6 interference management in mimo multicell

Interference Management in MIMO Multicell
Systems with variable CSIT
Giuseppe Abreu
g.abreu@jacobs-university.de
School of Engineering and Sciences
Jacobs University Bremen

November 7, 2013

Evolution

GSM
Application
Access
Latency
Switching Time

12 kbps
20 kbps
150 ms
Few seconds

3G
1 Mbps
24 kbps
50 ms
500 ms

4G
10 Mbps
300 Mbps
10 ms
200 ms

5G
1 Gbps
10 Gbps
1 ms
10 ms

Forecast

Mobile traﬃc volume trend: 1000x in 10 years.

Requirements

Forecast

The Internet of Things

Requirements

Forecast

The Internet of Things !

Requirements

Forecast

The Internet of Things !!!

Requirements

Forecast


Requirements
Improve energy eﬃciency

Forecast


Requirements
Improve QoS

Forecast


Requirements
Improve QoS
Improve spectrum eﬃciency

Past and Future
Drivers of cellular system improvement
2G:
3G:
4G:
5G:

Digitalization (GSM/CDMA)
Spread-spectrum (WCDMA)
LTE (OFDMA)
???

Past and Future
2G:
3G:
4G:
5G:

Digitalization (GSM/CDMA) → spectrum
Spread-spectrum (WCDMA)
LTE (OFDMA)
???

Past and Future
2G:
3G:
4G:
5G:

Spread-spectrum (WCDMA) → spectrum again...
LTE (OFDMA)
???

Past and Future
2G:
3G:
4G:
5G:

LTE (OFDMA) → and again...
???

Past and Future
2G:
3G:
4G:
5G:

“granularity”

Past and Future
2G:
3G:
4G:
5G:

“granularity” → devices

Past and Future
2G:
3G:
4G:
5G:

“granularity” → devices → antennas

Past and Future
2G:
3G:
4G:
5G:


Bottlenecked by Interference

Past and Future
2G:
3G:
4G:
5G:


Cooperation (Relaying)

Past and Future
2G:
3G:
4G:
5G:



Cooperation (Relaying) → security (?)

Past and Future
2G:
3G:
4G:
5G:



Het-Nets/Cognitive Radio

Past and Future
2G:
3G:
4G:
5G:



Het-Nets/Cognitive Radio → enough (?)

Past and Future
2G:
3G:
4G:
5G:



Interference Alignment

Past and Future
2G:
3G:
4G:
5G:



Interference Alignment → scalability (?)

Past and Future
2G:
3G:
4G:
5G:



Massive MIMO

Past and Future
2G:
3G:
4G:
5G:



Massive MIMO → revolutionary, expensive (!)

Past and Future
2G:
3G:
4G:
5G:



CoMP

Past and Future
2G:
3G:
4G:
5G:



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

Past and Future
2G:
3G:
4G:
5G:



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

Still lots to be done!!

CoMP’s System Model

B coordinating BSs
K users per cell
One BS per cell
Each BS with multiple antennas
User may have multiple antennas
Embedded power control
Desired Signal
Interference Signal

Inter-cell and intra-cell interference

Dissecting CoMP
Energy Eﬃciency

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

α

V {vk }

subject to |vn |2 ≤ α pn

SINRk ≥ γk ,

given

hk

K

TX : xN ×1 =

sk vk
k=1

SINRk =

RX : yk = hk · x + zk

|hk · vk |2
2
2
j=k |hk · vk | + σ

Dissecting CoMP
Energy Eﬃciency

minimize

α

V {vk }


SINRk ≥ γk ,

given

hk

K

TX : xN ×1 =

sk vk
k=1

SINRk =
N=

B
b=1


|hk · vk |2
2
2
j=k |hk · vk | + σ

Ntb TX antennas → MISO

Dissecting CoMP
Energy Eﬃciency

minimize

α

V {vk }


SINRk ≥ γk ,

given

hk

K

TX : xN ×1 =

sk vk
k=1

SINRk =
B


|hk · vk |2
2
2
j=k |hk · vk | + σ

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

Dissecting CoMP
Energy Eﬃciency

minimize

α

V {vk }


SINRk ≥ γk ,

given

hk

K

TX : xN ×1 =

sk vk
k=1

SINRk =
B


|hk · vk |2
2
2
j=k |hk · vk | + σ

Per user γk target SINRs → QoS balancing (?)

