6 interference management in mimo multicell

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6 interference management in mimo multicell

  1. 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
  2. 2. 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
  3. 3. Forecast Mobile traffic volume trend: 1000x in 10 years. Requirements
  4. 4. Forecast Mobile traffic volume trend: 1000x in 10 years. The Internet of Things Requirements
  5. 5. Forecast Mobile traffic volume trend: 1000x in 10 years. The Internet of Things The Internet of Things ! Requirements
  6. 6. Forecast Mobile traffic volume trend: 1000x in 10 years. The Internet of Things The Internet of Things ! The Internet of Things !!! Requirements
  7. 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. 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. 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. 10. Past and Future Drivers of cellular system improvement 2G: 3G: 4G: 5G: Digitalization (GSM/CDMA) Spread-spectrum (WCDMA) LTE (OFDMA) ???
  11. 11. Past and Future Drivers of cellular system improvement 2G: 3G: 4G: 5G: Digitalization (GSM/CDMA) → spectrum Spread-spectrum (WCDMA) LTE (OFDMA) ???
  12. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
  30. 30. Now let’s get serious...
  31. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
  59. 59. OK, so what if CSIT is not perfect ?
  60. 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. 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. 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. 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. 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. 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
  66. 66. Statistical SINR Constraint Towards a Closed-form Pr SINRk ≥ γk
  67. 67. Statistical SINR Constraint Towards a Closed-form Pr |hH vk |2 kk N i=1,i=k |hH vi |2 + σ 2 ki ≥ γk
  68. 68. Statistical SINR Constraint Towards a Closed-form  N Pr |hH vk |2 ≥ γk kk i=1,i=k  |hH vi |2 + γk · σ 2  ki
  69. 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. 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. 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. 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. 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. 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. 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
  76. 76. Statistical SINR Constraint Towards a Closed-form ∞ Pr (SINRk ≥ γk ) ≈ ∞ ... 0 ˜ Xkk ∼ χ2 2 0 N ˜ Pr Xkk ≥ =⇒ ckk δ 1+ kk 2 fXki (ti ) dti . . . dtN i=1,i=k ˜ Pr(Xkk ≥ x) = e−x/2
  77. 77. Statistical SINR Constraint Towards a Closed-form ∞ Pr (SINRk ≥ γk ) ≈ ∞ ... 0 ˜ Xkk ∼ χ2 2 0 exp − =⇒ ckk 1+ δkk 2 N fXki (ti ) dti . . . dtN i=1,i=k ˜ Pr(Xkk ≥ x) = e−x/2
  78. 78. Statistical SINR Constraint Towards a Closed-form ∞ Pr (SINRk ≥ γk ) ≈ ∞ ... 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 fXki (ti ) dti . . . dtN i=1,i=k ∞ 0 exp (−αki ti )fXki (ti ) dti 2 γk σki 2 σkk (1 + δkk 2 )
  79. 79. Statistical SINR Constraint Towards a Closed-form ∞ Pr (SINRk ≥ γk ) ≈ ∞ ... 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 = fXki (ti ) dti . . . dtN i=1,i=k ∞ i=1, i=k − N ckk 2 γk σki 2 σkk (1 + δkk 2 )
  80. 80. Statistical SINR Constraint Towards a Closed-form − Pr (SINRk ≥ γk ) ≈ e γk σ 2 2 (1+ δkk ) σ 2 kk N i=1,i=k E[e−αki ti ]
  81. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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
  94. 94. Statistical SINR Constraint Towards a Closed-form σ2 + N i=1,i=k H ˆ |hH vi |2 + vi Qki vi ki H vk Akk vk ≤ − ln(1 − ρk ) γk
  95. 95. Statistical SINR Constraint Towards a Closed-form H vk Akk vk σ2 + N i=1,i=k H ˆ |hH vi |2 + vi Qki vi ki ≥ −γk ln(1 − ρk )
  96. 96. Statistical SINR Constraint Towards a Closed-form H vk Akk vk σ2 N + i=1,i=k H ˆ |hH vi |2 + vi Qki vi ki ≥ −γk ln(1 − ρk ) ηk
  97. 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. 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
  99. 99. Now, how do we solve this ?
  100. 100. Some Tools...
  101. 101. 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
  102. 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. 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. 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. 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. 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. 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. 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. 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
  110. 110. Second-Order Cone 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
  111. 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)
  112. 112. Second-Order Cone Programming minimize V Tr(VH · V) 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)
  113. 113. Second-Order Cone Programming minimize V subject to given V Tr(VH · V) 1 1+ γk |hH vk |2 kk hH V kk ≥ σ2 2 hk [v1 , · · · , vK ] (Beamforming Matrix)
  114. 114. Second-Order Cone Programming 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 ] (Beamforming Matrix)
  115. 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
  116. 116. Some Results...
  117. 117. 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
  118. 118. 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
  119. 119. Thank You ! Questions ?

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