Degrees-of-Freedom Robust Transmission for the K-user Distributed Broadcast Channel

Size: px

Start display at page:

Download "Degrees-of-Freedom Robust Transmission for the K-user Distributed Broadcast Channel"

Robyn Hutchinson
5 years ago
Views:

1 /33 Degrees-of-Freedom Robust Transmission for the K-user Distributed Broadcast Channel Presented by Paul de Kerret Joint work with Antonio Bazco, Nicolas Gresset, and David Gesbert ESIT 2017 in Madrid, 10/05/2017

2 2/33 Our Focus: Decentralized Broadcast Channel with Imperfect CSIT Sharing sharing/caching of user s data symbols Imperfect CSI sharing x 1=w 1(H (1) )s x 3=w 3(H (3) )s x 2=w 2(H (2) )s

3 3/33 Broadcast Channel (JP-CoMP, Network MIMO) Some simplifying asumptions: (i) K single-antenna TXs and K single-antennas RXs (ii) Perfect CSI at the RX (iii) Gaussian data symbols (iv) Block fading channel A key asumption: User s data symbols are available at all TXs

4 3/33 Broadcast Channel (JP-CoMP, Network MIMO) Some simplifying asumptions: (i) K single-antenna TXs and K single-antennas RXs (ii) Perfect CSI at the RX (iii) Gaussian data symbols (iv) Block fading channel A key asumption: User s data symbols are available at all TXs Received signal at user i Received signal at user i {}}{ y i = h H i x + η i

5 3/33 Broadcast Channel (JP-CoMP, Network MIMO) Some simplifying asumptions: (i) K single-antenna TXs and K single-antennas RXs (ii) Perfect CSI at the RX (iii) Gaussian data symbols (iv) Block fading channel A key asumption: User s data symbols are available at all TXs Received signal at user i y i = Channel from all TXs to user i (1 K) {}}{ hi H x + η i

6 3/33 Broadcast Channel (JP-CoMP, Network MIMO) Some simplifying asumptions: (i) K single-antenna TXs and K single-antennas RXs (ii) Perfect CSI at the RX (iii) Gaussian data symbols (iv) Block fading channel A key asumption: User s data symbols are available at all TXs Received signal at user i y i = h H i Multi-user multi-tx transmit signal (K 1) {}}{ x +η i with {x} j sent from TX j

7 3/33 Broadcast Channel (JP-CoMP, Network MIMO) Some simplifying asumptions: (i) K single-antenna TXs and K single-antennas RXs (ii) Perfect CSI at the RX (iii) Gaussian data symbols (iv) Block fading channel A key asumption: User s data symbols are available at all TXs Received signal at user i y i = h H i x + Additive white Gaussian Noise N C (0, 1) {}}{ η i

8 3/33 Broadcast Channel (JP-CoMP, Network MIMO) Some simplifying asumptions: (i) K single-antenna TXs and K single-antennas RXs (ii) Perfect CSI at the RX (iii) Gaussian data symbols (iv) Block fading channel A key asumption: User s data symbols are available at all TXs Received signal at user i y i = h H i x + Additive white Gaussian Noise N C (0, 1) {}}{ η i For a given transmit power P, let C(P) denote the sum capacity Our figure of merit will be the Degrees-of-Freedom: C(P) DoF lim P log 2 (P)

9 4/33 Is DoF Useful? Rate First order approximation in the SNR R = DoF log 2 (SNR) + o(log 2 (SNR)) + Closed form results + New insights and new paradigms + First step towards capacity DoF SNR DoF Generalized DoF Finite Gap Results Capacity - Inaccurate if strong pathloss differences - Results not always relevant at finite SNR Very successful to discover new approaches/insights (MIMO [Telatar, 1999, ETC], IA [Cadambe and Jafar, 2008, TIT], delayed CSIT [Maddah-Ali and Tse, 2012, TIT],...)