Dissecting CoMP
Energy Eﬃciency

minimize

α

V {vk }


SINRk ≥ γk ,

given

hk

K

TX : xN ×1 =

sk vk
k=1

SINRk =
B


|hk · vk |2
2
2
j=k |hk · vk | + σ

Perfectly known hk for all users → overhead (!)

Dissecting CoMP
Energy Eﬃciency

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

minimize
p>0,V,U

wk p k
k=1

subject to SINRk ≥ γk

K

TX : xN ×1 =

given
√
pk s k v k

k=1

SINRk =

Hkj and wk
RX : yk = uH · Hkk · x + zk
k

pk |uH · Hkk · vk |2
k
H
2
2
j=k pj |uk · Hkj · vk | + σ

Dissecting CoMP
Energy Eﬃciency

K

minimize
p>0,V,U

wk p k
k=1


K

TX : xN ×1 =

given
√
pk s k v k

k=1

SINRk =
N=

B
b=1

Hkj and wk
k

k
H
2
2

Ntb TX antennas → MIMO

Dissecting CoMP
Energy Eﬃciency

K

minimize
p>0,V,U

wk p k
k=1


K

TX : xN ×1 =

given
√
pk s k v k

k=1

SINRk =
B

Hkj and wk
k

k
H
2
2

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

Dissecting CoMP
Energy Eﬃciency

K

minimize
p>0,V,U

wk p k
k=1


K

TX : xN ×1 =

given
√
pk s k v k

k=1

SINRk =
B

Hkj and wk
k

k
H
2
2


Dissecting CoMP
Energy Eﬃciency

K

minimize
p>0,V,U

wk p k
k=1


K

TX : xN ×1 =

given
√
pk s k v k

k=1

SINRk =
B

Hkj and wk
k

k
H
2
2

Perfectly known Hkj for all users → overhead (!)

