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Interference Management in MIMO Multicell Systems with variable CSIT Giuseppe Abreu [email protected] School of Engineering and Sciences Jacobs University Bremen November 7, 2013

6 interference management in mimo multicell

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Page 1: 6 interference management in mimo multicell

Interference Management in MIMO MulticellSystems with variable CSIT

Giuseppe [email protected]

School of Engineering and SciencesJacobs University Bremen

November 7, 2013

Page 2: 6 interference management in mimo multicell

Evolution

GSM 3G 4G 5G

Application Access Latency Switching Time

12 kbps 20 kbps 150 ms Few seconds

1 Mbps 24 kbps 50 ms 500 ms

10 Mbps 300 Mbps 10 ms 200 ms

1 Gbps 10 Gbps 1 ms 10 ms

Page 3: 6 interference management in mimo multicell

Forecast

I Mobile traffic volume trend: 1000x in 10 years.

I The Internet of ThingsI The Internet of Things !I The Internet of Things !!!

I Requirements

I Improve energy efficiencyI Improve QoSI Improve spectrum efficiency

Page 4: 6 interference management in mimo multicell

Forecast

I Mobile traffic volume trend: 1000x in 10 years.I The Internet of Things

I The Internet of Things !I The Internet of Things !!!

I Requirements

I Improve energy efficiencyI Improve QoSI Improve spectrum efficiency

Page 5: 6 interference management in mimo multicell

Forecast

I Mobile traffic volume trend: 1000x in 10 years.I The Internet of ThingsI The Internet of Things !

I The Internet of Things !!!

I Requirements

I Improve energy efficiencyI Improve QoSI Improve spectrum efficiency

Page 6: 6 interference management in mimo multicell

Forecast

I Mobile traffic volume trend: 1000x in 10 years.I The Internet of ThingsI The Internet of Things !I The Internet of Things !!!

I Requirements

I Improve energy efficiencyI Improve QoSI Improve spectrum efficiency

Page 7: 6 interference management in mimo multicell

Forecast

I Mobile traffic volume trend: 1000x in 10 years.I The Internet of ThingsI The Internet of Things !I The Internet of Things !!!

I RequirementsI Improve energy efficiency

I Improve QoSI Improve spectrum efficiency

Page 8: 6 interference management in mimo multicell

Forecast

I Mobile traffic volume trend: 1000x in 10 years.I The Internet of ThingsI The Internet of Things !I The Internet of Things !!!

I RequirementsI Improve energy efficiencyI Improve QoS

I Improve spectrum efficiency

Page 9: 6 interference management in mimo multicell

Forecast

I Mobile traffic volume trend: 1000x in 10 years.I The Internet of ThingsI The Internet of Things !I The Internet of Things !!!

I RequirementsI Improve energy efficiencyI Improve QoSI Improve spectrum efficiency

Page 10: 6 interference management in mimo multicell

Past and Future

I Drivers of cellular system improvementI 2G: Digitalization (GSM/CDMA)I 3G: Spread-spectrum (WCDMA)I 4G: LTE (OFDMA)I 5G: ???

“granularity”

I Bottlenecked by InterferenceI Cooperation (Relaying) → security (?)I Het-Nets/Cognitive Radio → enough (?)I Interference Alignment → scalability (?)I Massive MIMO → revolutionary, expensive (!)I CoMP → evolutionary, flexible, huge background, maturing...

Still lots to be done!!

Page 11: 6 interference management in mimo multicell

Past and Future

I Drivers of cellular system improvementI 2G: Digitalization (GSM/CDMA) → spectrumI 3G: Spread-spectrum (WCDMA)I 4G: LTE (OFDMA)I 5G: ???

“granularity”

I Bottlenecked by InterferenceI Cooperation (Relaying) → security (?)I Het-Nets/Cognitive Radio → enough (?)I Interference Alignment → scalability (?)I Massive MIMO → revolutionary, expensive (!)I CoMP → evolutionary, flexible, huge background, maturing...

Still lots to be done!!

Page 12: 6 interference management in mimo multicell

Past and Future

I Drivers of cellular system improvementI 2G: Digitalization (GSM/CDMA) → spectrumI 3G: Spread-spectrum (WCDMA) → spectrum again...I 4G: LTE (OFDMA)I 5G: ???

“granularity”

I Bottlenecked by InterferenceI Cooperation (Relaying) → security (?)I Het-Nets/Cognitive Radio → enough (?)I Interference Alignment → scalability (?)I Massive MIMO → revolutionary, expensive (!)I CoMP → evolutionary, flexible, huge background, maturing...

Still lots to be done!!

Page 13: 6 interference management in mimo multicell

Past and Future

I Drivers of cellular system improvementI 2G: Digitalization (GSM/CDMA) → spectrumI 3G: Spread-spectrum (WCDMA) → spectrum again...I 4G: LTE (OFDMA) → and again...I 5G: ???

“granularity”

I Bottlenecked by InterferenceI Cooperation (Relaying) → security (?)I Het-Nets/Cognitive Radio → enough (?)I Interference Alignment → scalability (?)I Massive MIMO → revolutionary, expensive (!)I CoMP → evolutionary, flexible, huge background, maturing...

Still lots to be done!!

Page 14: 6 interference management in mimo multicell

Past and Future

I Drivers of cellular system improvementI 2G: Digitalization (GSM/CDMA) → spectrumI 3G: Spread-spectrum (WCDMA) → spectrum again...I 4G: LTE (OFDMA) → and again...I 5G: “granularity”

I Bottlenecked by InterferenceI Cooperation (Relaying) → security (?)I Het-Nets/Cognitive Radio → enough (?)I Interference Alignment → scalability (?)I Massive MIMO → revolutionary, expensive (!)I CoMP → evolutionary, flexible, huge background, maturing...

Still lots to be done!!

Page 15: 6 interference management in mimo multicell

Past and Future

I Drivers of cellular system improvementI 2G: Digitalization (GSM/CDMA) → spectrumI 3G: Spread-spectrum (WCDMA) → spectrum again...I 4G: LTE (OFDMA) → and again...I 5G: “granularity” → devices

I Bottlenecked by InterferenceI Cooperation (Relaying) → security (?)I Het-Nets/Cognitive Radio → enough (?)I Interference Alignment → scalability (?)I Massive MIMO → revolutionary, expensive (!)I CoMP → evolutionary, flexible, huge background, maturing...

Still lots to be done!!

Page 16: 6 interference management in mimo multicell

Past and Future

I Drivers of cellular system improvementI 2G: Digitalization (GSM/CDMA) → spectrumI 3G: Spread-spectrum (WCDMA) → spectrum again...I 4G: LTE (OFDMA) → and again...I 5G: “granularity” → devices → antennas

I Bottlenecked by InterferenceI Cooperation (Relaying) → security (?)I Het-Nets/Cognitive Radio → enough (?)I Interference Alignment → scalability (?)I Massive MIMO → revolutionary, expensive (!)I CoMP → evolutionary, flexible, huge background, maturing...

Still lots to be done!!

Page 17: 6 interference management in mimo multicell

Past and Future

I Drivers of cellular system improvementI 2G: Digitalization (GSM/CDMA) → spectrumI 3G: Spread-spectrum (WCDMA) → spectrum again...I 4G: LTE (OFDMA) → and again...I 5G: “granularity” → devices → antennas

I Bottlenecked by Interference

I Cooperation (Relaying) → security (?)I Het-Nets/Cognitive Radio → enough (?)I Interference Alignment → scalability (?)I Massive MIMO → revolutionary, expensive (!)I CoMP → evolutionary, flexible, huge background, maturing...

Still lots to be done!!

