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• Statistical Properties of a Parametric

Channel Model for Multiple Antenna Systems

• S. Durrani∗, M. E. Bialkowski† and S. Latif∗

∗Department of Engineering,

The Australian National University, Canberra, Australia. Email: salman.durrani@anu.edu.au

†School of ITEE,

The University of Queensland, Brisbane, Australia.

IEEE PIMRC

• Sep. 2007

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• Outline

⊲ Introduction ⋄ MIMO Channel Models ⋄ Motivation ⊲ Reference Channel Model ⋄ Statistical Properties ⊲ Parametric Channel Model ⊲ Results ⋄ Temporal and Spatial Properties ⊲ Conclusions

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• Introduction

⊲ MIMO Channel Models can be classified as follows:†

NT TX antennas NR RX antennas 1 2 n 1 2 m ... ... MIMO CHANNEL Transmitter Receiver

†P. Almers et. al., “Survey of Channel and Radio Propagation Models for Wireless MIMO

Systems,” EURASIP Journal on Wireless Communications and Networking, 2007.

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• Introduction

⊲ MIMO Channel Models can be classified as follows:†

NT TX antennas NR RX antennas 1 2 n 1 2 m ... ... MIMO CHANNEL MODELS ANALYTICAL PHYSICAL Transmitter Receiver

†P. Almers et. al., “Survey of Channel and Radio Propagation Models for Wireless MIMO

Systems,” EURASIP Journal on Wireless Communications and Networking, 2007.

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• Introduction

⊲ MIMO Channel Models can be classified as follows:†

NT TX antennas NR RX antennas 1 2 n 1 2 m ... ... MIMO CHANNEL MODELS ANALYTICAL PHYSICAL DETERMINISTIC PARAMETRIC GEOMETRY-BASED Transmitter Receiver

†P. Almers et. al., “Survey of Channel and Radio Propagation Models for Wireless MIMO

Systems,” EURASIP Journal on Wireless Communications and Networking, 2007.

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• Parametric Channel Model

⊲ Parametric channel models use important physical parameters such as phases, delays, doppler frequency, angle of departure (AOD), angle of arrival (AOA) and angle spread to provide a description of the MIMO channel. Each path consists of (unresolvable) S subpaths that all have the same delay, but different angles of arrival and departures distributed around the mean angles. How many subpaths are sufficient to accurately capture the statis- tical properties of the MIMO wireless channel?

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• Parametric Channel Model

⊲ Parametric channel models use important physical parameters such as phases, delays, doppler frequency, angle of departure (AOD), angle of arrival (AOA) and angle spread to provide a description of the MIMO channel. ⊲ Each path consists of (unresolvable) S subpaths that all have the same delay, but different angles of arrival and departures distributed around the mean angles. How many subpaths are sufficient to accurately capture the statis- tical properties of the MIMO wireless channel?

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• Parametric Channel Model

⊲ Parametric channel models use important physical parameters such as phases, delays, doppler frequency, angle of departure (AOD), angle of arrival (AOA) and angle spread to provide a description of the MIMO channel. ⊲ Each path consists of (unresolvable) S subpaths that all have the same delay, but different angles of arrival and departures distributed around the mean angles. ⊲ How many subpaths are sufficient to accurately capture the statis- tical properties of the MIMO wireless channel?

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• Wireless Propagation Environment

⊲ We consider a MIMO system in an urban macro-cell environ- ment.