Dissecting CoMP
Quality of Service

Example 3: Min-max SINR problem [Huang et al. 2011]
maximize
p>0,V

subject to

min SINRk
∀k

p ≤P

vk = 1
given
K

TX : xN ×1 =

√

k=1

SINRk =

hk

p k s k vk


pk |hk · vk |2
2
2
j=k pj |hj · vk | + σ

Dissecting CoMP
Quality of Service

maximize
p>0,V

subject to

min SINRk
∀k

p ≤P

vk = 1
given
K

TX : xN ×1 =

√

k=1

SINRk =
N=

B
b=1

hk

p k s k vk


pk |hk · vk |2
2
2


Dissecting CoMP
Quality of Service

maximize
p>0,V

subject to

min SINRk
∀k

p ≤P

vk = 1
given
K

TX : xN ×1 =

√

k=1

SINRk =
B

hk

p k s k vk


pk |hk · vk |2
2
2


Dissecting CoMP
Quality of Service

maximize
p>0,V

subject to

min SINRk
∀k

p ≤P

vk = 1
given
K

TX : xN ×1 =

√

k=1

SINRk =
B

hk

p k s k vk


pk |hk · vk |2
2
2

Per user γk target SINRs → minimum QoS

Dissecting CoMP
Quality of Service

maximize
p>0,V

subject to

min SINRk
∀k

p ≤P

vk = 1
given
K

TX : xN ×1 =

√

k=1

SINRk =
B

hk

p k s k vk


pk |hk · vk |2
2
2

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

Dissecting CoMP
Quality of Service

Example 4: Min-max SINR problem [Cai et al. 2011]
maximize
p>0,V

min
∀k

SINRk
αk

subject to w · p ≤ P ,
given
K

TX : xN ×1 =

∈ L |L| < K

Hkj , wk and α
√
pk s k v k

k=1

SINRk =

k

k
pj |uH · Hkj · vk |2 + σ 2
j=k
k

Dissecting CoMP
Quality of Service

maximize
p>0,V

min
∀k

SINRk
αk

given
K

TX : xN ×1 =

Hkj , wk and α
√
pk s k v k

k=1

SINRk =
N=

B
b=1

∈ L |L| < K

k

k
pj |uH · Hkj · vk |2 + σ 2
j=k
k


Dissecting CoMP
Quality of Service

maximize
p>0,V

min
∀k

SINRk
αk

given
K

TX : xN ×1 =

Hkj , wk and α
√
pk s k v k

k=1

SINRk =
B

∈ L |L| < K

k

k
pj |uH · Hkj · vk |2 + σ 2
j=k
k

Known weight vectors per user wk and scores αk → how (?)

Dissecting CoMP
Spectral Eﬃciency

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

maximize
t,V

tk
k=1
1/αk

subject to SINRk ≥ tk
K

vk
k=1

given

2

−1

≤P

hk , P and α

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

Dissecting CoMP
Spectral Eﬃciency

K

maximize
t,V

tk
k=1
1/αk

K

vk
k=1

given

2

−1

≤P

hk , P and α

N=

B
b=1


Dissecting CoMP
Spectral Eﬃciency

K

maximize
t,V

tk
k=1
1/αk

K

vk
k=1

given

2

−1

≤P

hk , P and α

B

Fixed pn per-antenna target powers → optimized total pk
Known scores αk → how (?)

Dissecting CoMP
Spectral Eﬃciency

K

maximize
t,V

tk
k=1
1/αk

K

vk
k=1

given

2

−1

≤P

hk , P and α

B

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

Dissecting CoMP
Spectral Eﬃciency

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

maximize
V {Vk }

subject to

k=1

Hjk Vk
Vk

given

2
wk log2 |INr + (σk I + Φk )−1 Hkk Vk Vk HH |
kk

2

2

2
≤ αjk σj

≤ pk

Hjk , w, p and α
K
H
Hkj Vj Vj HH
kj

Φk =
k=1

Dissecting CoMP
Spectral Eﬃciency

K

maximize
V {Vk }

subject to

k=1

Hjk Vk
Vk

given

2
kk

2

2

2
≤ αjk σj

≤ pk

Hjk , w, p and α
K
H
Hkj Vj Vj HH
kj

Φk =
k=1

N=

B
b=1


Dissecting CoMP
Spectral Eﬃciency

K

maximize
V {Vk }

subject to

k=1

Hjk Vk
Vk

given

2
kk

2

2

2
≤ αjk σj

≤ pk

Hjk , w, p and α
K
H
Hkj Vj Vj HH
kj

Φk =
k=1
B
b=1

N=
Known weights w, target powers pk and scores αk → how (?)

tion of single beamforming vector) to address the practical issues of decentralized implementation based
on 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 with
per BS power for MIMO systems. Considering conditional eigenvalue problem with affine constraint and
non-linear Perron-Frobenius theory, authors in [52] optimized the physical layer link rate functions. Also
recently, in [57], a relaxed zero forcing method for MISO and MIMO interference channels is proposed for
the rate control problems with centralized and distributed heuristic approach.

Comprehensive Review in a Nutshell

Table 1: Literature Classification of Works on CoMP
MISO †
MIMO
P1

Instantaneous Perfect CSIT

Instantaneous Imperfect CSIT

Statistical CSIT

[26]

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

[43]

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

[60]

Covariance Information

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

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

[61]

[87]

P2

P3

[34] [35] [36]
[37]
[38] [39] [40] [41] [42]
[48] [49] [65] [51] [52]
[53] [54] [55] [56]
[57] [77]
[82] [58]
[59] [57] [78] [79]

†

[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 discussed
in Section (2.1.1). We have distinguished the key problems into MISO vs. MIMO with different channel
variations available at the transmitter. Despite the different tools and analytics present in the perfect channel
knowledge at the transmitter side, we have seen a significant need of work to be done in the the multiantenna receiver case. Also, as we progress from left to right of the table, the available literatures diminishes
in number. Within the cited literatures, we have pointed few key-holes for the key-problems which we state

OK, so what if CSIT is not perfect ?

