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

SLIDE 1

Ph.D. Defense, Aalborg University, 23 January 2008 Multiple-Input Multiple-Output Fading Channel Models and Their Capacity

Ph.D. student: Bjørn Olav Hogstad1,2 Main supervisor:

• Prof. Matthias P¨

atzold1 Second supervisor: Prof. Bernard H. Fleury2

1University of Agder, Grimstad, Norway 2Aalborg University, Aalborg, Denmark 1/50

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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

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Introduction 2 × 2 MIMO System:

h11(t)

Transmitter (Base station) (Mobile station)

h21(t) h12(t) s1(t) r1(t) r2(t)

Receiver

h22(t)

2×2 MIMO channel

s2(t)

• 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

SLIDE 4

• 1. Introduction

Typical Behaviour of the Channel Capacity

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 2 4 6 8 10 12

Time, t (s) Channel capacity, (bits/s/Hz) Channel capacity

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

• f 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

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Sum-of-Sinusoids Channel Simulators

The Reference Model

Rayleigh process: ζ(t) = |µ1(t) + jµ2(t)| where µi(t) ∼ N(0, σ2

0) (i = 1, 2).

Temporal ACF (isotropic scattering): rµiµi(τ) := E{µi(t)µi(t + τ)} = σ2

0J0(2πfmaxτ) .

Rice’s sum-of-sinusoids: µi(t) = lim

Ni→∞ Ni

• n=1

ci,n cos(2πfi,nt + θi,n) where ci,n = 2

• ∆fi Sµiµi(fi,n)

fi,n = n∆fi . The quantity ∆fi is the width of the frequency band associated with the nth component. The symbol Sµiµi(f) denotes the Doppler power spectral density.

Ph.D. Defense, Aalborg University, 23 January 2008 5/50

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Sum-of-Sinusoids Channel Simulators

The Simulation Model

ˆ µi(t) =

Ni

• n=1

ci,n cos(2πfi,nt + θi,n) Classes of sum-of-sinusoids channel simulators and their statistical properties Class Gains Frequencies Phases First-order Wide-sense Mean- Autocor.- ci,n fi,n θi,n stationary stationary ergodic ergodic I const. const. const. – – – – II const. const. RV yes yes yes yes III const. RV const. yes a yes a yes a no IV const. RV RV yes yes yes no V RV const. const. no no yes a no VI RV const. RV yes yes yes no VII RV RV const. yes a yes a yes a 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

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Generalized Concept of Deterministic Channel Modeling

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

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The MIMO Channel Capacity MIMO channel capacity: C(t) := log2

• det
• IMR + ρ

MT H(t)HH(t)

• [bits/s/Hz]

where H(t) = [hpq(t)]MR,MT

p,q=1

is the channel matrix. SIMO channel capacity: CSIMO(t) := log2(1 + ρhH(t)h(t)) [bits/s/Hz] where h(t) = [h1(t), . . . , hMR(t)]T is the MR × 1 complex channel gain vector. MISO channel capacity: CMISO(t) := log2

• 1 + ρ

MT hH(t)h(t)

• [bits/s/Hz]

where h(t) = [h1(t), . . . , hMT(t)]T is the MT × 1 complex channel gain vector. Capacity of OSTBC-MIMO systems: COSTBC(t) = log2

• 1 + ρ

MT hH(t)h(t)

• [bits/s/Hz]

where h(t) = [h1(t), . . . , hMT MR(t)]T is the MTMR × 1 complex channel gain vector.

Ph.D. Defense, Aalborg University, 23 January 2008 8/50

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The MIMO Channel Capacity The PDFs of the capacities can be expressed in closed forms as pC, SIMO(r) = ln 2 Γ(MR)ρMR2r(2r − 1)MR−1e −(2r−1)/ρ pC, MISO(r) = (MT)MT ln 2 Γ(MT)ρMT 2r · (2r − 1)MT −1e −MT (2r−1)/ρ pC, OSTBC(r) = ln 2(MT)MR2r/MT(2r/MT − 1)MR−1 Γ(MR)MTρMR e −MT (2r/MT −1)/ρ, r ≥ 0. The CDFs of the capacities can be expressed in closed forms as FC, SIMO(r) = 1 − ρ1−MRe −(2r−1)/ρ(2r − 1)MR−1

MR−1

• k=0

ρk Γ(MR − k)(2r − 1)k FC, MISO(r) = 1 − ρ MT 1−MT e −MT (2r−1)/ρ(2r − 1)MT −1

MT −1

• k=0

ρk Γ(MT − k)(MT)k(2r − 1)k FC, OSTBC(r) = 1 − ρ MT 1−MR e −MT (2r/MT −1)/ρ(2r/MT − 1)MR−1

MR−1

• k=0

ρk Γ(MR − k)(MT)k(2r/MT − 1)k, r ≥ 0.

Ph.D. Defense, Aalborg University, 23 January 2008 9/50

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The MIMO Channel Capacity The LCRs of the capacities can be obtained in closed forms as NC, SIMO(r) =

• 2ρβ(2r − 1)

Γ(MR)ρMR√π(2r − 1)MR−1e −(2r−1)/ρ NC, MISO(r) = (MT)MT −1/2 2ρβ(2r − 1) Γ(MT)ρMT√π (2r − 1)MT −1e−MT (2r−1)/ρ NC, OSTBC(r) = (MT)MR−1/2 2ρβ(2r/MT − 1) Γ(MR)ρMR√π (2r/MT − 1)MR−1e −MT (2r/MT −1)/ρ, r ≥ 0. The ADFs of the capacities can be obtained in closed forms as TC, SIMO(r) = FC, SIMO(r) NC, SIMO(r) TC, MISO(r) = FC, MISO(r) NC, MISO(r) TC, OSTBC(r) = FC, MIMO(r) NC, MIMO(r), r ≥ 0.

