Report

Aimé Lay-Ekuakille University of Salento Index: 1. 2. 3. 4. 5. 6. 7. 8. 9. Problem statement Main motivation FDM-Filter Diagonalization Method (mono) DSD-Decimate Signal Diagonalization (mono) Application for detection in pipeline FDM-Multidimensional DSD-Multidimensional Application for EEG Final outlook Problem statement 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook Many engineering and physicals issues deal with particles decaying problems, namely, NMR, fMRI, new resisting vegetables, industrial processes using radioactivity, light and photonics, etc. These issues can be modeled using special transforms and particular descriptions (Poisson) Main motivation 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook Traditional methods FFT and Laplace transform are not suitable for above problems. Recently, new methods have been introduced, termed Filter Diagonalization Method (FDM) and Decimated Signal Diagonalization (DSD), for obtaining the complete eigenspectra of arbitrarily large matrices that are theoretically generated with auto-correlation functions from time propagated wave packets. Using the equivalence between the auto-correlation functions and the exponentially damped signals spectrum is obtained as sums of pure Lorentzians FDM-Filter Diagonalization Method (mono) 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook In FDM, we would like to fit the diagonalization measured complex valued signal Cn as a sum of damped sinusoids: = exp(− =1 It can be represented as the time autocorrelation function of a fictitious dynamical system with non-Hermitian but symmetric Hamiltonian = 0 | − 0 ≡ 0 | 0 so that the highly nonlinear fitting problem is reduced to that of diagonalization , the evolution operator over a single time step FDM-Filter Diagonalization Method (mono) 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook Neither the explicit form of the Hamiltonian nor the ‘initial state’ 0 need be known, as only a matrix representation in specific basis is required for a numerical solution. The primitive basis is iteratively derived by letting act on 0 = 0 0 so that the overlap matrix element and the 1 matrix elements of are given by the measured data 0 = | = + 1 = | = ++1 FDM-Filter Diagonalization Method (mono) 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook The extraction of the eigenvalues, = exp(− , that determine line position and width, and the eigenvectors, , that determine amplitude and phase, results by solving a generalized eigenvalue problem of the form 1 = 0 FDM-Filter Diagonalization Method (mono) Frequency and amplitude are calculated from: 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook 1 = − ∡ 2 = FDM-Filter Diagonalization Method (mono) FDM Flowchart 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook DSD-Decimate Signal Diagonalization(mono) 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook The diagonalization procedure extracts all of relevant peak parameters, the complex frequencies and amplitudes, , , from window min , max . The DSD technique is restricted to signals that are given as sums of damped exponentials. Therefore, we model the band-limited decimated signal as: = − , Im < 0 =1 where the condition < 0 selects only those physically meaningful harmonics that decay exponentially with increasing time and k is the socalled local spectral rank which is equal to the number of Lorentzians generated by the equation above DSD-Decimate Signal Diagonalization(mono) 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook Like FDM, DSD uses windows to reduce a large data matrix to a number of simple ones before diagonalization. However, while FDM filters basis functions to create its windows, DSD filters the time signal. A time signal is processed to get a low-resolution spectrum by DFT. This spectrum is divided into M windows containing at most 200 data points to avoid an illposed problem. DSD-Decimate Signal Diagonalization(mono) 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook A new signal is created for each window setting to zero the content outside the window and then recentering the window at zero. The inverse DFT is performed to convert the frequency data back into the time domain. The decimation step occurs when this new time signal is sampled at M times greater than the original time step, creating a bandlimited decimated signal, which is diagonalized to extract the spectral parameters for the matrix overlapping U0d and U1d. The diagonalization procedure is realized for each of the M signals in this way = − , Im < 0 =1 DSD-Decimate Signal Diagonalization(mono) 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook The ωk are extracted from the eigenvalues u1k=e-jω1kτ of the operator = − 1 = − ∡ while the amplitude calculated as parameters 2 −1 = 0| 2 = 2 = =0 are DSD-Decimate Signal Diagonalization(mono) We normalize the as 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook 0 = ′ And we build spectrum from = =1 + 2 DSD-Decimate Signal Diagonalization(mono) DSD Flowchart 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook DSD-Decimate Signal Diagonalization(mono) DSD vs FFT 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook DSD-Decimate Signal Diagonalization(mono) DSD vs FFT 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook Application for detection in pipeline (mono) Experimental setup 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook Application for detection in pipeline (mono) 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook Application for detection in pipeline (mono) 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook Application for detection in pipeline (mono) 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook Application for detection in pipeline (mono) 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook Application for detection in pipeline (multidimensional) 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook Application for detection in pipeline (multidimensional) 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook Application for detection in pipeline 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook Application for detection in pipeline 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook FDM-(Multidimensional) 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook To create the multi-dimensional FDM, in this case twodimensional, it has been constructed in addition a new matrix of overlap 2 , as it has already been calculated for the matrices 1 and 0 . 2 = | 2 = ++2 With the new matrix, the new eigenvalues and eigenvectors will be calculated overlap, and from their processing we will derive new amplitudes 2 and pulse 2 of the second spectrum similar to the onedimensional case. 2 2 = 2 0 2 FDM-(Multidimensional) For the construction of bispectrum it will be calculated the value of: 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. 8. 9. CROSS AMPLITUDE = 1 0 2 1 2 BISPECTRUM DSD(multidimensional) Re 1,2 Application for EEG Final outlook 1,2 1 1 Im Im 1 − 21 2 − 22 FDM-(Multidimensional) FDM-2D Flowchart START 1. 2. Problem statement Main motivation Acquire N samples cn of the analog signal with sampling time τ and sampling frequency fs= 1/ τ Build square matrices U0,U1 e U2 3. M=N/2 FDM-(mono) Rank(U0)<M 4. DSD-(mono) ρ(U0)→+∞ 5. 6. 7. 8. Application for detection in pipeline FDM(multidimensional) DSD(multidimensional) Application for EEG Solve the generalized eigenvalue problem with DLSP (damped least squares pseudoinverse) Make normalization of eigenvector BK such that BKTU0BK=δkk Make normalization of eigenvector BK such that BKTU0BK=δkk Fix a smally and retain only eigenvalues u1k such that 1-ϒ<|u1k|<1-ϒ Fix a smally and retain only eigenvalues u1k such that 1-ϒ<|u2k|<1-ϒ Calculate ω1k=-1/τ Calculate ω2k=-1/τ Calculate d1k=(CTBK) Calculate d1k=(CTBK) Calculate cross amplitude 9. Final outlook Build bi-spectrum as END FDM-(Multidimensional) The figures below show an application of the FDM bispectrum 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook 3D-VIEW CONTOURVIEW DSD-(Multidimensional) 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook As for the FDM method and decimation step, a diagonalization procedure for each of the M signals is realized 1 = 1 −1 → 1 1 = 1 0 1 =1 2 = 2 −2 → 2 2 = 2 0 2 =1 DSD-(Multidimensional) 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. 9. The ω1k and ω2k are extracted from the eigenvalues u1k=e-jω1kτ and u2k=e-jω2kτ of the operator = − 1 1 = − ∡ 1 2 while the amplitude parameters are calculated as 2 −1 1 = 0|1 2 = 1 1 2 1 1 = Application for EEG Final outlook 1 = − ∡ 2 =0 2 −1 2 = 0|2 2 = 2 2 2 2 2 = =0 DSD-(Multidimensional) Then we calculate the cross amplitude as 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook = 1 0 2 1 2 And we build bi-spectrum as ′1,2 1 Im Im 1 − 21 2 − 22 (1 , 2 = 1,2 DSD-(Multidimensional) DSD-2D Flowchart 1. START Problem statement Acquire N samples of the analog signal with sampling time τ 2. Main motivation Compute FFT of the acquired samples obtaing the spectrum X Create a moving window W of size ND=N/M 3. FDM-(mono) K=1 Multiply X by W obtaing a small Nd-point spectrum XK 4. DSD-(mono) Center Xk to zero frequency (shift) IFFT-transform Xk obtaing a Nd-point signal Xk 5. 6. 7. Application for detection in pipeline FDM(multidimensional) DSD(multidimensional) 8. Application for EEG 9. Final outlook Build matrix overlapp U0,U1 e U2 Compute FDM analysis to obtain amplitudes and frequencies {(d1,ω1)}k of xk spectrum Compute FDM analysis to obtain amplitudes and frequencies {(d2,ω2)}k of xk spectrum Both compress frequencies and resize amplitudes by a factor M Both compress frequencies and resize amplitudes by a factor M Cut frequencies at right and left spectrum borders due to overlapping Cut frequencies at right and left spectrum borders due to overlapping Inverse shift xk spectrum Inverse shift xk spectrum Move forward W by (Nd-overlap)-point Move forward W by (Nd-overlap)-point k=k+1 yes k<=M ? no Calculate cross-amplitude Dkk Build spectrum as END DSD-(Multidimensional) The figures below show an application of the FDM bispectrum 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook 3D-VIEW CONTOURVIEW Application for EEG 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook Normal EEG Epileptic EEG Final outlook 1. Problem statement 2. Main motivation 3. FDM-(mono) 4. DSD-(mono) 5. Application for detection in pipeline 6. FDM(multidimensional) 7. DSD(multidimensional) 8. Application for EEG 9. Final outlook The FDM and DSD are parameter estimators which exhibit a 2-fold advantage over the most frequently applied spectral estimator, the Fast Fourier Transform (FFT). 1) FDM and DSD determine all the peak parameters (positions, magnitudes, relaxation times,phases, etc.) and then construct a spectrum in any desired mode. This includes absorption, which has a better resolving power than the corresponding magnitude spectrum. The absorption spectra are easily obtained without any additional experimental effort, as no phase problems exist. 2) When a spectrum is not too densely packed with spectral or noise features, remarkably good results can be achieved with shorter computation time than FFT.