Page 18: 6 interference management in mimo multicell

Past and Future

I Drivers of cellular system improvementI 2G: Digitalization (GSM/CDMA) → spectrumI 3G: Spread-spectrum (WCDMA) → spectrum again...I 4G: LTE (OFDMA) → and again...I 5G: “granularity” → devices → antennas

I Bottlenecked by InterferenceI Cooperation (Relaying)

→ security (?)I Het-Nets/Cognitive Radio → enough (?)I Interference Alignment → scalability (?)I Massive MIMO → revolutionary, expensive (!)I CoMP → evolutionary, flexible, huge background, maturing...

Still lots to be done!!

Page 19: 6 interference management in mimo multicell

Past and Future

I Drivers of cellular system improvementI 2G: Digitalization (GSM/CDMA) → spectrumI 3G: Spread-spectrum (WCDMA) → spectrum again...I 4G: LTE (OFDMA) → and again...I 5G: “granularity” → devices → antennas

I Bottlenecked by InterferenceI Cooperation (Relaying) → security (?)

I Het-Nets/Cognitive Radio → enough (?)I Interference Alignment → scalability (?)I Massive MIMO → revolutionary, expensive (!)I CoMP → evolutionary, flexible, huge background, maturing...

Still lots to be done!!

Page 20: 6 interference management in mimo multicell

Past and Future

I Drivers of cellular system improvementI 2G: Digitalization (GSM/CDMA) → spectrumI 3G: Spread-spectrum (WCDMA) → spectrum again...I 4G: LTE (OFDMA) → and again...I 5G: “granularity” → devices → antennas

I Bottlenecked by InterferenceI Cooperation (Relaying) → security (?)I Het-Nets/Cognitive Radio

→ enough (?)I Interference Alignment → scalability (?)I Massive MIMO → revolutionary, expensive (!)I CoMP → evolutionary, flexible, huge background, maturing...

Still lots to be done!!

Page 21: 6 interference management in mimo multicell

Past and Future

I Drivers of cellular system improvementI 2G: Digitalization (GSM/CDMA) → spectrumI 3G: Spread-spectrum (WCDMA) → spectrum again...I 4G: LTE (OFDMA) → and again...I 5G: “granularity” → devices → antennas

I Bottlenecked by InterferenceI Cooperation (Relaying) → security (?)I Het-Nets/Cognitive Radio → enough (?)

I Interference Alignment → scalability (?)I Massive MIMO → revolutionary, expensive (!)I CoMP → evolutionary, flexible, huge background, maturing...

Still lots to be done!!

Page 22: 6 interference management in mimo multicell

Past and Future

I Drivers of cellular system improvementI 2G: Digitalization (GSM/CDMA) → spectrumI 3G: Spread-spectrum (WCDMA) → spectrum again...I 4G: LTE (OFDMA) → and again...I 5G: “granularity” → devices → antennas

I Bottlenecked by InterferenceI Cooperation (Relaying) → security (?)I Het-Nets/Cognitive Radio → enough (?)I Interference Alignment

→ scalability (?)I Massive MIMO → revolutionary, expensive (!)I CoMP → evolutionary, flexible, huge background, maturing...

Still lots to be done!!

Page 23: 6 interference management in mimo multicell

Past and Future

I Drivers of cellular system improvementI 2G: Digitalization (GSM/CDMA) → spectrumI 3G: Spread-spectrum (WCDMA) → spectrum again...I 4G: LTE (OFDMA) → and again...I 5G: “granularity” → devices → antennas

I Bottlenecked by InterferenceI Cooperation (Relaying) → security (?)I Het-Nets/Cognitive Radio → enough (?)I Interference Alignment → scalability (?)

I Massive MIMO → revolutionary, expensive (!)I CoMP → evolutionary, flexible, huge background, maturing...

Still lots to be done!!

Page 24: 6 interference management in mimo multicell

Past and Future

I Drivers of cellular system improvementI 2G: Digitalization (GSM/CDMA) → spectrumI 3G: Spread-spectrum (WCDMA) → spectrum again...I 4G: LTE (OFDMA) → and again...I 5G: “granularity” → devices → antennas

I Bottlenecked by InterferenceI Cooperation (Relaying) → security (?)I Het-Nets/Cognitive Radio → enough (?)I Interference Alignment → scalability (?)I Massive MIMO

→ revolutionary, expensive (!)I CoMP → evolutionary, flexible, huge background, maturing...

Still lots to be done!!

Page 25: 6 interference management in mimo multicell

Past and Future

I Drivers of cellular system improvementI 2G: Digitalization (GSM/CDMA) → spectrumI 3G: Spread-spectrum (WCDMA) → spectrum again...I 4G: LTE (OFDMA) → and again...I 5G: “granularity” → devices → antennas

I Bottlenecked by InterferenceI Cooperation (Relaying) → security (?)I Het-Nets/Cognitive Radio → enough (?)I Interference Alignment → scalability (?)I Massive MIMO → revolutionary, expensive (!)

I CoMP → evolutionary, flexible, huge background, maturing...

Still lots to be done!!

Page 26: 6 interference management in mimo multicell

Past and Future

I Drivers of cellular system improvementI 2G: Digitalization (GSM/CDMA) → spectrumI 3G: Spread-spectrum (WCDMA) → spectrum again...I 4G: LTE (OFDMA) → and again...I 5G: “granularity” → devices → antennas

I Bottlenecked by InterferenceI Cooperation (Relaying) → security (?)I Het-Nets/Cognitive Radio → enough (?)I Interference Alignment → scalability (?)I Massive MIMO → revolutionary, expensive (!)I CoMP

→ evolutionary, flexible, huge background, maturing...

Still lots to be done!!

Page 27: 6 interference management in mimo multicell

Past and Future

I Drivers of cellular system improvementI 2G: Digitalization (GSM/CDMA) → spectrumI 3G: Spread-spectrum (WCDMA) → spectrum again...I 4G: LTE (OFDMA) → and again...I 5G: “granularity” → devices → antennas

I Bottlenecked by InterferenceI Cooperation (Relaying) → security (?)I Het-Nets/Cognitive Radio → enough (?)I Interference Alignment → scalability (?)I Massive MIMO → revolutionary, expensive (!)I CoMP → evolutionary, flexible, huge background, maturing...

Still lots to be done!!

Page 28: 6 interference management in mimo multicell

Past and Future

I Drivers of cellular system improvementI 2G: Digitalization (GSM/CDMA) → spectrumI 3G: Spread-spectrum (WCDMA) → spectrum again...I 4G: LTE (OFDMA) → and again...I 5G: “granularity” → devices → antennas

I Bottlenecked by InterferenceI Cooperation (Relaying) → security (?)I Het-Nets/Cognitive Radio → enough (?)I Interference Alignment → scalability (?)I Massive MIMO → revolutionary, expensive (!)I CoMP → evolutionary, flexible, huge background, maturing...

Still lots to be done!!

Page 29: 6 interference management in mimo multicell

CoMP’s System Model

Desired Signal

Interference Signal

I B coordinating BSs

I K users per cell

I One BS per cell

I Each BS with multiple antennas

I User may have multiple antennas

I Embedded power control

I Inter-cell and intra-cell interference

Page 30: 6 interference management in mimo multicell

Now let’s get serious...

Page 31: 6 interference management in mimo multicell

Dissecting CoMPEnergy Efficiency

I Example 1: Power minimization problem [Yu&Lan 2007]

minimizeV,{vk}

α

subject to |vn|2 ≤ αpnSINRk ≥ γk,

given hk

TX : xN×1 =

K∑

k=1

skvk RX : yk = hk · x + zk

SINRk =|hk · vk|2∑

j 6=k |hk · vk|2 + σ2

I N =∑B

b=1Ntb TX antennas → MISOI Fixed pn per-antenna target powers → how (?)I Per user γk target SINRs → QoS balancing (?)I Perfectly known hk for all users → overhead (!)

