6 interference management in mimo multicell

transcript

Interference Management in MIMO MulticellSystems with variable CSIT

Giuseppe Abreug.abreu@jacobs-university.de

School of Engineering and SciencesJacobs University Bremen

November 7, 2013

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

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

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

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

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

Forecast

I RequirementsI Improve energy efficiency

I Improve QoSI Improve spectrum efficiency

Forecast

I RequirementsI Improve energy efficiencyI Improve QoS

I Improve spectrum efficiency

Forecast

I RequirementsI Improve energy efficiencyI Improve QoSI Improve spectrum efficiency

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!!

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”

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”

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”

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”

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

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

Past and Future

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...

Past and Future

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...

Past and Future

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...

Past and Future

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...

Past and Future

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...

Past and Future

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...

Past and Future

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...

Past and Future

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...

Past and Future

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...

Past and Future

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...

Past and Future

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

Now let’s get serious...

Dissecting CoMPEnergy Efficiency

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

minimizeV,{vk}

subject to |vn|2 ≤ αpnSINRk ≥ γk,

given hk

TX : xN×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 (!)

minimizeV,{vk}

given hk

TX : xN×1 =

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 (!)

minimizeV,{vk}

given hk

TX : xN×1 =

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 (!)

minimizeV,{vk}

given hk

TX : xN×1 =

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 (!)

minimizeV,{vk}

given hk

TX : xN×1 =

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 (!)

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

minimizep>0,V,U

subject to SINRk ≥ γkgiven Hkj and wk

TX : xN×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 (!)

minimizep>0,V,U

TX : xN×1 =

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 (!)

minimizep>0,V,U

TX : xN×1 =

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 (!)

minimizep>0,V,U

TX : xN×1 =

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 (!)

minimizep>0,V,U

TX : xN×1 =

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 (!)

Dissecting CoMPQuality of Service

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

maximizep>0,V

min∀k

subject to ‖p‖ ≤ P‖vk‖ = 1

given hk

TX : xN×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 (!)

maximizep>0,V

min∀k

given hk

TX : xN×1 =

j 6=k pj |hj · vk|2 + σ2

I N =∑B

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

maximizep>0,V

min∀k

given hk

TX : xN×1 =

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 (!)

maximizep>0,V

min∀k

given hk

TX : xN×1 =

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 (!)

maximizep>0,V

min∀k

given hk

TX : xN×1 =

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 (!)

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∑

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 (!)

maximizep>0,V

min∀k

SINRkαk

TX : xN×1 =K∑

I N =∑B

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 (!)

maximizep>0,V

min∀k

SINRkαk

TX : xN×1 =K∑

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 (?)

maximizep>0,V

min∀k

SINRkαk

TX : xN×1 =K∑

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 (!)

Dissecting CoMPSpectral Efficiency

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

maximizet,V

subject to SINRk ≥ t1/αkk − 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 (!)

maximizet,V

‖vk‖2 ≤ P

given hk, P and α

I N =∑B

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

maximizet,V

‖vk‖2 ≤ P

given hk, P and α

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 (!)

maximizet,V

‖vk‖2 ≤ P

given hk, P and α

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 (!)

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

maximizeV,{Vk}

wk log2 |INr + (σ2kI + Φk)−1HkkVkVkH

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

given Hjk,w,p and α

HkjVjVHj HH

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 (!)

maximizeV,{Vk}

HkjVjVHj HH

I N =∑B

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 (!)

maximizeV,{Vk}

HkjVjVHj HH

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 (?)

maximizeV,{Vk}

HkjVjVHj HH

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 (!)

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

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

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

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

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

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

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

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

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

[60] [61]

† 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

OK, so what if CSIT is not perfect ?