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• Reference Channel Model

⊲ The channel impulse response between MS antenna m and BS antenna n for user k’s path l can be written as hm,n

k,l (t) = (hI)m,n k,l (t) + j(hQ)m,n k,l (t)

For isotropic scattering, the temporal correlation properties are summarized below: RhIhI(τ) = E[hI(t)hI(t + τ)] = J0(2πfDτ) RhQhQ(τ) = E[hQ(t)hQ(t + τ)] = J0(2πfDτ) RhIhQ(τ) = E[hI(t)hQ(t + τ)] = 0 Rhh(τ) = E[h(t)h∗(t + τ)] = J0(2πfDτ) R|h|2|h|2(τ) = 4 + 4J2

0(2πfDτ)

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• Reference Channel Model

⊲ The channel impulse response between MS antenna m and BS antenna n for user k’s path l can be written as hm,n

k,l (t) = (hI)m,n k,l (t) + j(hQ)m,n k,l (t)

⊲ For isotropic scattering, the temporal correlation properties are summarized below: RhIhI(τ) = E[hI(t)hI(t + τ)] = J0(2πfDτ) RhQhQ(τ) = E[hQ(t)hQ(t + τ)] = J0(2πfDτ) RhIhQ(τ) = E[hI(t)hQ(t + τ)] = 0 Rhh(τ) = E[h(t)h∗(t + τ)] = J0(2πfDτ) R|h|2|h|2(τ) = 4 + 4J2

0(2πfDτ)

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• Reference Channel Model

⊲ The Level Crossing rate is defined as the rate at which the fading envelope crosses a specified threshold in the positive slope L|h| = √ 2πfDρe−ρ2 ⊲ The Average Fade Duration is the average duration of time that the fading envelope remains below a specified T|h| = eρ2 − 1 ρ √ 2πfD

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• Reference Channel Model

⊲ We assume that the angular distribution of the subpaths at the MS can be modelled by a Uniform PDF over [−π, π]. Measurements have shown that the angular distribution of the sub- paths at the BS can be modelled by a Gaussian PDF. For urban macro-cellular environment, median angular spread: 5◦ − 20◦.

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• Reference Channel Model

⊲ We assume that the angular distribution of the subpaths at the MS can be modelled by a Uniform PDF over [−π, π]. ⊲ Measurements have shown that the angular distribution of the sub- paths at the BS can be modelled by a Gaussian PDF.† For urban macro-cellular environment, median angular spread: 5◦ − 20◦.

†K. I. Pedersen et. al., ”A stochastic model of the temporal and azimuth dispersion seen

at the base station in outdoor propagation environments,” IEEE Trans. VT, vol. 49, no. 2,

• pp. 437-447, Mar. 2000.

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• Reference Channel Model

⊲ We assume that the angular distribution of the subpaths at the MS can be modelled by a Uniform PDF over [−π, π]. ⊲ Measurements have shown that the angular distribution of the sub- paths at the BS can be modelled by a Gaussian PDF.† ⋄ For urban macro-cellular environment, median angular spread: 5◦ − 20◦.

†K. I. Pedersen et. al., ”A stochastic model of the temporal and azimuth dispersion seen

at the base station in outdoor propagation environments,” IEEE Trans. VT, vol. 49, no. 2,

• pp. 437-447, Mar. 2000.

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• Reference Channel Model

⊲ The spatial envelope correlation coefficient ρs, between the pth and qth antenna elements for a ULA, is given by ρs(p, q) = |Rs(p, q)|2 = |ℜ{Rs(p, q)} + jℑ{Rs(p, q)}|2 ⊲ Spatial Correlation at BS

ℜ{Rs(p, q)} = J0(zpq) + 2Cg

∞

• v=1

J2v(zpq) cos(2vθAOD)e(−2v2σ2

AOD)ℜ

• erf

π + j2vσ2

AOD

√ 2σAOD

• ℑ{Rs(p, q)}

= 2Cg

∞

• v=0

J2v+1(zpq) sin[(2v + 1)θAOD]e

• −(2v+1)2σ2

AOD 2

• ℜ
• erf

π + j(2v + 1)σ2

AOD

√ 2σAOD

• ⊲ Spatial Correlation at MS

ℜ{Rs(p, q)} = J0(zpq) + 2

∞

• v=1

J2v(zpq) cos(2vθAOA)sinc(2v∆) ℑ{Rs(p, q)} = 2

∞

• v=0

J2v+1(zpq) sin[(2v + 1)θAOA]sinc[(2v + 1)∆]

