Ph.D. Defense, Aalborg University, 23 January Multiple-Input Multiple-Output Fading Channel Models and Their Capacity

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1 Multiple-Input Multiple-Output Fading Channel Models and heir Capacity Ph.D. student: Bjørn Olav Hogstad,2 Main supervisor: Prof. Matthias Pätzold Second supervisor: Prof. Bernard H. Fleury 2 University of Agder, Grimstad, Norway 2 Aalborg University, Aalborg, Denmark /5

2 Contents Introduction Sum-of-Sinusoids Channel Simulators Generalized Concept of Deterministic Channel Modeling he MIMO Channel Capacity he One-ing MIMO Channel Model he wo-ing MIMO Channel Model he Elliptical MIMO Channel Model Summary 2/5

3 Introduction 2 2 MIMO System: h (t) s (t) h 2 (t) r (t) ransmitter (Base station) h 2 (t) eceiver (Mobile station) s 2 (t) h 22 (t) 2 2 MIMO channel r 2 (t) his 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 ice s sum-of-sinusoids. 3/5

4 . Introduction ypical Behaviour of the Channel Capacity Channel capacity, (bits/s/hz) Channel capacity ime, t (s) his Ph.D. project has the following contributions to the investigations of the MIMO channel capacity: Exact closed-form solutions for the probability density function (PDF), cumulative distribution function (CDF), level-crossing rate (LC), and average duration of fades (ADF) of the capacity of orthogonal space-time block code (OSBC) 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. 4/5

5 Sum-of-Sinusoids Channel Simulators he eference Model ayleigh process: where µ i (t) N(, σ 2 ) (i =, 2). ζ(t) = µ (t) + jµ 2 (t) emporal ACF (isotropic scattering): r µi µ i (τ) := E{µ i (t)µ i (t + τ)} ice s sum-of-sinusoids: where = σ 2 J (2πf max τ). c i,n = 2 f i S µi µ i (f i,n ) f i,n = n f i. µ i (t) = lim N i N i n= c i,n cos(2πf i,n t + θ i,n ) he quantity f i is the width of the frequency band associated with the nth component. he symbol S µi µ i (f) denotes the Doppler power spectral density. 5/5

6 Sum-of-Sinusoids Channel Simulators he Simulation Model ˆµ i (t) = N i n= c i,n cos(2πf i,n t + θ i,n ) 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. V yes yes yes yes III const. V const. yes a yes a yes a no IV const. V V yes yes yes no V V const. const. no no yes a no VI V const. V yes yes yes no VII V V const. yes a yes a yes a no VIII V V V yes yes yes no a If certain boundary conditions are fulfilled. 6/5

7 Generalized Concept of Deterministic Channel Modeling Generalized MEDS LPNM Parameter computation Fixed parameters Statistical properties Geometrical model eference model Stochastic simulation model Deterministic simulation model Simulation of sample functions Infinite complexity Finite complexity Finite complexity Non-realizable Infinite number of sample functions One (or some few) sample functions Non-realizable ealizable 7/5

8 he MIMO Channel Capacity MIMO channel capacity: C(t) := log 2 [det ( I M + ρ )] H(t)H H (t) M [bits/s/hz] where H(t) = [h pq (t)] M,M p,q= is the channel matrix. SIMO channel capacity: C SIMO (t) := log 2 ( + ρh H (t)h(t)) [bits/s/hz] where h(t) = [h (t),...,h M (t)] is the M complex channel gain vector. MISO channel capacity: ( C MISO (t) := log 2 + ρ ) h H (t)h(t) M [bits/s/hz] where h(t) = [h (t),...,h M (t)] is the M complex channel gain vector. Capacity of OSBC-MIMO systems: ( C OSBC (t) = log 2 + ρ ) h H (t)h(t) M [bits/s/hz] where h(t) = [h (t),...,h M M (t)] is the M M complex channel gain vector. 8/5

9 he MIMO Channel Capacity he PDFs of the capacities can be expressed in closed forms as p C, SIMO (r) = ln 2 Γ(M )ρ M 2r (2 r ) M e (2r )/ρ p C, MISO (r) = (M ) M ln 2 Γ(M )ρ M 2r (2 r ) M e M (2r )/ρ p C, OSBC (r) = ln 2(M ) M 2 r/m (2 r/m ) M Γ(M )M ρ M e M (2 r/m )/ρ, r. he CDFs of the capacities can be expressed in closed forms as F C, SIMO (r) = ρ M e (2r )/ρ (2 r ) M ( ρ F C, MISO (r) = M ( ρ F C, OSBC (r) = M M k= ) M M e M (2 r )/ρ (2 r ) M ρ k Γ(M k)(2 r ) k k= ) M M e M (2 r/m )/ρ (2 r/m ) M ρ k Γ(M k)(m ) k (2 r ) k k= ρ k Γ(M k)(m ) k (2 r/m ) k, r. 9/5

