Compressive Covariance Sensing

Transcription

1 EUSIPCO 2014 Lisbon, Portugal Compressive Covariance Sensing A New Flavor of Compressive Sensing Geert Leus Delft University of Technology g.j.t.leus@tudelft.nl Zhi Tian Michigan Technological University ztian@mtu.edu Daniel Romero University of Vigo dromero@gts.uvigo.es Acknowledgements: Dyonisius Dony Ariananda (TU Delft) Siavash Shakeri (TU Delft) Roberto López-Valcarce (UVigo) 1

2 Roadmap Introduction Compressive Covariance Sensing Covariance Estimation and Detection Sampler Design Advanced Techniques Open Issues Conclusions 2

3 Roadmap Introduction Compressive Covariance Sensing Covariance Estimation and Detection Sampler Design Advanced Techniques Open Issues Conclusions 3

4 Emerging Challenges (Ultra-)wideband signals Very large arrays Impulse radio Large Arrays Cognitive radio (CR) Massive MIMO Sampling rate issue Need for compressive techniques 4

5 Compressed Sensing Large bandwidths require high sampling rates Popular alternative is compressive sensing (CS) o Random linear projections of Nyquist rate samples [Donoho, 2006] [Candès et al, 2006] [Tropp, 2004] o Multiple sparse reconstruction techniques 5

6 Spectrum Estimation Most compressive spectrum estimation methods estimate the spectrum (or signal itself) using CS methods Underdetermined problems o High computational complexity o Difficult performance analysis sparsity constraint Observation: many applications just require second-order statistics (power spectrum) Overdetermined problems o Low computational complexity o Easy performance analysis o Sparsity or positivity constraints can also be included 6

7 Covariance and Spectrum Estimation Cognitive radio (CR) frequency spectrum Radar Doppler + angular spectra Radio astronomy spatial spectrum Medical Imaging resonance spectrum Seismic seismic design response spectrum 7

8 Acquisition of Wideband Signals RF circuit choices: multiple NB or single WB? multiple narrowband (NB) circuits Band 1 Band 2 Band N wideband (WB) circuit LNA LNA LNA LNA Fixed LO LO 1 LO 2 LO N AGC A/D AGC AGC AGC NB filter WB filter A/D A/D A/D SNR eff Wideband Sensing SNR eff Multiple, fixed RF chains Preset LO filter range Simple detection per BPF Single, flexible RF chain burden on A/D: f s ~ GHz complex wideband sensing Challenge: reduce the sampling rate without sacrificing bandwidth 8

9 Angular Spectrum Estimation Array processing o Imaging Optical/radar/ultrasound/acoustic Radio Astronomy Seismology o DoA estimation localization Source: IAI Inc. Acquisition and processing hardware prop. to #antenna elements Challenge: reduce number of antennas without sacrificing resolution 9

10 Roadmap Introduction Compressive Covariance Sensing o Problem definition o Covariance structures o Compression schemes o Modal Analysis Covariance Estimation and Detection Sampler Design Advanced Techniques Open Issues Conclusions 10

11 Compressive Covariance Sensing uncompressed signal compression compressed signal SOS: Problem definition: Estimate from, multiple s, or Structure: Compression operation: Remarks: NO SPARSITY NEEDED 11

12 Second-Order Statistics uncompressed signal compression compressed signal Stationary input: Auto-correlation: Power Spectral Density (PSD): Frequency domain PSD time frequency Angular domain PSD space angle 12

13 Covariance Structure All covariance matrices are Hermitian and positive semi-definite Typical structures o Toeplitz: Constant along diagonals Stationary time-signals Uniform linear arrays (ULA) Modal analysis o Circulant: Toeplitz + property diagonal in the freq. domain o -Banded Toeplitz + property Encompassing model: Basis Expansion Model (BEM) basis matrices 13

