Multi-Dimensional Array Signal Processing Applied to MIMO Systems

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Universidade de Brasília

Laboratório de Processamento de Sinais em Arranjos 1

Multi-Dimensional Array Signal Processing

Applied to MIMO Systems

Prof. Dr.-Ing. João Paulo C. Lustosa da Costa

University of Brasília (UnB)

Department of Electrical Engineering (ENE)

Laboratory of Array Signal Processing

PO Box 4386

Zip Code 70.919-970, Brasília - DF

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Outline

 Universidade de Brasília: A Short Overview  Motivation

 Antenna Array Based Systems

 Multi-Dimensional Array Signal Processing Multi-Dimensional Array Interpolation Model Order Selection

Prewhitening

 MIMO-OFDM System  Conclusions

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Outline

 Multi-Dimensional Array Signal Processing

Multi-Dimensional Array Interpolation

Model Order Selection

Prewhitening

 MIMO-OFDM System

 Conclusions

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Universidade de Brasília: A Short Overview (1)

 Universidade de Brasília (UNB)

One of the best federal universities in Brazil

The best university in the central-west region of Brazil

• Region with 12 million inhabitants UNB is located in Brasília

• capital of Brazil

– political influence and cooperation with the Federal Government 

• one of the most expensive cities in Brazil 

• one of the safest cities in Brazil 

• Great weather (avrg 20,6o_{C, avrg min 17}o_{C, avrg max 26,6}o_{C) } • Several amazing waterfalls around Brasília 

– Itiquira, Pirinópolis, Chapada dos Veadeiros and others

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Universidade de Brasília: A Short Overview (3)

 Universidade de Brasília (UNB)

In 2013, around 23234 candidates for 4219 places In 2013, 37465 students

In 2013, 2594 professors

(including all departments and all semesters)  Department of Electrical Engineering

composed of three bachelor courses

• Communication Network Engineering

• Mechatronics

• Electrical Engineering

• Computer Engineering

around 1560 bachelor students around 65 professors

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Outline

Prewhitening

 Conclusions

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Laboratório de Processamento de Sinais em Arranjos

Motivation (1)

5

[1] Ericsson Mobility Report, Nov 2014

 Mobile network subscribers forecast [1]

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Motivation (2)

5  Mobile network traffic forecast [1]

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Motivation (3)

5  5th_{generation mobile networks (5G) [2]}

Generation 1G 2G 3G 4G 5G

Year 1981 1991 2001 2011 2020 Throughput < 1 kbps 9,6 kpbs 144 – 2000 kpbs 100 – 1000 Mbps _{– 10 Gbps}100 Mbps

[2] J. G. Andrews, S. Buzzi, W. Choi, S. Hanly, A. Lozano, A.C.K. Soong, and J. Zhang, "What will 5G be?," IEEE Journal on Selected Areas in Communications, Vol. 32, No. 6, pp. 1065 - 1082, June 2014

millimeter wave wireless communications (up to 90 GHz)

• compensation using massive MIMO Massive Distributed MIMO

Multi-hop networks and device-to-device (D2D) communications Cognitive radio technology

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Outline

Prewhitening

 Conclusions

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Antenna Array Based Systems (1)

 Standard (Matrix) Array Signal Processing

5 Four gains: array gain, diversity gain, spatial multiplexing gain and

interference reduction gain

TX RX

Array gain: 3 for each side

Diversity gain: same information for each path

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Antenna Array Based Systems (2)

TX RX

Array gain: 3 for each side

Diversity gain: same information for each path

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Antenna Array Based Systems (3)

TX RX

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Antenna Array Based Systems (3)

TX RX

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Outline

 Multi-Dimensional Array Signal Processing Multi-Dimensional Array Interpolation

Prewhitening

 Conclusions

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Receive array: 1-D or 2-D

Frequency Time

Transmit array: 1-D or 2-D

Direction of Arrival (DOA)

Delay Doppler shift Direction of Departure (DOD)

Multi-Dimensional Array Signal Processing (1)

 MIMO channel model

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5 Dimensions depend on the type of application

• MIMO

– Received data: two spatial dimensions, frequency and time – Channel: 4 spatial dimensions, frequency and time

• Microphone array

– Received data: one spatial dimension and time

– After Time Frequency Analysis

• Space, time and frequency

• EEG (similarly as microphone array)

• Psychometrics

• Chemistry

• Food industry …

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 Multi-Dimensional (Tensor) Array Signal Processing

5 Advantages: increased identifiability, separation without imposing

additional constraints and improved accuracy (tensor gain)

RX: Uniform

Rectangular Array (URA)

1 2 3 1 2 3 1 2 3 m₂ 1 1 1 2 2 2 3 3 3 m₁ n ₁ ₂ ₃

 9 x 3 matrix: maximum rank is 3.

• Solve maximum 3 sources!

