1. 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
5. Forecast
Mobile traffic volume trend: 1000x in 10 years.
The Internet of Things
The Internet of Things !
Requirements
6. Forecast
Mobile traffic volume trend: 1000x in 10 years.
The Internet of Things
The Internet of Things !
The Internet of Things !!!
Requirements
7. Forecast
Mobile traffic volume trend: 1000x in 10 years.
The Internet of Things
The Internet of Things !
The Internet of Things !!!
Requirements
Improve energy efficiency
8. Forecast
Mobile traffic volume trend: 1000x in 10 years.
The Internet of Things
The Internet of Things !
The Internet of Things !!!
Requirements
Improve energy efficiency
Improve QoS
9. Forecast
Mobile traffic volume trend: 1000x in 10 years.
The Internet of Things
The Internet of Things !
The Internet of Things !!!
Requirements
Improve energy efficiency
Improve QoS
Improve spectrum efficiency
10. Past and Future
Drivers of cellular system improvement
2G:
3G:
4G:
5G:
Digitalization (GSM/CDMA)
Spread-spectrum (WCDMA)
LTE (OFDMA)
???
11. Past and Future
Drivers of cellular system improvement
2G:
3G:
4G:
5G:
Digitalization (GSM/CDMA) → spectrum
Spread-spectrum (WCDMA)
LTE (OFDMA)
???
12. Past and Future
Drivers of cellular system improvement
2G:
3G:
4G:
5G:
Digitalization (GSM/CDMA) → spectrum
Spread-spectrum (WCDMA) → spectrum again...
LTE (OFDMA)
???
13. Past and Future
Drivers of cellular system improvement
2G:
3G:
4G:
5G:
Digitalization (GSM/CDMA) → spectrum
Spread-spectrum (WCDMA) → spectrum again...
LTE (OFDMA) → and again...
???
14. Past and Future
Drivers of cellular system improvement
2G:
3G:
4G:
5G:
Digitalization (GSM/CDMA) → spectrum
Spread-spectrum (WCDMA) → spectrum again...
LTE (OFDMA) → and again...
“granularity”
15. Past and Future
Drivers of cellular system improvement
2G:
3G:
4G:
5G:
Digitalization (GSM/CDMA) → spectrum
Spread-spectrum (WCDMA) → spectrum again...
LTE (OFDMA) → and again...
“granularity” → devices
16. Past and Future
Drivers of cellular system improvement
2G:
3G:
4G:
5G:
Digitalization (GSM/CDMA) → spectrum
Spread-spectrum (WCDMA) → spectrum again...
LTE (OFDMA) → and again...
“granularity” → devices → antennas
17. Past and Future
Drivers of cellular system improvement
2G:
3G:
4G:
5G:
Digitalization (GSM/CDMA) → spectrum
Spread-spectrum (WCDMA) → spectrum again...
LTE (OFDMA) → and again...
“granularity” → devices → antennas
Bottlenecked by Interference
18. Past and Future
Drivers of cellular system improvement
2G:
3G:
4G:
5G:
Digitalization (GSM/CDMA) → spectrum
Spread-spectrum (WCDMA) → spectrum again...
LTE (OFDMA) → and again...
“granularity” → devices → antennas
Bottlenecked by Interference
Cooperation (Relaying)
19. Past and Future
Drivers of cellular system improvement
2G:
3G:
4G:
5G:
Digitalization (GSM/CDMA) → spectrum
Spread-spectrum (WCDMA) → spectrum again...
LTE (OFDMA) → and again...
“granularity” → devices → antennas
Bottlenecked by Interference
Cooperation (Relaying) → security (?)
20. Past and Future
Drivers of cellular system improvement
2G:
3G:
4G:
5G:
Digitalization (GSM/CDMA) → spectrum
Spread-spectrum (WCDMA) → spectrum again...
LTE (OFDMA) → and again...
“granularity” → devices → antennas
Bottlenecked by Interference
Cooperation (Relaying) → security (?)
Het-Nets/Cognitive Radio
21. Past and Future
Drivers of cellular system improvement
2G:
3G:
4G:
5G:
Digitalization (GSM/CDMA) → spectrum
Spread-spectrum (WCDMA) → spectrum again...
LTE (OFDMA) → and again...
“granularity” → devices → antennas
Bottlenecked by Interference
Cooperation (Relaying) → security (?)
Het-Nets/Cognitive Radio → enough (?)
22. Past and Future
Drivers of cellular system improvement
2G:
3G:
4G:
5G:
Digitalization (GSM/CDMA) → spectrum
Spread-spectrum (WCDMA) → spectrum again...
LTE (OFDMA) → and again...