Ph.D. Defense, Aalborg University, 23 January 2008 10/50

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The MIMO Channel Capacity

Confirmation of the Theory by Simulations

Simulation model: For example, 1 × MR SIMO channels.

(1)(t)

• (m)(t)
• R

(M )(t)

• 1

m MR

( ) ( ) 1,2 1,2 )

cos(2

• m

m

t

f

• 1

1

( ) ( ) 1, 1, )

cos(2

• m

m N N

t

f

• +

( ) 1 ( ) m t

• 1

2 N

• (

) 2 ( ) m t

• (

) ( ) ( ) 1 2

( ) ( ) |

( ) |

• m

m m

t j t

t

• +

2

2 N ( ) ( ) 2,2 2,2 )

cos(2

• m

m

t

f

• (

) ( ) 2,1 2,1 )

cos(2

• m

m

t

f

• 2

2

( ) ( ) 2, 2,

)

cos(2

• m

m N N

t

f

• (

) ( ) 1,1 1,1 )

cos(2

• m

m

t

f

• Parameters [1]: f(k)

i,n = fmax cos

π 2Ni

• n − 1

2

• + α(k)

i,0

• where α(k)

i,0 = (−1)(i−1) π

4Ni · k K.

[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

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The MIMO Channel Capacity

The PDFs of the SIMO/MISO Channel Capacities

2 4 6 8 10 0.2 0.4 0.6 0.8 1 Theory Simulation

(1 × 1) (1 × 2) (1 × 3) (1 × 5) (1 × 7) (1 × 9)

pC, SIMO(r)/fmax Level, r

The PDF of the (1 × MR) SIMO channel capacity.

2 4 6 8 10 0.2 0.4 0.6 0.8 1 Theory Simulation

(1 × 1) (2 × 1) (3 × 1) (5 × 1) (7 × 1) (9 × 1)

pC, MISO(r)/fmax Level, r

The PDF of the (MT × 1) MISO channel capacity.

Ph.D. Defense, Aalborg University, 23 January 2008 12/50

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The MIMO Channel Capacity

The LCRs of the SIMO/MISO Channel Capacities

2 4 6 8 10 0.2 0.4 0.6 0.8 1 1.2 Theory Simulation

(1 × 1) (1 × 2) (1 × 3)(1 × 5) (1 × 7) (1 × 9)

NC, SIMO(r)/fmax Level, r

The normalized LCR of the (1 × MR) SIMO channel capacity.

2 4 6 8 10 0.2 0.4 0.6 0.8 1 1.2 Theory Simulation

(1 × 1) (2 × 1) (3 × 1) (5 × 1) (7 × 1) (9 × 1)

NC, MISO(r)/fmax Level, r

The normalized LCR of the (MT × 1) MISO channel capacity.

Ph.D. Defense, Aalborg University, 23 January 2008 13/50

SLIDE 14

The MIMO Channel Capacity

The ADFs of the SIMO/MISO Channel Capacities

2 4 6 8 10 10

−2

10

−1

10 10

1

10

2

10

3

Theory Simulation

(1 × 1) (1 × 2) (1 × 3) (1 × 5) (1 × 7) (1 × 9)

TC, SIMO(r) · fmax Level, r

The normalized ADF of the (1 × MR) SIMO channel capacity.

2 4 6 8 10 10

−2

10

−1

10 10

1

10

2

10

3

Theory Simulation

(1 × 1) (3 × 1) (2 × 1) (5 × 1) (7 × 1) (9 × 1)

TC, MISO(r) · fmax Level, r

The normalized ADF of the (MT × 1) MISO channel capacity.

Ph.D. Defense, Aalborg University, 23 January 2008 14/50

SLIDE 15

The MIMO Channel Capacity

Gaussian Approximations to the Exact LCR

Assumption: The capacity COSTBC(t) is a continuous time Gaussian process. LCR: NC, OSTBC(r) =

• −¨

˜ rC(0) 2π e −(r−mC, OSTBC)2/(2σ2

C, OSTBC), ,

r ≥ 0. High SNR: After a simplification of one of the results obtain in [2], we have ¨ ˜ rC, OSTBC(0) = 2 (MTMR − 1) ˙ ψ(MTMR) ¨ rh(0) Low SNR: mC, OSTBC = MRρ log2 e σ2

C, OSTBC = MRρ2 log2

2 e

MT ¨ ˜ rC, OSTBC(0) = −4π2f2

max

(isotropic scattering) Hence, NC, OSTBC(r) = 2fmaxe −MT (r−MRρ log2 e)2/(2MRρ2 log2

2 e),

r ≥ 0.

[2] A. Giorgetti et al, “MIMO capacity, level crossing rates and fades: The impact of spatial/temporal channel corre- lation,” Journal of Communications and Networks, vol. 2, pp.789–793, Mar. 2002.

Ph.D. Defense, Aalborg University, 23 January 2008 15/50

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The MIMO Channel Capacity

Confirmation of the Theory by Simulations

Exact LCR: NC, OSTBC(r) = (MT)MR−1/2 2ρβ(2r/MT − 1) Γ(MR)ρMR√π (2r/MT − 1)MR−1e −MT (2r/MT −1)/ρ, r ≥ 0. Approximated LCR: NC, OSTBC(r) = 2fmaxe −MT (r−MRρ log2 e)2/(2MRρ2 log2

2 e),

r ≥ 0.

