Ph.D. Defense, Aalborg University, 23 January 2008 Multiple-Input - PowerPoint PPT Presentation

Ph.D. Defense, Aalborg University, 23 January 2008 Multiple-Input Multiple-Output Fading Channel Models and Their Capacity Bjørn Olav Hogstad 1 , 2 Ph.D. student: atzold 1 Prof. Matthias P¨ Main supervisor: Second supervisor: Prof. Bernard H. Fleury 2 1 University of Agder, Grimstad, Norway 2 Aalborg University, Aalborg, Denmark 1/50

Contents • Introduction • Sum-of-Sinusoids Channel Simulators • Generalized Concept of Deterministic Channel Modeling • The MIMO Channel Capacity • The One-Ring MIMO Channel Model • The Two-Ring MIMO Channel Model • The Elliptical MIMO Channel Model • Summary Ph.D. Defense, Aalborg University, 23 January 2008 2/50

Introduction 2 × 2 MIMO System: h 11 (t) s 1 (t) r 1 (t) h 21 (t) Transmitter Receiver h 12 (t) (Base station) (Mobile station) h 22 (t) s 2 (t) r 2 (t) 2×2 MIMO channel • This Ph.D. project has developed MIMO channel models based on the geometrical one-ring, two-ring, and elliptical scattering models. • All the developed MIMO channel models are based on Rice’s sum-of-sinusoids. Ph.D. Defense, Aalborg University, 23 January 2008 3/50

1. Introduction Typical Behaviour of the Channel Capacity 12 Channel capacity Channel capacity, (bits/s/Hz) 10 8 6 4 2 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Time, t (s) This Ph.D. project has the following contributions to the investigations of the MIMO channel ca- pacity: • Exact closed-form solutions for the probability density function (PDF), cumulative distribution function (CDF), level-crossing rate (LCR), and average duration of fades (ADF) of the capacity of orthogonal space-time block code (OSTBC) MIMO systems. • Upper bounds on the mean capacity. • Simulation results of the MIMO channel capacity by using the one-ring, two-ring, and elliptical MIMO channel models. Ph.D. Defense, Aalborg University, 23 January 2008 4/50

Sum-of-Sinusoids Channel Simulators The Reference Model Rayleigh process: ζ ( t ) = | µ 1 ( t ) + jµ 2 ( t ) | where µ i ( t ) ∼ N (0 , σ 2 0 ) ( i = 1 , 2) . Temporal ACF (isotropic scattering): r µ i µ i ( τ ) := E { µ i ( t ) µ i ( t + τ ) } = σ 2 0 J 0 (2 πf max τ ) . Rice’s sum-of-sinusoids: N i � µ i ( t ) = lim c i,n cos(2 πf i,n t + θ i,n ) N i →∞ n =1 where � c i,n = 2 ∆ f i S µ i µ i ( f i,n ) f i,n = n ∆ f i . The quantity ∆ f i is the width of the frequency band associated with the n th component. The symbol S µ i µ i ( f ) denotes the Doppler power spectral density. Ph.D. Defense, Aalborg University, 23 January 2008 5/50

Sum-of-Sinusoids Channel Simulators The Simulation Model N i � µ i ( t ) = ˆ c i,n cos(2 πf i,n t + θ i,n ) n =1 Classes of sum-of-sinusoids channel simulators and their statistical properties Class Gains Frequencies Phases First-order Wide-sense Mean- Autocor.- c i,n f i,n θ i,n stationary stationary ergodic ergodic I const. const. const. – – – – II const. const. RV yes yes yes yes yes a yes a yes a III const. RV const. no IV const. RV RV yes yes yes no yes a V RV const. const. no no no VI RV const. RV yes yes yes no yes a yes a yes a VII RV RV const. no VIII RV RV RV yes yes yes no a If certain boundary conditions are fulfilled. Ph.D. Defense, Aalborg University, 23 January 2008 6/50