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• Parametric Channel Model

⊲ The channel impulse response can be written as h(m,n)

k,l

(t)=

• Ωk,l

S S

• s=1

exp[j(φ(s)

k,l + 2πfDt cos θ(s) k,l,AOA)]

×exp[−jκdM(m − 1) sin θ(s)

k,l,AOA]

×exp[−jκdB(n − 1) sin θ(s)

k,l,AOD]

• δ(t − τk,l)

Temporal Parameters K = users; L = multipaths; S = sub-paths/path; Ωk,l = mean path power; τk,l = propagation delay; φ(s)

k,l = random phase;

fD = Doppler frequency; Spatial Parameters N = No. of antennas; d = inter-element distance; κ = 2π/λ; θ(s)

k,l,AOD = θk,AOD+ϑ(s) k,l,AOD

θ(s)

k,l,AOA=θk,AOA+ϑ(s) k,l,AOA

θk,AOA = Mean Angle of Arrival; θk,AOD = Mean Angle of Departure;

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• Parametric Channel Model

⊲ The channel impulse response can be written as h(m,n)

k,l

(t)=

• Ωk,l

S S

• s=1

exp[j(φ(s)

k,l + 2πfDt cos θ(s) k,l,AOA)]

×exp[−jκdM(m − 1) sin θ(s)

k,l,AOA]

×exp[−jκdB(n − 1) sin θ(s)

k,l,AOD]

• δ(t − τk,l)

Temporal Parameters K = users; L = multipaths; S = sub-paths/path; Ωk,l = mean path power; τk,l = propagation delay; φ(s)

k,l = random phase;

fD = Doppler frequency; Spatial Parameters N = No. of antennas; d = inter-element distance; κ = 2π/λ; θ(s)

k,l,AOD = θk,AOD+ϑ(s) k,l,AOD

θ(s)

k,l,AOA=θk,AOA+ϑ(s) k,l,AOA

θk,AOA = Mean Angle of Arrival; θk,AOD = Mean Angle of Departure;

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• Parametric Channel Model

⊲ The channel impulse response can be written as h(m,n)

k,l

(t)=

• Ωk,l

S S

• s=1

exp[j(φ(s)

k,l + 2πfDt cos θ(s) k,l,AOA)]

×exp[−jκdM(m − 1) sin θ(s)

k,l,AOA]

×exp[−jκdB(n − 1) sin θ(s)

k,l,AOD]

• δ(t − τk,l)

Temporal Parameters K = users; L = multipaths; S = sub-paths/path; Ωk,l = mean path power; τk,l = propagation delay; φ(s)

k,l = random phase;

fD = Doppler frequency; Spatial Parameters N = No. of antennas; d = inter-element distance; κ = 2π/λ; θ(s)

k,l,AOD = θk,AOD+ϑ(s) k,l,AOD

θ(s)

k,l,AOA=θk,AOA+ϑ(s) k,l,AOA

θk,AOA = Mean Angle of Arrival; θk,AOD = Mean Angle of Departure;

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• Parametric Channel Model

Aspect Parameter Value or Description Carrier frequency fc = 2 GHz General Number of channel samples T = 20000 Samples/wavelength 8 MS velocity v = 60 km/hr Temporal Number of paths L = 1 Number of subpaths S = 25 Antenna geometry ULA Antennas Number of antennas NB = 2, NM = 2 BS Inter-element distance dB = 5λ MS Inter-element distance dM = 0.5λ Mean Angle of Arrival θAOA = 60◦ pdf in Angle of Arrival Uniform [−π, π] Spatial Mean Angle of Departure θAOD = 0◦ pdf in Angle of Departure Gaussian BS Angle spread σAOD = 5◦, 10◦, 20◦

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• Results − Temporal Correlations

Autocorrelations

2 4 6 8 10 −0.5 0.5 1 Normalized Time Delay fDτ (s) Normalized Autocorrelations Reference Model Simulation R|h|

2 |h| 2(τ)

Rh

I h I

(τ)

Cross-correlation

2 4 6 8 10 −0.5 0.5 1 Normalized Time Delay fDτ (s) Cross−correlation: Rh

Ih Q

(τ) Reference Model Simulation

For S = 25, simulation results agree with reference results for 0 ≤ fDτ ≤ 3.