Page 32: 6 interference management in mimo multicell

Dissecting CoMPEnergy Efficiency

I Example 1: Power minimization problem [Yu&Lan 2007]

minimizeV,{vk}

α

subject to |vn|2 ≤ αpnSINRk ≥ γk,

given hk

TX : xN×1 =

K∑

k=1

skvk RX : yk = hk · x + zk

SINRk =|hk · vk|2∑

j 6=k |hk · vk|2 + σ2

I N =∑B

b=1Ntb TX antennas → MISO

I Fixed pn per-antenna target powers → how (?)I Per user γk target SINRs → QoS balancing (?)I Perfectly known hk for all users → overhead (!)

Page 33: 6 interference management in mimo multicell

Dissecting CoMPEnergy Efficiency

I Example 1: Power minimization problem [Yu&Lan 2007]

minimizeV,{vk}

α

subject to |vn|2 ≤ αpnSINRk ≥ γk,

given hk

TX : xN×1 =

K∑

k=1

skvk RX : yk = hk · x + zk

SINRk =|hk · vk|2∑

j 6=k |hk · vk|2 + σ2

I N =∑B

b=1Ntb TX antennas → MISOI Fixed pn per-antenna target powers → how (?)

I Per user γk target SINRs → QoS balancing (?)I Perfectly known hk for all users → overhead (!)

Page 34: 6 interference management in mimo multicell

Dissecting CoMPEnergy Efficiency

I Example 1: Power minimization problem [Yu&Lan 2007]

minimizeV,{vk}

α

subject to |vn|2 ≤ αpnSINRk ≥ γk,

given hk

TX : xN×1 =

K∑

k=1

skvk RX : yk = hk · x + zk

SINRk =|hk · vk|2∑

j 6=k |hk · vk|2 + σ2

I N =∑B

b=1Ntb TX antennas → MISOI Fixed pn per-antenna target powers → how (?)I Per user γk target SINRs → QoS balancing (?)

I Perfectly known hk for all users → overhead (!)

Page 35: 6 interference management in mimo multicell

Dissecting CoMPEnergy Efficiency

I Example 1: Power minimization problem [Yu&Lan 2007]

minimizeV,{vk}

α

subject to |vn|2 ≤ αpnSINRk ≥ γk,

given hk

TX : xN×1 =

K∑

k=1

skvk RX : yk = hk · x + zk

SINRk =|hk · vk|2∑

j 6=k |hk · vk|2 + σ2

I N =∑B

b=1Ntb TX antennas → MISOI Fixed pn per-antenna target powers → how (?)I Per user γk target SINRs → QoS balancing (?)I Perfectly known hk for all users → overhead (!)

Page 36: 6 interference management in mimo multicell

Dissecting CoMPEnergy Efficiency

I Example 2: Power minimization problem [Song et al. 2007]

minimizep>0,V,U

K∑

k=1

wk pk

subject to SINRk ≥ γkgiven Hkj and wk

TX : xN×1 =

K∑

k=1

√pk skvk RX : yk = uHk ·Hkk · x + zk

SINRk =pk|uHk ·Hkk · vk|2∑

j 6=k pj |uHk ·Hkj · vk|2 + σ2

I N =∑B

b=1Ntb TX antennas → MIMOI Fixed pn per-antenna target powers → optimized per user pkI Known weight per user wk → how (?)I Per user γk target SINRs → QoS balancing (?)I Perfectly known Hkj for all users → overhead (!)

Page 37: 6 interference management in mimo multicell

Dissecting CoMPEnergy Efficiency

I Example 2: Power minimization problem [Song et al. 2007]

minimizep>0,V,U

K∑

k=1

wk pk

subject to SINRk ≥ γkgiven Hkj and wk

TX : xN×1 =

K∑

k=1

√pk skvk RX : yk = uHk ·Hkk · x + zk

SINRk =pk|uHk ·Hkk · vk|2∑

j 6=k pj |uHk ·Hkj · vk|2 + σ2

I N =∑B

b=1Ntb TX antennas → MIMO

I Fixed pn per-antenna target powers → optimized per user pkI Known weight per user wk → how (?)I Per user γk target SINRs → QoS balancing (?)I Perfectly known Hkj for all users → overhead (!)

Page 38: 6 interference management in mimo multicell

Dissecting CoMPEnergy Efficiency

I Example 2: Power minimization problem [Song et al. 2007]

minimizep>0,V,U

K∑

k=1

wk pk

subject to SINRk ≥ γkgiven Hkj and wk

TX : xN×1 =

K∑

k=1

√pk skvk RX : yk = uHk ·Hkk · x + zk

SINRk =pk|uHk ·Hkk · vk|2∑

j 6=k pj |uHk ·Hkj · vk|2 + σ2

I N =∑B

b=1Ntb TX antennas → MIMOI Fixed pn per-antenna target powers → optimized per user pkI Known weight per user wk → how (?)

I Per user γk target SINRs → QoS balancing (?)I Perfectly known Hkj for all users → overhead (!)

Page 39: 6 interference management in mimo multicell

Dissecting CoMPEnergy Efficiency

I Example 2: Power minimization problem [Song et al. 2007]

minimizep>0,V,U

K∑

k=1

wk pk

subject to SINRk ≥ γkgiven Hkj and wk

TX : xN×1 =

K∑

k=1

√pk skvk RX : yk = uHk ·Hkk · x + zk

SINRk =pk|uHk ·Hkk · vk|2∑

j 6=k pj |uHk ·Hkj · vk|2 + σ2

I N =∑B

b=1Ntb TX antennas → MIMOI Fixed pn per-antenna target powers → optimized per user pkI Known weight per user wk → how (?)I Per user γk target SINRs → QoS balancing (?)

I Perfectly known Hkj for all users → overhead (!)

Page 40: 6 interference management in mimo multicell

Dissecting CoMPEnergy Efficiency

I Example 2: Power minimization problem [Song et al. 2007]

minimizep>0,V,U

K∑

k=1

wk pk

subject to SINRk ≥ γkgiven Hkj and wk

TX : xN×1 =

K∑

k=1

√pk skvk RX : yk = uHk ·Hkk · x + zk

SINRk =pk|uHk ·Hkk · vk|2∑

j 6=k pj |uHk ·Hkj · vk|2 + σ2

I N =∑B

b=1Ntb TX antennas → MIMOI Fixed pn per-antenna target powers → optimized per user pkI Known weight per user wk → how (?)I Per user γk target SINRs → QoS balancing (?)I Perfectly known Hkj for all users → overhead (!)

Page 41: 6 interference management in mimo multicell

Dissecting CoMPQuality of Service

I Example 3: Min-max SINR problem [Huang et al. 2011]

maximizep>0,V

min∀k

SINRk

subject to ‖p‖ ≤ P‖vk‖ = 1

given hk

TX : xN×1 =

K∑

k=1

√pk skvk RX : yk = hk · x + zk

SINRk =pk|hk · vk|2∑

j 6=k pj |hj · vk|2 + σ2

I N =∑B

b=1Ntb TX antennas → MISOI Fixed pn per-antenna target powers → optimized per user pkI Per user γk target SINRs → minimum QoSI Perfectly known hk for all → overhead (!)

Page 42: 6 interference management in mimo multicell

Dissecting CoMPQuality of Service

I Example 3: Min-max SINR problem [Huang et al. 2011]

maximizep>0,V

min∀k

SINRk

subject to ‖p‖ ≤ P‖vk‖ = 1

given hk

TX : xN×1 =

K∑

k=1

√pk skvk RX : yk = hk · x + zk

SINRk =pk|hk · vk|2∑

j 6=k pj |hj · vk|2 + σ2

I N =∑B

b=1Ntb TX antennas → MISO

I Fixed pn per-antenna target powers → optimized per user pkI Per user γk target SINRs → minimum QoSI Perfectly known hk for all → overhead (!)