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RX: Uniform

Rectangular Array (URA)

m₁ 1 2 3 m₂ 1 2 3 1 2 n 3

[3] J. B. Kruskal. Rank, decomposition, and uniqueness for 3-way and N-way arrays. Multiway Data Analysis, pages 7–18, 1989

3 x 3 x 3 tensor: maximum rank is 5 [3].

• Solve maximum 5 sources!

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=

+

• For matrix model, nonrealistic assumptions such as orthogonality (PCA) or independence (ICA) should be done.

• For tensor model, separation is unique up to scalar and permutation ambiguities.

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additional constraints and improved accuracy (tensor gain) • Array interpolation due to imperfections

– Application of tensor based techniques

• Estimation of number of sources d

– also known as model order selection

– multi-dimensional schemes: better accuracy

• Prewhitening schemes

– multi-dimensional schemes: better accuracy and lower complexity

• Parameter estimation

– Drastic reduce of computational complexity

• Multidimensional searches are decomposed into several one dimensional searches

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 Measurements or data from several applications, for instance,

 MIMO channels, EEG, stock markets, chemistry, pharmacology, medical imaging, radar, and sonar

 Array interpolation

 SPS, FBA, ESPRIT and multidimensional techniques

 Model order selection

 estimation of the number of the main components (total number of parameters)

 Parameter estimation techniques

 extraction of the parameters from the main components

 Subspace prewhitening schemes

 application of the noise statistics to improve the parameter estimation

Measurements Model order

selection Subspace Prewhitening Parameter Estimation Is the noise colored? Yes No Array Interpolation

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Outline

Prewhitening

 Conclusions

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Multi-Dimensional Array Interpolation (1)

 Data model

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Multi-Dimensional Array Interpolation (2)

 Power response for r-th dimension

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Multi-Dimensional Array Interpolation (3)

 Interpolation in each r-th mode

obtained from measurements. obtained from power response.  Interpolated data

Structure closer to PARAFAC structure

• Tensor is attenuated.

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Multi-Dimensional Array Interpolation (3)

 d = 3, 8 x 8 antenna array, N = 100

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Outline

Prewhitening

 Conclusions

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Model Order Selection (1)

 Noiseless case

Our objective is to estimate d from the noisy observations .

 Matrix data model

+

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1 2 3 4 5 6 7 8 0 2 4 6 8 10 Eigenvalue index i  i

Model Order Selection (2)

 Finite SNR, Finite N

M - d noise eigenvalues follow a Wishart distribution.

d signal plus noise eigenvalues

d = 2, M = 8, SNR = 0 dB, N = 10  The eigenvalues of the sample covariance matrix

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Model Order Selection (3)

 Observation is a superposition of noise and signal

 The noise eigenvalues still exhibit the exponential profile: EFT

 We can predict the profile of the noise eigenvalues to find the “breaking point”

 Let P denote the number

of candidate noise eigenvalues.

• choose the largest P such that the P noise eigenvalues can be fitted with a decaying exponential

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Multi-Dimensional Model Order Selection (1)

Noiseless data representation

Problem

where is the colored noise tensor.

=

+

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We can define global eigenvalues

 R-D exponential profile

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Comparison between the global eigenvalues profile and the profile of the last unfolding

 R-D exponential profile

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

 White Gaussian noise



Model Order Selection in Additive White Gaussian Noise Scenario Probability of correct Detection vs. SNR

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Outline

Prewhitening

 Conclusions

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Motivation

 Colored noise is encountered in a variety of signal processing applications, e.g., SONAR, communications, and speech processing.

 Without prewhitening the parameter estimation is severely degraded.

 Traditionally, stochastic prewhitening schemes are applied.

 By prewhitening the subspace via our proposed deterministic prewhitening scheme, an improvement of the parameter estimation is obtained compared to the stochastic prewhitening schemes.

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Noise Analysis

Stochastic

prewhitening schemes

With colored noise the d main

components are more affected.

 Analysis via SVD

Deterministic prewhitening scheme

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Simulations

The noise correlation is known.

SE – Standard ESPRIT

Subspace Prewhitening for Colored Noise with Structure RMSE vs. Correlation Level

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 These matrix based prewhitening schemes have a worse accuracy for

multidimensional colored noise or interference with Kronecker correlation structure,

 when applied in conjunction with the subspace-based parameter estimation techniques, such as R-D Standard ESPRIT and R-D Standard Tensor-ESPRIT

 Therefore, we propose the Sequential Generalized Singular Value Decomposition (S-GSVD) of the measurement tensor and of the multidimensional noise samples

 enables us to improve the subspace estimation

 based on the prewhitening correlation factors estimation

 has a low complexity and a high accuracy version

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Simulations

Subspace Prewhitening for Multi-dimensional Colored Noise RMSE vs. Number of Samples without Signal Components (N_l)

