Multi-Dimensional Array Signal Processing Applied to MIMO Systems

Transcrição

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 , Brasília - DF Homepage: 1

2 Outline : 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 2

4 : A Short Overview (1) (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 20,6 o C, avrg min 17 o C, avrg max 26,6 o C) Several amazing waterfalls around Brasília Itiquira, Pirinópolis, Chapada dos Veadeiros and others Cheap tickets to Rio de Janeiro and to the Northeast of Brazil 4

5 : A Short Overview (2) 5

6 : A Short Overview (3) (UNB) In 2013, around candidates for 4219 places In 2013, 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: 6

8 Motivation (1) Mobile network subscribers forecast [1] ~90% of mobile subscriptions will be for mobile broadband by the end of [1] Ericsson Mobility Report, Nov

9 Mobile network traffic forecast [1] Motivation (2) 5

10 Motivation (3) 5 th generation mobile networks (5G) [2] Generation 1G 2G 3G 4G 5G Year < 1 kbps 9,6 kpbs kpbs Mbps 100 Mbps Throughput 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 , June

12 Antenna Array Based Systems (1) Standard (Matrix) Array Signal Processing 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 Spatial multiplexing gain: different information for each path 5

13 Antenna Array Based Systems (2) Standard (Matrix) Array Signal Processing 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 Spatial multiplexing gain: different information for each path 5

14 Antenna Array Based Systems (3) Standard (Matrix) Array Signal Processing Four gains: array gain, diversity gain, spatial multiplexing gain and interference reduction gain TX RX Interferer 5

15 Antenna Array Based Systems (3) Standard (Matrix) Array Signal Processing Four gains: array gain, diversity gain, spatial multiplexing gain and interference reduction gain TX RX Interferer 5

17 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 Frequency Time Delay Doppler shift 5

18 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 5

19 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! m m n

20 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) RX: Uniform m m n Rectangular Array (URA) 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,

21 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. = + + 5

22 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 5

23 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 23

25 Multi-Dimensional Array Interpolation (1) Data model Unfoldings and sample covariance matrices 25

26 Multi-Dimensional Array Interpolation (2) Power response for r-th dimension 26

27 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. 27

28 Multi-Dimensional Array Interpolation (3) d = 3, 8 x 8 antenna array, N = 100 Standard deviation of the elements of : 28

30 Model Order Selection (1) Noiseless case = + + Matrix data model Our objective is to estimate d from the noisy observations. 30

31 Model Order Selection (2) The eigenvalues of the sample covariance matrix Finite SNR, Finite N 10 d = 2, M = 8, SNR = 0 db, N = 10 M - d noise eigenvalues follow a Wishart distribution. 8 6 d signal plus noise eigenvalues i Eigenvalue index i 31

32 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 32

33 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. 33

34 Multi-Dimensional Model Order Selection (2) R-D exponential profile We can define global eigenvalues 34

35 Multi-Dimensional Model Order Selection (3) R-D exponential profile Comparison between the global eigenvalues profile and the profile of the last unfolding 35

36 White Gaussian noise Multi-Dimensional Model Order Selection (4) Model Order Selection in Additive White Gaussian Noise Scenario Probability of correct Detection vs. SNR 36

38 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. 38

39 Noise Analysis Analysis via SVD Stochastic prewhitening schemes With colored noise the d main components are more affected. Deterministic prewhitening scheme 39

40 Simulations Subspace Prewhitening for Colored Noise with Structure RMSE vs. Correlation Level The noise correlation is known. SE Standard ESPRIT 40

41 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 41

42 Simulations Subspace Prewhitening for Multi-dimensional Colored Noise RMSE vs. Number of Samples without Signal Components (N l ) STE Standard Tensor-ESPRIT 42

43 Iterative S-GSVD In some multidimensional applications, the noise samples available without the presence of signal components are not 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) 43

44 Simulations Subspace Prewhitening for Multi-dimensional Colored Noise RMSE vs. Correlation Level STE Standard Tensor-ESPRIT 44

45 Simulations Subspace Prewhitening for Multi-dimensional Colored Noise RMSE vs. Number of Iterations STE Standard Tensor-ESPRIT 45

47 MIMO-OFDM System (1) Time dimensions: period and frames Communication system

48 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..

49 MIMO-OFDM System (3) Matrix representation Tensor representation Solved first! Pairing solved: first row of S known

50 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) 50

51 NMSE MIMO-OFDM Simulations (1) 10-1 Bit Error Rate vs. K=2, M=4, F=4, N=5, P=3 ALS ( 1 = 2 =0.0001) (P-)LS-KRF Parameter Settings: K=2, M=4, F=4, N=5, P= Channel estimate NMSE vs. K=2, M=4, F=4, N=5, P=3 ALS ( 1 = 2 =0.0001) (P-)LS-KRF SNR (db) SNR (db) 51

52 Number of Itertations Mean Processing Time (s) MIMO-OFDM Simulations (2) Mean Processing Time vs. K=2, M=4, F=4, N=5, P=3 ALS ( 1 = 2 =0.0001) LS-KRF P-LS-KRF SNR (db) Number of Iterations in ALS vs. K=2, M=4, F=4, N=5, P= No. of Iters. Outer ( 1 =0.0001) No. of Iters. Inner ( 2 =0.0001) SNR (db) 52

53 NMSE MIMO-OFDM Simulations (3) 10-1 Bit Error Rate vs. K=2, M=4, F=4, N=5, P=3 (P-)LS-KRF (w/ Overhead) S-CFP w/ Pairing (w/o Overhead) Parameter Settings I: K=2, M=4, F=4, N=5, P= Channel estimate NMSE vs. K=2, M=4, F=4, N=5, P=3 (P-)LS-KRF (w/ Overhead) S-CFP w/ Pairing (w/o Overhead) SNR (db) SNR (db) 53

55 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 In the MIMO-OFDM field: No availability of noise samples Simplified closed-form PARAFAC based scheme: no overhead (w/o pilots) 55

56 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 , Brasília - DF Homepage: 56