Power Minimization AgainFrom Perfect to Imperfect CSIT

maximizeV,{vk}

‖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

maximizeV,{vk}

‖vk‖2

N∑i=1,i 6=k

given {γ1, · · · , γK}

hki = hki + ehki

(|hHkkvk|2∑N

)≥ 1− ρk

maximizeV,{vk}

‖vk‖2

N∑i=1,i 6=k

given {γ1, · · · , γK}

hki = hki + ehki

(|hHkkvk|2∑N

)≥ 1− ρk

Power Minimization AgainRobust Formulation

maximizeV,{vk}

‖vk‖2

subject to Pr

(|hHkkvk|2∑N

)≥ 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

maximizeV,{vk}

‖vk‖2

subject to Pr

(|hHkkvk|2∑N

)≥ 1− ρk

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

hki = hki + ehki ehki ∼ CN (0,Qki)

maximizeV,{vk}

‖vk‖2

subject to Pr

(|hHkkvk|2∑N

)≥ 1− ρk

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

hki = hki + ehki ehki ∼ CN (0,Qki)

Statistical SINR ConstraintTowards a Closed-form

SINRk ≥ γk)

Xkk ,|hHkkvk|2vHk Qkkvk

Xki ,|hHkivi|2vHi Qkivi

σ2ki = vHi Qkivi

(|hHkkvk|2∑N

σ2ki = vHi Qkivi

|hHkkvk|2 ≥ γk

i=1,i 6=k|hHkivi|2 + γk · σ2

σ2ki = vHi Qkivi

|hHkkvk|2 ≥ γk

i=1,i 6=k|hHkivi|2 + γk · σ2

σ2ki = vHi Qkivi

σ2kkXkk − γk

i=1,i 6=kσ2iiXki ≥ γk · σ2

σ2ki = vHi Qkivi

∫ ∞

0. . .

∫ ∞

0Pr(Xkk ≥ ckk

) N∏

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

ckk =γkσ2kk

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.

∫ ∞

0. . .

∫ ∞

0Pr(Xkk ≥ ckk

) N∏

ckk =γkσ2kk

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.

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

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.

Model of XkkNon-central and Central

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

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

Pr (SINRk ≥ γk) ≈∫ ∞

0. . .

∫ ∞

0Pr(Xkk ≥ ckk

1+δkk2

) N∏

= e− γkσ

(1+δkk2 )

i=1,i6=k

∫ ∞

0exp (−αkiti)fXki(ti)dti

= e− γkσ

(1+δkk2 )

i=1,i 6=k

E[e−αkiti ]

0. . .

∫ ∞

0Pr(Xkk ≥ ckk

1+δkk2

) N∏

= e− γkσ

(1+δkk2 )

i=1,i6=k

∫ ∞

= e− γkσ

(1+δkk2 )

i=1,i 6=k

E[e−αkiti ]

Xkk ∼ χ22 =⇒ Pr(Xkk ≥ x) = e−x/2

0. . .

∫ ∞

(− ckk

1 + δkk2

= e− γkσ

(1+δkk2 )

i=1,i6=k

∫ ∞

= e− γkσ

(1+δkk2 )

i=1,i 6=k

E[e−αkiti ]

Xkk ∼ χ22 =⇒ Pr(Xkk ≥ x) = e−x/2

0. . .

∫ ∞

(− ckk

1 + δkk2

= e− γkσ

(1+δkk2 )

i=1,i6=k

∫ ∞

= e− γkσ

(1+δkk2 )

i=1,i 6=k

E[e−αkiti ]

αki =γkσ

σ2kk(1 +δkk2 )

0. . .

∫ ∞

(− ckk

1 + δkk2

= e− γkσ

(1+δkk2 )

i=1,i6=k

∫ ∞

= e− γkσ

(1+δkk2 )

i=1,i 6=k

E[e−αkiti ]

αki =γkσ

σ2kk(1 +δkk2 )

Pr (SINRk ≥ γk) ≈ e− γkσ

(1+δkk2 )

i=1,i 6=kE[e−αkiti ]

= e− γkσ

(1+δkk2 )

exp(∑N

i=1,i 6=k− αkiδki(1+2αki)

)∏Ni=1,i 6=k(1 + 2αki)

= e− γkσ

(1+δkk2 )

i=1,i 6=k

exp(− γkpki

σ2kk(1+

)+2γkσ2ki

1 +γkσ

σ2kk(1+δkk/2)