Generalized Concept of Deterministic Channel Modeling Generalized MEDS Parameter LPNM computation Fixed parameters Statistical properties Stochastic Deterministic Simulation of Geometrical model Reference model simulation model simulation model sample functions Infinite complexity Finite complexity Finite complexity Non-realizable Infinite number of One (or some few) sample functions sample functions Non-realizable Realizable Ph.D. Defense, Aalborg University, 23 January 2008 7/50

The MIMO Channel Capacity MIMO channel capacity: � � I M R + ρ �� H ( t ) H H ( t ) C ( t ) := log 2 det [bits/s/Hz] M T where H ( t ) = [ h pq ( t )] M R ,M T is the channel matrix. p,q =1 SIMO channel capacity: C SIMO ( t ) := log 2 (1 + ρ h H ( t ) h ( t )) [bits/s/Hz] where h ( t ) = [ h 1 ( t ) , . . . , h M R ( t )] T is the M R × 1 complex channel gain vector. MISO channel capacity: � � 1 + ρ h H ( t ) h ( t ) C MISO ( t ) := log 2 [bits/s/Hz] M T where h ( t ) = [ h 1 ( t ) , . . . , h M T ( t )] T is the M T × 1 complex channel gain vector. Capacity of OSTBC-MIMO systems: � � 1 + ρ h H ( t ) h ( t ) C OSTBC ( t ) = log 2 [bits/s/Hz] M T where h ( t ) = [ h 1 ( t ) , . . . , h M T M R ( t )] T is the M T M R × 1 complex channel gain vector. Ph.D. Defense, Aalborg University, 23 January 2008 8/50

The MIMO Channel Capacity The PDFs of the capacities can be expressed in closed forms as ln 2 Γ( M R ) ρ M R 2 r (2 r − 1) M R − 1 e − (2 r − 1) /ρ p C, SIMO ( r ) = p C, MISO ( r ) = ( M T ) M T ln 2 Γ( M T ) ρ M T 2 r · (2 r − 1) M T − 1 e − M T (2 r − 1) /ρ p C, OSTBC ( r ) = ln 2( M T ) M R 2 r/M T (2 r/M T − 1) M R − 1 e − M T (2 r/MT − 1) /ρ , r ≥ 0 . Γ( M R ) M T ρ M R The CDFs of the capacities can be expressed in closed forms as M R − 1 ρ k F C, SIMO ( r ) = 1 − ρ 1 − M R e − (2 r − 1) /ρ (2 r − 1) M R − 1 � Γ( M R − k )(2 r − 1) k k =0 � ρ M T − 1 � 1 − M T ρ k e − M T (2 r − 1) /ρ (2 r − 1) M T − 1 � F C, MISO ( r ) = 1 − Γ( M T − k )( M T ) k (2 r − 1) k M T k =0 � ρ M R − 1 � 1 − M R ρ k e − M T (2 r/MT − 1) /ρ (2 r/M T − 1) M R − 1 � F C, OSTBC ( r ) = 1 − Γ( M R − k )( M T ) k (2 r/M T − 1) k , r ≥ 0 . M T k =0 Ph.D. Defense, Aalborg University, 23 January 2008 9/50

The MIMO Channel Capacity The LCRs of the capacities can be obtained in closed forms as 2 ρβ (2 r − 1) � Γ( M R ) ρ M R √ π (2 r − 1) M R − 1 e − (2 r − 1) /ρ N C, SIMO ( r ) = 2 ρβ (2 r − 1) N C, MISO ( r ) = ( M T ) M T − 1 / 2 � (2 r − 1) M T − 1 e − M T (2 r − 1) /ρ Γ( M T ) ρ M T √ π N C, OSTBC ( r ) = ( M T ) M R − 1 / 2 � 2 ρβ (2 r/M T − 1) (2 r/M T − 1) M R − 1 e − M T (2 r/MT − 1) /ρ , Γ( M R ) ρ M R √ π r ≥ 0 . The ADFs of the capacities can be obtained in closed forms as T C, SIMO ( r ) = F C, SIMO ( r ) N C, SIMO ( r ) T C, MISO ( r ) = F C, MISO ( r ) N C, MISO ( r ) T C, OSTBC ( r ) = F C, MIMO ( r ) N C, MIMO ( r ) , r ≥ 0 . Ph.D. Defense, Aalborg University, 23 January 2008 10/50