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.2 0.4 0.6 0.8 1 1.2

Theory Approximation Simulation

(1 × 1) (2 × 2) (4 × 4)

NC, OSTBC(r)/fmax Level, r

The normalized LCR NC, OSTBC(r)/fmax of the (MT × MR) OSTBC-MIMO channel capacity(ρ = −30dB).

Ph.D. Defense, Aalborg University, 23 January 2008 16/50

SLIDE 17

The One-Ring MIMO Channel Model

The Geometrical One-Ring Scattering model

A(1)

R

D v A(1)

T

0T x

αT δR

A(MT )

T θmax

T

y RR

βR

A(MR)

R αR δT d1n dMT n dn1 ξ(n) S(n)

0R

dnMR φ(n)

T

φ(n)

R

• The local scatterers are laying on a ring around the receiver.
• If N → ∞, then the discrete AOA φ(n)

R tend to continuous RVs φR with given distribution p(φR).

• Assumption: D ≫ RR ≫ max{δT, δR} .

Ph.D. Defense, Aalborg University, 23 January 2008 17/50

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The One-Ring MIMO Channel Model

The Reference Model

Channel gains: hpq(t) = h

DIF

pq(t) + h

LOS

pq (t)

where h

DIF

pq(t) =

lim

N→∞

1

• (Kpq + 1)N

N

• n=1

an,q bn,p e j(2πf(n)

R t+θn)

an,q = e

jπ(MT −2q+1)δT

λ

• cos(αT )+φmax

T

sin(αT ) sin(φ(n)

R )

• bn,p = e

jπ(MR−2p+1)δR

λ

cos(φ(n)

R −αR)

f(n)

R

= fRmax cos(φ(n)

R − βR)

h

LOS

pq (t) =

• Kpq

Kpq + 1cq dp e j(2πfRt+θ0) cq = e

jπ(MT −2q+1)δT

λ

cos(αT )

dp = e

−jπ(MR−2p+1)δR

λ

cos(αR)

fR = fRmax cos(π − βR) θ0 = −2π λ D . Central limit theorem: hDIF

pq(t) ∼ CN(0, 1) as N → ∞. Ph.D. Defense, Aalborg University, 23 January 2008 18/50

SLIDE 19

The One-Ring MIMO Channel Model

Statistical Properties of the Reference Model

• Space-time CCF:

ρ11,22(δT, δR, τ) := Eθn,φ(n)

R {h11(t)h∗

22(t + τ)}

=

π

• −π

a2(δT, φR) b2(δR, φR) e −j2πf(φR)τp(φR) dφR where a(δT, φR) = e

jπδT

λ [ cos(αT )+φmax

T

sin(αT ) sin(φR)]

b(δR, φR) = e

jπδR

λ

cos(φR−αR)

f(φR) = fRmax cos(φR − βR) .

• Temporal ACF:

rhpq(τ) := ρ11,22(0, 0, τ) = E{hpq(t)h∗

pq(t + τ)}

=

π

• −π

e −j2πfRmax cos(φR−βR)τp(φR) dφR .

• 2D space CCF: ρ(δT, δR) := ρ11,22(δT, δR, 0).

Ph.D. Defense, Aalborg University, 23 January 2008 19/50

SLIDE 20

The One-Ring MIMO Channel Model

The Simulation Model

The Stochastic 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)

R as constants.

Channel gains: ˆ hpq(t) = 1 √ N

N

• n=1

an,q bn,p e j(2πf(n)

R t+θn)

⇒ The phases θn are i.i.d RVs ⇒ Stochastic process Channel capacity: ˆ C(t) := log2

• det
• IMR + PT,total

MTPN ˆ H(t) ˆ HH(t)

• [bits/s/Hz]

where ˆ H(t) = [ˆ hpq(t)]. The capacity ˆ C(t) is a stochastic process.

Ph.D. Defense, Aalborg University, 23 January 2008 20/50

SLIDE 21

The One-Ring MIMO Channel Model

Statistical Properties of the Stochastic Simulation Model

• Space-time CCF:

ˆ ρ11,22(δT, δR, τ) := Eθn

• ˆ

h11(t)ˆ h∗

22(t + τ)

• =

1 N

N

• n=1

a2

n,1 b2 n,1 e −j2πf(n)

R τ

• Temporal ACF:

ˆ rhpq(τ) := ˆ ρ11,22(0, 0, τ) = Eθn{ˆ hpq(t)ˆ h∗

pq(t + τ)}

= 1 N

N

• n=1

e −j2πfRmax cos(φ(n)

R −βR)τ

• 2D space CCF:

ˆ ρ(δT, δR) := ˆ ρ11,22(δT, δR, 0) = 1 N

N

• n=1

a2

n,1(δT) b2 n,1(δR) Ph.D. Defense, Aalborg University, 23 January 2008 21/50

SLIDE 22

The One-Ring MIMO Channel Model The Deterministic Simulation Model: Channel gains: ˜ hpq(t) = 1 √ N

N

• n=1

an,q bn,p e j(2πf(n)

R t+θn)

⇒ The phases θn are constant quantities ⇒ Deterministic process Channel capacity: ˜ C(t) := log2

• det
• IMR + PT,total

MTPN ˜ H(t) ˜ HH(t)

• [bits/s/Hz]

where ˜ H(t) = [˜ hpq(t)]. The capacity ˜ C(t) is a deterministic process. ⇒ The analysis of ˜ C(t) has to be performed by using time averages, e.g., < ˜ C(t) >= lim

T→∞

1 2T

T

• −T

˜ C(t) dt .