Z ∼ χ22(δ) =⇒ E[esZ ] =

sδ1−2s

1− 2s

(1+δkk2 )

= e− γkσ

(1+δkk2 )

exp(∑N

)∏Ni=1,i 6=k(1 + 2αki)

= e− γkσ

(1+δkk2 )

i=1,i 6=k

exp(− γkpki

σ2kk(1+

)+2γkσ2ki

1 +γkσ

σ2kk(1+δkk/2)

Z ∼ χ22(δ) =⇒ E[esZ ] =

sδ1−2s

1− 2s

(1+δkk2 )

= e− γkσ

(1+δkk2 )

exp(∑N

)∏Ni=1,i 6=k(1 + 2αki)

= e− γkσ

(1+δkk2 )

i=1,i 6=k

exp(− γkpki

σ2kk(1+

)+2γkσ2ki

1 +γkσ

σ2kk(1+δkk/2)

Z ∼ χ22(δ) =⇒ E[esZ ] =

sδ1−2s

1− 2s

(1+δkk2 )

= e− γkσ

(1+δkk2 )

exp(∑N

)∏Ni=1,i 6=k(1 + 2αki)

= e− γkσ

(1+δkk2 )

i=1,i 6=k

exp(− γkpki

σ2kk(1+

)+2γkσ2ki

1 +γkσ

σ2kk(1+δkk/2)

Z ∼ χ22(δ) =⇒ E[esZ ] =

sδ1−2s

1− 2s

Pr (SINRk ≥ γk) ≈e− γkσ

i=1,i 6=k

(−γk|h

Hkivi|

Akkvk+γkvHi

γkvHi

≥ 1− ρk

Akk , Qkk + hkkhHkk

ln(1 + xi) ≤N∑

i=1,i 6=k

(−γk|h

Hkivi|

Akkvk+γkvHi

γkvHi

≥ 1− ρk

Akk , Qkk + hkkhHkk

ln(1 + xi) ≤N∑

i=1,i 6=k

(−γk|h

Hkivi|

Akkvk+γkvHi

γkvHi

) ≥ 1− ρk

Akk , Qkk + hkkhHkk

ln(1 + xi) ≤N∑

e− γkσ

i=1,i6=k

(−γk|h

Hkivi|

Akkvk+γkvHi

γkvHi

) ≥ log(1− ρk)

Akk , Qkk + hkkhHkk

ln(1 + xi) ≤N∑

γkσ2

vHk Akkvk+

N∑i=1,i 6=k

γk|hHkivi|2

vHk Akkvk+γkvHi Qkivi

+N∑i=1,i 6=k

γkvHi Qkivi

vHk Akkvk

)≤− ln(1−ρk)

Akk , Qkk + hkkhHkk

ln(1 + xi) ≤N∑

γkσ2

vHk Akkvk+

N∑i=1,i 6=k

γk|hHkivi|2

+N∑i=1,i 6=k

γkvHi Qkivi

vHk Akkvk

)≤− ln(1−ρk)

Akk , Qkk + hkkhHkk

ln(1 + xi) ≤N∑

γkσ2

vHk Akkvk+

i=1,i6=k

(γk|hHkivi|

Hi Qkivi

vHk Akkvk

)≤ − ln(1− ρk)

Akk , Qkk + hkkhHkk

ln(1 + xi) ≤N∑

γkσ2

vHk Akkvk+

i=1,i6=k

(γk|hHkivi|

Hi Qkivi

vHk Akkvk

)≤ − ln(1− ρk)

Akk , Qkk + hkkhHkk

ln(1 + xi) ≤N∑

γkσ2

vHk Akkvk+

i=1,i 6=k

(γk|hHkivi|

)≤ − ln(1− ρk)

Akk , Qkk + hkkhHkk

ln(1 + xi) ≤N∑

γkσ2

vHk Akkvk+

i=1,i 6=k

(γk|hHkivi|

)≤ − ln(1− ρk)