Page 43: 6 interference management in mimo multicell

Dissecting CoMPQuality of Service

I Example 3: Min-max SINR problem [Huang et al. 2011]

maximizep>0,V

min∀k

SINRk

subject to ‖p‖ ≤ P‖vk‖ = 1

given hk

TX : xN×1 =

K∑

k=1

√pk skvk RX : yk = hk · x + zk

SINRk =pk|hk · vk|2∑

j 6=k pj |hj · vk|2 + σ2

I N =∑B

b=1Ntb TX antennas → MISOI Fixed pn per-antenna target powers → optimized per user pk

I Per user γk target SINRs → minimum QoSI Perfectly known hk for all → overhead (!)

Page 44: 6 interference management in mimo multicell

Dissecting CoMPQuality of Service

I Example 3: Min-max SINR problem [Huang et al. 2011]

maximizep>0,V

min∀k

SINRk

subject to ‖p‖ ≤ P‖vk‖ = 1

given hk

TX : xN×1 =

K∑

k=1

√pk skvk RX : yk = hk · x + zk

SINRk =pk|hk · vk|2∑

j 6=k pj |hj · vk|2 + σ2

I N =∑B

b=1Ntb TX antennas → MISOI Fixed pn per-antenna target powers → optimized per user pkI Per user γk target SINRs → minimum QoS

I Perfectly known hk for all → overhead (!)

Page 45: 6 interference management in mimo multicell

Dissecting CoMPQuality of Service

I Example 3: Min-max SINR problem [Huang et al. 2011]

maximizep>0,V

min∀k

SINRk

subject to ‖p‖ ≤ P‖vk‖ = 1

given hk

TX : xN×1 =

K∑

k=1

√pk skvk RX : yk = hk · x + zk

SINRk =pk|hk · vk|2∑

j 6=k pj |hj · vk|2 + σ2

I N =∑B

b=1Ntb TX antennas → MISOI Fixed pn per-antenna target powers → optimized per user pkI Per user γk target SINRs → minimum QoSI Perfectly known hk for all → overhead (!)

Page 46: 6 interference management in mimo multicell

Dissecting CoMPQuality of Service

I Example 4: Min-max SINR problem [Cai et al. 2011]

maximizep>0,V

min∀k

SINRkαk

subject to w` · p ≤ P`, ` ∈ L∣∣∣ |L| < K

given Hkj , wk and α

TX : xN×1 =K∑

k=1

√pk skvk RX : yk = uHk ·Hkk · x + zk

SINRk =pk|uHk ·Hkk · vk|2∑

j 6=k pj |uHk ·Hkj · vk|2 + σ2

I N =∑B

b=1Ntb TX antennas → MIMOI Fixed pn per-antenna target powers → optimized per user pkI Known weight vectors per user wk and scores αk → how (?)I Perfectly known Hkj for all users → overhead (!)

Page 47: 6 interference management in mimo multicell

Dissecting CoMPQuality of Service

I Example 4: Min-max SINR problem [Cai et al. 2011]

maximizep>0,V

min∀k

SINRkαk

subject to w` · p ≤ P`, ` ∈ L∣∣∣ |L| < K

given Hkj , wk and α

TX : xN×1 =K∑

k=1

√pk skvk RX : yk = uHk ·Hkk · x + zk

SINRk =pk|uHk ·Hkk · vk|2∑

j 6=k pj |uHk ·Hkj · vk|2 + σ2

I N =∑B

b=1Ntb TX antennas → MIMO

I Fixed pn per-antenna target powers → optimized per user pkI Known weight vectors per user wk and scores αk → how (?)I Perfectly known Hkj for all users → overhead (!)

Page 48: 6 interference management in mimo multicell

Dissecting CoMPQuality of Service

I Example 4: Min-max SINR problem [Cai et al. 2011]

maximizep>0,V

min∀k

SINRkαk

subject to w` · p ≤ P`, ` ∈ L∣∣∣ |L| < K

given Hkj , wk and α

TX : xN×1 =K∑

k=1

√pk skvk RX : yk = uHk ·Hkk · x + zk

SINRk =pk|uHk ·Hkk · vk|2∑

j 6=k pj |uHk ·Hkj · vk|2 + σ2

I N =∑B

b=1Ntb TX antennas → MIMOI Fixed pn per-antenna target powers → optimized per user pkI Known weight vectors per user wk and scores αk → how (?)

I Perfectly known Hkj for all users → overhead (!)

Page 49: 6 interference management in mimo multicell

Dissecting CoMPQuality of Service

I Example 4: Min-max SINR problem [Cai et al. 2011]

maximizep>0,V

min∀k

SINRkαk

subject to w` · p ≤ P`, ` ∈ L∣∣∣ |L| < K

given Hkj , wk and α

TX : xN×1 =K∑

k=1

√pk skvk RX : yk = uHk ·Hkk · x + zk

SINRk =pk|uHk ·Hkk · vk|2∑

j 6=k pj |uHk ·Hkj · vk|2 + σ2

I N =∑B

b=1Ntb TX antennas → MIMOI Fixed pn per-antenna target powers → optimized per user pkI Known weight vectors per user wk and scores αk → how (?)I Perfectly known Hkj for all users → overhead (!)

Page 50: 6 interference management in mimo multicell

Dissecting CoMPSpectral Efficiency

I Example 5: Sum-rate maximization problem [Tran et al. 2012]

maximizet,V

K∏

k=1

tk

subject to SINRk ≥ t1/αkk − 1

K∑

k=1

‖vk‖2 ≤ P

given hk, P and α

αk log2(1 + SINRk) −→ (1 + SINRk)αk −→ tk

I N =∑B

b=1Ntb TX antennas → MISOI Fixed pn per-antenna target powers → optimized total pkI Known scores αk → how (?)I Perfectly known hk for all users → overhead (!)

Page 51: 6 interference management in mimo multicell

Dissecting CoMPSpectral Efficiency

I Example 5: Sum-rate maximization problem [Tran et al. 2012]

maximizet,V

K∏

k=1

tk

subject to SINRk ≥ t1/αkk − 1

K∑

k=1

‖vk‖2 ≤ P

given hk, P and α

αk log2(1 + SINRk) −→ (1 + SINRk)αk −→ tk

I N =∑B

b=1Ntb TX antennas → MISO

I Fixed pn per-antenna target powers → optimized total pkI Known scores αk → how (?)I Perfectly known hk for all users → overhead (!)

Page 52: 6 interference management in mimo multicell

Dissecting CoMPSpectral Efficiency

I Example 5: Sum-rate maximization problem [Tran et al. 2012]

maximizet,V

K∏

k=1

tk

subject to SINRk ≥ t1/αkk − 1

K∑

k=1

‖vk‖2 ≤ P

given hk, P and α

αk log2(1 + SINRk) −→ (1 + SINRk)αk −→ tk

I N =∑B

b=1Ntb TX antennas → MISOI Fixed pn per-antenna target powers → optimized total pkI Known scores αk → how (?)

I Perfectly known hk for all users → overhead (!)

Page 53: 6 interference management in mimo multicell

Dissecting CoMPSpectral Efficiency

I Example 5: Sum-rate maximization problem [Tran et al. 2012]

maximizet,V

K∏

k=1

tk

subject to SINRk ≥ t1/αkk − 1

K∑

k=1

‖vk‖2 ≤ P

given hk, P and α

αk log2(1 + SINRk) −→ (1 + SINRk)αk −→ tk

I N =∑B

b=1Ntb TX antennas → MISOI Fixed pn per-antenna target powers → optimized total pkI Known scores αk → how (?)I Perfectly known hk for all users → overhead (!)

