Deterministic equivalents for Haar matrices Romain Couillet 1 , 2 , - PowerPoint PPT Presentation

Deterministic equivalents for Haar matrices Romain Couillet 1 , 2 , Jakob Hoydis 1 , M´ erouane Debbah 1 1 Alcatel-Lucent Chair on Flexible Radio, Sup´ elec, Gif sur Yvette, FRANCE 2 ST-Ericsson, Sophia-Antipolis, FRANCE romain.couillet@supelec.fr Random Matrix Theory Symposium R. Couillet (Sup´ elec) Deterministic equivalents for Haar matrices 13/10/2010 1 / 38

Outline Main Results 1 Deterministic Equivalent for a sum of independent Haar Comparison with the i.i.d. case Sketch of Proof 2 First deterministic equivalent Second deterministic equivalent Uniqueness and convergence of the det. eq. Simulation plots 3 Haar matrix with correlated columns 4 R. Couillet (Sup´ elec) Deterministic equivalents for Haar matrices 13/10/2010 2 / 38

Main Results Outline Main Results 1 Deterministic Equivalent for a sum of independent Haar Comparison with the i.i.d. case Sketch of Proof 2 First deterministic equivalent Second deterministic equivalent Uniqueness and convergence of the det. eq. Simulation plots 3 Haar matrix with correlated columns 4 R. Couillet (Sup´ elec) Deterministic equivalents for Haar matrices 13/10/2010 3 / 38

Main Results Problem statement We wish to characterize a deterministic equivalent for the following types of matrices “sum of correlated Haar” K � H k W k P k W H k H H B N = k k = 1 with H k ∈ C N × N k deterministic, W k ∈ C N k × n k unitary isometric, P k ∈ C n k × n k deterministic. Possible uses in wireless communications are multi-cell frequency selective CDMA/SDMA with a single user per cell single-cell downlink CDMA/SDMA with colored noise capacity and MMSE SINR “Haar matrices with a correlation profile” B N = XX H with X = [ x 1 , . . . , x n ] ∈ C N × n and 1 x k = R k w k 2 with R k deterministic and W = [ w 1 , . . . , w n ] ∈ C N × n isometric. Possible uses in wireless communications are single/multi-cell frequency selective CDMA/SDMA uplink and downlink beamforming with unitary precoding in frequency selective channels capacity and MMSE SINR R. Couillet (Sup´ elec) Deterministic equivalents for Haar matrices 13/10/2010 4 / 38

Main Results Problem statement We wish to characterize a deterministic equivalent for the following types of matrices “sum of correlated Haar” K � H k W k P k W H k H H B N = k k = 1 with H k ∈ C N × N k deterministic, W k ∈ C N k × n k unitary isometric, P k ∈ C n k × n k deterministic. Possible uses in wireless communications are multi-cell frequency selective CDMA/SDMA with a single user per cell single-cell downlink CDMA/SDMA with colored noise capacity and MMSE SINR “Haar matrices with a correlation profile” B N = XX H with X = [ x 1 , . . . , x n ] ∈ C N × n and 1 x k = R k w k 2 with R k deterministic and W = [ w 1 , . . . , w n ] ∈ C N × n isometric. Possible uses in wireless communications are single/multi-cell frequency selective CDMA/SDMA uplink and downlink beamforming with unitary precoding in frequency selective channels capacity and MMSE SINR R. Couillet (Sup´ elec) Deterministic equivalents for Haar matrices 13/10/2010 4 / 38

Main Results Problem statement We wish to characterize a deterministic equivalent for the following types of matrices “sum of correlated Haar” K � H k W k P k W H k H H B N = k k = 1 with H k ∈ C N × N k deterministic, W k ∈ C N k × n k unitary isometric, P k ∈ C n k × n k deterministic. Possible uses in wireless communications are multi-cell frequency selective CDMA/SDMA with a single user per cell single-cell downlink CDMA/SDMA with colored noise capacity and MMSE SINR “Haar matrices with a correlation profile” B N = XX H with X = [ x 1 , . . . , x n ] ∈ C N × n and 1 x k = R k w k 2 with R k deterministic and W = [ w 1 , . . . , w n ] ∈ C N × n isometric. Possible uses in wireless communications are single/multi-cell frequency selective CDMA/SDMA uplink and downlink beamforming with unitary precoding in frequency selective channels capacity and MMSE SINR R. Couillet (Sup´ elec) Deterministic equivalents for Haar matrices 13/10/2010 4 / 38

Main Results Problem statement We wish to characterize a deterministic equivalent for the following types of matrices “sum of correlated Haar” K � H k W k P k W H k H H B N = k k = 1 with H k ∈ C N × N k deterministic, W k ∈ C N k × n k unitary isometric, P k ∈ C n k × n k deterministic. Possible uses in wireless communications are multi-cell frequency selective CDMA/SDMA with a single user per cell single-cell downlink CDMA/SDMA with colored noise capacity and MMSE SINR “Haar matrices with a correlation profile” B N = XX H with X = [ x 1 , . . . , x n ] ∈ C N × n and 1 x k = R k w k 2 with R k deterministic and W = [ w 1 , . . . , w n ] ∈ C N × n isometric. Possible uses in wireless communications are single/multi-cell frequency selective CDMA/SDMA uplink and downlink beamforming with unitary precoding in frequency selective channels capacity and MMSE SINR R. Couillet (Sup´ elec) Deterministic equivalents for Haar matrices 13/10/2010 4 / 38