14 Covariance Structure BEM representation: Toeplitz real unknowns d-banded real unknowns Circulant real unknowns 14

15 Compression Schemes uncompressed signal compression compressed signal We consider throughout linear compression schemes Focus on: o Frequency PSD Time-domain autocorrelation Time compression o Angular PSD Space-domain autocorrelation Spatial compression 15

16 Periodic Compression # blocks Uncompressed domain Kronecker notation Compressed domain with Given and estimate 16

17 Compression in the Time Domain Compressive ADC (conceptual model) Nyquist-rate sampling MUX Periodic Acquisition 17

18 Compression in the Time Domain Implementations o Multi-coset sampling [Herley-Wong,1999][Venkataramani- Bresler,2000] o Random demodulator [Tropp et al, 2010] o Modulated wideband converter [Mishali-Eldar,2010] o Random modulator pre-integrator [Becker, 2011][Yoo et al, 2012] Modulated wideband converter [Mishali-Eldar,2010] 18

19 Compression in the Spatial Domain Uniform linear array (ULA) Periodic Acquisition 19

20 Compression in the Spatial Domain Implementation: o Sparse some antennas are active subarrays [Hoctor-Kassam, 1990], [Moffet, 1968] o Dense all antennas are active analog beamforming [Wang-Leus-Pandharipande, 2009],[Wang-Leus, 2010],[Venkateswaran-Van der Veen, 2010] [Wang-Leus, 2010] 20

21 PSD Estimation from Estimate from Dense PSD estimation o Fourier transform o Application: Frequency domain PSD estimation of time stationary signals Angular domain incoherent imaging (continuous source distribution) Sparse PSD estimation o o Application: Modal analysis Frequency domain frequency estimation of a sum of sinusoids in noise Angular domain direction of arrival (DoA) estimation (discrete source distribution) 21

22 Roadmap Introduction Compressive Covariance Sensing Covariance Estimation and Detection o Covariance Estimation Least Squares Maximum Likelihood Modal Analysis o Covariance Detection o Sample Statistics Pre-Processing Sampler Design Advanced Techniques Open Issues Conclusions 22

23 Roadmap Introduction Compressive Covariance Sensing Covariance Estimation and Detection o Covariance Estimation Least Squares Maximum Likelihood Modal Analysis o Covariance Detection o Sample Statistics Pre-Processing Sampler Design Advanced Techniques Open Issues Conclusions 23

24 Least Squares Estimation [Ariananda-Leus, 2012] [Leus-Ariananda, 2011] Major steps: 1. Identify relation between and 2. Identify relation between and 3. Estimate using (pre-processed) sample estimates 4. Invert the relation with least squares (LS) 5. Reconstruct 24

25 Least Squares Estimation [Ariananda-Leus, 2012] [Leus-Ariananda, 2011] Compressed domain Sample Estimate Least Squares Covariance Reconstruction Uncompressed domain Overdetermined system Unique reconstruction if full (column) rank Design of is critical!! 25

26 LS: Toeplitz/Circulant/Banded Matrices Toeplitz real unknowns d-banded real unknowns Circulant real unknowns 26

27 LS Estimation: PSD Power Spectrum: LS estimate: Improvements: OR OR PSD is non-negative PSD is sparse (can be relaxed) Covariance is pos. semidefinite 27

28 Simulations: Frequency PSD Estimation Least Squares reconstruction space of Hermitian Toeplitz d-banded matrices Complex baseband representation of OFDM signal: o 16-QAM data symbols o 8192 tones in band o 3072 active tones in o Cyclic prefix length of 1024 o SNR of 10 db and Start with length-42 minimal sparse ruler, Larger cases by randomly adding extra rows 28

29 MSE Simulations: Frequency PSD Estimation 10-1 Sparse Ruler (I=549) Sparse Ruler (I=1646) Sparse Ruler (I=3291) Sparse Ruler (I=5485) Nyquist (I=549) Nyquist (I=1646) Nyquist (I=3291) Nyquist (I=5485) Compression rate [M/N] MSE of the estimated PSD 29