“granularity” → devices → antennas
Bottlenecked by Interference
Cooperation (Relaying) → security (?)
Het-Nets/Cognitive Radio → enough (?)
Interference Alignment
23. Past and Future
Drivers of cellular system improvement
2G:
3G:
4G:
5G:
Digitalization (GSM/CDMA) → spectrum
Spread-spectrum (WCDMA) → spectrum again...
LTE (OFDMA) → and again...
“granularity” → devices → antennas
Bottlenecked by Interference
Cooperation (Relaying) → security (?)
Het-Nets/Cognitive Radio → enough (?)
Interference Alignment → scalability (?)
24. Past and Future
Drivers of cellular system improvement
2G:
3G:
4G:
5G:
Digitalization (GSM/CDMA) → spectrum
Spread-spectrum (WCDMA) → spectrum again...
LTE (OFDMA) → and again...
“granularity” → devices → antennas
Bottlenecked by Interference
Cooperation (Relaying) → security (?)
Het-Nets/Cognitive Radio → enough (?)
Interference Alignment → scalability (?)
Massive MIMO
25. Past and Future
Drivers of cellular system improvement
2G:
3G:
4G:
5G:
Digitalization (GSM/CDMA) → spectrum
Spread-spectrum (WCDMA) → spectrum again...
LTE (OFDMA) → and again...
“granularity” → devices → antennas
Bottlenecked by Interference
Cooperation (Relaying) → security (?)
Het-Nets/Cognitive Radio → enough (?)
Interference Alignment → scalability (?)
Massive MIMO → revolutionary, expensive (!)
26. Past and Future
Drivers of cellular system improvement
2G:
3G:
4G:
5G:
Digitalization (GSM/CDMA) → spectrum
Spread-spectrum (WCDMA) → spectrum again...
LTE (OFDMA) → and again...
“granularity” → devices → antennas
Bottlenecked by Interference
Cooperation (Relaying) → security (?)
Het-Nets/Cognitive Radio → enough (?)
Interference Alignment → scalability (?)
Massive MIMO → revolutionary, expensive (!)
CoMP
27. Past and Future
Drivers of cellular system improvement
2G:
3G:
4G:
5G:
Digitalization (GSM/CDMA) → spectrum
Spread-spectrum (WCDMA) → spectrum again...
LTE (OFDMA) → and again...
“granularity” → devices → antennas
Bottlenecked by Interference
Cooperation (Relaying) → security (?)
Het-Nets/Cognitive Radio → enough (?)
Interference Alignment → scalability (?)
Massive MIMO → revolutionary, expensive (!)
CoMP → evolutionary, flexible, huge background, maturing...
28. Past and Future
Drivers of cellular system improvement
2G:
3G:
4G:
5G:
Digitalization (GSM/CDMA) → spectrum
Spread-spectrum (WCDMA) → spectrum again...
LTE (OFDMA) → and again...
“granularity” → devices → antennas
Bottlenecked by Interference
Cooperation (Relaying) → security (?)
Het-Nets/Cognitive Radio → enough (?)
Interference Alignment → scalability (?)
Massive MIMO → revolutionary, expensive (!)
CoMP → evolutionary, flexible, huge background, maturing...
Still lots to be done!!
29. 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
31. 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
32. 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 =
N=
B
b=1
RX : yk = hk · x + zk
|hk · vk |2
2
2
j=k |hk · vk | + σ
Ntb TX antennas → MISO
33. 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 =
B
RX : yk = hk · x + zk
|hk · vk |2
2
2
j=k |hk · vk | + σ
N = b=1 Ntb TX antennas → MISO
Fixed pn per-antenna target powers → how (?)
34. 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 =
B
RX : yk = hk · x + zk
|hk · vk |2
2
2
j=k |hk · vk | + σ
N = b=1 Ntb TX antennas → MISO
Fixed pn per-antenna target powers → how (?)
Per user γk target SINRs → QoS balancing (?)
35. 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 =
B
RX : yk = hk · x + zk
|hk · vk |2
2
2
j=k |hk · vk | + σ
N = b=1 Ntb TX antennas → MISO
Fixed pn per-antenna target powers → how (?)
Per user γk target SINRs → QoS balancing (?)
Perfectly known hk for all users → overhead (!)
36. 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
37. 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 =
N=
B
b=1
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
Ntb TX antennas → MIMO
38. 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 =
B
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
N = b=1 Ntb TX antennas → MIMO
Fixed pn per-antenna target powers → optimized per user pk
Known weight per user wk → how (?)
39. 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 =
B
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
N = b=1 Ntb TX antennas → MIMO
Fixed pn per-antenna target powers → optimized per user pk
Known weight per user wk → how (?)