Power Minimization Again
From Perfect to Imperfect CSIT

K

maximize
V {vk }

subject to

vk

2

k=1

SINRk

N
i=1,
i=k

given

{γ1 , · · · , γK }

|hH vk |2
kk
|hH vi |2
ki

+

σ2

≥ γk


K

maximize
V {vk }

subject to

vk

2

k=1

SINRk

N
i=1,
i=k

given

|hH vk |2
kk
|hH vi |2
ki

{γ1 , · · · , γK }
ˆ
hki = hki + ehki

+

σ2

≥ γk


K

maximize
V {vk }

subject to

vk

2

k=1

SINRk

N
i=1,
i=k

given

|hH vk |2
kk
|hH vi |2
ki

+

σ2

≥ γk

{γ1 , · · · , γK }
ˆ
hki = hki + ehki

Pr (SINRk ≥ γk ) = Pr

|hH vk |2
kk

N
i=1,i=k

|hH vi |2 + σ 2
ki

≥ γk

≥ 1 − ρk

Robust Formulation

K

maximize
V {vk }

vk

2

k=1

subject to

given

Pr

|hH vk |2
kk

N
i=1,i=k

|hH vi |2 + σ 2
ki

≥ γk

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

ˆ
hki = hki + ehki

≥ 1 − ρk

Robust Formulation

K

maximize
V {vk }

vk

2

k=1

subject to

given

Pr

|hH vk |2
kk

N
i=1,i=k

|hH vi |2 + σ 2
ki

≥ γk

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

ˆ
hki = hki + ehki

ehki ∼ CN (0, Qki )

≥ 1 − ρk

Robust Formulation

K

maximize
V {vk }

vk

2

k=1

subject to

given

Pr

|hH vk |2
kk

N
i=1,i=k

|hH vi |2 + σ 2
ki

≥ γk

≥ 1 − ρk

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

ˆ
hki = hki + ehki
2
ˆ
|hH vi |2 ∼ χ2 (δki ; σki )
2
ki

ehki ∼ CN (0, Qki )
ˆ
δki

ˆ
|hH vi |2
ki

2
H
σki = vi Qki vi

Statistical SINR Constraint
Towards a Closed-form

Pr SINRk ≥ γk


Pr

|hH vk |2
kk

N
i=1,i=k

|hH vi |2 + σ 2
ki

≥ γk




N

Pr |hH vk |2 ≥ γk
kk

i=1,i=k



|hH vi |2 + γk · σ 2 
ki




2
Pr σkk Xkk − γk

Xkk



N
i=1,i=k

|hH vk |2
kk
HQ v
vk kk k

2
σii Xki ≥ γk · σ 2 

Xki

2
H
σki = vi Qki vi

|hH vi |2
ki
HQ v
vi ki i


∞
0

N

∞

...
0

Pr Xkk ≥ ckk

ckk =

[Kandukuri]



fXki (ti ) dti . . . dtN
i=1,i=k

γk  2
σ +
2
σkk

N

i=1,i=k



2
σki ti 

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


∞
0

N

∞

...
0

Pr Xkk ≥ ckk

ckk =



i=1,i=k

γk  2
σ +
2
σkk

N

i=1,i=k



2
σki ti 

ckk is non-central χ2
[Kandukuri]

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

Model of Xkk
Non-central and Central

Theorem (Cox&Reid):
˜
Let Z ∼ χ2 (δ; σ 2 ) and Z ∼ χ2 (0; σ 2 ), with δ/n small. Then,
n
n
˜
Pr (Z > γ) ≈ Pr Z >