10 /33 DoF and Pathloss A short Parenthesis (1) Example 2-user IC, single-antenna nodes, α 2 = 10 12, H i,j N C (0, 1) TX 1 TX 2 P P α H 21 α H 12 H 11 H 22 RX 1 RX 2 DoF analysis: DoF = 1 [Etkin et al., 2008, TIT] Not the expected behaviour

11 /33 Generalized DoF A short Parenthesis (2) With Generalized DoF, model the pathloss difference Generalized DoF (GDoF) then defined as Example continued: For P = 20dB, E[ H i,j 2 ]. = P γ i,j C(P, {γ i,j } i,j ) DoF ({γ i,j } i,j ) lim P log 2 (P) γ 1,2 = γ 2,1 = 6 and Expected behaviour! DoF ({γ i,j } i,j ) = 2 Remark GDoF not discussed here but extension for the 2 users case in [Bazco et al., 2017, ISIT]

12 7/33 Centralized VS Distributed CSI Centralized TX Independent : Conventional model Cloud RAN =H+σN Distributed TX Dependent : Our focus here H (1) =H+σ (1) N (1) H (2) =H+σ (2) N (2) H (3) =H+σ (3) N (3)

13 /33 Review of the Perfect CSIT Configuration Outline 1 Review of the Perfect CSIT Configuration 2 Review of the Centralized CSIT Configuration 3 Towards the Distributed CSIT Configuration 4 Warming up: The 2-User Case 5 The K-user Case

14 /33 Review of the Perfect CSIT Configuration DoF with Perfect CSIT Remark All TXs have perfect knowledge of H: Optimal DoF is DoF PCSI = K DoF-optimal transmission scheme is Zero Forcing: Received signal is H 1 x = P H 1 F y i = s 1.. s K P H 1 F s i + η i SNR scales in P: asymptotically possible to decode s i with the rate log 2 (P) bits Importantly, x can also be chosen as x = K i=1 beamformer/precoder t i C K 1 is P K t i = Π h 1,,h i 1,h i+1,...,h K h i t i t i s i where the

15 0/33 Review of the Centralized CSIT Configuration Outline 1 Review of the Perfect CSIT Configuration 2 Review of the Centralized CSIT Configuration 3 Towards the Distributed CSIT Configuration 4 Warming up: The 2-User Case 5 The K-user Case

16 11/33 Review of the Centralized CSIT Configuration Imperfect CSIT in the Centralized Case Conventional high SNR parameterization s1,s2,s3 hˆ h i i P δ i α [0, 1] is called the CSIT quality exponent. Intuitively, equal to the ratio between the available CSIT over the needed CSIT α = 0 no CSIT α = 1 perfect CSIT Some practical motivation: Quantization noise with VQ for B >> 1, with B = # quantization bits, If B = α(m 1) log 2 (P), α [0, 1] σ 2 2 M 1 B σ 2 P α

17 Review of the Centralized CSIT Configuration DoF Analysis of the Centralized Configuration DoF CCSI (α) = 1 + (K 1)α Outerbound recently proven in [Davoodi and Jafar, 2016, TIT] Achievable scheme in [Jindal, 2006, TIT][Hao et al., 2015, TCOM] K DoF / α

18 3/33 Review of the Centralized CSIT Configuration DoF-Optimal Scheme for the Centralized Case (1) [Jindal, 2006, TIT][Hao et al., 2015, TCOM] DoF-optimal scheme: Zero-Forcing (ZF) + Rate Splitting (RS) P TX 1 TX 2 P α P -α P -α y 1 = [ ] Ph1 H 1 s 0 0 }{{} RX 1. Common symbol = P RX 2 + P α h1 H t1 ZF s 1 }{{} +. private symbol = P α Pα h H 1 t ZF 2 s 2 }{{} interference. = P 0 with t ZF i = Π hī h i Π h hī i and with h1 H t2 ZF 2 = ĥh 1 t2 ZF + P }{{} α δ1 H t2 ZF 2 0 = P α δ H 1 t ZF 2 2

19 4/33 Review of the Centralized CSIT Configuration DoF-Optimal Scheme for the Centralized Case (2) [Jindal, 2006, TIT][Hao et al., 2015, TCOM] P TX 1 TX 2 P α P -α P -α y 1 = [ ] Ph1 H 1 s 0 0 }{{} RX 1. Common symbol = P RX 2 + P α h1 H t1 ZF s 1 }{{} +. private symbol = P α Pα h H 1 t ZF 2 s 2 }{{} interference. = P 0 Successive Decoding Decode first s 0 with rate (1 α) log 2 (P) bits (SNR. = P 1 α ) Decode then s 1 with rate α log 2 (P) bits (SNR. = P α ) Sum DoF is (1 α) + Kα

20 5/33 Towards the Distributed CSIT Configuration Outline 1 Review of the Perfect CSIT Configuration 2 Review of the Centralized CSIT Configuration 3 Towards the Distributed CSIT Configuration 4 Warming up: The 2-User Case 5 The K-user Case

