Random beamforming for multi cell multiple input multi output (MIMO) systems

Random Beamforming for Multi-cell Multi-Input Multi-Output (MIMO) Systems HIEU DUY NGUYEN (B Eng (First-Class Hons.), VNU) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2013 To my beloved parents RBF FOR MULTI-CELL MIMO SYSTEMS H D NGUYEN Declaration I hereby declare that this thesis is my original work and it has been written by me in its entirety I have duly acknowledged all the sources of information which have been used in the thesis This thesis has also not been submitted for any degree in any university previously Hieu Duy Nguyen 25 September 2013 DECLARATION RBF FOR MULTI-CELL MIMO SYSTEMS H D NGUYEN Acknowledgments I would like to express my sincere gratitude to my supervisor Assistant Professor Hon Tat Hui for his guidance and supervision during my Ph.D candidature He has supported me with enthusiastic encouragement and inspiration, without which I might not complete my degree on time I also would like to express my deepest appreciation to my co-supervisor Assistant Professor Rui Zhang, who has provided helpful discussions and insightful comments on my research topics It is my pleasure to work closely with him and benefit by his profound knowledge Last but not least, I would like to acknowledge my parents, who always support and encourage me to achieve my goals ACKNOWLEDGMENTS RBF FOR MULTI-CELL MIMO SYSTEMS H D NGUYEN Contents List of Figures xi List of Algorithms xv List of Acronyms xvii List of Notations xix Chapter Introduction 1.1 Motivation 1.2 Performance Measures 1.2.1 Output Signal-to-Noise Ratio and Signal-to-Interference-PlusNoise Ratio 1.2.2 Ergodic and Outage Capacity 1.2.3 Rate Region 1.2.4 1.3 Degrees of Freedom (DoF) and DoF Region Dissertation Overview and Major Contributions 10 i CONTENTS 1.3.1 Chapter - Transmission Schemes for Single- and Multi-Cell Downlink Systems 1.3.2 Chapter - Single-Cell MISO RBF 11 1.3.3 Chapter - Multi-Cell MISO RBF 11 1.3.4 Chapter - Multi-Cell MIMO RBF 12 Publications 13 1.4.1 Book Chapter 13 1.4.2 International Journal Papers 13 1.4.3 1.4 10 International Conference Papers 14 Chapter Transmission Schemes for Single- and Multi-Cell Downlink Systems 17 2.1 Single-Cell MIMO BC 18 2.1.1 Channel Model 18 2.1.2 Dirty-Paper Coding 19 2.1.3 Block Diagonalization 22 2.1.3.1 Channel Inversion for Single-Antenna Users 22 2.1.3.2 Block Diagonalization for Multi-antenna Users 24 2.1.3.3 Asymptotic Scaling Laws 27 Multi-Cell/Interference Channel: Interference Alignment 31 2.2.1 Channel Model 32 2.2.2 Asymptotic Interference Alignment with Symbol Extensions 33 2.2 ii RBF FOR MULTI-CELL MIMO SYSTEMS H D NGUYEN 2.2.2.1 35 2.2.2.2 Asymptotic Interference Alignment Scheme 36 2.2.2.3 Optimality of IA for the K-user SISO IC 38 Interference Alignment without Symbol Extensions 39 2.2.3.1 Minimizing the Interference Leakage 40 2.2.3.2 Maximizing the SINR 42 2.2.3.3 Maximizing the Sum of DoF 44 2.2.3.4 2.2.3 Interference Alignment Objectives Numerical Results and Discussions 47 Chapter Single-Cell MISO