γ
1+

δ
n

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

Model of Xkk
Non-central and Central
Comparison of the CDF of Non-central and Central χ2 Distribution

1

δ = 0.1

Cumulative Distribution Function: Pr(Z ≥ δ)

0.9
0.8
0.7

δ=2

0.6
0.5
0.4
0.3
0.2
0.1
0
0

Non-central χ2
Central χ2

1

2

3

4

5
6
7
8
9
10
Non-centrality Parameter: δ

11

12

13

14

15


∞

Pr (SINRk ≥ γk ) ≈

∞

...
0

0

˜
Pr Xkk ≥

N
ckk
δ
1+ kk
2

i=1,i=k


∞


∞

...
0

˜
Xkk ∼ χ2
2

0

N

˜
Pr Xkk ≥

=⇒

ckk
δ
1+ kk
2

i=1,i=k

˜
Pr(Xkk ≥ x) = e−x/2


∞


∞

...
0

˜
Xkk ∼ χ2
2

0

exp −

=⇒

ckk
1+

δkk
2

N

i=1,i=k

˜
Pr(Xkk ≥ x) = e−x/2


∞


∞

...
0
−

=e

0

exp −

γk σ 2
δ
σ 2 (1+ kk )
2
kk

δkk
2

1+

N
i=1,
i=k

αki =

ckk

N

i=1,i=k

∞
0

exp (−αki ti )fXki (ti ) dti

2
γk σki
2
σkk (1 +

δkk
2 )


∞


∞

...
0
−

=e

0

exp −

γk σ 2
δ
σ 2 (1+ kk )
2
kk

=e

γk σ 2
2 (1+ δkk )
σ
2
kk

δkk
2

1+

N
0

exp (−αki ti )fXki (ti ) dti

N

E[e−αki ti ]
i=1,
i=k

αki =

i=1,i=k

∞

i=1,
i=k

−

N

ckk

2
γk σki
2
σkk (1 +

δkk
2 )


−

Pr (SINRk ≥ γk ) ≈ e

γk σ 2
2 (1+ δkk )
σ
2
kk

N

i=1,i=k

E[e−αki ti ]


−


Z∼

χ2 (δ)
2

γk σ 2
2 (1+ δkk )
σ
2
kk

N

E[e−αki ti ]

i=1,i=k

=⇒

sZ

E[e ] =

exp

sδ
1−2s

1 − 2s


−

=e

Z∼

γk σ 2
2 (1+ δkk )
σ
2
kk

N

E[e−αki ti ]

i=1,i=k
γk σ 2
−
δ
σ 2 (1+ kk )
2
kk

χ2 (δ)
2

=⇒

exp

N
i=1,i=k
N
i=1,i=k (1

sZ

E[e ] =

exp

αki δ
− (1+2αki )
ki

+ 2αki )

sδ
1−2s

1 − 2s


−

=e

γk σ 2
2 (1+ δkk )
σ
2
kk

N

E[e−αki ti ]

i=1,i=k
γk σ 2
−
δ
σ 2 (1+ kk )
2
kk

−

=e

γk σ 2
δ
σ 2 (1+ kk )
2
kk

N
i=1,i=k

exp

N
i=1,i=k (1
N

Z∼

=⇒

sZ

+ 2αki )

