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
de Brasília
Homepage:Universidade
http://www.pgea.unb.br/~lasp
Laboratório de Processamento de Sinais em Arranjos
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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
UnB
Universidade de Brasília
Laboratório de Processamento de Sinais em Arranjos
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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
UnB
Universidade de Brasília
Laboratório de Processamento de Sinais em Arranjos
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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 20,6oC, avrg min 17oC, avrg max 26,6oC) 
• Several amazing waterfalls around Brasília 
– Itiquira, Pirinópolis, Chapada dos Veadeiros and others
• Cheap tickets to Rio de Janeiro and to the Northeast of Brazil 
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Universidade de Brasília: A Short Overview (2)
Universidade de Brasília
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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
Data retrieved from: http://www.dpo.unb.br/cursos_graduacao.php

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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
UnB
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Motivation (1)

Mobile network subscribers forecast [1]
 ~90% of mobile subscriptions will be for mobile broadband by the end of 2020.
[1] Ericsson Mobility Report, Nov 2014
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Motivation (2)

Mobile network traffic forecast [1]
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Motivation (3)

5th generation mobile networks (5G) [2]
Generation
1G
2G
3G
4G
5G
Year
1981
1991
2001
2011
2020
< 1 kbps
9,6 kpbs
144 – 2000 kpbs
Throughput
100 – 1000 Mbps
100 Mbps
– 10 Gbps
 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
 ….
[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
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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
UnB
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Antenna Array Based Systems (1)

Standard (Matrix) Array Signal Processing
 Four gains: array gain, diversity gain, spatial multiplexing gain and
interference reduction gain
RX
TX
 Array gain: 3 for each side
 Diversity gain: same information for each path
 Spatial multiplexing gain: different information for each path
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Antenna Array Based Systems (2)

Standard (Matrix) Array Signal Processing
 Four gains: array gain, diversity gain, spatial multiplexing gain and
interference reduction gain
RX
TX
 Array gain: 3 for each side
 Diversity gain: same information for each path
 Spatial multiplexing gain: different information for each path
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Antenna Array Based Systems (3)

Standard (Matrix) Array Signal Processing
 Four gains: array gain, diversity gain, spatial multiplexing gain and
interference reduction gain
RX
TX
Interferer
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Antenna Array Based Systems (3)

Standard (Matrix) Array Signal Processing
 Four gains: array gain, diversity gain, spatial multiplexing gain and
interference reduction gain
RX
TX
Interferer
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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
UnB
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Multi-Dimensional Array Signal Processing (1)

MIMO channel model
Direction of Departure (DOD)
Transmit array: 1-D or 2-D
Direction of Arrival (DOA)
Receive array: 1-D or 2-D
Delay
Frequency
Doppler shift
Time
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Multi-Dimensional Array Signal Processing (2)

Multi-Dimensional Array Signal Processing
 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 Array Signal Processing (3)

Multi-Dimensional (Tensor) Array Signal Processing
 Advantages: increased identifiability, separation without imposing
additional constraints and improved accuracy (tensor gain)
RX: Uniform
Rectangular Array (URA)
 9 x 3 matrix: maximum rank is 3.
• Solve maximum 3 sources!
m1
1
1
1
2
m2
1
2
3
1
2
2
3
3
3
2
3
1
2
3
n
1
2
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Multi-Dimensional Array Signal Processing (4)

Multi-Dimensional (Tensor) Array Signal Processing
 Advantages: increased identifiability, separation without imposing
additional constraints and improved accuracy (tensor gain)
1
2
m1
3
1
RX: Uniform
Rectangular Array (URA)
2 3 1
m2
2
3
n
 3 x 3 x 3 tensor: maximum rank is 5 [3].
• Solve maximum 5 sources!
[3] J. B. Kruskal. Rank, decomposition, and uniqueness for 3-way and N-way arrays.
Multiway Data Analysis, pages 7–18, 1989
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Multi-Dimensional Array Signal Processing (5)

Multi-Dimensional (Tensor) Array Signal Processing
 Advantages: increased identifiability, separation without imposing
additional constraints and improved accuracy (tensor gain)
• 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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+
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Multi-Dimensional Array Signal Processing (6)

Multi-Dimensional (Tensor) Array Signal Processing
 Advantages: increased identifiability, separation without imposing
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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Multi-Dimensional Array Signal Processing (7)
Measurements
Array
Interpolation
Model order
selection
Is the noise
colored?
No
Parameter
Estimation
Yes
Subspace
Prewhitening





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
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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
UnB
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Multi-Dimensional Array Interpolation (1)

Data model

Unfoldings and sample covariance matrices
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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.
• SPS, FBA and ESPRIT can be applied.
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Multi-Dimensional Array Interpolation (3)


d = 3, 8 x 8 antenna array, N = 100
Standard deviation of the elements of
:
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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
UnB
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Model Order Selection (1)

