# A comb-sampling method for enhanced mass analysis in linear electrostatic ion traps

**ABSTRACT** In this paper an algorithm for extracting spectral information from signals containing a series of narrow periodic impulses is presented. Such signals can typically be acquired by pickup detectors from the image-charge of ion bunches oscillating in a linear electrostatic ion trap, where frequency analysis provides a scheme for high-resolution mass spectrometry. To provide an improved technique for such frequency analysis, we introduce the CHIMERA algorithm (Comb-sampling for High-resolution IMpulse-train frequency ExtRAaction). This algorithm utilizes a comb function to generate frequency coefficients, rather than using sinusoids via a Fourier transform, since the comb provides a superior match to the data. This new technique is developed theoretically, applied to synthetic data, and then used to perform high resolution mass spectrometry on real data from an ion trap. If the ions are generated at a localized point in time and space, and the data is simultaneously acquired with multiple pickup rings, the method is shown to be a significant improvement on Fourier analysis. The mass spectra generated typically have an order of magnitude higher resolution compared with that obtained from fundamental Fourier frequencies, and are absent of large contributions from harmonic frequency components.

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- O Aviv, B Kafle, V Chandrasekaran, O Heber, M L Rappaport, H Rubinstein, D Schwalm, D Strasser, Y Toker, D Zajfman[Show abstract] [Hide abstract]

**ABSTRACT:**We describe a technique to measure absolute photo-induced cross sections for cluster anions stored in an electrostatic ion beam trap (EIBT) with a central deflector. The setup allows determination of total photo-destruction cross sections as well as partial cross sections for fragmentation and electron detachment. The unique properties of this special EIBT setup are investigated and illustrated using small Aln (-) clusters.The Review of scientific instruments 05/2013; 84(5):053106. · 1.52 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The Planar electrostatic ion trap expands the trapping space of the linear electrostatic ion trap, giving rise to higher tolerance to space charge. A rotational symmetrical design was made, which has a trapping field between two layers of concentric circular electrodes, and the ions are trapped to oscillate around the center plane between the electrodes. The oscillatory motions of the ions were simulated and the field distribution was optimized to achieve isochronous motion against energy spread in R, z, and φ directions. The image charge signal can be picked up by more than one circular electrode and using FFT the mass resolution for the optimized trap can reach 80,000 FWHM. While Fourier transform of the image charge signal generates many high harmonic peaks, the unwanted harmonic peaks can be eliminated by linear combination of image charge signals from multiple pick-up electrodes to give satisfactory results.Journal of the American Society for Mass Spectrometry 02/2013; · 3.59 Impact Factor -
##### Article: Fragmentation of Neutral Amino Acids and Small Peptides by Intense, Femtosecond Laser Pulses.

Martin J Duffy, Orla Kelly, Christopher R Calvert, Raymond B King, Louise Belshaw, Thomas J Kelly, John T Costello, David J Timson, William A Bryan, Thomas Kierspel, I C Edmond Turcu, Cephise M Cacho, Emma Springate, Ian D Williams, Jason B Greenwood[Show abstract] [Hide abstract]

**ABSTRACT:**High power femtosecond laser pulses have unique properties that could lead to their application as ionization or activation sources in mass spectrometry. By concentrating many photons into pulse lengths approaching the timescales associated with atomic motion, very strong electric field strengths are generated, which can efficiently ionize and fragment molecules without the need for resonant absorption. However, the complex interaction between these pulses and biomolecular species is not well understood. To address this issue, we have studied the interaction of intense, femtosecond pulses with a number of amino acids and small peptides. Unlike previous studies, we have used neutral forms of these molecular targets, which allowed us to investigate dissociation of radical cations without the spectra being complicated by the action of mobile protons. We found fragmentation was dominated by fast, radical-initiated dissociation close to the charge site generated by the initial ionization or from subsequent ultrafast migration of this charge. Fragments with lower yields, which are useful for structural determinations, were also observed and attributed to radical migration caused by hydrogen atom transfer within the molecule.Journal of the American Society for Mass Spectrometry 07/2013; · 3.59 Impact Factor

Page 1

REVIEW OF SCIENTIFIC INSTRUMENTS 82, 043103 (2011)