RBF 51 3.1 System Model 52 3.2 Achievable Rate 57 3.2.1 Rate Expression for (F1) Scheme 57 3.2.2 Rate Expression for (F2) Scheme 58 Asymptotic Analysis 61 3.3.1 Large Number of Users 62 3.3.2 Large System 64 3.4 Reduced and Quantized Feedback in OBF/RBF 65 3.5 Non-Orthogonal RBF and Grassmanian Line Packing Problem 66 3.6 User Scheduling Schemes 67 3.7 Other Studies 69 3.3 Chapter Multi-Cell MISO RBF 71 iii CONTENTS 4.1 System Model 4.2 74 Achievable Rate of Multi-Cell Random Beamforming: Finite-SNR Analysis 4.2.1 Single-Cell RBF 76 4.2.2 Multi-Cell RBF 77 4.2.3 4.3 76 Asymptotic Sum Rate as Kc → ∞ 82 Degrees of Freedom Region in Multi-Cell Random Beamforming: HighSNR Analysis 4.3.1 Single-Cell Case 86 4.3.2 Multi-Cell Case 90 4.3.3 Optimality of Multi-Cell RBF 93 4.3.3.1 Single-Cell Case 94 4.3.3.2 4.4 84 Multi-Cell Case 95 Conclusions 96 Chapter Multi-Cell MIMO RBF 5.1 99 System Model 101 5.1.1 5.1.2 5.2 Multi-Cell RBF 102 DoF Region 106 SINR Distribution 108 5.2.1 RBF-MMSE 108 5.2.2 RBF-MF 110 iv APPENDIX B Therefore, from (B.9), we conclude e−s/ηc e−s/ηc A= A0 = ηc ηc (1 + s)Mc −1 C l=1,l=c (1 + µl,c M s) l ηc dA ds By differentiating A in (B.9) with respect to s, we obtain (B.13) A = − ηc − B , ηc i.e., (c) fS (s) = A + B = −ηc dA Combining this result and (B.13), (4.10) and (4.11) are ds obtained This completes the proof of Lemma 4.2.2 B.3 Proof of Theorem 4.2.1 Using the similar derivation as in Appendix B.1, we obtain (c) RRBF Mc = log Mc = log Kc (−1)n n=1 Kc (−1) n=1 n ∞ Kc n exp(−nx/η)dx (1 + x)n(Mc −1)+1 C Kc n l=1,l=c c àl ì C l=1,l=c àl,c x c nMl ì exp(nx/)dx (x + nMl +1 1)n(Mc −1)+1 C l=1,l=c x+ ηc µl,c nMl (B.14) By applying the partial fractional decomposition given in (4.13) and using [20, 2.234.2] for each term therein, we arrive at (4.12) This completes the proof of Theorem 4.2.1 B.4 Proof of Proposition 4.2.1 Due to the similarity between (4.8) and (4.11), the original approach in [64, Theorem 1] can be applied to prove this proposition with minor modifications For the completeness, only a sketch proof is presented here 162 RBF FOR MULTI-CELL MIMO SYSTEMS H D NGUYEN K →∞ (c) (c) (c) To show that RRBF − c− → Mc log2 log Kc , we first note that fS (s) and FS (s) −− satisfy the von Mises condition for the Gumbel-type limiting distributions (see, e.g., [13, Theorem 10.5.2.c]) Therefore, as Kc → ∞, there exist constants aKc and bKc (c) such that FS (aKc x + bKc ) Kc → exp (−e−x ) The value of bKc can be found to be1 C bKc = ηc log Kc − ηc l=1 Ml − log log Kc + O(log log log Kc ) (c) (B.15) (c) (c) − FS (s) /fS (s) for Furthermore, the growth function, defined as gS (s) = s ≥ 0, is given by (c) gS (s) = ηc + Mc −1 s+1 + C Ml η l=1,l=c s+ µ c (B.16) l,c It is easy to verify the followings: (c) • lims→∞ gS (s) = ηc > 0, • bKc = O(log Kc ) as Kc → ∞, and (c) • The derivative of gS (s) satisfies (c) dn gS (s) dsn =O s=bKc bn+1 Kc (B.17) Hence, by applying [64, Corollary A.1], we have C Pr ηc log Kc − ηc ≤ ηc log Kc − ηc Ml l=1 C l=1 log log Kc + O(log log log Kc ) ≤ Ml − log log Kc + O(log log log Kc ) max