21 6/33 Towards the Distributed CSIT Configuration Distributed CSIT Configuration With imperfect CSIT sharing extends to s 1,s 2,s 3 hˆ (1) h P (1) δ (1) hˆ (3) h s 1,s 2,s 3 P (3) δ (3) hˆ (2) h P (2) s 1,s 2,s 3 CSIT configuration characterized by 1 α (1) α (2)... α (K) 0 Remark Arbitrary CSIT configuration δ (2)

22 7/33 Towards the Distributed CSIT Configuration An Intuitive Outerbound [de Kerret and Gesbert, 2016, ISIT] Theorem (The Centralized Outerbound) DoF DCSI (α) 1 + (K 1) max j {1,...,K} α(j) }{{} =α (1) DoF upperbounded by DoF achieved by full CSIT exchange Having Ĥ α(1),...,..., Ĥ α(k) doesn t help over having just best CSIT Ĥ α(1) K DoF 2 1 Centralized outer-bound 0 1 α (1)

23 Towards the Distributed CSIT Configuration Conventional Zero Forcing [de Kerret and Gesbert, 2012, TIT] First idea: Use ZF (DoF-optimal for Centralized CSIT) DoF ZF = 1 + (K 1) min j {1,...,K} }{{} =α (K) Very inefficient! Why? Goal is to design T 1 and T 2 such that [ ] T1 0, (Zero Forcing constraint at RX 1) h H 1 T 2 i.e., find a vector orthogonal to h H 1 α (j) T 2 * T 2 *+n T 2 18/33 h 1 H T 1 * T 1

24 19/33 Towards the Distributed CSIT Configuration Problem Statement K DoF 2 Optimal DoF? 1 Centralized outer-bound ZF 0 1 α (1)

25 0/33 Warming up: The 2-User Case Outline 1 Review of the Perfect CSIT Configuration 2 Review of the Centralized CSIT Configuration 3 Towards the Distributed CSIT Configuration 4 Warming up: The 2-User Case 5 The K-user Case

26 21/33 Warming up: The 2-User Case Active-Passive Zero-Forcing (AP-ZF) [de Kerret and Gesbert, 2012, TIT] Main Idea: Less informed TX generates interference, more informed TX removes it {(h (1) 1 )H } 1T 1 + {(h (1) 1 )H } 2T 2 = 0 T 1 = {(h(1) 1 )H } 2 T {(h (1) 2 1 )H } 1 T 2 * T 2 T 2 APZF =c T 1 APZF T 1 * T 1 Achieves the DoF DoF APZF = 1 + α (1)

27 2/33 Warming up: The 2-User Case Active-Passive Zero-Forcing (AP-ZF) Achieves the DoF DoF APZF = 1 + α (1) DoF 2 1 Centralized outer-bound AP-ZF 0 1 α (1) Remark In fact achieves also Generalized DoF [Bazco et al., 2017]

28 3/33 The K-user Case Outline 1 Review of the Perfect CSIT Configuration 2 Review of the Centralized CSIT Configuration 3 Towards the Distributed CSIT Configuration 4 Warming up: The 2-User Case 5 The K-user Case

29 4/33 The K-user Case Generalization of AP-ZF? Problem: AP-ZF doesn t help much with more users Need for a different approach Main Idea DoF APZF = 1 + (K 1)α (K 1) Exploit interference as side information: Interference useful for both the interfered user and the desired user Analogy to the use of delayed CSIT [Maddah-Ali and Tse, 2012, TIT]

30 25/33 The K-user Case A Multi-layer Transmission Scheme [de Kerret and Gesbert, 2016, ISIT] 1 All TXs serve all users with power P α(1) using Active-Passive ZF Generate interferences of power P α(1) K x = P α(1) T APZF i s i i=1 TX 1 TX 2 TX 3 AP-ZF Interferences P α(1) RX 1 RX 2 RX 3

= P α(1) T APZF i s i i=1 Estimate & quantize

31 26/33 The K-user Case A Multi-layer Transmission Scheme [de Kerret and Gesbert, 2016, ISIT] 2 TX 1 estimates and quantizes the interference terms before their generations K x = P α(1) T APZF i s i i=1 Estimate & quantize at TX 1 TX 1 TX 2 TX 3 AP-ZF Interferences P α(1) RX 1 RX 2 RX 3

[ ] 1 K P s 0 0 + P α(1) T APZF i s i K 1 i=1 Estimate & quantize Broadcast at TX 1 TX 1 TX 2 TX