10 he MIMO Channel Capacity he LCs of the capacities can be obtained in closed forms as 2ρβ(2r ) N C, SIMO (r) = Γ(M )ρ M π (2r ) M e (2r )/ρ N C, MISO (r) = (M ) M /2 2ρβ(2 r ) Γ(M )ρ M (2 r ) M e M (2 r )/ρ π N C, OSBC (r) = (M ) M /2 2ρβ(2 r/m ) Γ(M )ρ M (2 r/m ) M e M (2 r/m )/ρ, r. π he ADFs of the capacities can be obtained in closed forms as C, SIMO (r) = F C, SIMO(r) N C, SIMO (r) C, MISO (r) = F C, MISO(r) N C, MISO (r) C, OSBC (r) = F C, MIMO(r) N C, MIMO (r), r. /5

11 he MIMO Channel Capacity Confirmation of the heory by Simulations Simulation model: For example, M SIMO channels. cos(2 f ( m) ( m) ), t, () (t) cos(2 f ( m) ( m) ),2 t,2 2 N + ( m ) ( t) (m) (t) (M ) (t) m M cos(2 f cos(2 f cos(2 f ( m) ( m), N t, N ) ( m) ( m) ) 2, t 2, ( m) ( m) ) 2,2 t 2,2 + 2 N 2 ( ) m 2 ( t) ( m) ( m) ( m) ( t) 2 ( t) j ( t) cos(2 f ( m) ( m) 2, N t 2, N ) 2 2 ( Parameters []: f (k) i,n = f max cos π 2Ni (n ) 2 ) + α (k) i, where α (k) i, = ( )(i ) π 4N i kk. [] M. Pätzold et al, wo new methods for the generation of multiple uncorrelated ayleigh fading waveforms, in Proc. 63th IEEE Semiannual Vehicular echnology Conference, VC 26-Spring Melbourne, Australia, May 26, vol. 6, pp /5

12 he MIMO Channel Capacity he PDFs of the SIMO/MISO Channel Capacities.8 heory Simulation ( 7) ( 9).8 heory Simulation (9 ) (7 ) p C, SIMO (r)/fmax ( ) ( 2) ( 3) ( 5) p C, MISO (r)/fmax (5 ) (3 ) (2 ) ( ) Level, r he PDF of the ( M ) SIMO channel capacity Level, r he PDF of the (M ) MISO channel capacity. 2/5

13 he MIMO Channel Capacity he LCs of the SIMO/MISO Channel Capacities N C, SIMO (r)/fmax heory Simulation ( ) ( 2) ( 3)( 5)( 7) ( 9) N C, MISO (r)/fmax ( ) (2 ) (3 ) (5 ) (7 ) (9 ) heory Simulation Level, r he normalized LC of the ( M ) SIMO channel capacity Level, r he normalized LC of the (M ) MISO channel capacity. 3/5

14 he MIMO Channel Capacity he ADFs of the SIMO/MISO Channel Capacities C, SIMO (r) fmax 3 heory Simulation 2 ( ) ( 2) ( 3) ( 5) ( 7) ( 9) C, MISO (r) fmax 3 heory Simulation 2 ( ) (2 ) (3 ) (5 ) (7 ) (9 ) Level, r he normalized ADF of the ( M ) SIMO channel capacity Level, r he normalized ADF of the (M ) MISO channel capacity. 4/5

15 he MIMO Channel Capacity Gaussian Approximations to the Exact LC Assumption: he capacity C OSBC (t) is a continuous time Gaussian process. LC: N C, OSBC (r) = r C () 2π e (r m C, OSBC) 2 /(2σ 2 C, OSBC),, r. High SN: After a simplification of one of the results obtain in [2], we have Low SN: r C, OSBC () = 2 (M M ) ψ(m M ) r h() m C, OSBC = M ρ log 2 e σ 2 = M ρ 2 log 2 2 e C, OSBC M r C, OSBC () = 4π 2 fmax 2 (isotropic scattering) Hence, N C, OSBC (r) = 2f max e M (r M ρ log 2 e) 2 /(2M ρ 2 log 2 2 e), r. [2] A. Giorgetti et al, MIMO capacity, level crossing rates and fades: he impact of spatial/temporal channel correlation, Journal of Communications and Networks, vol. 2, pp , Mar /5