k∈{1,··· ,Kc } (c) SINRk,m ≥1−O log Kc (B.18) This completes the proof of Proposition 4.2.1 Let f (Kc ) be a function of Kc f (Kc ) = O(log log log Kc ) means that f (Kc )/ log log log Kc < ∞ as Kc → ∞ 163 APPENDIX B B.5 Proof of Lemma 4.3.1 For convenience, we denote the following auxiliary random variable Rk,m = log2 (1 + SINRk,m) (B.19) From(4.8), the CDF of Rk,m is obtained as r e−(2 −1)/η FR (r) = − r(M −1) To prove Lemma 4.3.1, we first show that Pr α α log2 η + log2 log η ≥ max Rk,1 ≥ log2 η − log2 log η k∈{1,··· ,K} M −1 M −1 η→∞ − − 1, if < α ≤ M − 1, −→ Pr log2 η + log2 log η + log2 α ≥ max k∈{1,··· ,K} Rk,1 ≥ log2 η + log2 log η + log2 β η→∞ − − 1, if α > M − 1, −→ in which the constant β is define as β = (B.20) α−M +1 ; (B.21) hence α > β > when α > M − Considering (B.20), the upper-bound probability can be expressed as α log2 η + log2 log η ≥ max Rk,1 k∈{1,··· ,K} M −1 K  α K exp −η M −1 −1 log η exp (η −1 ) α  = FR log2 η + log2 log η = 1 − M −1 η α (log η)M −1 Pr (B.22) 164 RBF FOR MULTI-CELL MIMO SYSTEMS H D NGUYEN Using the asymptotic relation log(1 − x) = −x + O(x2 ) when x is small, we get   α exp −η M −1 −1 log η exp (η −1 )  K log 1 − η α (log η)M −1 =− +O K α η α (log η)M −1 exp −η M −1 −1 log η exp (η −1 ) K α η 2α (log η)2(M −1) exp −2η M −1 −1 log η exp (2η −1 ) η→∞ − − 0, −→ (B.23) in which we have used the assumptions K = Θ(η α ), and < α ≤ M − As a consequence, the upper-bound probability converges to when η → ∞ To show the convergence of the lower-bound probability in (B.20), we can utilize the same technique described above by showing Pr α log2 η − log2 log η ≥ max Rk,1 k∈{1,··· ,K} M −1 = FR  = 1 − α log2 η − log2 log η M −1 K α exp − log η η M −1 −1 exp (η −1)(log η)M −1 ηα Note that   α exp − log η η M −1 −1 exp (η −1 )(log η)M −1  K log 1 − ηα =− K  (B.24) α K (log η)M −1 exp − η M −1 −1 exp (η −1 ) α η log η +O Mα −1 K (log η)2(M −1) exp − η −1 exp (2η −1 ) 2α η log η η→∞ − − −∞, −→ (B.25) since, when η → ∞, the first term in (B.25) goes to −∞, while the second term goes to (B.24) thus converges to and the lower-bound probability is confirmed The 165 APPENDIX B proof of (B.21) follows similar arguments as the above, and is omitted for brevity This completes the proof of Lemma 4.3.1 B.6 Proof of Proposition 4.3.1 According to the main text, it is sufficient to show the DoF upper bound to be NT as follows Note that in a single-cell MISO-BC, the DPC yields the optimal sum-rate, RDPC From [30, Theorem 1], we have ρ→∞ log2 ρ denoted by RDPC Therefore, d∗ (α) = lim RDPC ≤ NT E log2 + η max k∈{1,··· ,K} ||hk ||2 (B.26) Note that η = PT /(NT σ ) is the SNR per beam, and ||hk ||2 ’s are i.i.d chi-square random variables with 2NT degrees of freedom, denoted by χ2 (2NT ) Thus, if we denote Rk = log2 (1 + η||hk ||2 ), the CDF of R = Rk is FR (r) = γ Γ(NT ) r −1 NT , η , where Γ(·) and γ(·, ·) are the gamma and the incomplete gamma function, respectively The same reasoning as in the proof of Lemma 4.3.1 can be reused here to show that Pr log2 η + log2 log η + log2 (α + 1) ≥ This completes the proof