Ph.D. Defense, Aalborg University, 23 January 2008 22/50

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The One-Ring MIMO Channel Model

Statistical Properties of the Deterministic Simulation Model

• Space-time CCF:

˜ ρ11,22(δT, δR, τ) := < ˜ h11(t)˜ h∗

22(t + τ) >

= 1 N

N

• n=1

a2

n,1 b2 n,1 e −j2πf(n)

R τ

= ˆ ρ11,22(δT, δR, τ)

• Temporal ACF:

˜ rhpq(τ) := ˜ ρ11,22(0, 0, τ) = < ˜ hpq(t)˜ h∗

pq(t + τ) >

= 1 N

N

• n=1

e −j2πfRmax cos(φ(n)

R −βR)τ

= ˆ rhpq(τ)

• 2D space CCF:

˜ ρ(δT, δR) := ˜ ρ11,22(δT, δR, 0) = 1 N

N

• n=1

a2

n,1(δT) b2 n,1(δR)

= ˆ ρ(δT, δR)

Ph.D. Defense, Aalborg University, 23 January 2008 23/50

SLIDE 24

The One-Ring MIMO Channel Model

Parameter Computation Methods

Parameters: The model parameters to be determined are the discrete AOAs φ(n)

R (n = 1, . . . , N).

Problem: Determine the model parameters φ(n)

R such that

ρ11,22(δT, δR, τ) ≈ ˜ ρ11,22(δT, δR, τ)

• r, alternatively,

rhpq(τ) ≈ ˜ rhpq(τ) and ρ(δT, δR) ≈ ˜ ρ(δT, δR) . Solutions:

• Generalized method of exact Doppler spread (MEDS).
• Lp-norm method (LPNM).

Performance measure: Absolute errors |rhpq(τ) − ˜ rhpq(τ)| and |ρ(δT, δR) − ˜ ρ(δT, δR)| .

Ph.D. Defense, Aalborg University, 23 January 2008 24/50

SLIDE 25

The One-Ring MIMO Channel Model

Generalized Method of Exact Doppler Spread (MEDS)

Generalized MEDS: Closed-form solution φ(n)

R = qπ

2N

• n − 1

2

• + φ0

R,

n = 1, . . . , N where q ∈ {1, 2, 3, 4} and φ0

R is called the angle of rotation.

In this Ph.D. project, the following closed-form solution has been used φ(n)

R = 2π

N

• n − 1

2

• + π

2N , n = 1, . . . , N . It can be shown that ˜ rhpq(τ) → rhpq(τ) if N → ∞ ˜ ρ(δT, δR) → ρ(δT, δR) if N → ∞ . Advantages:

• Simple and closed-form solution.
• Very high performance.

Disadvantage: Only valid for isotropic scattering.

Ph.D. Defense, Aalborg University, 23 January 2008 25/50

SLIDE 26

The One-Ring MIMO Channel Model

Lp-norm method (LPNM)

LPNM: The AOAs φ(n)

R have to be determined by minimizing the following two error norms:

E1 :=    1 τmax

τmax

• |rhpq(τ) − ˜

rhpq(τ)|2 dτ   

1/2

E2 :=      1 δT

maxδR max δT

max

• δR

max

• |ρ(δT, δR) − ˜

ρ(δT, δR)|2 dδR dδT     

1/2

An optimization of E1 and E2 can be carried out by using the Fletcher-Powell algorithm. Advantages:

• General solution (isotropic and non-isotropic scattering).
• Very high performance.

Disadvantage:

• No closed-form solution.
• High complexity.

Ph.D. Defense, Aalborg University, 23 January 2008 26/50

SLIDE 27

The One-Ring MIMO Channel Model

Performance Evaluation of the LPNM and Generalized MEDS

2 4 6 8 10 12 −0.5 0.5 1

Normalized time lag τ · fmax Temporal autocorrelation function Reference model Simulation model (generalized MEDS, q = 4, N = 25) Simulation model (LPNM, N = 25)

N 4

The temporal ACFs rhpq(τ) (reference model) and ˜ rhpq(τ) (simulation model) for isotropic scattering environments.

Ph.D. Defense, Aalborg University, 23 January 2008 27/50

SLIDE 28

The One-Ring MIMO Channel Model

Performance Evaluation of the Generalized MEDS

10 20 30 1 2 3 −0.5 0.5 1

δT /λ δR/λ 2D space cross-correlation function

The 2D space CCF ρ(δT, δR) of the reference model for isotropic scattering environments. Reference model

10 20 30 1 2 3 −0.5 0.5 1

δT /λ δR/λ 2D space cross-correlation function

The 2D space CCF ˆ ρ(δT, δR) of the simulation model for isotropic scattering environments (generalized MEDS q = 4, N = 25). Simulation model

Ph.D. Defense, Aalborg University, 23 January 2008 28/50

SLIDE 29

The One-Ring MIMO Channel Model

A New Tight Upper Bound on the 2 × 2 MIMO Channel Capacity

By using Jensen’s inequality and the concavity of log2 function, we obtain E{C(t)} ≤ Cup = log2

• E
• det
• I2 + PBS,total

2PN HHH(t)

• = log2
• 1 + 2PT,total

PN + Ptotal 2NPN 2

N

• m=1

N

• n=1
• 2 − a2

n,1

a2

m,1

− a2

m,1

a2

n,1

1 − b2

n,1

b2

m,1

• .