16 he MIMO Channel Capacity Exact LC: Confirmation of the heory by Simulations N C, OSBC (r) = (M ) M /2 2ρβ(2 r/m ) Γ(M )ρ M (2 r/m ) M e M (2 r/m )/ρ, r. π Approximated LC: N C, OSBC (r) = 2f max e M (r M ρ log 2 e) 2 /(2M ρ 2 log 2 2 e), r..2 ( ) (2 2) (4 4) heory Approximation Simulation N C, OSBC (r)/fmax Level, r he normalized LC N C, OSBC (r)/f max of the (M M ) OSBC-MIMO channel capacity(ρ = 3dB). 6/5

17 he One-ing MIMO Channel Model he Geometrical One-ing Scattering model δ A () α φ (n) θ max v A () x d M n d n ξ (n) d nm S (n) d n A (M ) y φ (n) β α δ A (M ) D he local scatterers are laying on a ring around the receiver. If N, then the discrete AOA φ (n) tend to continuous Vs φ with given distribution p(φ ). Assumption: D max{δ, δ }. 7/5

18 he One-ing MIMO Channel Model he eference Model Channel gains: h pq (t) = h DIF pq (t) + hlos pq (t) where h DIF pq (t) = lim N (Kpq + )N a n,q = e jπ(m 2q+) δ λ [ N n= cos(α )+φ max b n,p = e jπ(m 2p+) δ λ cos(φ (n) α ) f (n) = f max cos(φ (n) β ) a n,q b n,p e j(2πf(n) t+θ n) sin(α ) sin(φ (n) ) ] h LOS pq (t) = K pq K pq + c q d p e j(2πf t+θ ) c q = e jπ(m 2q+) δ λ cos(α ) d p = e jπ(m 2p+) δ λ cos(α ) f = f max cos(π β ) θ = 2π λ D. Central limit theorem: h DIF pq(t) CN(, ) as N. 8/5

19 he One-ing MIMO Channel Model Space-time CCF: Statistical Properties of the eference Model ρ,22 (δ, δ, τ) := E θn,φ (n) = π {h (t)h 22(t + τ)} a 2 (δ, φ ) b 2 (δ, φ ) e j2πf(φ )τ p(φ ) dφ where π a(δ, φ ) = e jπδ λ [cos(α )+φ max sin(α ) sin(φ )] emporal ACF: b(δ, φ ) = e jπδ λ cos(φ α ) f(φ ) = f max cos(φ β ). r hpq (τ) := ρ,22 (,, τ) = E{h pq (t)h pq (t + τ)} = π π e j2πf max cos(φ β )τ p(φ ) dφ. 2D space CCF: ρ(δ, δ ) := ρ,22 (δ, δ, ). 9/5

20 he One-ing MIMO Channel Model he Stochastic Simulation Model: he Simulation Model From the reference model, a stochastic simulation model is obtained by: using finite values for the number N of scatterers, considering the AOA φ (n) as constants. Channel gains: ĥ pq (t) = N N n= a n,q b n,p e j(2πf(n) t+θ n) he phases θ n are i.i.d Vs Stochastic process Channel capacity: Ĉ(t) := log 2 [det ( I M + P )],total M P Ĥ(t)ĤH (t) N [bits/s/hz] where Ĥ(t) = [ĥpq(t)]. he capacity Ĉ(t) is a stochastic process. 2/5

21 he One-ing MIMO Channel Model Statistical Properties of the Stochastic Simulation Model Space-time CCF: } ˆρ,22 (δ, δ, τ) := E θn {ĥ (t)ĥ 22(t + τ) emporal ACF: 2D space CCF: = N ˆr hpq (τ) := ˆρ,22 (,, τ) N n= = E θn {ĥpq(t)ĥ pq(t + τ)} = N N n= a 2 n, b2 n, e j2πf(n) τ e j2πf max cos(φ(n) β )τ ˆρ(δ, δ ) := ˆρ,22 (δ, δ, ) = N a 2 n, N (δ ) b 2 n, (δ ) n= 2/5

22 he One-ing MIMO Channel Model he Deterministic Simulation Model: Channel gains: h pq (t) = N N n= a n,q b n,p e j(2πf(n) t+θ n) he phases θ n are constant quantities Deterministic process Channel capacity: C(t) := log 2 [det ( I M + P )],total H(t) M P HH (t) N [bits/s/hz] where H(t) = [ h pq (t)]. he capacity C(t) is a deterministic process. he analysis of C(t) has to be performed by using time averages, e.g., < C(t) >= lim 2 C(t) dt. 22/5