Main Results Deterministic Equivalent for a sum of independent Haar Outline Main Results 1 Deterministic Equivalent for a sum of independent Haar Comparison with the i.i.d. case Sketch of Proof 2 First deterministic equivalent Second deterministic equivalent Uniqueness and convergence of the det. eq. Simulation plots 3 Haar matrix with correlated columns 4 R. Couillet (Sup´ elec) Deterministic equivalents for Haar matrices 13/10/2010 5 / 38

Main Results Deterministic Equivalent for a sum of independent Haar Fundamental equations Theorem (Theorem 1) Let P i ∈ C n i × n i and R i ∈ C N × N be Hermitian nonnegative matrices and ¯ c 1 , . . . , ¯ c K be positive scalars. Then the following system of equations in (¯ e 1 , . . . , ¯ e K ) e i = 1 � − 1 � ¯ N tr P i e i P i + [¯ c i − e i ¯ e i ] I n i − 1   K e i = 1 � N tr R i ¯ e j R j − z I N . (1)   j = 1 e K ( z )) with z �→ e i ( z ) , C \ R + → C , the Stieltjes has a unique functional solution (¯ e 1 ( z ) , . . . , ¯ transform of a distribution function with support on R + . R. Couillet (Sup´ elec) Deterministic equivalents for Haar matrices 13/10/2010 6 / 38

Main Results Deterministic Equivalent for a sum of independent Haar Point-wise uniqueness Theorem (Theorem 2) For each z real negative, the system of equations (1) has a unique scalar-valued solution e ( t ) e ( t ) (¯ e 1 , . . . , ¯ e K ) with ¯ e i = lim t →∞ ¯ , where ¯ is the unique solution of i i = 1 � − 1 � e ( t ) e ( t ) c i − e ( t ) e ( t ) ¯ P i + [¯ ¯ N tr P i ] I n i (2) i i i i c i / e ( t ) ) , e ( 0 ) can take any positive value and e ( t ) within the interval [ 0 , c i ¯ is recursively defined by: i i i − 1   K = 1 e ( t + 1 ) � e ( t ) ¯ R j − z I N . N tr R i   i j j = 1 e ( t ) The solution ¯ of (2) is explicitly given by i e ( t ) e ( t , k ) ¯ ¯ = lim , i i k →∞ e ( t , 0 ) c i / e ( t ) with ¯ ∈ [ 0 , c i ¯ ) and, for k ≥ 1 , i i = 1 � − 1 � e ( t , k ) e ( t ) c i − e ( t ) e ( t , k − 1 ) ¯ P i + [¯ ¯ N tr P i ] I n i . i i i i R. Couillet (Sup´ elec) Deterministic equivalents for Haar matrices 13/10/2010 7 / 38

Main Results Deterministic Equivalent for a sum of independent Haar Convergence in distribution Theorem (Theorem 3) Let P i ∈ C n i × n i be a Hermitian nonnegative matrix with spectral norm bounded uniformly along n i and W i ∈ C N i × n i be the n i ≤ N i columns of a unitary Haar distributed random matrix. We also i ∈ C N × N has uniformly bounded consider H i ∈ C N × N i a random matrix such that R i � H i H H spectral norm along N, almost surely. Denote K � H i W i P i W H i H H B N = i . i = 1 c i � N i Then, as N, N 1 , . . . , N K , n 1 , . . . , n K grow to infinity with ¯ N satisfying c i < ∞ and 0 ≤ n i 0 < lim inf ¯ c i ≤ lim sup ¯ N i � c i ≤ 1 for all i, we have F B N − F N ⇒ 0 almost surely, where F N is the distribution function with Stieltjes transform m N ( z ) defined by − 1   K m N ( z ) = 1 � N tr ¯ e i ( z ) R i − z I N , (3)   i = 1 e i ( z ) , C \ R + → C , defined in Theorem 1. with z �→ ¯ R. Couillet (Sup´ elec) Deterministic equivalents for Haar matrices 13/10/2010 8 / 38

Main Results Deterministic Equivalent for a sum of independent Haar Deterministic equivalent of the Shannon transform Theorem (Theorem 4) Let B N ∈ C N × N be defined as in Theorem 3 with z = − 1 / x for some x > 0 . Denoting V B N ( x ) = 1 N log det ( x B N + I N ) the Shannon-transform of F B N , we have V B N ( x ) − V N ( x ) a . s . − → 0 , (4) as N, N 1 , . . . , N K , n 1 , . . . , n K grow to infinity with 0 < lim inf ¯ c i ≤ lim sup ¯ c i < ∞ , where   K V N ( x ) = 1 � ¯ N log det  I N + x e i R i  i = 1 � 1 K � � � � + N log det [¯ c i − e i ¯ e i ] I n i + e i P i + ( 1 − c i )¯ c i log (¯ c i − e i ¯ e i ) − ¯ c i log (¯ c i ) . (5) i = 1 R. Couillet (Sup´ elec) Deterministic equivalents for Haar matrices 13/10/2010 9 / 38