10 20 30 40 50 5 10 15 20 25

SNR, (dB) Mean capacity (bits/s/Hz) New upper bound Cup Simulation Ph.D. Defense, Aalborg University, 23 January 2008 29/50

SLIDE 30

The Two-Ring MIMO Channel Model

The Geometrical Two-Ring Scattering Model

RR x 0T D αT 0R dmn αR φ(m)

T

d1m A(1)

T

φ(n)

R

dn1 S(n)

R

A(1)

R

S(m)

T

y RT δT δR A(MR)

R

A(MT )

T

dMT m vR vT βT βR dnMR

• The local scatterers are laying on rings around the transmitter and the receiver.
• If M → ∞ and N → ∞, then the discrete AOD φ(m)

T

and AOA φ(n)

R tend to continuous RVs φT

and φR with given distribution p(φT) and p(φR), respectively.

• Assumptions: max{RT, RR} ≪ D and max{δT, δR} ≪ min{rT, RR}.

Ph.D. Defense, Aalborg University, 23 January 2008 30/50

SLIDE 31

The Two-Ring MIMO Channel Model

The Reference Model

Channel gains: hpq(t) = lim

M→∞ N→∞

1 √ MN

M

• m=1

N

• n=1

gpqmne j[2π(f(m)

T

+f(n)

R )t+θmn]

where gpqmn = am,q bn,p cm,n am,q = e

jπ(MT −2q+1)δT

λ

cos(φ(m)

T

−αT )

bn,p = e

jπ(MR−2p+1)δR

λ

cos(φ(n)

R −αR)

cm,n = e

j2π

λ

(RT cos φ(m)

T

−RR cos φ(n)

R )

f(m)

T

= fTmax cos(φ(m)

T

− βT) f(n)

R

= fRmax cos(φ(n)

R − βR) .

Central limit theorem: hpq(t) ∼ CN(0, 1) as M → ∞ and N → ∞.

Ph.D. Defense, Aalborg University, 23 January 2008 31/50

SLIDE 32

The Two-Ring MIMO Channel Model

Statistical Properties of the Reference Model

• Space-time CCF:

ρ11,22(δT, δR, τ) := Eθmn,φ(m)

T

,φ(n)

R {h11(t)h∗

22(t + τ)}

= ρT(δT, τ) · ρR(δR, τ) where ρT(δT, τ) =

π

• −π

a2(δT, φT)e −j2πfT (φT )τpφT(φT) dφT (transmit CF) ρR(δR, τ) =

π

• −π

b2(δR, φR)e −j2πfR(φR)τpφR(φR) dφR (receive CF) with a(δT, φT) = e jπ(δT /λ) cos(φT −αT ) b(δR, φR) = e jπ(δR/λ) cos(φR−αR) fT(φT) = fTmax cos(φT − βT) fR(φR) = fRmax cos(φR − βR) . Remark: The space-time CCF ρ11,22(δT, δR, τ) can be expressed as the product of the CF ρT(δT, τ) and the receive CF ρR(δR, τ).

Ph.D. Defense, Aalborg University, 23 January 2008 32/50

SLIDE 33

The Two-Ring MIMO Channel Model

• Temporal ACF:

rhpq(τ) := ρ11,22(0, 0, τ) = Eθmn,φ(m)

T

,φ(n)

R {hpq(t)h∗

pq(t + τ)}

= ρT(0, τ) · ρR(0, τ) where ρT(0, τ) =

π

• −π

e −j2πfT (φT )τpφT(φT) dφT ρR(0, τ) =

π

• −π

e −j2πfR(φR)τpφR(φR) dφR Remark: The temporal ACF rhpq(τ) is independent of p and q.

• 2D space CCF:

ρ(δT, δR) := ρ11,22(δT, δR, 0) = ρT(δT, 0) · ρR(δR, 0)

Ph.D. Defense, Aalborg University, 23 January 2008 33/50

SLIDE 34

The Two-Ring MIMO Channel Model

The Simulation Model

The 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)

T

and AOA φ(n)

R as constants.

Channel gains: ˆ hpq(t) = 1 √ MN

M

• m=1

N

• n=1

gpqmne j[(2π(f(m)

T

+f(n)

R )t+θmn)]

⇒ The phases θmn are i.i.d RVs ⇒ Stochastic process Channel capacity: ˆ C(t) := log2

• det
• IMR + PT,total

MTPN ˆ H(t) ˆ HH(t)

• [bits/s/Hz]

where ˆ H(t) = [ˆ hpq(t)]. The capacity ˆ C(t) is a stochastic process.

Ph.D. Defense, Aalborg University, 23 January 2008 34/50

SLIDE 35

The Two-Ring MIMO Channel Model

Statistical Properties of the Stochastic Simulation Model

• Space-time CCF:

ˆ ρ11,22(δT, δR, τ) := Eθmn

• ˆ

h11(t)ˆ h∗

22(t + τ)

• =

ˆ ρT(δT, τ) · ˆ ρR(δR, τ) where ˆ ρT(δT, τ) = 1 M

M

• m=1

a2

m,1(δT)e −j2πf(m)

T

τ

ˆ ρR(δR, τ) = 1 N

N

• n=1

b2

n,1(δR)e −j2πf(n)

R τ .

• Temporal ACF:

ˆ rhpq(τ) := Eθn{ˆ hpq(t)ˆ h∗

pq(t + τ)}

= 1 N

N

• n=1

e −j2πfRmax cos(φ(n)

R −βR)τ

• 2D space CCF:

ˆ ρ(δT, δR) := ˆ ρ11,22(δT, δR, 0) = 1 MN

M

• m=1

N

• n=1

a2

m,1(δT)b2 n,1(δR) Ph.D. Defense, Aalborg University, 23 January 2008 35/50

SLIDE 36

The Two-Ring MIMO Channel Model The Deterministic Simulation Model: Channel gains: ˜ hpq(t) = 1 √ MN

M

• m=1

N

• n=1

gpqmne j[(2π(f(m)

T

+f(n)

R )t+θmn)]

⇒ The phases θmn are constant quantities ⇒ Deterministic process Channel capacity: ˜ C(t) := log2

• det
• IMR + PT,total

MTPN ˜ H(t) ˜ HH(t)

• [bits/s/Hz]

where ˜ H(t) = [˜ hpq(t)]. The capacity ˜ C(t) is a deterministic process. ⇒ The analysis of ˜ C(t) has to be performed by using time averages, e.g., < ˜ C(t) >= lim

T→∞

1 2T

T

• −T

˜ C(t) dt .

