Channel Models, Favorable Propagation and Multi-Stage Linear - - PowerPoint PPT Presentation

2020 IEEE International Symposium on Information Theory (ISIT), 21-26 June 2020

Roya Gholami Dirk Slock Laura Cottatellucci

Channel Models, Favorable Propagation and Multi-Stage Linear Detection in Cell-Free Massive MIMO

Outline

• I. Objectives and Motivations
• II. System and Channel Model
• III. Analytical Conditions of Favorable Propagation
• IV. Mathematical Results for DAS and CF massive MIMO Analysis
• V. Favorable Propagation in Cell-Free Massive MIMO
• VI. Performance Analysis of Multistage Detectors
• VII. Simulation Results
• VIII. Summary and Conclusions
• R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi-Stage Linear Detection in CF Massive MIMO”

• I. Objectives and Motivations

1

Advantages:

 Allowing accommodation of more users  Higher data rates  Reducing users’ energy consumption  Improving transmission quality

• R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi-Stage Linear Detection in CF Massive MIMO”

Challenge:

• High receiver complexity

Distributed Antenna Systems (DASs)

• Access Points (APs) geographically distributed
• APs connected to a CPU performing joint

decoding 𝑂𝑈 users 𝑂𝑆 antennas

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• R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi-Stage Linear Detection in CF Massive MIMO”

 DAS comprising a massive number of APs jointly serving a much smaller number of users In Massive MIMO Systems, as 𝑂𝑆 → ∞ and 𝑂𝑈 remains constant, the users’ channels become orthogonal

Favorable Propagation

Low complexity Matched Filters are optimum

Cell-Free Massive MIMO Systems Does Favorable Propagation hold in CF massive MIMO ?

• I. Objectives and Motivations

• Users and APs randomly and independently distributed
• Users′ intensity 𝜍T →

𝑂𝑈 = 𝜍T 𝑀2 users

• APs′ intensity 𝜍R

→ 𝑂R = 𝜍R 𝑀2 APs

• All users transmit with same power 𝑄

Received signal at the processing unit

• II. System Model

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• R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi-Stage Linear Detection in CF Massive MIMO”

Channel Matrix 𝑂𝑆 × 𝑂𝑼 Transmit Symbol Vector 𝑂𝑼 × 1 Noise 𝑂𝑺 × 1

𝑀

𝒛 = 𝑄 𝑯 𝒚 + 𝒐

 (𝑗, 𝑘)-th element of the Path Loss Matrix ෠

𝐇 ො 𝑕𝑗,𝑘 = ො 𝑕(𝑒𝑗𝑘) = ቐ

𝑒0

𝛽

𝑒𝑗𝑘𝛽

if 𝑒𝑗𝑘> 𝑒0 1

• therwise
• 2𝛽 : Path Loss exponent
• 𝑒0 : Reference distance
• 𝑒𝑗𝑘 = 𝒔𝑗 − 𝒖𝑘

2 : Euclidean distance between user 𝑘 and AP 𝑗

• 𝒔𝑗 = (𝑠𝑗,𝑦 , 𝑠𝑗,𝑧)
• 𝒖𝑘 = (𝑢𝑘,𝑦 , 𝑢𝑘,𝑧)

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• R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi-Stage Linear Detection in CF Massive MIMO”
• II. Channel Model

𝑒0

(

𝑒0 𝑒𝑗𝑘)𝛽

𝑒 ො 𝑕(𝑒)

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• R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi-Stage Linear Detection in CF Massive MIMO”

𝑕𝑗,𝑘 = 𝑕(𝑒𝑗𝑘) = ො 𝑕(𝑒𝑗𝑘) 𝑓−i 2𝜌 𝜇−1𝑒𝑗𝑘

• 𝜇 : Radio signal wavelength

𝑕𝑗,𝑘 = 𝑕(𝑒𝑗𝑘) = ො 𝑕(𝑒𝑗𝑘) ℎ𝑗,𝑘

• ℎ𝑗,𝑘 : i.i.d complex Gaussian variables with

zero mean and unit variance

Channel Coefficients for Path Loss and Rayleigh Fading

Path Loss Phase Rotation Path Loss Rayleigh fading

Channel Coefficients for Path Loss and LoS

• II. Channel Model

• III. Analytical Conditions of Favorable Propagation

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• R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi-Stage Linear Detection in CF Massive MIMO”

Eigenvalue moment of order 𝑚 of the channel covariance matrix 𝑫 = 𝐇𝐼 𝐇 𝑛𝑫

(𝑚) = ׬ 𝜈𝑚 𝑒𝐺𝑫(𝜈) = 𝔽{ 1 𝑂𝑈 trace (𝑫𝑚)}

• 𝜈: eigenvalue, 𝐺𝑫(𝜈): empirical eigenvalue distribution of matrix 𝑫

Analytical conditions of Favorable propagation Moment Ratio : MR(𝑚)=

𝑛𝑫

(𝑚)

trace{ ( diag 𝑫 )𝑚 }/𝑂𝑈 ≈ 1

∀𝑚 ∈ 𝑂+ Do Favorable Propagation Conditions Hold in CF Massive MIMO ? For LoS Channels ? And for Path Loss and Rayleigh fading ?

