S. A. Cooke
Purchase College SUNY
Frequency
8,388,608 (223) data points
120 nozzle pulses
14-bit vertical resolution
Fast Fourier Transform
Zoom
Cooley - Tukey
FT-ICR
Frequency
Local data sets:




800,000 points (free induction decay)
1975 Ekkers-Flygare
25 ps/pt
spectrometer
8-bit vertical resolution
used a 1-bit comparator
Hardware limit of 10,000 averages per
acquisitiona
a Might be
a very local problem!

Average multiple acquisition sets in
the time-domain

One acquisition
set is often out
of phase with a
second
acquisition set

Lissajous plots
provide a
convenient way
to test phase
coherence
https://commons.wikimedia.org/wiki/User:Krishnavedala

Generally we use one zero fill to produce 220
(1,048,576) points.

Rectangular window

Subroutine REALFT from Numerical Recipes
FFT has real and imaginary components
1
𝑅𝑒 𝜔 =
𝑁
𝑓 𝑘 cos 2𝜋𝜔𝑘
𝑘
𝑖
𝑖𝐼𝑚 𝜔 =
𝑁
𝑓 𝑘 sin 2𝜋𝜔𝑘
𝑘
for (int j = 2; j <= (count/2); j++ ){
pwr[j]=(float) ((2.0*((rl[j]*rl[j])+(im[j]*im[j])))/(m*m));
}
Absorption spectrum = 𝐴𝑏𝑠(𝜔)=𝑅𝑒′(𝜔)=𝑅𝑒 𝜔 cos 𝜃 − 𝐼𝑚 𝜔 sin 𝜃
Dispersion spectrum = 𝐷𝑖𝑠𝑝(𝜔)=𝐼𝑚′ 𝜔 = 𝑅𝑒 𝜔 sin 𝜃 − 𝐼𝑚 𝜔 cos 𝜃
𝑅𝑒′(𝜔)=𝑅𝑒 𝜔 cos 𝜃 − 𝐼𝑚 𝜔 sin 𝜃
q = 1.7386 rad
Transition in absorption spectrum
Transition in dispersion spectrum
“Removing” the dispersion contribution results in a narrower line width
Also tackled by E.J. Campbell and R. D. Suenram
and most recently
D. Plusquellic and K. O. Douglass
• The DISPA method easily
provides the phase correction
angle
• Improved results with more zero
fills prior to the FFT (provides
more data points for the DISPA
circle)
Unphased
DISPA
Phased
Magnitude
Difficulty: Phase correction is a (quadratic?) function of frequency and experimental
parametersa. This is very well known in FT-ICR.
a In FTMW the phase is
likely also affected by the microwave components and…
Voigt profile may be used which is
a convolution of the Lorentzian
and Gaussian profiles
ACM Transactions on Mathematical Software, Vol
38, No. 2, Article 15(2011),22 pages
Available from arXiv.org

Wavelet transformations: useful for denoising. Didn’t pursue.

Hankel Singular Value Decomposition
method.

Maximum Entropy Method.
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
yn (t )   ck cos2vk tn  fk e
k 1
t
 bn
k
Problems
Time / s
Hankel matrix dimension

Used xevlmem and memcof from Numerical
Recipes
𝑎0
1+

𝑀
𝑘 2
𝑎
𝑧
𝑘=1 𝑘
≈ Power Spectrum
M is the number of “poles” which should be
set to a few times the number of spectral
features expected.
Relatively fast:
Set N = 800000, M = 300000
Approx 10 minutes computation

DISPA will provide the phase angle for any
line in your spectrum

The functional dependence of phase angle on
frequency is more complex than NMR/FT-ICR

Power spectra estimates by the maximum
entropy method can show transitions more
clearly than the FFT algorithm

Prof. Alan Marshall (U.Florida/NHMFL – Sample FT-ICR data
set)

Prof. Elliot Burnell (UBC – Sample FT-NMR data sets)

Profs. Stew Novick and Pete Pringle (Wesleyan)

Prof. Andrea Minei (College of Mount St. Vincent)

ACS-PRF 53451-UR6

Algorithm
Use a function to model the transient emission(s):
K
yn (t )   ck cos2vk tn  fk e
t
 bn
k
k 1
ck, vk, fk, and bk represent amplitude, frequency, phase and damping
factor for the kth signal, tn = nDt, with Dt = 25 ps.
X=
x1
x2
x3 x4
x2
x3
x4
x3
x4
. . . . . .
. . . .
. . . .
.
.
.
The Hankel Matrix
xn
Singular value decomposition:
X  U V
T
 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:

U  UZ
Diagonalize Z to obtain K signal poles, or roots, zK
zk  expbk  i 2v Dt 
Then create the Vandermonde matrix:
 1 z1

 1 z2
   1 z3


 1 zk
z12
z22
z32
zk2
 z1M 1 

 z2M 1 
 z3M 1 


 zkM 1 
Then perform linear least squares fit:
ck '  xn
ck '  ck exp(ifk )