A comb-sampling method for enhanced mass analysis in linear

electrostatic ion traps

J. B. Greenwood,1,a)O. Kelly,1C. R. Calvert,1M. J. Duffy,1R. B. King,1L. Belshaw,1

L. Graham,1J. D. Alexander,1I. D. Williams,1W. A. Bryan,2I. C. E. Turcu,3C. M. Cacho,3

and E. Springate3

1Centre for Plasma Physics, School of Mathematics and Physics, Queen’s University Belfast,

Belfast BT7 1NN, United Kingdom

2Department of Physics, Swansea University, Swansea SA2 8PP, United Kingdom

3Central Laser Facility, STFC Rutherford Appleton Laboratory, Didcot, Oxfordshire OX11 0QX,

United Kingdom

(Received 16 December 2010; accepted 9 March 2011; published online 8 April 2011)

In this paper an algorithm for extracting spectral information from signals containing a series of nar-

row periodic impulses is presented. Such signals can typically be acquired by pickup detectors from

the image-charge of ion bunches oscillating in a linear electrostatic ion trap, where frequency analy-

sis provides a scheme for high-resolution mass spectrometry. To provide an improved technique for

suchfrequencyanalysis,weintroducetheCHIMERAalgorithm(Comb-samplingforHigh-resolution

IMpulse-train frequency ExtRAaction). This algorithm utilizes a comb function to generate frequency

coefficients, rather than using sinusoids via a Fourier transform, since the comb provides a superior

match to the data. This new technique is developed theoretically, applied to synthetic data, and then

used to perform high resolution mass spectrometry on real data from an ion trap. If the ions are

generated at a localized point in time and space, and the data is simultaneously acquired with mul-

tiple pickup rings, the method is shown to be a significant improvement on Fourier analysis. The

mass spectra generated typically have an order of magnitude higher resolution compared with that

obtained from fundamental Fourier frequencies, and are absent of large contributions from harmonic

frequency components. © 2011 American Institute of Physics. [doi:10.1063/1.3572331]

I. INTRODUCTION

Electrostatic devices are increasingly being used to

store ions for scientific studies and for mass spectrometry

applications.1Unlike ion traps which use magnetic or time-

varying electric fields, the storage conditions in an electro-

static trap are independent of mass for ions of the same en-

ergy per charge; this allows molecules as large as proteins to

be studied. While the electrostatic Kingdon trap2has been in

use for nearly 90 years, it is only in the last 20 years that a

new range of electrostatic devices have been developed.

The electrostatic ion storage ring in Aarhus (ELISA)3

was one of the first of these instruments to be developed; it is

much smaller than the magnetic equivalent, and has inspired

the further development of cryogenic4,5and miniature6stor-

age rings. Around the same time, the Kingdon trap was mod-

ified to create the Orbitrap,7for which the mass resolution

reduces more slowly as a function of mass than with other

high performance mass spectrometers, such as ion cyclotron

resonance (ICR) devices.

The linear electrostatic ion trap (LEIT), which acts as

an analogue to a laser cavity, is another innovation in elec-

trostatic storage,8–10with useful applications for ion tar-

get preparation11and the potential for mass spectrometry

schemes.12Ions are stored along a linear trajectory, oscillat-

ing back and forth within an electrostatic cavity. Unlike sig-

nals acquired from ICRs or Orbitraps, the signal from LEITs

a)Electronic mail: j.greenwood@qub.ac.uk.

is nonsinusoidal in the time domain. Due to this, the corre-

sponding Fourier transform inherently contains a rich spec-

trum of harmonics. Through the analysis of these harmonics13

LEITs have demonstrated a higher resolution capability than

is achievable for a similar trapping time in an Orbitrap.7

However, mass analysis of LEIT data is complicated by

this existence of multiple harmonics in the Fourier frequency

spectra, such that the generation of a mass spectrum is non-

trivial, even where there are relatively few ion species.

In this paper an alternative method is proposed for the

frequency analysis of anharmonic data acquired from a LEIT.