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � The MIMO Channel Capacity Confirmation of the Theory by Simulations Simulation model: For example, 1 × M R SIMO channels. ( m ) ( m ) cos(2 f t 1,1 ) 1,1 2 1 ( m ) ( m ) cos(2 f t 1,2 ) N 1 (1) (t) 1,2 + m t ( ) 1 ( ) ( m ) ( m ) cos(2 f t 1, ) 1, N N 1 1 m ( m ) ( m ) ( m ) ( ) | t ( ) t j ( ) | t (m) (t) 1 2 ( m ) ( m ) cos(2 f t 2,1 ) 2,1 2 N ( m ) ( m ) 2 cos(2 f t 2,2 ) 2,2 (M ) (t) + R m t ( ) 2 ( ) M R ( m ) ( m ) cos(2 f t ) 2, N 2, N 2 2 � π � � � n − 1 i, 0 = ( − 1) ( i − 1) π 4 N i · k Parameters [1]: f ( k ) + α ( k ) where α ( k ) i,n = f max cos K . 2 2 N i i, 0 [1] M. P¨ atzold et al, “Two new methods for the generation of multiple uncorrelated Rayleigh fading waveforms,” in Proc. 163th IEEE Semiannual Vehicular Technology Conference, VTC 2006-Spring Melbourne, Australia, May 2006, vol. 6, pp.2782-2786. Ph.D. Defense, Aalborg University, 23 January 2008 11/50

The MIMO Channel Capacity The PDFs of the SIMO/MISO Channel Capacities 1 1 Theory Theory Simulation Simulation (1 × 9) (9 × 1) 0.8 0.8 (1 × 7) (7 × 1) (1 × 5) (5 × 1) p C, MISO ( r ) /f max 0.6 0.6 p C, SIMO ( r ) /f max (1 × 3) (3 × 1) 0.4 0.4 (1 × 2) (2 × 1) (1 × 1) (1 × 1) 0.2 0.2 0 0 0 2 4 6 8 10 0 2 4 6 8 10 Level, r Level, r The PDF of the ( M T × 1) MISO channel The PDF of the (1 × M R ) SIMO channel capacity. capacity. Ph.D. Defense, Aalborg University, 23 January 2008 12/50

The MIMO Channel Capacity The LCRs of the SIMO/MISO Channel Capacities 1.2 1.2 Theory Theory (1 × 1) Simulation Simulation (1 × 2) (1 × 3)(1 × 5) (1 × 7) 1 1 (1 × 9) 0.8 N C, MISO ( r ) /f max N C, SIMO ( r ) /f max 0.8 (1 × 1) (2 × 1) 0.6 0.6 (3 × 1) 0.4 (5 × 1) 0.4 (7 × 1) (9 × 1) 0.2 0.2 0 0 0 2 4 6 8 10 0 2 4 6 8 10 Level, r Level, r The normalized LCR of the ( M T × 1) MISO The normalized LCR of the (1 × M R ) SIMO channel capacity. channel capacity. Ph.D. Defense, Aalborg University, 23 January 2008 13/50

The MIMO Channel Capacity The ADFs of the SIMO/MISO Channel Capacities 3 3 10 10 Theory Theory (1 × 1) Simulation Simulation (1 × 1) (2 × 1) 2 2 10 (1 × 2) (3 × 1) 10 (1 × 3) (5 × 1) (1 × 5) (7 × 1) T C, SIMO ( r ) · f max T C, MISO ( r ) · f max 1 (1 × 7) (9 × 1) 10 1 10 (1 × 9) 0 10 0 10 −1 10 −1 10 −2 10 0 2 4 6 8 10 −2 10 0 2 4 6 8 10 Level, r Level, r The normalized ADF of the (1 × M R ) SIMO The normalized ADF of the ( M T × 1) MISO channel capacity. channel capacity. Ph.D. Defense, Aalborg University, 23 January 2008 14/50