23 he One-ing MIMO Channel Model Statistical Properties of the Deterministic Simulation Model Space-time CCF: emporal ACF: 2D space CCF: ρ,22 (δ, δ, τ) := < h (t) h 22(t + τ) > = N a 2 n, b 2 n, e j2πf(n) τ N r hpq (τ) := ρ,22 (,, τ) n= = ˆρ,22 (δ, δ, τ) = < h pq (t) h pq (t + τ) > = N N n= = ˆr hpq (τ) e j2πf max cos(φ(n) β )τ ρ(δ, δ ) := ρ,22 (δ, δ, ) = N a 2 N n,(δ ) b 2 n,(δ ) n= = ˆρ(δ, δ ) 23/5

24 he One-ing MIMO Channel Model Parameter Computation Methods Parameters: he model parameters to be determined are the discrete AOAs φ (n) Problem: Determine the model parameters φ (n) or, alternatively, Solutions: such that ρ,22 (δ, δ, τ) ρ,22 (δ, δ, τ) r hpq (τ) r hpq (τ) and ρ(δ, δ ) ρ(δ, δ ). Generalized method of exact Doppler spread (MEDS). L p -norm method (LPNM). (n =,..., N). Performance measure: Absolute errors r hpq (τ) r hpq (τ) and ρ(δ, δ ) ρ(δ, δ ). 24/5

25 he One-ing MIMO Channel Model Generalized Method of Exact Doppler Spread (MEDS) Generalized MEDS: Closed-form solution where q {, 2, 3, 4} and φ φ (n) = qπ 2N ( n ) 2 + φ, n =,..., N is called the angle of rotation. In this Ph.D. project, the following closed-form solution has been used φ (n) = 2π ( n ) + π, n =,...,N. N 2 2N It can be shown that Advantages: r hpq (τ) r hpq (τ) if N ρ(δ, δ ) ρ(δ, δ ) if N. Simple and closed-form solution. Very high performance. Disadvantage: Only valid for isotropic scattering. 25/5

26 he One-ing MIMO Channel Model L p -norm method (LPNM) LPNM: he AOAs φ (n) have to be determined by minimizing the following two error norms: τ max /2 E := r hpq (τ) r hpq (τ) 2 dτ E 2 := τ max δ maxδ max δ max δ max ρ(δ, δ ) ρ(δ, δ ) 2 dδ dδ An optimization of E and E 2 can be carried out by using the Fletcher-Powell algorithm. /2 Advantages: General solution (isotropic and non-isotropic scattering). Very high performance. Disadvantage: No closed-form solution. High complexity. 26/5

27 he One-ing MIMO Channel Model Performance Evaluation of the LPNM and Generalized MEDS emporal autocorrelation function.5 eference model Simulation model (generalized MEDS, q = 4, N = 25) Simulation model (LPNM, N = 25) N 4 Normalized time lag τ f max he temporal ACFs r hpq (τ) (reference model) and r hpq (τ) (simulation model) for isotropic scattering environments. 27/5

28 he One-ing MIMO Channel Model eference model δ /λ δ /λ he 2D space CCF ρ(δ, δ ) of the reference model for isotropic scattering environments. 2D space cross-correlation function 2D space cross-correlation function Performance Evaluation of the Generalized MEDS Simulation model δ /λ 3 δ /λ he 2D space CCF ρ (δ, δ ) of the simulation model for isotropic scattering environments (generalized MEDS q = 4, N = 25). 28/5

29 he One-ing MIMO Channel Model A New ight Upper Bound on the 2 2 MIMO Channel Capacity By using Jensen s inequality and the concavity of log 2 function, we obtain E{C(t)} C up = log 2 [E { = log 2 { ( det I 2 + P BS,total 2P N ) 2 N + 2 P (,total Ptotal + P N 2NP N )}] HH H (t) ( N m= n= 2 a2 n, a 2 m, a2 m, a 2 n, ) ( b2 n, b 2 m, )}. Mean capacity (bits/s/hz) New upper bound C up Simulation SN, (db) 29/5

30 he wo-ing MIMO Channel Model he Geometrical wo-ing Scattering Model y S (n) v v β δ A () α d m φ (m) S (m) d mn d nm d n φ (n) β A () α δ x d M m A (M ) A (M ) D he local scatterers are laying on rings around the transmitter and the receiver. If M and N, then the discrete AOD φ (m) and AOA φ (n) and φ with given distribution p(φ ) and p(φ ), respectively. Assumptions: max{, } D and max{δ, δ } min{r, }. tend to continuous Vs φ 3/5