Page 54: 6 interference management in mimo multicell

Dissecting CoMPSpectral Efficiency

I Example 6: Sum-rate maximization problem [Park et al. 2013]

maximizeV,{Vk}

K∑

k=1

wk log2 |INr + (σ2kI + Φk)−1HkkVkVkH

Hkk|

subject to ‖HjkVk‖2 ≤ αjkσ2j‖Vk‖2 ≤ pk

given Hjk,w,p and α

Φk =

K∑

k=1

HkjVjVHj HH

kj

I N =∑B

b=1Ntb TX antennas → MIMOI Fixed pn per-antenna target powers → optimized per user pkI Known weights w, target powers pk and scores αk → how (?)I Perfectly known Hkj for all users → overhead (!)

Page 55: 6 interference management in mimo multicell

Dissecting CoMPSpectral Efficiency

I Example 6: Sum-rate maximization problem [Park et al. 2013]

maximizeV,{Vk}

K∑

k=1

wk log2 |INr + (σ2kI + Φk)−1HkkVkVkH

Hkk|

subject to ‖HjkVk‖2 ≤ αjkσ2j‖Vk‖2 ≤ pk

given Hjk,w,p and α

Φk =

K∑

k=1

HkjVjVHj HH

kj

I N =∑B

b=1Ntb TX antennas → MIMO

I Fixed pn per-antenna target powers → optimized per user pkI Known weights w, target powers pk and scores αk → how (?)I Perfectly known Hkj for all users → overhead (!)

Page 56: 6 interference management in mimo multicell

Dissecting CoMPSpectral Efficiency

I Example 6: Sum-rate maximization problem [Park et al. 2013]

maximizeV,{Vk}

K∑

k=1

wk log2 |INr + (σ2kI + Φk)−1HkkVkVkH

Hkk|

subject to ‖HjkVk‖2 ≤ αjkσ2j‖Vk‖2 ≤ pk

given Hjk,w,p and α

Φk =

K∑

k=1

HkjVjVHj HH

kj

I N =∑B

b=1Ntb TX antennas → MIMOI Fixed pn per-antenna target powers → optimized per user pkI Known weights w, target powers pk and scores αk → how (?)

I Perfectly known Hkj for all users → overhead (!)

Page 57: 6 interference management in mimo multicell

Dissecting CoMPSpectral Efficiency

I Example 6: Sum-rate maximization problem [Park et al. 2013]

maximizeV,{Vk}

K∑

k=1

wk log2 |INr + (σ2kI + Φk)−1HkkVkVkH

Hkk|

subject to ‖HjkVk‖2 ≤ αjkσ2j‖Vk‖2 ≤ pk

given Hjk,w,p and α

Φk =

K∑

k=1

HkjVjVHj HH

kj

I N =∑B

b=1Ntb TX antennas → MIMOI Fixed pn per-antenna target powers → optimized per user pkI Known weights w, target powers pk and scores αk → how (?)I Perfectly known Hkj for all users → overhead (!)

Page 58: 6 interference management in mimo multicell

Comprehensive Review in a Nutshell

low iteration algorithm for solving P3 is presented in [55], and a SDP formulation for a multi-cell approach tothe problem for bounded CSI noise model via translating the problem to convex formulations thereby usingSDP is considered in [60]. In [56], a connection is established between P3 and virtual SINR (VSINR) (func-tion of single beamforming vector) to address the practical issues of decentralized implementation basedon local channel information. Utilization of downlink-uplink duality to formulate noise covariance in uplink,GP to characterize downlink power, and an alternating optimization problem in [59], authors solved P3 withper BS power for MIMO systems. Considering conditional eigenvalue problem with affine constraint andnon-linear Perron-Frobenius theory, authors in [52] optimized the physical layer link rate functions. Alsorecently, in [57], a relaxed zero forcing method for MISO and MIMO interference channels is proposed forthe rate control problems with centralized and distributed heuristic approach.

Table 1: Literature Classification of Works on CoMP

MISO †

MIMO Instantaneous Perfect CSIT Instantaneous Imperfect CSIT Statistical CSIT Covariance Information

P1

[20] [21] [22] [23][24] [25]

[87]

[26] [27] [28] [29][30]

[89] [31] [32][33]

P2

[34] [35] [36][37]

[38] [39] [40] [41] [42]

[43] [44] [45] [46][47] [42]

P3

[48] [49] [65] [51] [52][53] [54] [55] [56][57] [77]

[82] [58][59] [57] [78] [79]

[60] [61]

[81]

† Works on the upper diagonal portion of each cell are on MISO, while those on the lower diagonal are on MIMO.

Table 1 summarizes the the key-problems in the literatures, to the best of our knowledge, we have discussedin Section (2.1.1). We have distinguished the key problems into MISO vs. MIMO with different channelvariations available at the transmitter. Despite the different tools and analytics present in the perfect channelknowledge at the transmitter side, we have seen a significant need of work to be done in the the multi-antenna receiver case. Also, as we progress from left to right of the table, the available literatures diminishesin number. Within the cited literatures, we have pointed few key-holes for the key-problems which we stateas-

• Significant work has been performed for MISO case and few addressed MIMO case.

• Apart from instantaneous channel feedback (ideal case), other variations of CSIT for MIMO systemsare not addressed.

• The solution to the optimization problems with strict constrains may vary with relaxed constrainedproblems, and the comparison of those solutions are found lacking.

• An analogy between the centralized or the distributed approach for the similar mathematical tools andits complexity has not been discussed in most literatures.

• Comparison of all key problems to address an issue of energy efficiency with respect to QoS deliveredhas not been addressed till date.

Upon visualizing the key-holes in the problems we have discussed so far, we frame this project withinMIMO multi-cell systems under various available channel states at the transmitter side, avoiding the workdone in the MISO systems. Note that the design criteria for a MIMO channel is more challenging than

7

Page 59: 6 interference management in mimo multicell

OK, so what if CSIT is not perfect ?

Page 60: 6 interference management in mimo multicell

Power Minimization AgainFrom Perfect to Imperfect CSIT

maximizeV,{vk}

K∑

k=1

‖vk‖2

subject to SINRk ,|hHkkvk|2

N∑i=1,i 6=k

|hHkivi|2 + σ2≥ γk

given {γ1, · · · , γK}

hki = hki + ehki

Pr (SINRk ≥ γk) = Pr

(|hHkkvk|2∑N

i=1,i 6=k |hHkivi|2 + σ2≥ γk

)≥ 1− ρk

Page 61: 6 interference management in mimo multicell

Power Minimization AgainFrom Perfect to Imperfect CSIT

maximizeV,{vk}

K∑

k=1

‖vk‖2

subject to SINRk ,|hHkkvk|2

N∑i=1,i 6=k

|hHkivi|2 + σ2≥ γk

given {γ1, · · · , γK}

hki = hki + ehki

Pr (SINRk ≥ γk) = Pr

(|hHkkvk|2∑N

i=1,i 6=k |hHkivi|2 + σ2≥ γk

)≥ 1− ρk

Page 62: 6 interference management in mimo multicell

Power Minimization AgainFrom Perfect to Imperfect CSIT

maximizeV,{vk}

K∑

k=1

‖vk‖2

subject to SINRk ,|hHkkvk|2

N∑i=1,i 6=k

|hHkivi|2 + σ2≥ γk

given {γ1, · · · , γK}

hki = hki + ehki

Pr (SINRk ≥ γk) = Pr

(|hHkkvk|2∑N

i=1,i 6=k |hHkivi|2 + σ2≥ γk

)≥ 1− ρk

Page 63: 6 interference management in mimo multicell

Power Minimization AgainRobust Formulation

maximizeV,{vk}

K∑

k=1

‖vk‖2

subject to Pr

(|hHkkvk|2∑N

i=1,i 6=k |hHkivi|2 + σ2≥ γk

)≥ 1− ρk

given {γ1, · · · , γK} and {ρ1, · · · , ρK}

hki = hki + ehki

ehki ∼ CN (0,Qki)