Ph.D. Defense, Aalborg University, 23 January 2008 36/50

SLIDE 37

The Two-Ring MIMO Channel Model

Statistical Properties of the Deterministic Simulation Model

• Space-time CCF:

˜ ρ11,22(δT, δR, τ) := < ˜ h11(t)˜ h∗

22(t + τ) >

= 1 MN

M

• m=1

N

• n=1

a2

m,1(δT)b2 n,1(δR)e −j2π(f(m)

T

+f(n)

R )τ

= ˆ ρ11,22(δT, δR, τ) ⇒ The simulation model is ergodic w.r.t. the space-time CCF .

• Temporal ACF:

˜ rhpq(τ) = ˆ rhpq(τ)

• 2D space CCF:

˜ ρ(δT, δR) = ˆ ρ(δT, δR)

Ph.D. Defense, Aalborg University, 23 January 2008 37/50

SLIDE 38

The Two-Ring MIMO Channel Model Parameters: The model parameters to be determined are the discrete AODs φ(m)

T

(m = 1, . . . , M) and the discrete AOAs φ(n)

R (n = 1, . . . , N).

Problem: Determine the model parameters φ(m)

T

and φ(n)

R such that

ρ11,22(δT, δR, τ) ≈ ˜ ρ11,22(δT, δR, τ)

• r, alternatively,

ρT(δT, τ) ≈ ˜ ρT(δT, τ) and ρR(δR, τ) ≈ ˜ ρR(δR, τ) . Solutions:

• Generalized method of exact Doppler spread (MEDS).
• Lp-norm method (LPNM).

Performance measure: Absolute errors |ρT(δT, τ) − ˜ ρT(δT, τ)| and |ρR(δR, τ) − ˜ ρR(δR, τ)| .

Ph.D. Defense, Aalborg University, 23 January 2008 38/50

SLIDE 39

The Two-Ring MIMO Channel Model Generalized MEDS: Closed-form solution φ(m)

T

= qπ 2M

• m − 1

2

• + φ0

T,

m = 1, . . . , M φ(n)

R

= qπ 2N

• n − 1

2

• + φ0

R,

n = 1, . . . , N where q ∈ {1, 2, 3, 4}, φ0

T and φ0 R are called the angles of rotation.

In this Ph.D. project, the following closed-form solution has been used φ(m)

T

= 2π M

• m − 1

2

• + π

2M , m = 1, . . . , M φ(n)

R

= 2π N

• n − 1

2

• + π

2N , n = 1, . . . , N . It can be shown that ˜ ρT(δT, τ) → ρT(δT, τ) if M → ∞ ˜ ρR(δR, τ) → ρR(δR, τ) if N → ∞ . Advantages: Simple and closed-form solution, very high performance. Disadvantage: Only valid for isotropic scattering.

Ph.D. Defense, Aalborg University, 23 January 2008 39/50

SLIDE 40

The Two-Ring MIMO Channel Model

Performance Evaluation of the Generalized MEDS

2 4 6 8 2 4 6 8 −0.5 0.5 1

τ · fTmax δT /λ Transmit correlation function, ρT (δT , τ)

The transmit CF ρT(δT, τ) of the 2 × 2 MIMO mobile-to-mobile reference channel model for isotropic scattering environments. Reference model

2 4 6 8 2 4 6 8 −0.5 0.5 1

τ · fTmax δT /λ Transmit correlation function, ˜ ρT (δT , τ)

The transmit CF ˆ ρT(δT, τ) of the 2 × 2 MIMO mobile-to-mobile channel simulator designed by applying the generalized MEDS (q = 4, M = 40). Simulation model

Ph.D. Defense, Aalborg University, 23 January 2008 40/50

SLIDE 41

The Two-Ring MIMO Channel Model

Lp-norm method (LPNM)

LPNM: The AODs φ(m)

T

AOAs φ(n)

R have to be determined by minimizing the following two error

norms: E(p)

1

:=      1 δT,maxτT,max

δT,max

• τT,max
• |ρT(δT, τ) − ˜

ρT(δT, τ)|pdδTdτ     

1/p

E(p)

2

:=      1 δR,maxτR,max

δR,max

• τR,max
• |ρR(δR, τ) − ˜

ρR(δR, τ)|pdδRdτ     

1/p

Note that the error norms E(p)

1

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.