By using comb functions with different time offsets to sam-

ple data obtained from several different pickup detectors, and

by comparing with a standard fast Fourier transform, a spec-

trum free of harmonics can be obtained. This CHIMERA

(comb-sampling for high-resolution impulse-train frequency-

extraction) algorithm allows frequencies to be easily con-

verted into a mass spectrum with substantially higher res-

olution than can be achieved through Fourier analysis. The

implementation of this technique is demonstrated using ex-

perimental data obtained from the electrostatic trapping of

charged species created by an intense femtosecond laser in-

teraction.

To provide the context for our new analysis approach, we

describe our ion trap in Sec. II. A discussion of the Fourier

treatment of the signal from the instrument and its draw-

backs are given in Sec. III. In Sec. IV the CHIMERA al-

gorithm is presented, while its application to synthetically

generated datasets is detailed in Sec. V. The application of

0034-6748/2011/82(4)/043103/12/$30.00© 2011 American Institute of Physics

82, 043103-1

Page 2

043103-2Greenwood et al.Rev. Sci. Instrum. 82, 043103 (2011)

the CHIMERA algorithm to experimental results is reported

in Sec. VI, for both isotope identification in atoms and for

molecular fragmentation spectra. In Sec. VII the results are

summarized, the performance of the algorithm relative to

Fourier analysis is discussed, and suggestions for applications

in other devices are made. Appendices are used to detail some

of the mathematical derivations.

II. THE KEIRA ELECTROSTATIC ION TRAP

The potential energy surface for our kilovolt electrostatic

ion reflection analyzer (KEIRA) device is shown in Fig. 1.

The electrostatic mirrors at each end of the trap provide axial

confinement (z) of the ion motion. The mirror regions consist

of six 3 mm thick plates (R1–R6), with apertures of 16 mm

diameter, separated by 7 mm, and one end plate (R7) situated

a further 14 mm from R6. The potentials that are applied to

these electrodes increase incrementally up to a maximum of

5 kV.

Radial confinement (r) is provided by two sets of four

plates, which are also separated by 7 mm with 16 mm diame-

ter apertures. In each set, the outer plates are earthed and the

two central plates are held at a negative potential to form an

electrostatic lens. Ions are generated by laser ionization of a

gasjet,whichemergescolinearwiththetrapaxisfroma1mm

hole in one end plate. The laser is aligned perpendicular to the

z axis, to enter the 14 mm gap between R6 and R7 and is fo-

cused into the gas jet. The ions are generated in a well defined

spatial region within a couple of mm of the trap z axis, with

effectively zero initial velocity.

They are immediately accelerated by the electrostatic

field into the field free region at the center of the trap. With the

appropriate electrode potentials, a large fraction of these ions

oscillate between the mirrors on stable trajectories. The ions

travel a total distance of approximately L = 410 mm from

mirror to mirror.

KEIRA has three image-charge detectors (pickup rings),

one at the geometric center of the trap and one either side,

separated by 48 mm center-to-center. The pickup itself is a

cylinder of 16 mm length and 9 mm inner diameter (shown

in more detail elsewhere15). It is situated within an earthed

housing which has 8 mm entrance and exit apertures which

are separated from the pickup by 1 mm. An ion bunch at the

center will have 97% of its charge imaged on the pickup.15

FIG. 1. (Color online) Potential energy surface14for the KEIRA ion trap,

with the mirror regions defined by the electrodes R7–R1, and two sets of four

electrodes (the inner two held at a negative potential) defining the lenses.

Three isolated pickup rings are located in the central, field free region of the

trap. The trap is cylindrically symmetric, with ion propagation along the z

axis direction.

0 1.02.0 3.04.05.0

0

0.2

0.4

time (ms)

amplitude

00.0050.0100.015

0

0.5

1.0

00.0050.0100.015

22.005

time (ms)

2.0102.015

0

0.04

0.08

2 2.005

time (ms)

2.010 2.015

Offset PickupCentre Pickup

(a)

(b)

(c)

(e)

(d)

FIG.2. (Coloronline)(a)Temporalspectraacquiredusingthecentralpickup

from ionization of H2O present in background gas. (b)–(d) show 15 μs win-

dows at different trapping times for the central pickup ring (b, d) and a pickup

ring closer to the ionization region (c, e).