Euclidean matrix ෡ 𝐇 decomposable as ෡

𝐇 = 𝛚𝑆 ෡ 𝑼 𝛚𝑈

𝐼

• ෡

𝑼 (𝜄2 × 𝜄2) : deterministic depending on ො 𝑕(𝑒)

• 𝛚𝑆 (𝑂𝑆 × 𝜄2) : depending on Rx locations
• 𝛚𝑈 (𝑂𝑼 × 𝜄2) : depending on Tx locations

How matrices 𝛚𝑆 , 𝛚𝑈 , and ෡ 𝑼 are obtained?

Consider a 𝜄2 × 𝜄2 channel matrix ෡ 𝓗 of a system with 𝜄2 transmit and receive nodes regularly spaced on a grid.  ෡ 𝓗 is a symmetric block Toeplitz matrix of 𝜄 × 𝜄 blocks  ෡ 𝓗 admits an eigenvalue decomposition based on the 𝜄2 × 𝜄2 Fourier matrix F as 𝜄2 → ∞ : ෡

𝓗 = F ෡ 𝑼 F 𝐼

𝛚R and 𝛚T obtained by extracting uniformly at random 𝑂R and 𝑂T rows from matrix F.

• IV. Mathematical Results for DAS and CF massive MIMO Analysis

7

• R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi-Stage Linear Detection in CF Massive MIMO”

𝜐 𝑀 = 𝜐 𝜄

• Decomposition of channel matrix for path loss and LoS channel: 𝐇 = 𝛚𝑆 𝑼 𝛚𝑈

𝐼

• Approximation by ෩

𝐇 = 𝛠𝑆 𝑼 𝛠𝑈

𝐼

where 𝛠𝑆 and 𝛠𝑈 consist of i.i.d. Gaussian elements of zero mean and variance 𝜄−2 .

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• R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi-Stage Linear Detection in CF Massive MIMO”

Fundamental Results

• Recursive algorithm to compute the eigenvalue moments of ෩

𝑫= ෩ 𝐇𝐼 ෩ 𝐇 , as 𝜄2, 𝑂𝑆, 𝑂𝑈 → ∞ with

𝑂𝑆 𝜄2 → 𝛾𝑆 and 𝑂𝑈 𝜄2 → 𝛾𝑈

• Dependence of eigenvalue moments 𝑛෩

𝑫 (𝑚) only on 𝛾𝑆, 𝛾𝑈 , and 𝑛𝑼

• ሚ

𝐷𝑙𝑙

(𝑚) = 𝑛෩ 𝑫 (𝑚) ,∀𝑙,𝑚 , ሚ

𝐷𝑙𝑙

(𝑚) being the diagonal elements of matrix ෩

𝑫𝑚

• IV. Mathematical Results: Channel Matrix with Path Loss and LoS

9

• R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi-Stage Linear Detection in CF Massive MIMO”

Eigenvalue Moments

As L → ∞ , the eigenvalue moments of matrix ෩ 𝑫= ෩ 𝐇𝐼 ෩ 𝐇 converge to a deterministic value

𝑛෩

𝑫 (𝑚) =

𝑛෡

𝑼 2 𝑚

σ𝑙=0

𝑚−1 1 𝑙+1

𝑚 − 1 𝑙 𝑚 𝑙 𝛾𝑈

𝑙 𝛾𝑆 𝑚−𝑙

; 𝑚 ≥ 1 The eigenvalue moments 𝑛෩

𝑫 (𝑚) depend only on 𝛾𝑈, 𝛾𝑆 , and 𝑛෡ 𝑼 2

• IV. Mathematical Results: Channel with Path Loss and Rayleigh Fading

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• R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi-Stage Linear Detection in CF Massive MIMO”

For any 𝑚, as 𝛾𝑆→ ∞ and 𝛾𝑈 is kept constant, i.e., for Τ 𝛾𝑈 𝛾𝑆 → 0 , and 𝛾𝑈 > 0

MR(𝑚) =

𝑛෩

𝑫 (𝑚)

trace{ ( 𝑒𝑗𝑏𝑕 ෩ 𝑫 )𝑚}/𝑂𝑈 𝑀→∞

1 Favorable Propagation Conditions are satisfied in CF Massive MIMO with Path Loss and Rayleigh fading

• V. Favorable Propagation Conditions in CF massive MIMO with Rayleigh fading

Low Complexity Matched Filters are almost optimum

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• R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi-Stage Linear Detection in CF Massive MIMO”