Noiseless case
=

+
+
Matrix data model
Our objective is to estimate d from the noisy observations
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Model Order Selection (2)

The eigenvalues of the sample covariance matrix
d = 2, M = 8, SNR = 0 dB, N = 10
10
Finite SNR, Finite N
 M - d noise eigenvalues follow a
Wishart distribution.
 d signal plus noise eigenvalues
8
6
i

4
2
0
1
2
3
4
5
6
7
Eigenvalue index i
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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
d = 3, M = 8, SNR = 20 dB, N = 10
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Multi-Dimensional Model Order Selection (1)
Noiseless data representation
=
+
+
Problem
where
is the colored noise tensor.
Our objective is to estimate d from the noisy observations
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Multi-Dimensional Model Order Selection (2)

R-D exponential profile
 We can define global eigenvalues
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Multi-Dimensional Model Order Selection (3)

R-D exponential profile
 Comparison between the global eigenvalues profile and the profile
of the last unfolding
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Multi-Dimensional Model Order Selection (4)
Model Order Selection in Additive White Gaussian Noise Scenario
Probability of correct Detection vs. SNR


White Gaussian noise

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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
UnB
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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

Analysis via SVD
Stochastic
prewhitening schemes
With colored
noise the d main
components are
more affected.
Deterministic
prewhitening scheme
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Simulations
Subspace Prewhitening for Colored Noise with Structure
RMSE vs. Correlation Level
The noise correlation
is known.
SE – Standard ESPRIT
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Sequential GSVD

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 (Nl)
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
STE – Standard
Tensor-ESPRIT
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Simulations
Subspace Prewhitening for Multi-dimensional Colored Noise
RMSE vs. Number of Iterations
STE – Standard
Tensor-ESPRIT
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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
UnB
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MIMO-OFDM System (1)

Time dimensions: period and frames

Communication system
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MIMO-OFDM System (2)
M: number of transmit antennas.
N: number of time-slots in the whole time frame.
P: number of symbol periods in each time-slot.
Known.
K: number of receive antennas.
Known.
F: number of subcarriers
Our objective is to estimate S and H from the noisy observations Y.
.
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MIMO-OFDM System (3)

Matrix representation

Tensor representation
Solved first!
 Pairing solved: first row of S known
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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)
Bit Error Rate vs. SNR @ K=2, M=4, F=4, N=5, P=3
Parameter Settings:
K=2, M=4, F=4, N=5, P=3
ALS (1= 2=0.0001)
(P-)LS-KRF
-1
10
-2
Channel estimate NMSE vs. SNR @ K=2, M=4, F=4, N=5, P=3
10
ALS (1= 2=0.0001)
(P-)LS-KRF
0
10
-3
10
-1
10
-4
NMSE
10
-15
-10
-5
0
5
10
SNR (dB)
15
20
25
30
-2
10
-3
10
-15
-10
-5
0
10
5
SNR (dB)
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Mean Processing Time vs. SNR @ K=2, M=4, F=4, N=5, P=3
0.06
ALS (1= 2=0.0001)
LS-KRF
P-LS-KRF
0.04
0.02
Number of Itertations
Mean Processing Time (s)
MIMO-OFDM Simulations (2)
0
-15
-10
-5
0
5
10
15
20
25
SNR (dB)
Number of Iterations in ALS vs. SNR @ K=2, M=4, F=4, N=5, P=3
30
No. of Iters. Outer (1=0.0001)
20
No. of Iters. Inner (2=0.0001)
15
10
5
-15
-10
-5
0
5
10
SNR (dB)
15
20
25
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MIMO-OFDM Simulations (3)
Bit Error Rate vs. SNR @ K=2, M=4, F=4, N=5, P=3
Parameter Settings I:
K=2, M=4, F=4, N=5, P=3
(P-)LS-KRF (w/ Overhead)
S-CFP w/ Pairing (w/o Overhead)
-1
10
Channel estimate NMSE vs. SNR @ K=2, M=4, F=4, N=5, P=3
1
10
(P-)LS-KRF (w/ Overhead)
S-CFP w/ Pairing (w/o Overhead)
-2
10
0
10
-3
NMSE
10
-15
-10
-5
0
5
10
SNR (dB)
15
20
25
30
-1
10
-2
10
-3
10
-15
-10
-5
0
10
5
SNR (dB)
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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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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:
 Simplified closed-form PARAFAC based scheme: no overhead (w/o pilots)
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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
de Brasília
Homepage:Universidade
http://www.pgea.unb.br/~lasp
Laboratório de Processamento de Sinais em Arranjos
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