of Proposition 4.3.1 166 max k∈{1,··· ,K} Rk η→∞ − − −→ (B.27) RBF FOR MULTI-CELL MIMO SYSTEMS H D NGUYEN Appendix C Proofs of Chapter C.1 Proof of Corollary 5.2.1 (c) The interference-plus-noise covariance matrix W k given in (5.2) can be written as (c) Wk = lim N →∞ PT ˜ (c,c) ˜ (c,c) H k,−m H Mc k,−m H C + l=1,l=c PT γl,c ˜ (l,c) ˜ (l,c) Hk Hk Ml H + σ2 HN HH N N , (C.1) where H N ∈ CNR ×N consists of i.i.d random variables each distributed as ∼ (MMSE,c) CN (0, 1) To find the PDF of SINRk,m in (5.3), we apply Theorem 5.2.1 with ˜ (c,c) h := hk,m , ˜ (C,c) ˜ (l,c) ˜ (c,c) ˜ (1,c) X := H k,−m , H k , · · · , H k , · · · , H k , H N , 167 (C.2) APPENDIX C and    µl,c µl,c µC,c 1  µC,c  Ψ := diag 1, · · · , 1, · · · , ,··· , ,··· , ,··· , , ,··· ,  ηc ηc ηc ηc Nηc Nηc  Mc −1 Ml MC N (C.3) (MMSE,c) The PDF of S := SINRk,m FS (s) = − lim N →∞ can thus be expressed as NR −1 θi si i=0 1+ s N ηc N (1 + s)Mc −1 Mc −1 l=1,l=c (1 µl,c M s) l ηc + where θi is the coefficient of si in the polynomial expansion of Mc −1 l=1,l=c (1 s)Mc −1 + , 1+ (C.4) s N ηc N (1 + µl,c Ml s) ηc Next, by letting N → ∞, in the denominator in (C.4), the term (1 + converges to es/ηc , while the nominator converges to NR −1 ζ i si , i=0 s N ) N ηc where ζi ’s are defined in Corollary 5.2.1 We thus obtain (5.14) This completes the proof of Corollary 5.2.1 C.2 Proof of Theorem 5.2.2 We first note that ∞ (c) fS (s) = fS|V (s|v)fV (v)dv ∞ ∞ = −∞ where j = √ sNR −1 2πΓ(NR ) 1 NR −(v+ ηc )s −jωv ) e e ηc C µl,c Mc −1 (v + (1 − jω) −1 168 l=1,l=c 1−j ηc Ml ω dvdω, (C.5) RBF FOR MULTI-CELL MIMO SYSTEMS H D NGUYEN Therefore, ∞ ∞ (v + s FS (s) = −∞ 0 1 NR −jωv NR −1 −(v+ ηc )x ) e x e ηc C µl,c Mc −1 2πΓ(NR ) (1 − jω) l=1,l=c 1−j ηc Ml dxdvdω (C.6) ω Now by using [20, (3.351.1)], we can write (C.6) as NR −1 FS (s) = − k=0 NR −1 =1− k=0 ∞ e−s/ηc sk 2πk! −∞ (v + ∞ (1 − jω) k −(s+jω) ) e ηc C µl,c ω 1−j ηc Mc −1 l=1,l=c Ml dvdω (C.7) k e−s/ηc sk × k−m (k − m)!ηc m=0 × ∞ 2π −∞ dω C (s + jω)m+1 (1 − jω)Mc −1 l=1,l=c µl,c ω 1−j ηc Ml , Tm (s) (C.8) where we have used the binomial expansion and the result in [20, (3.351.3)] to obtain (C.7) From (B.6)-(B.13), we see that T0 (s) can be expressed as in (5.19) It is also easy to show that Tm (s) = (−1)m dm T0 (s) m! dsm Combining this result, (5.19), and (C.8), we obtain (5.18) This completes the proof of Theorem 5.2.2 C.3 C.3.1 Proof of Lemma 5.3.1 RBF-MMSE We first investigate the DoF with RBF-MMSE Consider the following two cases 169 APPENDIX C Case 1, NR ≤ M − C.3.1.1 (MMSE) Denote Rk,m (MMSE) We first show that the following two := log2 + SINRk,m probabilities are true Pr α α (MMSE) log2 η + log2 log η ≥ max Rk,1 ≥ log2 η − log2 log η k∈{1,··· ,K} M − NR M − NR η→∞ − − 1, if < α ≤ M − NR , −→ Pr log2 η + log2 log η + log2 α ≥ max (MMSE) k∈{1,··· ,K} Rk,1 (C.9) ≥ log2 η + log2 log η + log2 β1 η→∞ − − 1, if α > M − NR , −→ where β1 = α−M +NR ; (C.10) hence, α > β1 > when α > M − NR (MMSE) From Corollary 5.2.1, the CDF of the single-cell RBF-MMSE S := SINRk,1 can be expressed as −s/η FS (s) = − e NR −1 (M −1)! si i=1 i!