30 Power/Frequency (db/rad/sample) Simulations: Frequency PSD Estimation Theoretical Noisy PSD Sparse Ruler ( samples, M/N=0.5) Normalized Frequency ( p rad/sample) Reconstructed PSD 30

31 Maximum Likelihood If is well designed: independent independent It suffices to estimate Maximum Likelihood (ML) estimate: 31

32 Maximum Likelihood Gaussian Gaussian Sample estimate Numerical solution Covariance matching Inverse Iteration Algorithm (IIA) Trading performance with computation [Burg et al, 1982] Pre-processed sample estimates [Romero-Leus,2013a] Asymptotic approximations: COMET [Ottersten et al, 1998], SPICE [Stoica et al, 2011], etc. 32

33 Simulations: Wideband Spectrum Sensing [Romero-Leus, 2013a] primary transmitters -th transmitter normalized PSD Estimators: received power: o LS estimators: Weighted LS and constrained & weighted LS o Approx. ML estimation: SIIA C-ADC: bandpass Gaussian signals with disjoint support white Gaussian noise 33

34 Simulations: Wideband Spectrum Sensing Strong compression ratios may result in a small performance loss 34

35 Modal Analysis: Observation Model # sources/sinusoids Time domain Space domain 35

36 Modal Analysis Uncompressed domain Compressed domain What kind of sources do we have? o Uncorrelated sources: is diagonal is Toeplitz [Pillai et al,1985] [Abramovich et al,1998,1999] [Pal-Vaidyanathan, 2010, 2011] [Shakeri-Ariananda-Leus, 2012] [Yen-Tsai-Wang, 2013], [Krieger-Kochman-Wornell, 2013] o Correlated sources? [Ariananda-Leus, 2012, 2013] 36

37 Modal Analysis: Standard Methods Correlation matrix of : More measurements than sources MUSIC: [Bresler, 2008], [Mishali-Eldar, 2009] MVDR: [Wang-Pandharipande-Leus, 2010] 37

38 Modal Analysis: Gridding-Based Methods Grid of frequencies/angles and virtual sources/sinusoids on this grid source vector is overcomplete basis for but is sparse Sparse reconstruction [Malioutov et al, 2005] [Mishali-Eldar, 2009] [Tropp et al, 2010] More measurements than sources!! 38

39 Modal Analysis: Virtual Sampler/Array Uncorrelated sources: represents a virtual sampler/array of virtual samples/antennas receiving virtual sources/sinusoids equations unknowns 39

40 Modal Analysis: Virtual Sampler/Array Problem: virtual sources are constant or fully coherent o Gridding [Shakeri-Ariananda-Leus, 2012] : LS : sparsity/positivity Sparse sampling (antenna selection): o MUSIC or MVDR with spatial smoothing Virtual sampler/array should be uniform! sparse ruler o Translate problem into circulant covariance matrix Uniform gridding required with grid points circular sparse ruler 40

41 Simulations: DoA Estimation Least Squares and MUSIC reconstruction space of Hermitian Toeplitz matrices ULA of and available antenna positions (aperture) o Minimal sparse ruler array Virtual ULA of 2x36-1 antennas o Two-level nested array Inner array of 5 and outer array of 6 antennas Virtual ULA of 2x36-1 antennas o Co-prime array 9 antennas spacing 2 and 3 antennas spacing 9 Virtual ULA of only 2x20-1 antennas 1600 time samples SNR of 0 db 41

42 Simulations: DoA Estimation Normalized Spectrum [db] 0 Least Squares Method vs. MUSIC Method LS method MUSIC method Direction of Arrival [sin( )] Reconstructed spectrum using LS and MUSIC for the minimal sparse ruler array (S=17 sources with 10 degrees of separation; for LS S=71) 42

43 Normalized Spectrum [db] Simulations: DoA Estimation 0-5 LS method MUSIC method Least Squares Method vs. MUSIC Method Direction of Arrival [sin( )] Reconstructed spectrum using LS and MUSIC for the minimal sparse ruler array (continuous source from 30 to 40 degrees; for LS S=71) 43