31 he wo-ing MIMO Channel Model he eference Model Channel gains: where h pq (t) = lim M N M MN N m= n= g pqmn e j[2π(f(m) +f (n) )t+θ mn] g pqmn = a m,q b n,p c m,n a m,q = e jπ(m 2q+) δ λ cos(φ (m) α ) b n,p = e jπ(m 2p+) δ λ cos(φ (n) α ) c m,n = e j2π λ ( cosφ (m) cosφ (n) ) f (m) f (n) = f max cos(φ (m) β ) = f max cos(φ (n) β ). Central limit theorem: h pq (t) CN(, ) as M and N. 3/5

32 he wo-ing MIMO Channel Model Space-time CCF: Statistical Properties of the eference Model ρ,22 (δ, δ, τ) := E θmn,φ (m) {h (t)h 22(t + τ)},φ(n) = ρ (δ, τ) ρ (δ, τ) where ρ (δ, τ) = π a 2 (δ, φ )e j2πf (φ )τ p φ (φ ) dφ (transmit CF) ρ (δ, τ) = π π b 2 (δ, φ )e j2πf (φ )τ p φ (φ ) dφ (receive CF) with π a(δ, φ ) = e jπ(δ /λ) cos(φ α ) b(δ, φ ) = e jπ(δ /λ) cos(φ α ) f (φ ) = f max cos(φ β ) f (φ ) = f max cos(φ β ). emark: he space-time CCF ρ,22 (δ, δ, τ) can be expressed as the product of the CF ρ (δ, τ) and the receive CF ρ (δ, τ). 32/5

33 he wo-ing MIMO Channel Model emporal ACF: r hpq (τ) := ρ,22 (,, τ) = E θmn,φ (m) {h pq (t)h pq(t + τ)},φ(n) = ρ (, τ) ρ (, τ) where ρ (, τ) = ρ (, τ) = π π π π e j2πf (φ )τ p φ (φ ) dφ e j2πf (φ )τ p φ (φ ) dφ emark: he temporal ACF r hpq (τ) is independent of p and q. 2D space CCF: ρ(δ, δ ) := ρ,22 (δ, δ, ) = ρ (δ, ) ρ (δ, ) 33/5

34 he wo-ing MIMO Channel Model he Simulation Model he Stochastic Simulation Model: From the reference model, a stochastic simulation model is obtained by: using finite values for the numbers of scatterers (M, N), considering the AOD φ (m) and AOA φ (n) as constants. Channel gains: ĥ pq (t) = M MN N m= n= g pqmn e j[(2π(f(m) +f (n) )t+θ mn)] he phases θ mn are i.i.d Vs Stochastic process Channel capacity: Ĉ(t) := log 2 [det ( I M + P )],total M P Ĥ(t)ĤH (t) N [bits/s/hz] where Ĥ(t) = [ĥpq(t)]. he capacity Ĉ(t) is a stochastic process. 34/5

35 he wo-ing MIMO Channel Model Statistical Properties of the Stochastic Simulation Model Space-time CCF: } ˆρ,22 (δ, δ, τ) := E θmn {ĥ (t)ĥ 22(t + τ) where emporal ACF: 2D space CCF: ˆρ (δ, τ) = M ˆρ (δ, τ) = N = ˆρ (δ, τ) ˆρ (δ, τ) M m= N n= a 2 m,(δ )e j2πf(m) τ b 2 n,(δ )e j2πf(n) τ. ˆr hpq (τ) := E θn {ĥpq(t)ĥ pq(t + τ)} = N N n= e j2πf max cos(φ(n) β )τ ˆρ(δ, δ ) := ˆρ,22 (δ, δ, ) = MN M m= n= 35/5 N a 2 m,(δ )b 2 n,(δ )

36 he wo-ing MIMO Channel Model he Deterministic Simulation Model: Channel gains: h pq (t) = M MN N m= n= g pqmn e j[(2π(f(m) +f (n) )t+θ mn)] he phases θ mn are constant quantities Deterministic process Channel capacity: C(t) := log 2 [det ( I M + P )],total H(t) M P HH (t) N [bits/s/hz] where H(t) = [ h pq (t)]. he capacity C(t) is a deterministic process. he analysis of C(t) has to be performed by using time averages, e.g., < C(t) >= lim 2 C(t) dt. 36/5