(M −1−i)! (1 + s)M −1 = − e−s/η Θ (s + 1)M −NR +O (s + 1)M −NR +1 (MMSE) as s and/or η → ∞ Therefore, the CDF of Yk := Rk,1 , (C.11) has the following asymp- totic form y −1)/η FYk (y) = − e−(2 Θ 2(M −NR )y 170 +O 2(M −NR +1)y , RBF FOR MULTI-CELL MIMO SYSTEMS H D NGUYEN as y and/or η → ∞ In (C.9), the upper-bound probability can thus be given as Pr α log2 η + log2 log η ≥ max Yk k∈{1,··· ,K} M − NR = FYk  K α log2 η + log2 log η M − NR  α −1 = 1 − exp −η M −NR log η + ×  η   × Θ  ηα (log η) M −NR   K    , M −NR +1    +O α M −NR η log η (C.12) as η → ∞ Note that when x is small, we have the following asymptotic relation log(1 − x) = −x + O(x2 ) We thus have    α  −1 Θ K log 1 − exp −η M −NR log η +  η   = −Θ K α  K  +O α η M −NR log η K +O Θ +O −1 α M −NR log η log η + η      M −NR +1   α  −1  exp −η M −NR log η + M −NR +1 η K η log η η exp −η M −NR  η 2α (log η)2n α n η α (log η)M −NR  +O η α (log η)M −NR α exp −2η n −1 log η + α 2(n+1) exp −2η n −1 log η + η→∞ − − 0, −→ η η (C.13) since K = Θ(η α ), and < α ≤ M − NR As a consequence, the upper-bound 171 APPENDIX C probability converges to when η → ∞ To prove the convergence to of the lower-bound probability in (C.12), we observe that Pr α log2 η − log2 log η ≥ max Yk k∈{1,··· ,K} M − NR = FYk = K α log2 η − log2 log η M − NR α − exp −η M −NR × −1 Θ 1 × + log η η (log η)M −NR ηα (log η)M −NR +1 +O η α(M −NR +1) M −NR K (C.14) Note that α K log − exp −η M −NR × = −Θ Θ K(log η)M −NR ηα +O 1 × + log η η (log η)M −NR ηα +O Θ α exp −η M −NR K (log η)M −NR +1 η +O −1 α(M −NR +1) M −NR K (log η)2(M −NR +1) η 2α(M −NR +1) M −NR η −1 α −1 1 + log η η α exp −2η M −NR α exp −2η M −NR η→∞ − − −∞, −→ α(M −NR +1) M −NR 1 + log η η exp −η M −NR K(log η)2(M −NR ) η 2α (log η)M −NR +1 +O −1 −1 + log η η + log η η (C.15) since, when η → ∞, the first term goes to −∞, while the second term goes to (C.14) thus converges to and the lower-bound probability is confirmed We omit the proof of (C.10) since it follows similar arguments With (C.9) and (C.10), the results in (5.22a) and (5.22b) follow immediately 172 RBF FOR MULTI-CELL MIMO SYSTEMS C.3.1.2 H D NGUYEN Case 2, NR ≥ M Suppose that M receive antennas are used Then the DoF is M from Case above Therefore, dRBF-MMSE (α, m) ≥ M Also note that in a single-cell MIMO RBF with M transmit beams, the BS can be considered as having M transmit antennas only Proposition 5.3.1 thus leads to dRBF-MMSE (α, m) ≤ M We thus conclude that dRBF-MMSE (α, m) = M C.3.2 RBF-MF/AS To obtain the DoF of RBF-MF/AS, we first show that Pr α α (MF/AS) log2 η + log2 log η ≥ max Rk,1 ≥ log2 η − log2 log η k∈{1,··· ,K} M −1 M −1 η→∞ − − 1, if < α ≤ M − 1, −→ Pr log2 η + log2 log η + log2 α ≥ max (MF/AS) k∈{1,··· ,K} Rk,1 ≥ log2 η + log2 log η + log2 β η→∞ − − 1, if α > M − 1, −→ where β2 = α−M +1 ; (C.16) (C.17) hence, α > β2 > when α > M − From Theorem 5.2.2, the (MF) CDF of the single-cell RBF-MF S := SINRk,m is NR −1 −s/η FS (s) = − e (MF) Denote Zk := Rk,1 k=0 (M +m−2)! k sk (M −2)! k−m (s + 1)M +m−1 (k − m)!m!