|hHkivi|2 ∼ χ22(δki;σ

2ki) δki , |hHkivi|2 σ2ki = vHi Qkivi

Page 64: 6 interference management in mimo multicell

Power Minimization AgainRobust Formulation

maximizeV,{vk}

K∑

k=1

‖vk‖2

subject to Pr

(|hHkkvk|2∑N

i=1,i 6=k |hHkivi|2 + σ2≥ γk

)≥ 1− ρk

given {γ1, · · · , γK} and {ρ1, · · · , ρK}

hki = hki + ehki ehki ∼ CN (0,Qki)

|hHkivi|2 ∼ χ22(δki;σ

2ki) δki , |hHkivi|2 σ2ki = vHi Qkivi

Page 65: 6 interference management in mimo multicell

Power Minimization AgainRobust Formulation

maximizeV,{vk}

K∑

k=1

‖vk‖2

subject to Pr

(|hHkkvk|2∑N

i=1,i 6=k |hHkivi|2 + σ2≥ γk

)≥ 1− ρk

given {γ1, · · · , γK} and {ρ1, · · · , ρK}

hki = hki + ehki ehki ∼ CN (0,Qki)

|hHkivi|2 ∼ χ22(δki;σ

2ki) δki , |hHkivi|2 σ2ki = vHi Qkivi

Page 66: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

Pr(

SINRk ≥ γk)

Xkk ,|hHkkvk|2vHk Qkkvk

Xki ,|hHkivi|2vHi Qkivi

σ2ki = vHi Qkivi

Page 67: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

Pr

(|hHkkvk|2∑N

i=1,i 6=k |hHkivi|2 + σ2≥ γk

)

Xkk ,|hHkkvk|2vHk Qkkvk

Xki ,|hHkivi|2vHi Qkivi

σ2ki = vHi Qkivi

Page 68: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

Pr

|hHkkvk|2 ≥ γk

N∑

i=1,i 6=k|hHkivi|2 + γk · σ2

Xkk ,|hHkkvk|2vHk Qkkvk

Xki ,|hHkivi|2vHi Qkivi

σ2ki = vHi Qkivi

Page 69: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

Pr

|hHkkvk|2 ≥ γk

N∑

i=1,i 6=k|hHkivi|2 + γk · σ2

Xkk ,|hHkkvk|2vHk Qkkvk

Xki ,|hHkivi|2vHi Qkivi

σ2ki = vHi Qkivi

Page 70: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

Pr

σ2kkXkk − γk

N∑

i=1,i 6=kσ2iiXki ≥ γk · σ2

Xkk ,|hHkkvk|2vHk Qkkvk

Xki ,|hHkivi|2vHi Qkivi

σ2ki = vHi Qkivi

Page 71: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

∫ ∞

0. . .

∫ ∞

0Pr(Xkk ≥ ckk

) N∏

i=1,i 6=kfXki(ti)dti . . . dtN

ckk =γkσ2kk

σ2 +

N∑

i=1,i 6=kσ2kiti

ckk is non-central χ2

[Kandukuri] S. Kandukuri and S. Boyd, “Optimal power control in interference-limited fading wireless channelswith outage-probability specifications,” IEEE Transactions on Wireless Communications, vol. 1,no. 1, pp. 46–55, Jan 2002.

Page 72: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

∫ ∞

0. . .

∫ ∞

0Pr(Xkk ≥ ckk

) N∏

i=1,i 6=kfXki(ti)dti . . . dtN

ckk =γkσ2kk

σ2 +

N∑

i=1,i 6=kσ2kiti

ckk is non-central χ2

[Kandukuri] S. Kandukuri and S. Boyd, “Optimal power control in interference-limited fading wireless channelswith outage-probability specifications,” IEEE Transactions on Wireless Communications, vol. 1,no. 1, pp. 46–55, Jan 2002.

Page 73: 6 interference management in mimo multicell

Model of XkkNon-central and Central

Theorem (Cox&Reid):

Let Z ∼ χ2n(δ;σ

2) and Z ∼ χ2n(0;σ

2), with δ/n small. Then,

Pr (Z > γ) ≈ Pr

(Z >

γ

1 + δn

)

[Cox&Reid] D. R. Cox and N. Reid, “Approximations to noncentral distributions,” The Canadian Journal ofStatistics / La Revue Canadienne de Statistique, vol. 15, no. 2, pp. 105–114, 1987.

Page 74: 6 interference management in mimo multicell

Model of XkkNon-central and Central

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Comparison of the CDF of Non-central and Central χ2 DistributionCumulative

Distribution

Function:Pr(Z

≥δ)

Non-centrality Parameter: δ

δ = 0.1

δ = 2

Non-central χ2

Central χ2

Page 75: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

Pr (SINRk ≥ γk) ≈∫ ∞

0. . .

∫ ∞

0Pr(Xkk ≥ ckk

1+δkk2

) N∏

i=1,i 6=kfXki(ti)dti . . . dtN

= e− γkσ

2

σ2kk

(1+δkk2 )

N∏

i=1,i6=k

∫ ∞

0exp (−αkiti)fXki(ti)dti

= e− γkσ

2

σ2kk

(1+δkk2 )

N∏

i=1,i 6=k

E[e−αkiti ]

Page 76: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

Pr (SINRk ≥ γk) ≈∫ ∞

0. . .

∫ ∞

0Pr(Xkk ≥ ckk

1+δkk2

) N∏

i=1,i 6=kfXki(ti)dti . . . dtN

= e− γkσ

2

σ2kk

(1+δkk2 )

N∏

i=1,i6=k

∫ ∞

0exp (−αkiti)fXki(ti)dti

= e− γkσ

2

σ2kk

(1+δkk2 )

N∏

i=1,i 6=k

E[e−αkiti ]

Xkk ∼ χ22 =⇒ Pr(Xkk ≥ x) = e−x/2

Page 77: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

Pr (SINRk ≥ γk) ≈∫ ∞

0. . .

∫ ∞

0exp

(− ckk

1 + δkk2

)N∏

i=1,i 6=kfXki(ti)dti . . . dtN

= e− γkσ

2

σ2kk

(1+δkk2 )

N∏

i=1,i6=k

∫ ∞

0exp (−αkiti)fXki(ti)dti

= e− γkσ

2

σ2kk

(1+δkk2 )

N∏

i=1,i 6=k

E[e−αkiti ]

Xkk ∼ χ22 =⇒ Pr(Xkk ≥ x) = e−x/2

Page 78: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

Pr (SINRk ≥ γk) ≈∫ ∞

0. . .

∫ ∞

0exp

(− ckk

1 + δkk2

)N∏

i=1,i 6=kfXki(ti)dti . . . dtN

= e− γkσ

2

σ2kk

(1+δkk2 )

N∏

i=1,i6=k

∫ ∞

0exp (−αkiti)fXki(ti)dti

= e− γkσ

2

σ2kk

(1+δkk2 )

N∏

i=1,i 6=k

E[e−αkiti ]

αki =γkσ

2ki

σ2kk(1 +δkk2 )

Page 79: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

Pr (SINRk ≥ γk) ≈∫ ∞

0. . .