37 he wo-ing MIMO Channel Model Statistical Properties of the Deterministic Simulation Model Space-time CCF: ρ,22 (δ, δ, τ) := < h (t) h 22(t + τ) > M N = a 2 MN m,(δ )b 2 n,(δ )e j2π(f(m) +f (n) )τ m= n= = ˆρ,22 (δ, δ, τ) he simulation model is ergodic w.r.t. the space-time CCF. emporal ACF: r hpq (τ) = ˆr hpq (τ) 2D space CCF: ρ(δ, δ ) = ˆρ(δ, δ ) 37/5

38 he wo-ing MIMO Channel Model Parameters: he model parameters to be determined are the discrete AODs φ (m) and the discrete AOAs φ (n) (n =,...,N). (m =,...,M) Problem: Determine the model parameters φ (m) or, alternatively, and φ (n) such that ρ,22 (δ, δ, τ) ρ,22 (δ, δ, τ) ρ (δ, τ) ρ (δ, τ) and ρ (δ, τ) ρ (δ, τ). Solutions: Generalized method of exact Doppler spread (MEDS). L p -norm method (LPNM). Performance measure: Absolute errors ρ (δ, τ) ρ (δ, τ) and ρ (δ, τ) ρ (δ, τ). 38/5

39 he wo-ing MIMO Channel Model Generalized MEDS: Closed-form solution φ (m) = qπ ( m ) + φ 2M 2, m =,..., M φ (n) = qπ ( n ) + φ 2N 2, n =,..., N where q {, 2, 3, 4}, φ and φ are called the angles of rotation. In this Ph.D. project, the following closed-form solution has been used φ (m) = 2π ( m ) + π M 2 2M, m =,..., M φ (n) = 2π ( n ) + π N 2 2N, n =,..., N. It can be shown that ρ (δ, τ) ρ (δ, τ) if M ρ (δ, τ) ρ (δ, τ) if N. Advantages: Simple and closed-form solution, very high performance. Disadvantage: Only valid for isotropic scattering. 39/5

40 he wo-ing MIMO Channel Model eference model Simulation model δ /λ τ fmax he transmit CF ρ (δ, τ ) of the 2 2 MIMO mobile-to-mobile reference channel model for isotropic scattering environments. ransmit correlation function, ρ (δ, τ ) ransmit correlation function, ρ (δ, τ ) Performance Evaluation of the Generalized MEDS δ /λ τ fmax he transmit CF ρ (δ, τ ) of the 2 2 MIMO mobile-to-mobile channel simulator designed by applying the generalized MEDS (q = 4, M = 4). 4/5

41 he wo-ing MIMO Channel Model LPNM: he AODs φ (m) norms: E (p) := E (p) 2 := AOAs φ (n) δ,max τ,max δ,max τ,max L p -norm method (LPNM) have to be determined by minimizing the following two error δ,max δ,max τ,max τ,max ρ (δ, τ) ρ (δ, τ) p dδ dτ ρ (δ, τ) ρ (δ, τ) p dδ dτ /p /p Note that the error norms E (p) and E (p) 2 can be minimized independently. Advantages: General solution (isotropic and non-isotropic scattering). Very high performance. Disadvantage: No closed-form solution. High complexity. 4/5

42 he wo-ing MIMO Channel Model Performance Evaluation of the LPNM eference model Simulation model ρ (δ, τ ) ρ (δ, τ ) δ /λ τ fmax Absolute value of the transmit CF ρ (δ, τ ) of the 2 2 MIMO mobile-to-mobile reference channel model under non-isotropic scattering conditions δ /λ τ fmax Absolute value of the transmit CF ρ (δ, τ ) of the 2 2 MIMO mobile-to-mobile channel simulator designed by applying the LPNM (M = 5, p = ). 42/5

43 he wo-ing MIMO Channel Model he stochastic simulation model: he MIMO Channel Capacity Ĉ(t) := log 2 [det ( I M + P )],total M P Ĥ(t)ĤH (t) N where Ĥ(t) = [ĥpq(t)]. Statistical average: Ĉs := E θmn {Ĉ(t)}. he deterministic simulation model: C(t) := log 2 [det where H(t) = [ h pq (t)]. ime Average: C t := lim Mean capacities Ĉs and Ct (bits/s/hz) Ĉ s (statistical average) C t (time average) ( I M + P )],total H(t) M P HH (t) N 2 C(t) dt. M = M = 6 M = M = 4 M = M = SN (db) 43/5

44 he Elliptical MIMO Channel Model he Geometric Elliptical Scattering Model S (n) d n d n δ A () D (n) d M n d nm D (n) δ A () v BS α φ (n) φ (n) MS α β 2b A (M ) 2f A (M ) 2a he transmitter and the receiver are located at the focal points of an ellipse. he local scatterers S (n) are laying on the ellipse around the transmitter and the receiver. he AOD φ (n) is determined by the φ (n) and the location of S(n). 44/5