44 Normalized Spectrum [db] Simulations: DoA Estimation Minimal Sparse Ruler Array Two-Level Nested Array Coprime Array Direction of Arrival [sin( )] Reconstructed spectrum using MUSIC (S=21 sources with 7 degrees separation) 44

45 Roadmap Introduction Compressive Covariance Sensing Covariance Estimation and Detection o Covariance Estimation Least Squares Maximum Likelihood Modal Analysis o Covariance Detection o Sample Statistics Pre-Processing Sampler Design Advanced Techniques Open Issues Conclusions 45

46 Covariance Detection Binary hypothesis test Noise covariance Problem statement: Signal covariance Given decide or Depending on prior information: Neyman-Pearson, generalized likelihood ratio test (GLRT), Bayesian, etc GLRT: [Kay, 1998] 46

47 Covariance Detection [Romero-Leus,2013a] [VázquezVilar-LópezValcarce, 2011] GLRT: alternative formulation Exact computation of ML estimates expensive Alleviate computational cost o Replace ML estimates by approximations o Smoothing/cropping sample statistics BEM formulation: given 47

48 Covariance Detection: Spectrum Sensing Wideband spectrum sensing formulation: #signals Power -th signal [Romero-Leus,2013a] [VázquezVilar-LópezValcarce, 2011] Is the -th primary user transmitting GLRT: ML estimator under Efficiency approximate ML estimators ML estimator under 48

49 Simulation: Wideband Spectrum Sensing Test on the user with primary transmitters -th transmitter normalized PSD Estimators: o LS estimators: WLS, CWLS o ML estimator: LIKES received power: o Approx. ML estimation: SSPICE, SIIA, SLIKES C-ADC: bandpass Gaussian signals with disjoint support white Gaussian noise [Romero-Leus,2013a] 49

50 Simulation: Wideband Spectrum Sensing Approximate ML estimators achieve a similar performance at a much reduced cost 50

51 Roadmap Introduction Compressive Covariance Sensing Covariance Estimation and Detection o Covariance Estimation Least Squares Maximum Likelihood Modal Analysis o Covariance Detection o Sample Statistics Pre-processing Sampler Design Advanced Techniques Open Issues Conclusions 51

52 Sample Statistics Pre-Processing LS and ML work on Sometimes several observations of are available Dual domain averaging Spatial auto-correlation average along time Temporal auto-correlation average along space 52

53 Sample Statistics Pre-Processing Dual domain averaging 53

54 Sample Statistics Pre-Processing MUX MUX MUX Dual domain averaging 54

55 Sample Statistics Pre-Processing If a single observation of is given o Raw sample estimate may result in poor performance o Smoothing exploiting periodic structure: stationary Rows of stationary Smoothed estimate Controlling the bias/variance trade-off windowing/cropping o Given blocks make be with o May also help to control complexity 55

56 Roadmap Introduction Compressive Covariance Sensing Covariance Estimation and Detection Sampler Design o Design criteria o Sparse samplers o Dense samplers Advanced Techniques Open Issues Conclusions 56

57 Sampler Design: Compression Model General structure for o Temporal compression o Spatial compression o Model can represent o Periodic sampling o Non-periodic sampling o Sparse sampling o Dense sampling 57

58 Design Problem # blocks Uncompressed domain Compression Ratio Compressed domain Goals Find conditions for to allow estimation of from Maximize the compression ratio among the admissible samplers 58

59 Covariance Structure Uncompressed domain Compressed domain Dimension: Modal analysis Multi-band signal Toeplitz subspace Banded subspace Circulant subspace 59

60 Design Criteria A matrix defines an -covariance sampler if the associated function is invertible. The identifiability of is preserved linearly independent linearly independent Focus on Toeplitz subspaces banded subspaces Circulant subspaces Universal cov. samplers 60