η m=0 (MF) := log2 + SINRk,1 NR −1 −z/η FZk (z) = − e k=0 (C.18) The CDF of Zk is thus k (2z − 1)k (M + m − 2)! (k − m)!m!(M − 2)! η k−m 2(M +m−1)z m=0 173 (C.19) APPENDIX C In (C.16), the upper-bound probability can thus be given as Pr α log2 η + log2 log η ≥ max Zk k∈{1,··· ,K} M −1 = − exp −η M −1 −1 log η + α NR −1 × = − exp −η k=0 = FZk × η k K α (η M −1 log η − 1)k (M + m − 2)! (k − m)!m!(M − 2)! η k−m η α(M +m−1) (log η)(M +m−1) M −1 m=0 α −1 M −1 log η + η Θ η α (log η)M −1 K α log2 η + log2 log η M −1 +O K αM η M −1 , (C.20) as η → ∞, which is quite similar to (C.12) with NR = in this case Now following the same reasoning as in (C.13), we can prove that the upper-bound probability (C.20) → as η → ∞ To prove the convergence of the lower-bound, we note that Pr α log2 η − log2 log η ≥ max Zk k∈{1,··· ,K} M −1 = − exp −η M −1 −1 α NR −1 × = − exp −η k=0 = FZk 1 × + log η η α −1 M −1 1 + log η η K α (η M −1 log η − 1)k (log η)M +m−1 (M + m − 2)! α(M +m−1) (k − m)!m!(M − 2)! η k−mη M −1 m=0 k Θ (log η)M −1 ηα K α log2 η − log2 log η M −1 +O (log η)M −1 αM η M −1 K , (C.21) which is quite similar to (C.14) Now following the same reasoning as in (C.15), we can prove that (C.21) → as η → ∞ Thus we confirm (C.16) The proof of (C.17) follows similarly and is thus omitted On the other hand, for the case of RBF-AS, note that RBF-AS scheme consists of two selection processes: antenna selection at each MS with NR antennas and user 174 RBF FOR MULTI-CELL MIMO SYSTEMS H D NGUYEN selection at the BS with K users The rate performance of RBF-AS is therefore equivalent to that of MISO RBF with NR K single-antenna users in the cell Thus, we obtain (C.16) and (C.17) for the case of RBF-AS With (C.16) and (C.17), the results in (5.23a) and (5.23b) follow immediately This thus completes the proof of Lemma 5.3.1 C.4 Proof of Proposition 5.3.1 In a single-cell MIMO-BC, DPC yields the maximum sum-rate, denoted by RDPC RDPC From [30, Theorem ρ→∞ log2 ρ Therefore, the single-cell DoF can be bounded as d ≤ lim 1], we have max ||H k ||2 max T r HHHk k RDPC ≤ NT E log2 + η k∈{1,··· ,K} ≤ NT E log2 + η k∈{1,··· ,K} (C.22) Denote Rk := log2 + ηT r H H H k Note that T r H H H k is distributed as k k χ2 (2NT NR ) Similarly to (C.10) and (C.17), we can show that Pr log2 η + log2 log η + log2 (α + 1) ≥ max k∈{1,··· ,K} Rk η→∞ − − −→ (C.23) Combining (C.22) and (C.23), we obtain d ≤ NT , where the equality is achieved by, e.g., the DPC scheme The proof of Proposition 5.3.1 is thus completed 175 APPENDIX C 176 ... Matched Filter MIMO Multiple Input Multiple Output MISO Multiple Input Single Output MMSE Minimum-Mean-Square Error xvii H D NGUYEN LIST OF ACRONYMS MU Multi- User OBF Orthogonal Beamforming PDF Probability... spatial diversity on the rate performance of RBF is not yet fully characterized even in a single -cell setup We thus study a multicell multiple- input multiple- output (MIMO) broadcast system with RBF... 175 vi RBF FOR MULTI- CELL MIMO SYSTEMS H D NGUYEN Abstract Random beamforming (RBF) is a practically favourable transmission scheme for multiuser multi- antenna downlink systems since it