Per user γk target SINRs → QoS balancing (?)
40. 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 =
B
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
N = b=1 Ntb TX antennas → MIMO
Fixed pn per-antenna target powers → optimized per user pk
Known weight per user wk → how (?)
Per user γk target SINRs → QoS balancing (?)
Perfectly known Hkj for all users → overhead (!)
41. 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
RX : yk = hk · x + zk
pk |hk · vk |2
2
2
j=k pj |hj · vk | + σ
42. 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 =
N=
B
b=1
hk
p k s k vk
RX : yk = hk · x + zk
pk |hk · vk |2
2
2
j=k pj |hj · vk | + σ
Ntb TX antennas → MISO
43. 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 =
B
hk
p k s k vk
RX : yk = hk · x + zk
pk |hk · vk |2
2
2
j=k pj |hj · vk | + σ
N = b=1 Ntb TX antennas → MISO
Fixed pn per-antenna target powers → optimized per user pk
44. 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 =
B
hk
p k s k vk
RX : yk = hk · x + zk
pk |hk · vk |2
2
2
j=k pj |hj · vk | + σ
N = b=1 Ntb TX antennas → MISO
Fixed pn per-antenna target powers → optimized per user pk
Per user γk target SINRs → minimum QoS
45. 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 =
B
hk
p k s k vk
RX : yk = hk · x + zk
pk |hk · vk |2
2
2
j=k pj |hj · vk | + σ
N = b=1 Ntb TX antennas → MISO
Fixed pn per-antenna target powers → optimized per user pk
Per user γk target SINRs → minimum QoS
Perfectly known hk for all → overhead (!)
46. 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 =
RX : yk = uH · Hkk · x + zk
k
pk |uH · Hkk · vk |2
k
pj |uH · Hkj · vk |2 + σ 2
j=k
k
47. 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 =
Hkj , wk and α
√
pk s k v k
k=1
SINRk =
N=
B
b=1
∈ L |L| < K
RX : yk = uH · Hkk · x + zk
k
pk |uH · Hkk · vk |2
k
pj |uH · Hkj · vk |2 + σ 2
j=k
k
Ntb TX antennas → MIMO
48. 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 =
Hkj , wk and α
√
pk s k v k
k=1
SINRk =
B
∈ L |L| < K
RX : yk = uH · Hkk · x + zk
k
pk |uH · Hkk · vk |2
k
pj |uH · Hkj · vk |2 + σ 2
j=k
k
N = b=1 Ntb TX antennas → MIMO
Fixed pn per-antenna target powers → optimized per user pk
Known weight vectors per user wk and scores αk → how (?)
49. 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 =
Hkj , wk and α
√
pk s k v k
k=1
SINRk =
B
∈ L |L| < K
RX : yk = uH · Hkk · x + zk
k
pk |uH · Hkk · vk |2
k
pj |uH · Hkj · vk |2 + σ 2
j=k
k
N = b=1 Ntb TX antennas → MIMO
Fixed pn per-antenna target powers → optimized per user pk
Known weight vectors per user wk and scores αk → how (?)
Perfectly known Hkj for all users → overhead (!)
50. 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
51. 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
N=
B
b=1
Ntb TX antennas → MISO
52. 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
B
N = b=1 Ntb TX antennas → MISO
Fixed pn per-antenna target powers → optimized total pk
Known scores αk → how (?)
53. 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
B
N = b=1 Ntb TX antennas → MISO
Fixed pn per-antenna target powers → optimized total pk
Known scores αk → how (?)
Perfectly known hk for all users → overhead (!)
54. 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
55. 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
N=
B
b=1
Ntb TX antennas → MIMO
56. 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
B
b=1
N=
Ntb TX antennas → MIMO
Fixed pn per-antenna target powers → optimized per user pk
Known weights w, target powers pk and scores αk → how (?)
57. 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
B
b=1
N=
Ntb TX antennas → MIMO
Fixed pn per-antenna target powers → optimized per user pk
Known weights w, target powers pk and scores αk → how (?)
Perfectly known Hkj for all users → overhead (!)
58. 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
60. 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
61. 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
|hH vk |2
kk
|hH vi |2
ki
{γ1 , · · · , γK }
ˆ
hki = hki + ehki
+
σ2
≥ γk
62. 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
|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
63. Power Minimization Again
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
64. Power Minimization Again
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
65. Power Minimization Again
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
69. Statistical SINR Constraint
Towards a Closed-form
Pr |hH vk |2 ≥ γk
kk
Xkk
N
i=1,i=k
|hH vk |2
kk
HQ v
vk kk k
|hH vi |2 + γk · σ 2
ki
Xki
2
H
σki = vi Qki vi
|hH vi |2
ki
HQ v
vi ki i
70. Statistical SINR Constraint
Towards a Closed-form
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
71. Statistical SINR Constraint
Towards a Closed-form
∞
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.