61 Sparse Samplers Architectures: o Space domain sampling Sub-array o Time domain sampling C-ADC Set representation: 61

62 (Linear) Sparse Rulers Difference set: w/o repetition Sparse ruler: is a length- sparse ruler Minimal sparse ruler [Rédei-Rényi, 1949] [Leech, 1956] [Wichmann, 1963] [Moffet, 1968] [Miller, 1971] [Wild, 1987] [Pearson et al, 1990] [Linebarger et al, 1993] [Ariananda-Leus, 2012] Suboptimal designs: nested, co-prime, [Wichmann, 1963] [Pearson et al, 1990] [Linebarger et al, 1993] [Pumphrey, 1993] [Pal-Vaidyanathan, 2010] [Pal-Vaidyanathan, 2011] 62

63 Circular Sparse Rulers Modular difference set: w/o repetition Circular sparse ruler: is a length- circular sparse ruler Minimal circular sparse ruler [Singer, 1938] [Miller, 1971] [Ariananda-Leus, 2012] [Romero-Leus, 2013b] [Krieger-Kochman-Wornell, 2013] [Romero-LópezValcarce-Leus, 2014] 63

64 Sparse Samplers: Toeplitz Subspace Sparse sampler covariance sampler Toeplitz subspace Optimum Sampler minimal linear sparse ruler linear sparse ruler [Rédei-Rényi, 1949] [Leech, 1956] [Pearson et al, 1990][Romero-LópezValcarce-Leus, 2014] 64

65 Sparse Samplers: Circulant Subspace Sparse sampler covariance sampler Circulant subspace Optimum Sampler minimal circular sparse ruler circular sparse ruler [Romero-LópezValcarce-Leus, 2014] 65

66 Sparse Samplers: Banded Subspace Sparse sampler -banded subspace linear sparse ruler covariance sampler circular sparse ruler incomp. sparse ruler [Ariananda-Leus, 2012][Romero-LópezValcarce-Leus, 2014] 66

67 Dense Samplers Architectures: o Space domain sampling analog beamforming o Time domain sampling C-ADC Design: o Similar to CS random designs o Existing random designs Continuous distributions Cov. samplers with probability one Attain compression limits [Romero-LópezValcarce-Leus, 2014] 67

68 Random Sampling: Compression Limits drawn from a cont. distrib. -covariance sampler a.s. iff Toeplitz subspace: Banded subspace: Circulant subspace: [Romero-LópezValcarce-Leus, 2014] 68

69 Optimal Samplers: Summary (transpose the table) 69

70 Sampler Design: Summary 70

71 Roadmap Introduction Compressive Covariance Sensing Covariance Estimation and Detection Sampler Design Advanced Techniques o Cyclostationarity o Multiband spectrum estimation o Distributed Compressive Covariance Sensing o Dynamic Sampling Open Issues Conclusions 71

72 Roadmap Introduction Compressive Covariance Sensing Covariance Estimation and Detection Sampler Design Advanced Techniques o Cyclostationarity Estimation Sampler Design o Multiband spectrum estimation o Distributed Compressive Covariance Sensing o Dynamic Sampling Open Issues Conclusions 72

73 Cyclostationary modulated signals Cyclic features reveal critical signal parameters: o carrier frequency o symbol rate o modulation type o timing, phase etc. Non-cyclic signals (e.g. noise) do not possess cycle frequencies 2x Fourier Transform Periodic autocorrelation Cyclic spectrum 73

74 Low SNR High SNR Noise Suppression in the Cyclic Domain Energy detection vs. cyclic feature detection e.g., [Sahai-Cabric, 2005] Power spectrum density (PSD) (a = 0) Spectral correlation density (SCD) Multi-harmonics peaks at no noise components when 74

75 Source Separation in Cyclic Domain Spectral correlation density (SCD) e.g., [Gardner, 1988] BPSK in noise plus five AM interferers BPSK signal alone Overlapping in PSD, separable in SCD White noise plus five AM interferences Cyclostationarity-based approach to detection o resilient against Gaussian noise o robust to multipath o can differentiate modulation types and separate interference o insensitive to unknown signal parameters 75