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• Results − Temporal Correlations

Autocorrelations

2 4 6 8 10 −0.5 0.5 1 Normalized Time Delay fDτ (s) Normalized Autocorrelations Reference Model Simulation R|h|

2 |h| 2(τ)

Rh

I h I

(τ)

Cross-correlation

2 4 6 8 10 −0.5 0.5 1 Normalized Time Delay fDτ (s) Cross−correlation: Rh

Ih Q

(τ) Reference Model Simulation

⊲ For S = 25, simulation results agree with reference results for 0 ≤ fDτ ≤ 3.

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• Results − LCR & AFD

MS velocity v = 60 km/hr (fD = 111.11 Hz)

−25 −20 −15 −10 −5 5 10 10

−3

10

−2

10

−1

10 10

1

10

2

Normalised Level Crossing Rate Normalised Threshold ρ (dB)

−25 −20 −15 −10 −5 5 10 10

−3

10

−2

10

−1

10 10

1

10

2

Normalised Average Fade Duration

LCR (Reference Model) LCR (Simulation) AFD (Reference Model) AFD (Simulation)

⊲ For S = 25, simulation results deviate from reference results only for very low threshold values.

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• Results − MS Spatial Correlation

Spatial Correlation Coefficient vs. distance dM/λ

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalised inter−element distance dM/λ MS Spatial Envelope Correlation Coefficient ρs(1,2) Reference Model Simulation

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• Results − BS Spatial Correlation

Spatial Correlation Coefficient vs. distance dB/λ Mean AOD = 0◦

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalised inter−element distance dB/λ BS Spatial Envelope Correlation Coefficient ρs(1,2)

σAOD = 5° (Reference Model) σAOD = 5° (Simulation) σAOD = 10° (Reference Model) σAOD = 10° (Simulation) σAOD = 20° (Reference Model) σAOD = 20° (Simulation)

Mean AOD = 60◦

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalised inter−element distance dB/λ BS Spatial Envelope Correlation Coefficient ρs(1,2)

σAOD = 5° (Reference Model) σAOD = 5° (Simulation) σAOD = 10° (Reference Model) σAOD = 10° (Simulation) σAOD = 20° (Reference Model) σAOD = 20° (Simulation)

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• Conclusions

⊲ In this paper, we have analysed the statistical properties of a parametric channel model for MIMO systems in an urban macrocell environment. The proposed channel model can accurately represent the tempo- ral correlations for time delays 0 ≤ fDτ ≤ 3 and spatial corre- lations at MS and BS for inter-element spacings 0 ≤ d/λ ≤ 3. The obtained results have shown that S = 25 subpaths is sufficient to capture the important statistical properties of MIMO wireless channel.

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• Conclusions

⊲ In this paper, we have analysed the statistical properties of a parametric channel model for MIMO systems in an urban macrocell environment. ⊲ The proposed channel model can accurately represent the tempo- ral correlations for time delays 0 ≤ fDτ ≤ 3 and spatial corre- lations at MS and BS for inter-element spacings 0 ≤ d/λ ≤ 3. The obtained results have shown that S = 25 subpaths is sufficient to capture the important statistical properties of MIMO wireless channel.

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• Conclusions

⊲ In this paper, we have analysed the statistical properties of a parametric channel model for MIMO systems in an urban macrocell environment. ⊲ The proposed channel model can accurately represent the tempo- ral correlations for time delays 0 ≤ fDτ ≤ 3 and spatial corre- lations at MS and BS for inter-element spacings 0 ≤ d/λ ≤ 3. ⊲ The obtained results have shown that S = 25 subpaths is sufficient to capture the important statistical properties of MIMO wireless channel.

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