32 27/33 The K-user Case A Multi-layer Transmission Scheme [de Kerret and Gesbert, 2016, ISIT] 3 TX 1 then transmits them via a common data symbol at the same time as the private data symbols x = [ ] 1 K P s P α(1) T APZF i s i K 1 i=1 Estimate & quantize Broadcast at TX 1 TX 1 TX 2 TX 3 AP-ZF Interferences Multicast using superposition coding Common signal P P α(1) RX 1 RX 2 RX 3

33 28/33 The K-user Case Signal Processing at TX 1 1 Interference estimation at TX 1: α(1) P (ĥ(1) 1 )H T APZF 2 s 2 = P = α(1) (ĥ(1) 1 + P α(1) h H 1 T APZF P α(1) δ (1) 2 s 2 + P α(1) (δ (1) 1 )H T APZF 2 s 2 1 )H T APZF 2 s 2 } {{ } O(1) TX 1 can compute DoF-perfect estimate of the interference terms! 2 Interference quantization: Use α (1) log 2 (P) bits to quantize the signal scaling in P α(1) Quantization error scaling in P 0 [Cover and Thomas, 2006] 3 Transmit 3α (1) log 2 (P) bits to all users

34 9/33 The K-user Case Signal Processing at RX 1 (w.l.o.g.) User 1 has received y 1 = [ ] Ph1 H 1 s 0 0 K 1 }{{}.=P + P α(1) h1 H T APZF 1 s 1 }{{}.=P α(1) + P α(1) h1 H T APZF 2 s 2 P }{{} α(1) h1 H T APZF 3 s 3 }{{}.=P α(1) + User 1 decodes s 0 and obtains then α(1) P (ĥ(1) 1 )H T APZF 2 s 2, Useful: Remove interference α(1) P (ĥ(1) 2 )H T APZF 3 s 3, Useless (for RX 1) α(1) P (ĥ(1) 3 )H T APZF 1 s 1, Useful: Desired data Achieved DoF: If 3α (1) 1 α (1), achieves DoF DoF = 6α (1) }{{} 2α (1) per user + (1 α (1) ) 3α (1) }{{} multicast DoF after retransmitting interference.=p 0

35 DoF The K-user Case Weak CSIT Regime Theorem If max j {1,...,K} α (j) 1 (weak CSIT regime), 1+K(K 2) DoF DCSI (α) 1 + (K 1) max j {1,...,K} α(j) 3 Weak CSIT Regime [ISIT 16] 2 1 Conventional ZF Active-Passive ZF [ISIT 16] Centralized outerbound [ISIT 16] α1 0/33 Figure: DoF as a function of α (1) for α (2) = 2 3 α(1) and α (3) = 0

36 31/33 DoF The K-user Case Take Home Message DoF analysis allows to develop new schemes/insights with simple linear algebra Role of each TX adapts to the full multi-tx CSIT configuration Multi-layer transmission scheme: Estimate, Qantize & transmit interference at the most informed user Many extensions: Developed a new Hierarchical Zero-Forcing to extend optimality region Extend further? Improving the centralized-outerbound Going beyond the DoF 3 Weak CSIT Regime [ISIT 16] 2 1 Conventional ZF Active-Passive ZF [ISIT 16] Centralized outerbound [ISIT 16] α1

37 32/33 The K-user Case References I Antonio. Bazco, P. de Kerret, D. Gesbert, and N. Gresset. Generalized Degrees-of-Freedom of the 2-User Case MISO Broadcast Channel with Distributed CSIT. to be published IEEE International Symposium on Information Theory (ISIT), V. R. Cadambe and S. A. Jafar. Interference alignment and degrees of freedom of the K-user interference channel. IEEE Trans. Inf. Theory, 54(8): , Aug T. Cover and A. Thomas. Elements of information theory. Wiley-Interscience, Jul A. Gholami Davoodi and S. A. Jafar. Aligned image sets under channel uncertainty: Settling conjectures on the collapse of Degrees of Freedom under finite precision CSIT. IEEE Trans. Inf. Theo., 62(10): , Oct P. de Kerret and D. Gesbert. Degrees of freedom of the network MIMO channel with distributed CSI. IEEE Trans. Inf. Theory, 58(11): , Nov P. de Kerret and D. Gesbert. Network MIMO: Transmitters with no CSI Can Still be Very Useful. In Proc. IEEE International Symposium on Information Theory (ISIT), P. de Kerret, Bazco A., and D. Gesbert. Enforcing Coordination in Network MIMO with Unequal CSIT. In Proc. IEEE Asilomar Conference on Signals, Systems and Computers (ACSSC), 2016a. P. de Kerret, D. Gesbert, J. Zhang, and P. Elia. Optimally bridging the gap from delayed to perfect CSIT in the K-user MISO BC. In Proc. IEEE Information Theory Workshop (ITW), 2016b. R.H. Etkin, D.N.C. Tse, and H. Wang. Gaussian interference channel capacity to within one bit. IEEE Trans. Inf. Theory, 54(12): , Dec C. Hao, Y. Wu, and B. Clerckx. Rate analysis of two-receiver MISO Broadcast Channel with finite rate feedback: A rate-splitting approach. IEEE Trans. on Commun., 63(9): , Sept N. Jindal. MIMO Broadcast Channels with finite-rate feedback. IEEE Trans. Inf. Theory, 52(11): , Nov M.A. Maddah-Ali and D. Tse. Completely stale transmitter channel state information is still very useful. IEEE Trans. Inf. Theory, 58(7): , Jul I. Emre Telatar. Capacity of multi-antenna Gaussian channels. European Transaction on Communications, 10: , 1999.