∫ ∞

0exp

(− ckk

1 + δkk2

)N∏

i=1,i 6=kfXki(ti)dti . . . dtN

= e− γkσ

2

σ2kk

(1+δkk2 )

N∏

i=1,i6=k

∫ ∞

0exp (−αkiti)fXki(ti)dti

= e− γkσ

2

σ2kk

(1+δkk2 )

N∏

i=1,i 6=k

E[e−αkiti ]

αki =γkσ

2ki

σ2kk(1 +δkk2 )

Page 80: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

Pr (SINRk ≥ γk) ≈ e− γkσ

2

σ2kk

(1+δkk2 )

N∏

i=1,i 6=kE[e−αkiti ]

= e− γkσ

2

σ2kk

(1+δkk2 )

exp(∑N

i=1,i 6=k− αkiδki(1+2αki)

)∏Ni=1,i 6=k(1 + 2αki)

= e− γkσ

2

σ2kk

(1+δkk2 )

N∏

i=1,i 6=k

exp(− γkpki

σ2kk(1+

δkk2

)+2γkσ2ki

)

1 +γkσ

2ki

σ2kk(1+δkk/2)

Z ∼ χ22(δ) =⇒ E[esZ ] =

exp(

sδ1−2s

)

1− 2s

Page 81: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

Pr (SINRk ≥ γk) ≈ e− γkσ

2

σ2kk

(1+δkk2 )

N∏

i=1,i 6=kE[e−αkiti ]

= e− γkσ

2

σ2kk

(1+δkk2 )

exp(∑N

i=1,i 6=k− αkiδki(1+2αki)

)∏Ni=1,i 6=k(1 + 2αki)

= e− γkσ

2

σ2kk

(1+δkk2 )

N∏

i=1,i 6=k

exp(− γkpki

σ2kk(1+

δkk2

)+2γkσ2ki

)

1 +γkσ

2ki

σ2kk(1+δkk/2)

Z ∼ χ22(δ) =⇒ E[esZ ] =

exp(

sδ1−2s

)

1− 2s

Page 82: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

Pr (SINRk ≥ γk) ≈ e− γkσ

2

σ2kk

(1+δkk2 )

N∏

i=1,i 6=kE[e−αkiti ]

= e− γkσ

2

σ2kk

(1+δkk2 )

exp(∑N

i=1,i 6=k− αkiδki(1+2αki)

)∏Ni=1,i 6=k(1 + 2αki)

= e− γkσ

2

σ2kk

(1+δkk2 )

N∏

i=1,i 6=k

exp(− γkpki

σ2kk(1+

δkk2

)+2γkσ2ki

)

1 +γkσ

2ki

σ2kk(1+δkk/2)

Z ∼ χ22(δ) =⇒ E[esZ ] =

exp(

sδ1−2s

)

1− 2s

Page 83: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

Pr (SINRk ≥ γk) ≈ e− γkσ

2

σ2kk

(1+δkk2 )

N∏

i=1,i 6=kE[e−αkiti ]

= e− γkσ

2

σ2kk

(1+δkk2 )

exp(∑N

i=1,i 6=k− αkiδki(1+2αki)

)∏Ni=1,i 6=k(1 + 2αki)

= e− γkσ

2

σ2kk

(1+δkk2 )

N∏

i=1,i 6=k

exp(− γkpki

σ2kk(1+

δkk2

)+2γkσ2ki

)

1 +γkσ

2ki

σ2kk(1+δkk/2)

Z ∼ χ22(δ) =⇒ E[esZ ] =

exp(

sδ1−2s

)

1− 2s

Page 84: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

Pr (SINRk ≥ γk) ≈e− γkσ

2

vHk

Akkvk

N∏

i=1,i 6=k

exp

(−γk|h

Hkivi|

2

vHk

Akkvk+γkvHi

Qkivi

)(1+

γkvHi

Qkivi

vHk

Akkvk

)

≥ 1− ρk

Akk , Qkk + hkkhHkk

N∑

i=1

ln(1 + xi) ≤N∑

i=1

xi

Page 85: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

Pr (SINRk ≥ γk) ≈e− γkσ

2

vHk

Akkvk

N∏

i=1,i 6=k

exp

(−γk|h

Hkivi|

2

vHk

Akkvk+γkvHi

Qkivi

)(1+

γkvHi

Qkivi

vHk

Akkvk

)

≥ 1− ρk

Akk , Qkk + hkkhHkk

N∑

i=1

ln(1 + xi) ≤N∑

i=1

xi

Page 86: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

Pr (SINRk ≥ γk) ≈e− γkσ

2

vHk

Akkvk

N∏

i=1,i 6=k

exp

(−γk|h

Hkivi|

2

vHk

Akkvk+γkvHi

Qkivi

)(1+

γkvHi

Qkivi

vHk

Akkvk

) ≥ 1− ρk

Akk , Qkk + hkkhHkk

N∑

i=1

ln(1 + xi) ≤N∑

i=1

xi

Page 87: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

log

e− γkσ

2

vHk

Akkvk

N∏

i=1,i6=k

exp

(−γk|h

Hkivi|

2

vHk

Akkvk+γkvHi

Qkivi

)(1+

γkvHi

Qkivi

vHk

Akkvk

) ≥ log(1− ρk)

Akk , Qkk + hkkhHkk

N∑

i=1

ln(1 + xi) ≤N∑

i=1

xi

Page 88: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

γkσ2

vHk Akkvk+

N∑i=1,i 6=k

γk|hHkivi|2

vHk Akkvk+γkvHi Qkivi

+N∑i=1,i 6=k

ln

(1+

γkvHi Qkivi

vHk Akkvk

)≤− ln(1−ρk)

Akk , Qkk + hkkhHkk

N∑

i=1

ln(1 + xi) ≤N∑

i=1

xi

Page 89: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

γkσ2

vHk Akkvk+

N∑i=1,i 6=k

γk|hHkivi|2

vHk Akkvk+γkvHi Qkivi

+N∑i=1,i 6=k

ln

(1+

γkvHi Qkivi

vHk Akkvk

)≤− ln(1−ρk)

Akk , Qkk + hkkhHkk

N∑

i=1

ln(1 + xi) ≤N∑

i=1

xi

Page 90: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

γkσ2

vHk Akkvk+

N∑

i=1,i6=k

(γk|hHkivi|

2

vHk Akkvk+γkvHi Qkivi

+γkv

Hi Qkivi

vHk Akkvk

)≤ − ln(1− ρk)

Akk , Qkk + hkkhHkk

N∑

i=1

ln(1 + xi) ≤N∑

i=1

xi

Page 91: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

γkσ2

vHk Akkvk+

N∑

i=1,i6=k

(γk|hHkivi|

2

vHk Akkvk+γkvHi Qkivi

+γkv

Hi Qkivi

vHk Akkvk

)≤ − ln(1− ρk)

Akk , Qkk + hkkhHkk

N∑

i=1

ln(1 + xi) ≤N∑

i=1

xi

Page 92: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

γkσ2

vHk Akkvk+

N∑

i=1,i 6=k

(γk|hHkivi|

2

vHk Akkvk+γkvHi Qkivi

)≤ − ln(1− ρk)

Akk , Qkk + hkkhHkk

N∑

i=1

ln(1 + xi) ≤N∑

i=1

xi

Page 93: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

γkσ2

vHk Akkvk+

N∑

i=1,i 6=k

(γk|hHkivi|

2

vHk Akkvk+γkvHi Qkivi

)≤ − ln(1− ρk)

Akk , Qkk + hkkhHkk

N∑

i=1

ln(1 + xi) ≤N∑

i=1

xi

Page 94: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

σ2 +N∑

i=1,i 6=k|hHkivi|2 + vHi Qkivi

vHk Akkvk≤ − ln(1− ρk)

γk

Page 95: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

vHk Akkvk

σ2 +N∑

i=1,i 6=k|hHkivi|2 + vHi Qkivi

≥ −γkln(1− ρk)

Page 96: 6 interference management in mimo multicell

Statistical SINR ConstraintTowards a Closed-form

vHk Akkvk

σ2 +N∑

i=1,i 6=k|hHkivi|2 + vHi Qkivi

≥ −γkln(1− ρk)

, ηk

Page 97: 6 interference management in mimo multicell

Power Minimization AgainRobust Formulation with Closed-form Constraint

maximizeV,{vk}

K∑

k=1

‖vk‖2

subject to Pr

(|hHkkvk|2∑N

i=1,i 6=k |hHkivi|2 + σ2≥ γk

)≥ 1− ρk

given {γ1, · · · , γK} and {ρ1, · · · , ρK}

given hki

Page 98: 6 interference management in mimo multicell

Power Minimization AgainRobust Formulation with Closed-form Constraint

maximizeV,{vk}

K∑

k=1

‖vk‖2

subject toσ2+

∑Ni=1,i 6=k |hHkivi|

2+vHi Qkivi

vHk Akkvk≤ − ln(1− ρk)

γk

given {γ1, · · · , γK} and {ρ1, · · · , ρK}

given hki and Qki

Page 99: 6 interference management in mimo multicell

Now, how do we solve this ?