exp −

i=1,i=k

χ2 (δ)
2

αki δ
− (1+2αki )
ki

E[e ] =

2
σkk (1+

1+
exp

γk pki
δkk
2
)+2γk σki
2

2
γk σki
2 (1+δ /2)
σkk
kk

sδ
1−2s

1 − 2s


Pr (SINRk ≥ γk ) ≈e

−

γk σ 2
vH Akk vk
k

N
i=1,
i=k

exp

ˆ
−γk |hH vi |2
ki
vH Akk vk +γk vH Qki vi
i
k
γ vH Q v
1+ kHi ki i
v Akk vk
k



−

γk σ 2
vH Akk vk
k

N
i=1,
i=k

Akk

exp

ˆ
−γk |hH vi |2
ki
i
k
γ vH Q v
1+ kHi ki i
v Akk vk
k

ˆ ˆ
Qkk + hkk hH
kk



−

γk σ 2
vH Akk vk
k

N
i=1,
i=k

Akk

exp

ˆ
−γk |hH vi |2
ki
i
k
γ vH Q v
1+ kHi ki i
v Akk vk
k

ˆ ˆ
Qkk + hkk hH
kk

≥ 1 − ρk




γ σ2
− Hk
v Akk vk
k


log e

N
i=1,
i=k

exp

ˆ
−γk |hH vi |2
ki
i
k
γ vH Q v
1+ kHi ki i
v Akk vk
k

Akk

ˆ ˆ
Qkk + hkk hH
kk




 ≥ log(1 − ρk )


ˆ
N
N
γk |hH vi |2
γk vH Qki vi
γk σ 2
ki
ln 1+ Hi
+
+
H
H
H
vk Akk vk i=1, vk Akk vk +γk vi Qki vi i=1,
vk Akk vk
i=k

i=k

Akk

ˆ ˆ
Qkk + hkk hH
kk

≤− ln(1−ρk )


ˆ
N
N
γk |hH vi |2
γk vH Qki vi
γk σ 2
ki
ln 1+ Hi
+
+
H
H
H
vk Akk vk i=1, vk Akk vk +γk vi Qki vi i=1,
vk Akk vk
i=k

i=k

Akk

ˆ ˆ
Qkk + hkk hH
kk

N
i=1

N

ln(1 + xi ) ≤

xi
i=1

≤− ln(1−ρk )


γk σ 2
H
vk Akk vk

N

+
i=1,
i=k

ˆ
γk |hH vi |2
ki
H
H
vk Akk vk +γk vi Qki vi

Akk

+

ˆ ˆ
Qkk + hkk hH
kk

N
i=1

H
γk vi Qki vi
H
vk Akk vk

N

ln(1 + xi ) ≤

xi
i=1

≤ − ln(1 − ρk )


γk σ 2
H
vk Akk vk

N

+
i=1,
i=k

ˆ
γk |hH vi |2
ki
H
H
vk Akk vk +γk vi Qki vi

Akk

ˆ ˆ
Qkk + hkk hH
kk

N
i=1

≤ − ln(1 − ρk )

N

ln(1 + xi ) ≤

xi
i=1


σ2 +

N
i=1,i=k

H
ˆ
|hH vi |2 + vi Qki vi
ki

H
vk Akk vk

≤

− ln(1 − ρk )
γk


H
vk Akk vk

σ2 +

N
i=1,i=k

H
ˆ
ki

≥

−γk
ln(1 − ρk )


H
vk Akk vk

σ2

N

+
i=1,i=k

H
ˆ
ki

≥

−γk
ln(1 − ρk )

ηk

Robust Formulation with Closed-form Constraint

K

maximize
V {vk }

subject to

vk

2

k=1

Pr

|hH vk |2
kk

N
i=1,i=k

|hH vi |2 + σ 2
ki

≥ γk

given

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

given

hki

≥ 1 − ρk

Robust Formulation with Closed-form Constraint

K

maximize
V {vk }

subject to

vk

2

k=1
σ2 +

N
H
ˆH 2
i=1,i=k |hki vi | +vi Qki vi
HA v
vk kk k

≤

− ln(1 − ρk )
γk

given

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

given

ˆ
hki and Qki

Semi-Deﬁnite Programming

K

minimize
V {vk }

vk

2

k=1
N

subject to |hH vk |2 − γk
kk
given

hk

i=1,
i=k

|hH vi |2 ≥ γk σ 2
ki


K

minimize
V {vk }

vk

2

k=1
N

kk
given
Vk

i=1,
i=k

|hH vi |2 ≥ γk σ 2
ki

hk
H
vk vk

(Positive Semi-Deﬁnite Matrix)