STE – Standard Tensor-ESPRIT

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Iterative S-GSVD

 In some multidimensional applications,

 the noise samples without the presence of signal components are not

available

 For these cases, we propose the Iterative Sequential GSVD (I-S-GSVD)

 jointly estimation of the signal data and of the noise statistics via a proposed

iterative algorithm in conjunction with the S-GSVD

 low computational complexity of the S-GSVD

 for intermediate and high SNR regimes similar accuracy as the S-GSVD, where

is required

 convergence with two or three iterations

 applied in conjunction with the subspace-based parameter estimation techniques, e.g., R-D Standard Tensor-ESPRIT (R-D STE)

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Simulations

Subspace Prewhitening for Multi-dimensional Colored Noise RMSE vs. Correlation Level

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Simulations

Subspace Prewhitening for Multi-dimensional Colored Noise RMSE vs. Number of Iterations

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Outline

Prewhitening

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MIMO-OFDM System (1)

 Time dimensions: period and frames

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MIMO-OFDM System (2)

K: number of receive antennas. M: number of transmit antennas.

N: number of time-slots in the whole time frame.

P: number of symbol periods in each time-slot.

F: number of subcarriers

Our objective is to estimate S and H from the noisy observations Y. .

Known.

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MIMO-OFDM System (3)

 Matrix representation

 Tensor representation

Solved first!

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State-of-the-art MIMO-OFDM Schemes

 Existing Solution: Alternating Least Squares (ALS) Receiver

Drawback: iterative, higher complexity, requires pilot symbols (loss in transmission efficiency)

 Proposed Solution I: Least Squares Khatri-Rao factorization (LS-KRF) Closed-form, lower complexity for medium-to-high SNRs, requires

pilot symbols (loss in transmission efficiency)

 Proposed Solution II: Simplified Closed-form PARAFAC

Avoid the knowledge on the first row in the symbol matrix

Closed-form, lower complexity, same performance of the pilot symbols based schemes for intermediate and high SNR regimes (high transmission efficiency)

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MIMO-OFDM Simulations (1)

51 -15 -10 -5 0 5 10 15 20 25 30 10-4 10-3 10-2 10-1 SNR (dB) B it E rr o r R a te

Bit Error Rate vs. SNR @ K=2, M=4, F=4, N=5, P=3

ALS (₁=₂=0.0001) (P-)LS-KRF -15 -10 -5 0 5 10 15 20 25 30 10-3 10-2 10-1 100 SNR (dB) N M S E

Channel estimate NMSE vs. SNR @ K=2, M=4, F=4, N=5, P=3

ALS (₁=₂=0.0001)

(P-)LS-KRF

Parameter Settings:

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-15 -10 -5 0 5 10 15 20 25 30 0 0.02 0.04 0.06 SNR (dB) M e a n P ro c e s s in g T im e ( s

) Mean Processing Time vs. SNR @ K=2, M=4, F=4, N=5, P=3

-15 -10 -5 0 5 10 15 20 25 30 5 10 15 20 SNR (dB) N u m b e r o f It e rt a ti o n s

Number of Iterations in ALS vs. SNR @ K=2, M=4, F=4, N=5, P=3

ALS (

1=2=0.0001)

LS-KRF P-LS-KRF

No. of Iters. Outer (

1=0.0001)

No. of Iters. Inner (₂=0.0001)

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Parameter Settings I:

K=2, M=4, F=4, N=5, P=3

-15 -10 -5 0 5 10 15 20 25 30 10-3 10-2 10-1 SNR (dB) B it E rr o r R a te

Bit Error Rate vs. SNR @ K=2, M=4, F=4, N=5, P=3 (P-)LS-KRF (w/ Overhead) S-CFP w/ Pairing (w/o Overhead)

-15 -10 -5 0 5 10 15 20 25 30 10-3 10-2 10-1 100 101 SNR (dB) N M S E

Channel estimate NMSE vs. SNR @ K=2, M=4, F=4, N=5, P=3 (P-)LS-KRF (w/ Overhead) S-CFP w/ Pairing (w/o Overhead)

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Outline

Prewhitening

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Conclusions

 In this presentation, we have present our state-of-the-art proposed schemes for

 array interpolation

 model order selection (MOS)

 subspace prewhitening

 Multi-Dimensional Array Interpolation: measurements fit to PARAFAC structure

 Important contributions in the MOS field

 Modified Exponential Fitting Test (M-EFT): Matrix data contaminated by white noise

 R-D EFT: Tensor data contaminated by white noise

 Important contributions in the subspace prewhitening field

 Deterministic prewhitening: Matrix data and noise with correlation structure

 Sequential GSVD: Tensor data and noise with tensor structure

 Iterative Sequential GSVD: Tensor data and noise with tensor structure No availability of noise samples

 In the MIMO-OFDM field:

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Thank you for your attention!

Prof. Dr.-Ing. João Paulo C. Lustosa da Costa

University of Brasília (UnB)

Department of Electrical Engineering (ENE)

Laboratory of Array Signal Processing

PO Box 4386

Zip Code 70.919-970, Brasília - DF