Singular Value Decomposition-Based Modeling of Time Domain Signals in Broadband

Microwave Spectroscopy

A. J. MineiCollege of Mount St. Vincent

S. A. CookePurchase College SUNY

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?

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

Motivation:

This is reference 7:

K

kknkkn

kbnt

etvcty1

2cos)(

ck, vk, fk, and bk represent amplitude, frequency, phase and damping factor for the kth signal, tn = nDt, with Dt = 25 ps.

The Algorithm:

Use a function to model the transient emission:

x1 x2 x3 x4

x2 x3x4 . . . .

. . . . . . xn

x3 x4 . . . .

.

.

.

X =

The Hankel Matrix

TVUX Singular value decomposition:

S 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.

Then find Z which satisfies:

ZUU

tvibz kk 2exp

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, zK

Then create the Vandermonde matrix:

12

13

233

12

222

11

211

1

1

1

1

Mkkk

M

M

M

zzz

zzz

zzz

zzz

Then perform linear least squares fit:

nk xc ' )exp(' kkk icc

do 88 k = 1,kfit ampl(k) = cdabs (x(k)) fase(k) = datan2(dimag(x(k)),dble(x(k))) 88 continue

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.

k JK-1K+1 – JK-1K+1 ck bk / ms

fk / deg. vk / MHz vk / MHz (1024k FFT)

1 321 – 211 0.00111 1.42 -29.2 8753.1008 8753.2290

2 312 – 202 0.00222 63.84 23.6 8756.5952 8756.6013

3 330 – 220 0.00171 2.39 -96.2 9265.7422 9265.7423

4 331 – 221 0.00372 13.73 63.6 9369.4766 9369.4739

5 422 – 312 0.00195 12.30 58.5 11595.9084 11565.8884

6 423 – 313 0.00146 5.35 -147.1 11903.3664 11903.3749

7 432 – 322 0.00121 8.21 -39.0 12075.4803 12075.4601

8 441 – 331 0.00103 12.34 -140.8 12549.3485 12549.3555

K

kknkkn

kbnt

etvcty1

2cos)(

Data pts Time ck bk / ms vk / MHz vkFFT – vk / kHz fk / deg.

512 1 0.00390 0.12 9369.4188 55.1 62.7

1024 2 0.00371 5.41 9369.4901 -16.2 62.5

2048 9 0.00369 6.42 9369.6226 -148.7 61.8

4096 61 0.00368 3.25 9369.5171 -43.2 62.8

8192 513 0.00372 13.72 9369.4766 -2.7 63.6

Parameters of the model function for the 331 ← 221 transition for 1H, 2H perfluorocyclobutane for different numbers of FID data points

K

kknkkn

kbnt

etvcty1

2cos)(

Performance as a function of transition amplitude

Performance as a function of transition frequency

Possible applications:

1. Signal reconstruction2. Identifying non-molecular signals3. Alternative to zero-filling4. Time domain signal filtering5. Peak picking6. De-noising (!)7. Experimental rate improvement?8. …?

Matrix dimension

Time / s

Problems

Intel Core i7-3610QM @ 2.30 GHz16 Gb RAMgfortran compiled code (LAPACK/BLAS)

Future work:

1. Make it go faster

Thank you!

Acknowledgements:Thanks to Dirk van Ormondt – Delft USouthern New England Microwave Spectroscopy Consortium

Different compilerMulti-threading (ScaLAPACK, MPI)