45 he Elliptical MIMO Channel Model he eference Model Channel gains: where h pq (t) = lim N N N n= a n,q b n,p e j(2πf(n) t+θ n) a n,q = e jπ(m 2q+) δ λ cos(φ (n) α ) b n,p = e jπ(m 2p+) δ λ cos(φ (n) α ) f (n) = f max cos(φ (n) β ). Central limit theorem: h pq (t) CN(, ) as N. 45/5

46 Summary Contributions A detailed study of the stationary and ergodic properties of sum-of-sinusoids-based ayleigh fading channel simulators is presented. Exact closed-form expressions for PDF, CDF, LC, and ADF of the capacity of OSBC-MIMO systems are derived. A new tight upper bound on the MIMO channel capacity is derived. An efficient simulation model, by using the geometrical one-ring scateering model, for MIMO frequency nonselective and frequency selective ayleigh mobile fading channels is proposed. A MIMO channel model for mobile-to-mobile communications is proposed. A reference model for a wideband MIMO channel model is presented. he reference model is based on the geometric elliptical scattering model. From the reference model, an efficient simulation model has been obtained. he channel simulator enables the performance evaluation of MIMO-OFDM systems. he MIMO channel capacity has been studied under various propagation conditions imposed by the geometry of the one-ring, two-ring, and elliptical scattering models. 46/5

47 Summary Future Works Change the proposed reference models with channel measurements. Develop mobile-to-mobile channel models that are frequency-selective. Exact-closed form expressions for the PDF, CDF, LC, and ADF of the SIMO/MISO channel capacity when the sub-channels are correlated. If possible, find exact closed-form expressions for the PDF, CDF, LC, and ADF of the general MIMO channel capacity for both uncorrelated and correlated sub-channels. 47/5

48 Summary Publications During Ph.D. Studies [] M. Pätzold, B. O. Hogstad, and N. Youssef, Modeling, analysis, and simulation of MIMO mobile-to-mobile fading channels, IEEE rans. Wireless Commun., accepted for publication, 27. [2] M. Pätzold, B. O. Hogstad, and D. Kim, A New Design Concept for High-Performance Fading Channel Simulators Using Set Partitioning, Wireless Personal Communications, vol. 4, no. 2, pp , Feb. 27. [3] M. Pätzold and B. O. Hogstad, Classes of Sum-of-Sinusoids ayleigh Fading Channel Simulators and heir Stationary and Ergodic Properties Part I, WSEAS ransactions on Mathematics, Issue 2, Volume 5, February 26, pp [4] M. Pätzold and B. O. Hogstad, Classes of Sum-of-Sinusoids ayleigh Fading Channel Simulators and heir Stationary and Ergodic Properties Part II, WSEAS ransactions on Mathematics, Issue 4, Volume 4, October 25, pp [5] C. E. D. Sterian, H. Singh, M. Pätzold, B. O. Hogstad, Super-Orthogonal Space-ime Codes with ectangular Constellations and wo ransmit Antennas for High Data ate Wireless Communications, IEEE rans. Wireless Commun., vol. 5, no. 7, Jul. 26, pp [6] M. Pätzold and B. O. Hogstad, A Space-ime Channel Simulator for MIMO Channels Based on the Geometrical One-ing Scattering Model, Wireless Communications and Mobile Computing, Special Issue on Multiple-Input Multiple-Output (MIMO) Communications, vol. 4, no. 7, Nov. 24, pp [7] B. O. Hogstad and M. Pätzold, On the Stationarity of Sum-of Cisoids-Based Mobile Fading Channel Simulators, Proc. 67th IEEE Vehicular echnology Conference, VC28-Spring, Singapore, May. 28, accepted for publication. [8] B. O. Hogstad and M. Pätzold, Exact Closed-Form Expressions for the Distribution, Level-Crossing ate, and Average Duration of Fades of the Capacity of MIMO Channels, Proc. 65th Semiannual Vehicular Vehicular echnology Conference, VC 27-Spring, Dublin, Ireland, Apr. 27, pp [9] M. Pätzold and B. O. Hogstad, A Wideband Space-ime MIMO Channel Simulator Based on the Geometrical One-ing Model, Proc. 64th IEEE Semiannual Vehicular echnology Conference, IEEE VC 26-Fall, Montreal, Canada, Sept. 26. [] M. Pätzold and B. O. Hogstad, A Wideband MIMO Channel Model Derived From the Geometric Elliptical Scattering Model, Proc. 3rd International Symposium on Wireless Communication System, ISWCS 6, Valencia, Spain, Sept. 26, pp [] B. O. Hogstad, M. Pätzold, and A. Chopra, A Study on the Capacity of Narrow- and Wideband MIMO Channel Models, Proc. 5th IS Mobile & Communications Summit, IS 26, Myconos, Greece, June 26. [2] M. Pätzold and B. O. Hogstad, wo New Methods for the Generation of Multiple Uncorrelated ayleigh Fading Waveforms, Proc. 63rd Semiannual Vehicular echnology Conference, IEEE VC 26-Spring, Melbourne, Australia, May 26, vol. 6, pp /5