K

minimize
{Vk } 0

Tr(Vk )
k=1
N

kk
given
Vk

i=1,
i=k

|hH vi |2 ≥ γk σ 2
ki

hk
H
vk vk



K

minimize
{Vk } 0

Tr(Vk )
k=1
N

kk
given

i=1,
i=k

|hH vi |2 ≥ γk σ 2
ki

hk

Vk

H
vk vk


Hi

hki hH
ki



K

minimize
{Vk } 0

Tr(Vk )
k=1
N

subject to Tr(Vk Hk ) − γk
given

Hk

i=1,
i=k

Tr(Vi Hi ) ≥ γk σ 2

0

Vk

H
vk vk


Hi

hki hH
ki


Power Minimization with Imperfect CSIT
SDP Formulation

K

minimize
V {vk }

subject to

vk

2

k=1
σ2 +

H
vk Akk vk
N
H
ˆH 2

≥

γk
− ln(1 − ρk )

1
ηk

SDP Formulation

K

minimize
{Vk } 0

subject to

Tr(Vk )
k=1
σ2 +

H
vk Akk vk
N
H
ˆH 2

≥

γk
− ln(1 − ρk )

1
ηk

SDP Formulation

K

minimize
{Vk } 0

Tr(Vk )
k=1
N

subject to

H
ηk vk Akk vk ≥ σ 2 +

i=1,i=k

H
ˆ
ki

SDP Formulation

K

minimize
{Vk } 0

Tr(Vk )
k=1
N

ˆ ˆ
subject to Tr Vk (hkk hH + Qkk ) ≥ σ 2 +
kk

ˆ ˆ
Tr Vi (hki hH + Qki )
ki

i=1,i=k

Second-Order Cone Programming

K

minimize
V {vk }

vk

2

k=1
N

kk

given

hk

i=1,
i=k

|hH vi |2 ≥ γk σ 2
ki


K

minimize
V {vk }

vk

2

k=1
N

kk

given
V

i=1,
i=k

|hH vi |2 ≥ γk σ 2
ki

hk

[v1 , · · · , vK ]

(Beamforming Matrix)


minimize
V

Tr(VH · V)
N

kk

given
V

i=1,
i=k

|hH vi |2 ≥ γk σ 2
ki

hk

[v1 , · · · , vK ]



minimize
V

subject to

given
V

Tr(VH · V)
1
1+
γk

|hH vk |2
kk

hH V
kk
≥
σ2

2

hk

[v1 , · · · , vK ]



minimize

α

V

subject to

given
V

1
1+
γk

|hH vk |2
kk

hH V
kk
≥
σ2

2

Tr(VH · V) ≤ α
hk

[v1 , · · · , vK ]


SOCP Formulation

minimize

α

V

subject to

hH V
kk
hH U
ki
σ2

ˆ
ηk |hH vk |2 ≥
kk

2

Tr(VH · V) ≤ α
ˆ
hk and Qki

given

V

[v1 , · · · , vK ]
Uk


[uk1 , · · · , ukK ]

uki

ˆ
hH Qki
ki

CoMP with Imperfect CSIT
Higher Channel Estimation Error
Performance of the Proposed Scheme with Low Ch. Estimation Error
30

SDP method
SOCP method
Perfect CSI

25

Minimum Power Required in dB

20

15

ρ = 0.05
10

ρ = 0.1
5

ρ = 0.3

0

−5

−10

−15

−20
0

2

4

6

Target SINR in dB

8

10

12

CoMP with CSIT
Higher Channel Estimation Error
Performance of the Proposed Scheme with High Ch. Estimation Error
30

15

wi

(in dB)

SDP Method
SDP Method
Perfect CSIT

20

2

25

Minimum Required Power:

10

ρ = 0.05

ρ = 0.1

5

0

ρ = 0.3
−5

−10

−15

−20
−4

−2

0

2

4

Target SINR: γ (in dB)

6

8

10

6 interference management in mimo multicell

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