Ph.D. Defense, Aalborg University, 23 January 2008 41/50

SLIDE 42

The Two-Ring MIMO Channel Model

Performance Evaluation of the LPNM

1 2 3 4 5 1 2 3 4 5 0.2 0.4 0.6 0.8 1

τ · fTmax δT /λ |ρT (δT , τ)|

Absolute value of the transmit CF |ρT(δT, τ)|

• f the 2 × 2 MIMO mobile-to-mobile reference

channel model under non-isotropic scattering conditions. Reference model

1 2 3 4 5 1 2 3 4 5 0.2 0.4 0.6 0.8 1

τ · fTmax δT /λ |˜ ρT (δT , τ)|

Absolute value of the transmit CF |ˆ ρT(δT, τ)|

• f the 2 × 2 MIMO mobile-to-mobile chan-

nel simulator designed by applying the LPNM (M = 50, p = 100). Simulation model

Ph.D. Defense, Aalborg University, 23 January 2008 42/50

SLIDE 43

The Two-Ring MIMO Channel Model

The MIMO Channel Capacity

The stochastic simulation model: ˆ C(t) := log2

• det
• IMR + PT,total

MTPN ˆ H(t) ˆ HH(t)

• where ˆ

H(t) = [ˆ hpq(t)]. Statistical average: ˆ Cs := Eθmn{ ˆ C(t)} . The deterministic simulation model: ˜ C(t) := log2

• det
• IMR + PT,total

MTPN ˜ H(t) ˜ HH(t)

• where ˜

H(t) = [˜ hpq(t)]. Time Average: ˜ Ct := lim

T→∞

1 2T

T

• −T

˜ C(t) dt .

10 20 30 40 50 10 20 30 40 50 60 70 80 90

SNR (dB) Mean capacities ˆ Cs and ˜ Ct (bits/s/Hz) MT = MR = 2 MT = MR = 4 MT = MR = 6 ˆ Cs (statistical average) ˜ Ct (time average)

Ph.D. Defense, Aalborg University, 23 January 2008 43/50

SLIDE 44

The Elliptical MIMO Channel Model

The Geometric Elliptical Scattering Model

2b 2a 2f BS MS S(n)

δT αT δR φ(n)

T

A(MR)

R

A(1)

R

A(MT )

T

A(1)

T

φ(n)

R

dMT n dn1 dnMR d1n αR D(n)

T

D(n)

R

βR

v

• The transmitter and the receiver are located at the focal points of an ellipse.
• The local scatterers S(n) are laying on the ellipse around the transmitter and the receiver.
• The AOD φ(n)

T

is determined by the φ(n)

R and the location of S(n). Ph.D. Defense, Aalborg University, 23 January 2008 44/50

SLIDE 45

The Elliptical MIMO Channel Model

The Reference Model

Channel gains: hpq(t) = lim

N→∞

1 √ N

N

• n=1

an,qbn,pe j(2πf(n)

R t+θn)

where an,q = e

jπ(MT −2q+1)δT

λ

cos(φ(n)

T −αT )

bn,p = e

jπ(MR−2p+1)δR

λ

cos(φ(n)

R −αR)

f(n)

R

= fRmax cos(φ(n)

R − βR) .

Central limit theorem: hpq(t) ∼ CN(0, 1) as N → ∞.

Ph.D. Defense, Aalborg University, 23 January 2008 45/50

SLIDE 46

Summary

Contributions

• A detailed study of the stationary and ergodic properties of sum-of-sinusoids-based Rayleigh

fading channel simulators is presented.

• Exact closed-form expressions for PDF

, CDF , LCR, and ADF of the capacity of OSTBC-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 Rayleigh mobile fading channels is pro- posed.

• A MIMO channel model for mobile-to-mobile communications is proposed.
• A reference model for a wideband MIMO channel model is presented. The reference model

is based on the geometric elliptical scattering model. From the reference model, an effi- cient simulation model has been obtained. The channel simulator enables the performance evaluation of MIMO-OFDM systems.

• The MIMO channel capacity has been studied under various propagation conditions imposed

by the geometry of the one-ring, two-ring, and elliptical scattering models.

Ph.D. Defense, Aalborg University, 23 January 2008 46/50

SLIDE 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 , LCR, 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 , LCR, and ADF of the general MIMO channel capacity for both uncorrelated and correlated sub-channels.

Ph.D. Defense, Aalborg University, 23 January 2008 47/50

SLIDE 48

Summary

Publications During Ph.D. Studies

[1] M. P¨ atzold, B. O. Hogstad, and N. Youssef, Modeling, analysis, and simulation of MIMO mobile-to-mobile fading channels, IEEE

• Trans. Wireless Commun., accepted for publication, 2007.

[2] M. P¨ atzold, B. O. Hogstad, and D. Kim, A New Design Concept for High-Performance Fading Channel Simulators Using Set Parti- tioning, Wireless Personal Communications, vol. 40, no. 2, pp. 267–279, Feb. 2007. [3] M. P¨ atzold and B. O. Hogstad, Classes of Sum-of-Sinusoids Rayleigh Fading Channel Simulators and Their Stationary and Ergodic Properties Part I, WSEAS Transactions on Mathematics, Issue 2, Volume 5, February 2006, pp. 222–230. [4] M. P¨ atzold and B. O. Hogstad, Classes of Sum-of-Sinusoids Rayleigh Fading Channel Simulators and Their Stationary and Ergodic Properties Part II, WSEAS Transactions on Mathematics, Issue 4, Volume 4, October 2005, pp. 441–449. [5] C. E. D. Sterian, H. Singh, M. P¨ atzold, B. O. Hogstad, Super-Orthogonal Space-Time Codes with Rectangular Constellations and Two Transmit Antennas for High Data Rate Wireless Communications, IEEE Trans. Wireless Commun., vol. 5, no. 7, Jul. 2006, pp. 1857–1865. [6] M. P¨ atzold and B. O. Hogstad, A Space-Time Channel Simulator for MIMO Channels Based on the Geometrical One-Ring Scattering Model, Wireless Communications and Mobile Computing, Special Issue on Multiple-Input Multiple-Output (MIMO) Communications,

• vol. 4, no. 7, Nov. 2004, pp. 727–737.

