme of signal-space alignment for net-coding to

the MIMO Y-channel with asymmetric DoF allocation

  • As a result, we characterized the DoF of the MIMO Y-channel with

arbitrary number of antennas Future work:

  • Extension to networks with more users

2 3 R 1

Thank

u!

Anas Chaaban, Karlheinz Ochs, and Aydin Sezgin DoF of the MIMO Y-Channel 22