49 Summary [3] M. Pätzold and B. O. Hogstad, Classes of Sum-of-Sinusoids ayleigh Fading Channel Simulators and heir Stationary and Ergodic Properties, Proc. of the 4th WSEAS International Conference on Information Security, Communications and Computers, enerife, Spain, December 25, pp [4] B. O. Hogstad, M. Pätzold, A. Chopra, D. Kim, and K. B. Yeom, A Wideband MIMO Channel Simulation Model Based On the Geometrical Elliptical Scattering Model, Proc. 5th Wireless World esearch Forum Meeting, WWF5, December 25, Paris, France. [5] M. Pätzold, B. O. Hogstad, D. Kim, and S. Kim, A New Design Concept for High-Performance Fading Channel Simulators Using Set Partitioning, Proc. 8th International Symposium on Wireless Personal Multimedia Communications, WPMC 25, Aalborg, Denmark, September 25, pp [6] B. O. Hogstad and M. Pätzold, A Study of the MIMO Channel Capacity When Using the Geometrical wo-ing Scattering Model, Proc. 8th International Symposium on Wireless Personal Multimedia Communications, WPMC 25, Aalborg, Denmark, September 25, pp [7] M. Pätzold, B. O. Hogstad, N. Youssef, and D. Kim, A MIMO Mobile-to-Mobile Channel Model: Part I - he eference Model, Proc. 6th IEEE International Symposium on Personal, Indoor and Mobile adio Communications, PIMC 25, Berlin, Germany, September 25, vol., pp [8] B. O. Hogstad, M. Pätzold, N. Youssef, and D. Kim, A MIMO Mobile-to-Mobile Channel Model: Part II - he Simulation Model, Proc. 6th IEEE International Symposium on Personal, Indoor and Mobile adio Communications, PIMC 25, Berlin, Germany, September 25, vol., pp [9] M. Pätzold and B. O. Hogstad, Design and Performance of MIMO Channel Simulators Derived From the wo-ing Scattering Model, Proc. 4th IS Mobile & Communications Summit, IS 25, Dresden, Germany, June 25, paper no. 2. [2] M. Pätzold and B. O. Hogstad, A Space-ime Channel Simulator for MIMO Channels Based on the Geometrical One-ing Scattering Model, Proc. 6th IEEE Semiannual Vehicular echnology Conference, IEEE VC 24-Fall, Los Angeles, CA, USA, Sept. 24. [2] B. O. Hogstad and M. Pätzold, Capacity Studies of MIMO Channel Models Based on the Geometrical One-ing Scattering Model, Proc. 5th IEEE International Symposium on Personal, Indoor and Mobile adio Communications, IEEE PIMC 24, Barcelona, Spain, Sept. 24, vol. 3, pp [22] B. O. Hogstad and M. Pätzold, New ight Upper Bounds on the MIMO Channel Capacity, Proc. Nordic adio Symposium (NS) 24, including the Finnish Wireless Communications Workshop (FWCW) 24, Oulu, Finland, August 24. [23] M. Pätzold and B. O. Hogstad, A General Concept for the Design of MIMO Channel Simulators, Proc. Nordic adio Symposium (NS) 24, including the Finnish Wireless Communications Workshop (FWCW) 24, Oulu, Finland, August /5

50 Summary Acknowledgements Ph.D. supervisors: Main supervisor: Prof. Matthias Pätzold from Agder University, Norway. Second supervisor: Prof. Bernard H. Fleury from Aalborg University, Denmark. Co-operators: he Communications Section, Electrical Engineering Department, CINVESAV-IPN, Mexico. Prof. Neji Youssef from Ecole Superieure des Communications de unis, unis. Prof. Corneliu E. D. Sterian from Politechnica University of Bucharest, omania. Colleagues at University of Agder and Aalborg University. Ph.D. evaluation committee members (Prof. Per Høeg, Prof. Valeri Kontorovitch, and Prof. Serguei Primak). My family and Ana. 5/5