[7] B. O. Hogstad and M. P¨ atzold, On the Stationarity of Sum-of Cisoids-Based Mobile Fading Channel Simulators, Proc. 67th IEEE Vehicular Technology Conference, VTC2008-Spring, Singapore, May. 2008, accepted for publication. [8] B. O. Hogstad and M. P¨ atzold, Exact Closed-Form Expressions for the Distribution, Level-Crossing Rate, and Average Duration of Fades of the Capacity of MIMO Channels, Proc. 65th Semiannual Vehicular Vehicular Technology Conference, VTC 2007-Spring, Dublin, Ireland, Apr. 2007, pp. 455–460. [9] M. P¨ atzold and B. O. Hogstad, A Wideband Space-Time MIMO Channel Simulator Based on the Geometrical One-Ring Model, Proc. 64th IEEE Semiannual Vehicular Technology Conference, IEEE VTC 2006-Fall, Montreal, Canada, Sept. 2006. [10] M. P¨ atzold 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’06, Valencia, Spain, Sept. 2006, pp. 138–143. [11] B. O. Hogstad, M. P¨ atzold, and A. Chopra, A Study on the Capacity of Narrow- and Wideband MIMO Channel Models, Proc. 15th IST Mobile & Communications Summit, IST 2006, Myconos, Greece, June 2006. [12] M. P¨ atzold and B. O. Hogstad, Two New Methods for the Generation of Multiple Uncorrelated Rayleigh Fading Waveforms, Proc. 63rd Semiannual Vehicular Technology Conference, IEEE VTC 2006-Spring, Melbourne, Australia, May 2006, vol. 6, pp. 2782–2786. Ph.D. Defense, Aalborg University, 23 January 2008 48/50

SLIDE 49

Summary

[13] M. P¨ atzold and B. O. Hogstad, Classes of Sum-of-Sinusoids Rayleigh Fading Channel Simulators and Their Stationary and Ergodic Properties, Proc. of the 4th WSEAS International Conference on Information Security, Communications and Computers, Tenerife, Spain, 16. - 18. December 2005, pp. 488–504. [14] B. O. Hogstad, M. P¨ atzold, A. Chopra, D. Kim, and K. B. Yeom, A Wideband MIMO Channel Simulation Model Based On the Geo- metrical Elliptical Scattering Model, Proc. 15th Wireless World Research Forum Meeting, WWRF15, 8. - 9. December 2005, Paris, France. [15] M. P¨ atzold, 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 2005, Aalborg, Denmark,

• 18. - 22. September 2005, pp. 496–502.

[16] B. O. Hogstad and M. P¨ atzold, A Study of the MIMO Channel Capacity When Using the Geometrical Two-Ring Scattering Model,

• Proc. 8th International Symposium on Wireless Personal Multimedia Communications, WPMC 2005, Aalborg, Denmark, 18. - 22.

September 2005, pp. 1790–1794. [17] M. P¨ atzold, B. O. Hogstad, N. Youssef, and D. Kim, A MIMO Mobile-to-Mobile Channel Model: Part I - The Reference Model, Proc. 16th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, PIMRC 2005, Berlin, Germany, 11. - 14. September 2005, vol. 1, pp. 573–578. [18] B. O. Hogstad, M. P¨ atzold, N. Youssef, and D. Kim, A MIMO Mobile-to-Mobile Channel Model: Part II - The Simulation Model, Proc. 16th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, PIMRC 2005, Berlin, Germany, 11. - 14. September 2005, vol. 1, pp. 562–567. [19] M. P¨ atzold and B. O. Hogstad, Design and Performance of MIMO Channel Simulators Derived From the Two-Ring Scattering Model,

• Proc. 14th IST Mobile & Communications Summit, IST 2005, Dresden, Germany, 19. - 23. June 2005, paper no. 121.

[20] M. P¨ atzold and B. O. Hogstad, A Space-Time Channel Simulator for MIMO Channels Based on the Geometrical One-Ring Scattering Model, Proc. 60th IEEE Semiannual Vehicular Technology Conference, IEEE VTC 2004-Fall, Los Angeles, CA, USA, 26. - 29. Sept. 2004. [21] B. O. Hogstad and M. P¨ atzold, Capacity Studies of MIMO Channel Models Based on the Geometrical One-Ring Scattering Model,

• Proc. 15th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, IEEE PIMRC 2004, Barcelona,

Spain, 05. - 08. Sept. 2004, vol. 3, pp. 1613–1617. [22] B. O. Hogstad and M. P¨ atzold, New Tight Upper Bounds on the MIMO Channel Capacity, Proc. Nordic Radio Symposium (NRS) 2004, including the Finnish Wireless Communications Workshop (FWCW) 2004, Oulu, Finland, 16. - 18. August 2004. [23] M. P¨ atzold and B. O. Hogstad, A General Concept for the Design of MIMO Channel Simulators, Proc. Nordic Radio Symposium (NRS) 2004, including the Finnish Wireless Communications Workshop (FWCW) 2004, Oulu, Finland, 16. - 18. August 2004. Ph.D. Defense, Aalborg University, 23 January 2008 49/50

SLIDE 50

Summary

Acknowledgements

• Ph.D. supervisors:
• Main supervisor: Prof. Matthias P¨

atzold from Agder University, Norway.

• Second supervisor: Prof. Bernard H. Fleury from Aalborg University, Denmark.
• Co-operators:
• The Communications Section, Electrical Engineering Department, CINVESTAV-IPN,

Mexico.

• Prof. Neji Youssef from Ecole Superieure des Communications de Tunis, Tunis.
• Prof. Corneliu E. D. Sterian from Politechnica University of Bucharest, Romania.
• 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.

Ph.D. Defense, Aalborg University, 23 January 2008 50/50