76 Second-order Cyclic Statistics uncompressed signal compression compressed signal Suppose is periodic in t with period T T=1: x[t] stationary T>1: x[t] cyclostationary Periodic autocorrelation and cyclic spectrum Cyclic frequency: Stack samples over a block of multiple ( o # samples per block: Frequency: f ) cyclic periods 76

77 Covariance Structure Cyclostationary signals with period T Covariance structure o Within one cyclic period: dense, with DoF = T 2 e.g. o Over a block of N cyclic periods block Toeplitz DoF 77

78 Compressive samples Compressive measurements: M samples per block Corss-correlation of compressive samples v.s. It can be shown that 78

79 Vector-form Relationships: Covariances Assuming limited support we obtain is a block-circulant matrix, which allows for block-diagonalization 79

80 Vector-form Relationships: Cyclic Spectrum Since, delays in Hence, uniquely determined by samples in f Since is related to by DFT and DTFT operations: is a invertible matrix, so 80

81 Cyclic Spectrum Reconstruction If then we can use simple least squares (LS) When this is only possible if (no compression) When (e.g., when ) this is possible even for When the period of the sampler is larger than the period of the cyclostationarity, we can reconstruct the cyclic spectrum by LS Advantages Computational simplicity Performance guarantees (e.g., probability of false alarm) 81

82 power Cyclic Spectrum Estimation for CR C-ADC (sub-nyquist) Sparse cyclic spectrum reconstruction Cyclic feature detection Cyclic feature classification Wideband aspect: multiple signal sources frequency 82

83 Example: Robustness to Rate Reduction Probability of detection vs. compression ratio (P FA = 0.1, #blocks=32, 200) 50% compression 50% compression Monitored band f max < 300 MHz 2 sources (noise-free): PU 1 - BPSK at 150MHz; PU 2 - QPSK at 225MHz; Ts= μs Cisco DSSS Spread spectrum 83

84 Example: Robustness to Noise Uncertainty Receiver Operating Characteristic (ROC): P D vs P FA (SNR=5dB, 50% compression) cyclic Energy detection (noise uncertainty = 0, 1, 2, 3dB) Outperforms energy detection and insensitive to noise uncertainty 84

85 Roadmap Introduction Compressive Covariance Sensing Covariance Estimation and Detection Sampler Design Advanced Techniques o Cyclostationarity Estimation Sampler Design o Multiband spectrum estimation o Distributed Compressive Covariance Sensing o Dynamic Sampling Open Issues Conclusions 85

86 Sampler Design for Cyclostationary Signals Sampler design for reconstructing cyclic spectrum o Multiple samplers used periodically across N blocks [Leus-Tian, 2011] Original signal x[n] Structure of R x Compressed signal y[n] Stationary Toeplitz Cyclostationary Cyclostationary (period T) Block Toeplitz (block size NT ) Cyclostationary (period M) Proposition: it is possible to reconstruct nonsparse 2D cyclic spectrum in closed form when Circular sparse ruler 86

87 Cyclostationary Signals: Intuition Stationary signal Input: stationary with cross-correlations Output: cyclostationary with period cross-correlations Increase of degrees of freedom! 87

88 Cyclostationary Signals: Intuition Cyclostationary signal Input: cyclostationary with period cross-correlations Output: cyclostationary with period Q cross-correlations No increase of degrees of freedom! 88

89 Cyclostationary Signals: Intuition [Leus-Tian, 2011] Cyclostationary signal Input: cyclostationary with period cross-correlations Output: cyclostationary with period cross-correlations To increase degrees of freedom the period of the sampler needs to be larger than the period of the cyclostationarity! 89

90 System of Equations selection matrix collection of correlations of at blocks-lag collection of correlations of at blocks-lag Our goals: reconstruct from using least squares (LS) must have full column rank. 90