For 𝑚 = 2 , 3 : as 𝛾𝑆→ ∞ and 𝛾𝑈 > 0 , i.e., for Τ 𝛾𝑈 𝛾𝑆 → 0 MR(2)

𝑀→∞ 1 + 𝛾𝑈 𝑛𝑈

(4)

( 𝑛𝑈

(2))2

MR(3)

𝑀→∞ 1 + 3𝛾𝑈 𝑛𝑈

(4)

( 𝑛𝑈

(2))2 + 𝛾𝑈

2 𝑛𝑈

(6)

( 𝑛𝑈

(2))3

Favorable propagation does not hold in CF massive MIMO with LoS Linear multistage detectors are expected to provide substantial gains compared to Matched Filter

• V. Favorable Propagation Conditions in CF massive MIMO with Path Loss and LoS

• VI. Multistage Detectors

12

Asymptotic analysis and design based on ሚ 𝐷𝑙𝑙

(𝑚) and 𝑛෩ 𝑫 (𝑚), Cottatellucci et al. ‘05

• For an 𝑁-stage detector define matrix

SM (𝑌) = 𝑌(2) + 𝜏2𝑌(1) … 𝑌(𝑁+1) + 𝜏2𝑌(𝑁) 𝑌(3) + 𝜏2𝑌(2) … 𝑌(𝑁+2) + 𝜏2𝑌(𝑁+1) ⋮ ⋱ ⋮ 𝑌(𝑁+1) + 𝜏2𝑌(𝑁) … 𝑌(2𝑁) + 𝜏2𝑌(2𝑁−1)

and a vector sM (𝑌) = [𝑌 1 , 𝑌 2 , … , 𝑌(𝑁)]𝑈

• Polynomial Expansion detectors, Moshavi et al. ‘96: 𝑌= 𝑛෩

𝑫

• Multistage Wiener filters (MSWF), Goldstein et al. ‘98 : 𝑌= ሚ

𝐷𝑙𝑙 In DASs, MSWF and Polynomial Expansion Detectors are equivalent with SINR𝑁=

𝒕𝑁

𝑈

𝑛෩

𝑫 𝑻𝑁 −1 𝑛෩ 𝑫 𝒕𝑁 𝑈

𝑛෩

𝑫

1−𝒕𝑁

𝑈

𝑛෩

𝑫 𝑻𝑁 −1 𝑛෩ 𝑫 𝒕𝑁 𝑈

𝑛෩

𝑫

• R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi-Stage Linear Detection in CF Massive MIMO”

13

Favorable Propagation Conditions via Moment Ratios (MR)

• MR(3) for channels with Path Loss and LoS

channel or Rayleigh fading channel.

• LoS

: lim

Τ 𝛾𝑈 𝛾𝑆→0 MR(𝑚) ≠ 1 and

favorable propagation conditions are not satisfied.

• Rayleigh fading channel offers favorable

propagation.

• R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi-Stage Linear Detection in CF Massive MIMO”
• VII. Simulation Results

14

Gain of a M-stage Wiener filter over a matched filter G =

SINR𝑁−SINR1 SINR1

• R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi-Stage Linear Detection in CF Massive MIMO”

Path Loss plus Rayleigh fading

• Gain G becomes negligible, as 𝜍R increases
• Matched filter achieves almost optimal performance

for 𝜍R → ∞

Path Loss plus LoS

• Gain G ≫ 0 as 𝜍R → ∞
• SINR of a multistage detector is 240% the SINR of a

matched filter

Multistage Detector versus Matched Filter

• VII. Simulation Results

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• Increasing

Τ 𝜍𝑈 𝜍𝑆, the SINR of a 2-stage detector increases enormously.

• At high

Τ 𝜍𝑈 𝜍𝑆, the gains of higher order multistage detectors over a 2-stage detector become significant.

• R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi-Stage Linear Detection in CF Massive MIMO”

Effect of the system load per unit area Τ 𝜍𝑈 𝜍𝑆 in the case of LoS

• VII. Simulation Results

• VIII. Summary and Conclusions
• Analysis of DASs and CF massive MIMO based on channel eigenvalue moments
• In CF massive MIMO, favorable propagation conditions
• are not satisfied in channels with Path Loss and LoS
• are satisfied with Path Loss plus Rayleigh fading
• Analysis of multistage detectors performance in DASs
• Low complexity multistage detectors outperform considerably matched filter in

channels with Path Loss and LoS

• Matched filters achieve almost optimal performance in CF massive MIMO in Path

Loss plus Rayleigh fading but only at extremely low Τ 𝛾𝑈 𝛾𝑆 ratios

• Equivalence of MSWF and polynomial expansion detector

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• R. Gholami, L. Cottatellucci, D. Slock, “Channel Models, Favorable Propagation and Multi-Stage Linear Detection in CF Massive MIMO”

2020 IEEE International Symposium on Information Theory (ISIT), 21-26 June 2020

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