38 33/33 The K-user Case Conventional precoding 05 décembre 2013.gwb - 1/4-13 déc :49:58

39 34/33 DoF The K-user Case Extension of the Weak CSIT Regime for K = 3 [de Kerret et al., 2016a, Asilomar] Improved scheme building on a new Hierarchical ZF precoding paradigm 3 Weak CSIT Regime [ISIT 16] Extension of Weak CSIT Regime [Asilomar 16] 2 1 Conventional ZF Multi-layer scheme [ISIT 16] Multi-layer with HZF [Asilomar 16] Centralized outerbound [ISIT 16] α 1 Figure: DoF as a function of α (1) for α (2) = 2 3 α(1) and α (3) = 0

40 35/33 The K-user Case Beyond the Weak CSIT Regime Definition (Weak CSIT regime) 1 α (1) α(2) 0.75 Definition (Heterogeneous CSIT regime) α (1) > min (2α (2), ) α(2) α (2) 0.5 Extension of weak CSIT regime Intermediate CSIT regime Definition (Intermediate CSIT regime) weak CSIT regime [ISIT16] Heterogeneous CSIT regime α(2) < α (1) 2α (2) α (1)

41 6/33 The K-user Case Achievable DoF [de Kerret et al., 2016a, Asilomar] Theorem In the 3-user MIMO BC with D-CSIT, it holds that 1 + 2α (1) (Weak CSIT) DoF DCSI 3 (α) (1 + 2 α(2) ) (Intermediate CSIT) 1+α (1) + 3α(1) (1 α (1) )+α (2) (5α (1) 3α (2) 1) (Heterogeneous CSIT) 9α (1) 8α (2) Can be achieved building on new precoding scheme: Hierarchical zero-forcing

42 7/33 The K-user Case Hierarchical ZF with K = 3: Main Property Lemma Let t HZF 3 be the HZF beamformer towards user 3 with average power P. Then: TX 1 TX 2 TX 3 h H 1 t HZF 3 2 P 1 α(1) h H 2 t HZF 3 2 P 1 α(2) 1- α(1) P P 1-α(2) compared with conventional ZF RX 1 RX 2 RX 3 TX 1 TX 2 TX 3 P 1-α(2) P 1-α(2) RX 1 RX 2 RX 3

43 8/33 The K-user Case Roadmap of Hierarchical ZF 1 Make CSIT hierarchical 2 Split precoding in layers 3 Design layer k to reduce interference at user k without reducing interference reduction already realized

44 9/33 The K-user Case (1) Make CSIT Hierarchical Example Example for two transmitters TX1, TX2 with α (1) α (2) Let Q α (2) be our Hierarchical quantizer using α (2) log 2 (P) bits Let us define Ĥ (1) (Ĥ(1) ) Q α (2) α (2) Ĥ (2) α (2) Q α (2) (Ĥ(2) ) Then, there exists a quantizer Q α (2) such that [de Kerret et al., 2016b, ITW] : lim {Ĥ Pr (1) = Ĥ (2) = 1 P α (2) [ ] E Ĥ (j) H 2 α (2) F P α(2), j = 1, 2 α (2) } TX 1 can obtain Ĥ (2) α (2) : CSIT configuration has been made hierarchical More generally, TX i knows what TX i + 1 knows (post quantizing)

45 40/33 The K-user Case (1) Make CSIT Hierarchical (2) H Ĥ (2) Ĥ (1) (1) (1) (2) Q α(1) Q α(2) Q α(3) (1) (2) (3)