91 Minimal Sparse Ruler Sampling One possible sampling pattern that leads to a full column rank minimal sparse ruler based design Example: Other are set to identity matrix Minimal sparse ruler based design compression for the case when each empty matrix obtain the minimum is equal to either [] or 91

92 Example Consider have: based on length-5 minimal sparse ruler, we and All have full column rank 92

93 Exploiting Correlations Between -blocks 93

94 Exploiting Correlations Between -blocks System of equations: correlations of at blocks-lag When is not constrained to either [] or greedy algorithm [Ariananda-Leus, 2014] to design sub-optimal compression rate 94

95 Exploiting Correlations Between -blocks Example:, and as well as full column rank! full column rank! full column rank! full column rank! 95

96 Exploiting Correlations Between -blocks constrained to or for each Maximum compression minimal circular sparse ruler Example: Other are set to [Ariananda-Leus, 2014] 96

97 Achieved Compression Rate Compression Rate Minimal Circular Sparse Ruler (all T) Greedy Algorithm T=18 Greedy Algorithm T=22 Greedy Algorithm T=26 Greedy Algorithm T= The value of N 97

98 Roadmap Introduction Compressive Covariance Sensing Covariance Estimation and Detection Sampler Design Advanced Techniques o Cyclostationarity o Multiband spectrum estimation o Distributed Compressive Covariance Sensing o Dynamic Sampling Open Issues Conclusions 98

99 Temporal Compression analog-to-information converter (AIC) MUX Nyquistrate samples 99

100 Spatial Compression uniform linear array (ULA) [Krieger et al, 2013] 100

101 Data Model Compute the DTFT: and Nyquist-rate case vector With compression vector: We then have: 101

102 Multi-bin Approach Consider the multiband model and divide into [Mishali-Eldar, 2009] bins... Write and collect in We have an IDFT matrix 102

103 Diagonal/Circulant Correlation Matrix The correlation matrix of : If the bin size is equal to the largest band is diagonal [Yen-Tsai-Wang, 2013] is circulant [Ariananda-Romero-Leus, 2013, 2014] 103

104 Diagonal/Circulant Correlation Matrix Given If has full column rank, we can perform LS ( ) ML methods can also be employed [Romero-Leus, 2013] Multicoset sampling/array full column rank universal and M/N > 1/2 [Yen-Tsai-Wang, 2013] full column rank circular sparse ruler [Ariananda-Romero-Leus, 2013, 2014] 104

105 Sample Estimates Q: Over which domain do we average? A: The dual domain! Spatial-domain Compression

106 Sample Estimates Time-domain Compression MUX MUX MUX

107 Simulations Number of samples: block size: Number of blocks: Sensing based on length- minimal circular sparse ruler => active cosets: Averaged over 200 independent realizations Six bands (user signals) => details can be found in the paper. White Gaussian noise for each realization; variance: 107

108 Normalized Magnitude/Frequency (mw/radian/sample) Simulations 200 M/N=0.278 Nyquist rate p ( 2p) (radian) 108

109 Roadmap Introduction Compressive Covariance Sensing Covariance Estimation and Detection Sampler Design Advanced Techniques o Cyclostationarity o Multiband spectrum estimation o Distributed Compressive Covariance Sensing o Dynamic Sampling Open Issues Conclusions 109

110 Distributed Sensing: Partial Observation Each sensor cluster may estimate a subset of the total lags Sampling rate reduction LS estimation [Ariananda-Romero-Leus, 2014] 110

111 Distributed Sensing: Sampler Design Necessary condition Number of sensor clusters Modular dif. set Non-overlapping circular Golumb rulers [Ariananda-Romero-Leus, 2014] 111

112 Simulation 112

113 Roadmap Introduction Compressive Covariance Sensing Covariance Estimation and Detection Sampler Design Advanced Techniques o Cyclostationarity o Multiband spectrum estimation o Distributed Compressive Covariance Sensing o Dynamic Sampling Open Issues Conclusions 113