Page 100: 6 interference management in mimo multicell

Some Tools...

Page 101: 6 interference management in mimo multicell

Semi-Definite Programming

minimizeV,{vk}

K∑

k=1

‖vk‖2

subject to |hHkkvk|2 − γkN∑

i=1,i6=k

|hHkivi|2 ≥ γkσ2

given hk

Vk , vkvHk (Positive Semi-Definite Matrix)

Hi , hkihHki (Positive Semi-Definite Matrix)

Page 102: 6 interference management in mimo multicell

Semi-Definite Programming

minimizeV,{vk}

K∑

k=1

‖vk‖2

subject to |hHkkvk|2 − γkN∑

i=1,i6=k

|hHkivi|2 ≥ γkσ2

given hk

Vk , vkvHk (Positive Semi-Definite Matrix)

Hi , hkihHki (Positive Semi-Definite Matrix)

Page 103: 6 interference management in mimo multicell

Semi-Definite Programming

minimize{Vk}�0

K∑

k=1

Tr(Vk)

subject to |hHkkvk|2 − γkN∑

i=1,i6=k

|hHkivi|2 ≥ γkσ2

given hk

Vk , vkvHk (Positive Semi-Definite Matrix)

Hi , hkihHki (Positive Semi-Definite Matrix)

Page 104: 6 interference management in mimo multicell

Semi-Definite Programming

minimize{Vk}�0

K∑

k=1

Tr(Vk)

subject to |hHkkvk|2 − γkN∑

i=1,i6=k

|hHkivi|2 ≥ γkσ2

given hk

Vk , vkvHk (Positive Semi-Definite Matrix)

Hi , hkihHki (Positive Semi-Definite Matrix)

Page 105: 6 interference management in mimo multicell

Semi-Definite Programming

minimize{Vk}�0

K∑

k=1

Tr(Vk)

subject to Tr(VkHk)− γkN∑

i=1,i 6=k

Tr(ViHi) ≥ γkσ2

given Hk � 0

Vk , vkvHk (Positive Semi-Definite Matrix)

Hi , hkihHki (Positive Semi-Definite Matrix)

Page 106: 6 interference management in mimo multicell

Power Minimization with Imperfect CSITSDP Formulation

minimizeV,{vk}

K∑

k=1

‖vk‖2

subject tovHk Akkvk

σ2+∑Ni=1,i 6=k |hHkivi|2+vHi Qkivi

≥ γk− ln(1− ρk)

,1

ηk

Page 107: 6 interference management in mimo multicell

Power Minimization with Imperfect CSITSDP Formulation

minimize{Vk}�0

K∑

k=1

Tr(Vk)

subject tovHk Akkvk

σ2+∑Ni=1,i 6=k |hHkivi|2+vHi Qkivi

≥ γk− ln(1− ρk)

,1

ηk

Page 108: 6 interference management in mimo multicell

Power Minimization with Imperfect CSITSDP Formulation

minimize{Vk}�0

K∑

k=1

Tr(Vk)

subject to ηkvHk Akkvk ≥ σ2 +

N∑

i=1,i 6=k|hHkivi|2 + vHi Qkivi

Page 109: 6 interference management in mimo multicell

Power Minimization with Imperfect CSITSDP Formulation

minimize{Vk}�0

K∑

k=1

Tr(Vk)

subject to Tr(Vk(hkkh

Hkk + Qkk)

)≥ σ2 +

N∑

i=1,i 6=kTr(Vi(hkih

Hki + Qki)

)

Page 110: 6 interference management in mimo multicell

Second-Order Cone Programming

minimizeV,{vk}

K∑

k=1

‖vk‖2

subject to |hHkkvk|2 − γkN∑

i=1,i6=k

|hHkivi|2 ≥ γkσ2

Tr(~VH · ~V) ≤ α

given hk

~V , [v1, · · · ,vK ] (Beamforming Matrix)

Page 111: 6 interference management in mimo multicell

Second-Order Cone Programming

minimizeV,{vk}

K∑

k=1

‖vk‖2

subject to |hHkkvk|2 − γkN∑

i=1,i6=k

|hHkivi|2 ≥ γkσ2

Tr(~VH · ~V) ≤ α

given hk

~V , [v1, · · · ,vK ] (Beamforming Matrix)

Page 112: 6 interference management in mimo multicell

Second-Order Cone Programming

minimize~V

Tr(~VH · ~V)

subject to |hHkkvk|2 − γkN∑

i=1,i6=k

|hHkivi|2 ≥ γkσ2

Tr(~VH · ~V) ≤ α

given hk

~V , [v1, · · · ,vK ] (Beamforming Matrix)

Page 113: 6 interference management in mimo multicell

Second-Order Cone Programming

minimize~V

Tr(~VH · ~V)

subject to

(1 +

1

γk

)|hHkkvk|2 ≥

∥∥∥∥∥hHkk

~Vσ2

∥∥∥∥∥

2

Tr(~VH · ~V) ≤ α

given hk

~V , [v1, · · · ,vK ] (Beamforming Matrix)

Page 114: 6 interference management in mimo multicell

Second-Order Cone Programming

minimize~V

α

subject to

(1 +

1

γk

)|hHkkvk|2 ≥

∥∥∥∥∥hHkk

~Vσ2

∥∥∥∥∥

2

Tr(~VH · ~V) ≤ αgiven hk

~V , [v1, · · · ,vK ] (Beamforming Matrix)

Page 115: 6 interference management in mimo multicell

Power Minimization with Imperfect CSITSOCP Formulation

minimize~V

α

subject to ηk|hHkkvk|2 ≥

∥∥∥∥∥∥

hHkk~V

hHki~U

σ2

∥∥∥∥∥∥

2

Tr(~VH · ~V) ≤ α

given hk and Qki

~V , [v1, · · · ,vK ] (Beamforming Matrix)

~Uk , [uk1, · · · ,ukK ] uki , hHkiQki

Page 116: 6 interference management in mimo multicell

Some Results...

Page 117: 6 interference management in mimo multicell

CoMP with Imperfect CSITHigher Channel Estimation Error

0 2 4 6 8 10 12−20

−15

−10

−5

0

5

10

15

20

25

30

Performance of the Proposed Scheme with Low Ch. Estimation ErrorMinim

um

Pow

erRequired

indB

Target SINR in dB

ρ = 0.3

ρ = 0.1

ρ = 0.05

SDP methodSOCP methodPerfect CSI

Page 118: 6 interference management in mimo multicell

CoMP with CSITHigher Channel Estimation Error

−4 −2 0 2 4 6 8 10−20

−15

−10

−5

0

5

10

15

20

25

30

Performance of the Proposed Scheme with High Ch. Estimation ErrorMinim

um

Required

Pow

er:∑

‖wi‖

2(indB)

Target SINR: γ (in dB)

ρ = 0.3

ρ = 0.1ρ = 0.05

SDP MethodSDP MethodPerfect CSIT

Page 119: 6 interference management in mimo multicell

Thank You !

Questions ?