1. Aimé Lay-Ekuakille University of Salento

2. Index: 1.Problem statement 2.Main motivation 3.FDM-Filter Diagonalization Method (mono) 4.DSD-Decimate Signal Diagonalization (mono) 5.Application for detection in pipeline 6.FDM-Multidimensional 7.DSD-Multidimensional 8.Application for EEG 9.Final outlook

3. 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)

4. 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 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 Traditional methods FFT and Laplace transform are not suitable for above problems.

5. 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:

6. 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

7. 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

8. 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 Frequency and amplitude are calculated from:

9. 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 FDM Flowchart

10. 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

11. 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 ill- posed problem.

12. 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 U 0d and U 1d. The diagonalization procedure is realized for each of the M signals in this way

13. 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 while the amplitude parameters are calculated as

14. 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 And we build spectrum from

15. 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 DSD Flowchart

16. 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 DSD vs FFT

17. 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 DSD vs FFT

18. 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 Experimental setup

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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 For the construction of bispectrum it will be calculated the value ​​ of:

29. 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 FDM-2D Flowchart

30. 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 The figures below show an application of the FDM bispectrum 3D-VIEW CONTOUR- VIEW

31. 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

32. 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 while the amplitude parameters are calculated as

33. 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 Then we calculate the cross amplitude as And we build bi-spectrum as

34. 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 DSD-2D Flowchart

35. 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 The figures below show an application of the FDM bispectrum 3D-VIEW CONTOUR- VIEW

36. 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

37. 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.