Akk , Qkk + hkkhHkk

ln(1 + xi) ≤N∑

σ2 +N∑

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

vHk Akkvk≤ − ln(1− ρk)

vHk Akkvk

σ2 +N∑

≥ −γkln(1− ρk)

vHk Akkvk

σ2 +N∑

≥ −γkln(1− ρk)

Power Minimization AgainRobust Formulation with Closed-form Constraint

maximizeV,{vk}

‖vk‖2

subject to Pr

(|hHkkvk|2∑N

)≥ 1− ρk

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

given hki

Power Minimization AgainRobust Formulation with Closed-form Constraint

maximizeV,{vk}

‖vk‖2

subject toσ2+

∑Ni=1,i 6=k |hHkivi|

2+vHi Qkivi

vHk Akkvk≤ − ln(1− ρk)

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

given hki and Qki

Now, how do we solve this ?

Some Tools...

Semi-Definite Programming

minimizeV,{vk}

‖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)

minimizeV,{vk}

‖vk‖2

i=1,i6=k

given hk

minimize{Vk}�0

Tr(Vk)

i=1,i6=k

given hk

minimize{Vk}�0

Tr(Vk)

i=1,i6=k

given hk

minimize{Vk}�0

Tr(Vk)

subject to Tr(VkHk)− γkN∑

i=1,i 6=k

Tr(ViHi) ≥ γkσ2

given Hk � 0

Power Minimization with Imperfect CSITSDP Formulation

minimizeV,{vk}

‖vk‖2

subject tovHk Akkvk

σ2+∑Ni=1,i 6=k |hHkivi|2+vHi Qkivi

≥ γk− ln(1− ρk)

minimize{Vk}�0

Tr(Vk)

subject tovHk Akkvk

σ2+∑Ni=1,i 6=k |hHkivi|2+vHi Qkivi

≥ γk− ln(1− ρk)

minimize{Vk}�0

Tr(Vk)

subject to ηkvHk Akkvk ≥ σ2 +

minimize{Vk}�0

Tr(Vk)

subject to Tr(Vk(hkkh

Hkk + Qkk)

)≥ σ2 +

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

Hki + Qki)

Second-Order Cone Programming

minimizeV,{vk}

‖vk‖2

i=1,i6=k

Tr(~VH · ~V) ≤ α

given hk

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

minimizeV,{vk}

‖vk‖2

i=1,i6=k

Tr(~VH · ~V) ≤ α

given hk

minimize~V

Tr(~VH · ~V)

i=1,i6=k

Tr(~VH · ~V) ≤ α

given hk

minimize~V

Tr(~VH · ~V)

subject to

)|hHkkvk|2 ≥

∥∥∥∥∥hHkk

∥∥∥∥∥

Tr(~VH · ~V) ≤ α

given hk

minimize~V

subject to

)|hHkkvk|2 ≥

∥∥∥∥∥hHkk

∥∥∥∥∥

Tr(~VH · ~V) ≤ αgiven hk

Power Minimization with Imperfect CSITSOCP Formulation

minimize~V

subject to ηk|hHkkvk|2 ≥

∥∥∥∥∥∥

hHkk~V

hHki~U

∥∥∥∥∥∥

Tr(~VH · ~V) ≤ α

given hk and Qki

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

Some Results...

CoMP with Imperfect CSITHigher Channel Estimation Error

0 2 4 6 8 10 12−20

Performance of the Proposed Scheme with Low Ch. Estimation ErrorMinim

erRequired

Target SINR in dB

ρ = 0.3

ρ = 0.1

ρ = 0.05

SDP methodSOCP methodPerfect CSI

CoMP with CSITHigher Channel Estimation Error

−4 −2 0 2 4 6 8 10−20

Performance of the Proposed Scheme with High Ch. Estimation ErrorMinim

Required

er:∑

‖wi‖

2(indB)

Target SINR: γ (in dB)

ρ = 0.3

ρ = 0.1ρ = 0.05

SDP MethodSDP MethodPerfect CSIT

Thank You !

Questions ?

6 interference management in mimo multicell

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