72. Statistical SINR Constraint
Towards a Closed-form
∞
0
N
∞
...
0
Pr Xkk ≥ ckk
ckk =
fXki (ti ) dti . . . dtN
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.
73. 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.
74. 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
75. Statistical SINR Constraint
Towards a Closed-form
∞
Pr (SINRk ≥ γk ) ≈
∞
...
0
0
˜
Pr Xkk ≥
N
ckk
δ
1+ kk
2
fXki (ti ) dti . . . dtN
i=1,i=k
81. Statistical SINR Constraint
Towards a Closed-form
−
Pr (SINRk ≥ γk ) ≈ e
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
82. Statistical SINR Constraint
Towards a Closed-form
−
Pr (SINRk ≥ γk ) ≈ e
=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
83. Statistical SINR Constraint
Towards a Closed-form
−
Pr (SINRk ≥ γk ) ≈ e
=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
84. Statistical SINR Constraint
Towards a Closed-form
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
85. Statistical SINR Constraint
Towards a Closed-form
Pr (SINRk ≥ γk ) ≈e
−
γk σ 2
vH Akk vk
k
N
i=1,
i=k
Akk
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
ˆ ˆ
Qkk + hkk hH
kk
86. Statistical SINR Constraint
Towards a Closed-form
Pr (SINRk ≥ γk ) ≈e
−
γk σ 2
vH Akk vk
k
N
i=1,
i=k
Akk
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
ˆ ˆ
Qkk + hkk hH
kk
≥ 1 − ρk
87. Statistical SINR Constraint
Towards a Closed-form
γ σ2
− Hk
v Akk vk
k
log e
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
Akk
ˆ ˆ
Qkk + hkk hH
kk
≥ log(1 − ρk )
88. Statistical SINR Constraint
Towards a Closed-form
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 )
89. Statistical SINR Constraint
Towards a Closed-form
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 )
90. Statistical SINR Constraint
Towards a Closed-form
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 )
91. Statistical SINR Constraint
Towards a Closed-form
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 )
92. Statistical SINR Constraint
Towards a Closed-form
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
93. Statistical SINR Constraint
Towards a Closed-form
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
97. Power Minimization Again
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
98. Power Minimization Again
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
102. Semi-Definite Programming
K
minimize
V {vk }
vk
2
k=1
N
subject to |hH vk |2 − γk
kk
given
Vk
i=1,
i=k
|hH vi |2 ≥ γk σ 2
ki
hk
H
vk vk
(Positive Semi-Definite Matrix)
103. Semi-Definite Programming
K
minimize
{Vk } 0
Tr(Vk )
k=1
N
subject to |hH vk |2 − γk
kk
given
Vk
i=1,
i=k
|hH vi |2 ≥ γk σ 2
ki
hk
H
vk vk
(Positive Semi-Definite Matrix)
104. Semi-Definite Programming
K
minimize
{Vk } 0
Tr(Vk )
k=1
N
subject to |hH vk |2 − γk
kk
given
i=1,
i=k
|hH vi |2 ≥ γk σ 2
ki
hk
Vk
H
vk vk
(Positive Semi-Definite Matrix)
Hi
hki hH
ki
(Positive Semi-Definite Matrix)
105. Semi-Definite Programming
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
(Positive Semi-Definite Matrix)
Hi
hki hH
ki
(Positive Semi-Definite Matrix)
106. 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
i=1,i=k |hki vi | +vi Qki vi
≥
γk
− ln(1 − ρk )
1
ηk
107. Power Minimization with Imperfect CSIT
SDP Formulation
K
minimize
{Vk } 0
subject to
Tr(Vk )
k=1
σ2 +
H
vk Akk vk
N
H
ˆH 2
i=1,i=k |hki vi | +vi Qki vi
≥
γk
− ln(1 − ρk )
1
ηk
108. Power Minimization with Imperfect CSIT
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
|hH vi |2 + vi Qki vi
ki
109. Power Minimization with Imperfect CSIT
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
111. Second-Order Cone Programming
K
minimize
V {vk }
vk
2
k=1
N
subject to |hH vk |2 − γk
kk
given
V
i=1,
i=k
|hH vi |2 ≥ γk σ 2
ki
hk
[v1 , · · · , vK ]
(Beamforming Matrix)
115. Power Minimization with Imperfect CSIT
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
(Beamforming Matrix)
[uk1 , · · · , ukK ]
uki
ˆ
hH Qki
ki