46 1/33 The K-user Case (2) Split Precoding in Layers t HZF 3 aimed at user 3 decomposed as t HZF t3 HZF 3 (1) = (2)} 1 3 (2)} {t HZF {t HZF {t3 HZF (3)} 1 {t3 HZF (3)} 2 {t3 HZF (3)} 3 e.g., TX 2 needs to be able to compute the 2th row: t HZF t3 HZF 3 (1) {t3 HZF (2)} 1 {t = 0 + {t 3 HZF 3 HZF (3)} 1 (2)} 2 + {t3 HZF (3)} {t3 HZF (3)} 3

47 42/33 The K-user Case (3) Hierarchical ZF for K = 3 First layer (at TX 1, TX 2 and TX 3) t HZF 3 (3) = λ HZF Ĥ (3)H (Ĥ (3) (Ĥ (3) ) H + 1 P I3 ) 1 e 3 TX 1 TX 2 TX 3 Interfered Users Served user RX 1 RX 2 RX 3

48 42/33 The K-user Case (3) Hierarchical ZF for K = 3 First layer (at TX 1, TX 2 and TX 3) t HZF 3 (3) = λ HZF Ĥ (3)H (Ĥ (3) (Ĥ (3) ) H + 1 P I3 ) 1 e 3 Second layer (at TX 1 and TX 2) TX 1 TX 2 TX 3 t3 HZF (2)= Ĥ (2)H [1:2,1:2] (Ĥ (2) [1:2,1:2] Ĥ(2)H [1:2,1:2] + 1 ) 1Ĥ P I2 (2) [1:2,1:3] thzf 3 (3) P 1-α(2) P 1-α(2) RX 1 RX 2 RX 3

49 42/33 The K-user Case (3) Hierarchical ZF for K = 3 First layer (at TX 1, TX 2 and TX 3) t HZF 3 (3) = λ HZF Ĥ (3)H (Ĥ (3) (Ĥ (3) ) H + 1 P I3 ) 1 e 3 Second layer (at TX 1 and TX 2) t3 HZF (2)= Ĥ (2)H [1:2,1:2] (Ĥ (2) [1:2,1:2] Ĥ(2)H [1:2,1:2] + 1 ) 1Ĥ P I2 (2) [1:2,1:3] thzf 3 (3) TX 1 TX 2 TX 3 Third layer (at TX 1) ( t3 HZF (1)= Ĥ (1)H 1,1 Ĥ (1) 1, ) 1 ([ ] ĥ (1)H t HZF 3 (2) 1 P 0 ) +t3 HZF (3) 1- α(1) P RX 1 P 1-α(2) RX 2 RX 3

50 The K-user Case (3) Hierarchical ZF for K = 3 First layer (at TX 1, TX 2 and TX 3) t HZF 3 (3) = λ HZF Ĥ (3)H (Ĥ (3) (Ĥ (3) ) H + 1 P I3 ) 1 e 3 Second layer (at TX 1 and TX 2) t3 HZF (2)= Ĥ (2)H [1:2,1:2] (Ĥ (2) [1:2,1:2] Ĥ(2)H [1:2,1:2] + 1 ) 1Ĥ P I2 (2) [1:2,1:3] thzf 3 (3) TX 1 TX 2 TX 3 Third layer (at TX 1) ( t3 HZF (1)= Ĥ (1)H 1,1 Ĥ (1) 1, ) 1 ([ ] ĥ (1)H t HZF 3 (2) 1 P 0 ) +t3 HZF (3) 1- α(1) P RX 1 P 1-α(2) RX 2 RX 3 Main Intuition of the Proof Does not increase interference at user 2 when reducing interference at user 1 because 42/33 t HZF 3 (1) 2 P 1 α(2)

51 43/33 The K-user Case Transmission Scheme P Common data symbols HZF data symbols P α(1) P α(1)-α(2) BC s1 s2 s3 RX1 BC s1 s2 s3 RX2 BC s1 s2 s3 Interferences TextText RX3 Less interference bits to convey in second layer More information bits can be squeezed in first layer

Generalized Degrees-of-Freedom of the 2-User Case MISO Broadcast Channel with Distributed CSIT

207 IEEE IEEE International Symposium on Information Theory (ISIT), Aachen, Germany, June 207 DOI: 009/ECCE2078096480 Generalized Degrees-of-Freedom of the 2-User Case MISO Broadcast Channel with Distributed