1. Singular Value Decomposition-Based Modeling of Time Domain Signals in Broadband Microwave Spectroscopy A. J. Minei College of Mount St. Vincent S. A. Cooke Purchase College SUNY

2. Motivation: Summer 2012 recorded some strong free induction decays pertaining to the rotational spectrum of 1H,2H-perfluorocyclobutane. 800,000 points. 25 ps/point. Could we do something with the time domain signals?

3. Comments -Very useful for resolving overlapping lines -Implemented in FTMW++ But: -Need to know how many lines in the FID -Performed iteratively -Non-linear least squares fit, needs starting parameters

4. Motivation: This is reference 7:

5. c k, v k,  k, and b k represent amplitude, frequency, phase and damping factor for the kth signal, t n = n  t, with  t = 25 ps. The Algorithm: Use a function to model the transient emission:

6. x1x1 x2x2 x3x3 x4x4 x2x2 x3x3 x 4....... xnxn x3x3 x4x4....... X = The Hankel Matrix

7. Singular value decomposition:  is a diagonal vector, with the singular values along the diagonal, U and V are matrices for which columns contain the left and right singular vectors.

8. Then find Z which satisfies: do 77 k = 1,kfit damp(k) = dlog(cdabs(root(k))) freq(k) = datan2(dimag(root(k)),dble(root(k))) / (2.0d0*pi) 77 continue Diagonalize Z to obtain K signal poles, or roots, z K

9. Then create the Vandermonde matrix: Then perform linear least squares fit: do 88 k = 1,kfit ampl(k) = cdabs (x(k)) fase(k) = datan2(dimag(x(k)),dble(x(k))) 88 continue

10. Output: Parameters of the model function for several transitions of 1H,2H-perfluorocyclobutane using the HSVD-method. Only the first 8192 data points (1% or 200ns) of the FID were used. kJ K-1K+1 – J K-1K+1 ckck b k /  s  k / deg. v k / MHzv k / MHz (1024k FFT) 13 21 – 2 11 0.001111.42-29.28753.10088753.2290 23 12 – 2 02 0.0022263.8423.68756.59528756.6013 33 30 – 2 20 0.001712.39-96.29265.74229265.7423 43 31 – 2 21 0.0037213.7363.69369.47669369.4739 54 22 – 3 12 0.0019512.3058.511595.908411565.8884 64 23 – 3 13 0.001465.35-147.111903.366411903.3749 74 32 – 3 22 0.001218.21-39.012075.480312075.4601 84 41 – 3 31 0.0010312.34-140.812549.348512549.3555

12. Data ptsTimeckck b k /  s v k / MHzv k FFT – v k / kHz  k / deg. 51210.003900.129369.418855.162.7 102420.003715.419369.4901-16.262.5 204890.003696.429369.6226-148.761.8 4096610.003683.259369.5171-43.262.8 81925130.0037213.729369.4766-2.763.6 Parameters of the model function for the 3 31 ← 2 21 transition for 1H, 2H perfluorocyclobutane for different numbers of FID data points

13. Performance as a function of transition amplitude

14. Performance as a function of transition frequency

16. Possible applications: 1.Signal reconstruction 2.Identifying non-molecular signals 3.Alternative to zero-filling 4.Time domain signal filtering 5.Peak picking 6.De-noising (!) 7.Experimental rate improvement? 8.…?

17. Matrix dimension Time / s Problems Intel Core i7-3610QM @ 2.30 GHz 16 Gb RAM gfortran compiled code (LAPACK/BLAS)

18. Future work: 1. Make it go faster Thank you! Acknowledgements: Thanks to Dirk van Ormondt – Delft U Southern New England Microwave Spectroscopy Consortium Different compiler Multi-threading (ScaLAPACK, MPI)