114 Dynamic Sampling Changing sub-array configurations allows detecting more correlated sources than antennas Uncorrelated sources: represents a virtual sampler/array of virtual samples/antennas receiving virtual sources/sinusoids equations unknowns 114

115 Dynamic Sampling Assume the sources are correlated [Ariananda-Leus, 2012] Problem: degrees of freedom are not increased Solution: periodic compression with slots per period o Periodic (in time) antenna selection o Periodic (in space) sample selection 115

116 Correlated Sources Combining the correlations from slots represents a virtual sampler/array of virtual samples/antennas receiving virtual sources Spatial smoothing not directly possible, so gridding left: Over-determined: LS Under-determined: sparsity or positivity constraints Universality: every pair of samples appears in a period 116

117 Roadmap Introduction Compressive Covariance Sensing Covariance Estimation and Detection Sampler Design Advanced Techniques Open Issues Conclusions 117

118 Open Issues Super resolution Big data Combining spatial and temporal domains Extensions to Doppler spectrum, imaging, Applications to radar, MRI, seismic, radio astronomy, 118

119 Open Issues Large-scale networks Internet backbone network (Abilene) Disease gene network Sample size issue sparsity-enforcing regularization 119

120 Open Issues Sparse event detection using large-scale wireless sensor nets o o Local phenomena induce spatially sparse signals Desiderata: energy efficiency, scalability, robustness Distributed optimization [Ling-Tian, 2010, 2011] Structural Health Monitoring (13,1) (13,2) (13,3) (13,4) (13,5) (13,6) (13,7) (13,8) (13,9) (13,10) Floor 12 (12,1) (12,2) (12,3) (12,4) (12,5) (12,6) (12,7) (12,8) (12,9) (12,10) Floor 11 (11,1) (11,2) (11,3) (11,4) (11,5) (11,6) (11,7) (11,8) (11,9) (11,10) Floor 10 (10,1) (10,2) (10,3) (10,4) (10,5) (10,6) (10,7) (10,8) (10,9) (10,10) Floor 9 (9,1) (9,2) (9,3) (9,4) (9,5) (9,6) (9,7) (9,8) (9,9) (9,10) Floor 8 (8,1) (8,2) (8,3) (8,4) (8,5) (8,6) (8,7) (8,8) (8,9) (8,10) Centralized FUSION CENTER Scalability Robustness Infrastructureless Decentralized FUSION CENTER Floor 7 Floor 6 Floor 5 Floor 4 Floor 3 (7,1) (7,2) (7,3) (7,4) (7,5) (7,6) (7,7) (7,8) (7,9) (7,10) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) (6,7) (6,8) (6,9) (6,10) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (5,7) (5,8) (5,9) (5,10) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (4,7) (4,8) (4,9) (4,10) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (3,7) (3,8) (3,9) (3,10) Floor 2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (2,7) (2,8) (2,9) (2,10) Floor 1 Bay 1 Bay 2 Bay 3 Bay 4 Bay 5 Bay 6 Bay 7 Bay 8 Bay 9 Bay 10 [Ling-Tian et al, 2009] 120

121 Roadmap Introduction Compressive Covariance Sensing Covariance Estimation and Detection Sampler Design Advanced Techniques Open Issues Conclusions 121

122 Conclusions: CS vs. CCS Compressed Covariance Sensing Compressed Sensing Aims at recovering statistics Aims at recovering the signal Lossy Lossless Use sparse sampling Use random sampling No sparsity is required Sparsity is required Linear/non-linear reconstruction Non-linear reconstruction Overdetermined Underdetermined 122

123 Conclusions Aiming at reconstructing the second-order statistics o Compressive samples o Linear compression o Linear/non-linear covariance structures Estimation/detection: o LS o ML o GLRT Sampler design o Sparse samplers o Dense samplers Advanced techniques 123

124 Thank you! 124

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