Figure 2 shows time signal obtained from photoioniza-

tion of background gas at a pressure of 4 × 10−8mbar ob-

tained from the central and one offset pickup ring in KEIRA.

The ion signal is almost exclusively due to H2O+, and was

produced using the Artemis16femtosecond laser at the Cen-

tral Laser Facility. The laser pulse had a central wavelength

of 790 nm, pulse length of 15 fs, and a maximum intensity

of about 5 × 1014W cm−2. The trap potentials (in kV) were

set at R1 = 0.5, R2 = 1.0, R3 = 1.5, R4 = 2.0, R5 = 2.5,

R6 = 3.0, R7 = 3.55, Lens = −2.8. From a previous cal-

ibration of the pickup ring,15we estimate that about 3.2 ×

104ions were initially produced by the laser, with 1.5 × 104

being captured on stable trajectories. The data was acquired

for 5 ms and, to improve the statistics, was averaged over 104

laser shots. The decay in the amplitude of the signal seen in

Fig. 2(a) is due to ion losses from collisions with background

gas, and an increase in the bunch length due to slight differ-

ences in oscillation times for the ions. In Figs. 2(b) and 2(c)

the peak widths correspond to the time taken to traverse the

pickup, while the signals at 2 ms [Figs. 2(d) and 2(e)] indicate

that the bunch length now exceeds that of the pickup.

The evolution of ion bunches in LEITs has been exten-

sively studied by the Weizmann group.17They identified three

types of behavior corresponding to

coherent diffusion : t2

w= t2

w= t2

w= constant,

0+ n2?T2,

0+ n?T2,

non-coherent diffusion : t2

(1)

self-bunching : t2

where tw is the bunch length in time, t0is the initial bunch

length, n is the number of full oscillations, and ?T is the

Page 3

043103-3Greenwood et al.Rev. Sci. Instrum. 82, 043103 (2011)

increase in bunch length per oscillation. In coherent diffusion,

the ions are weakly perturbed by external or internal (ion–ion)

interactions, and differences in ion oscillation times are solely

determined by the differences in the initial ion conditions. For

noncoherent diffusion, the ion trajectories are disturbed at a

rate greater than the oscillation frequency, resulting in the ion

having no “memory” of the previous oscillation.

For the data in Fig. 2, the ion density is too low to sup-

port the self-bunching phenomenon, but we have observed the

self-bunching in KEIRA for larger ion numbers. The present

data set exhibits coherent diffusion with ?T ≈ 1 ns, a value

which can be reduced by tuning the trap potentials.

III. LEIT MASS RESOLUTION: FOURIER ANALYSIS

A. Constant ion bunch length

The mass resolution is defined as R = m0/?m, where

m0is the ion mass and ?m is the minimum resolvable mass

difference. We define the following quantities: f0is the ion

oscillation frequency, U is the total energy of the ion, ?v? is

the ion velocity averaged over one oscillation,and L is the

length of the trap. The frequency of oscillation can be given

by

f0=?v?

2L=

β

2L

?2U

m0

?1

2

,

(2)

where β is a constant that depends on the trap geometry and

applied potentials (for KEIRA β ≈ 0.75). In terms of fre-

quency, the resolution is given by

R =

m0

?m≈

hf0

2?f,

(3)

where ?f is the full width half maximum (FWHM) of the hth

harmonic of the fundamental frequency f0.

The signal shown in Fig. 2 can be modeled by a comb

function convolved (⊗) with a distribution function and mul-

tiplied by an exponential decay term of lifetime τ. The time

distribution generated from one pass of the ion bunch is it-

self a convolution of the instrumental response of the pickup

with the ion density per unit length, which we represent by

a Lorentzian with FWHM tw, so that the signal in time (t) is

given by

s (t) ∝ e−|t|/τ

tw

t2

w/4 + t2⊗

?n +1

?

∞

?

??

n=−∞

δ

?

t −(n + c)

f0

?

+δ

?

t −

2− c?

f0

,

(4)

where c/f0is the time taken for the ions to travel from the

geometric center of the trap to the center of the pickup used

to acquire the data. We formally consider a signal symmetric

in time, for mathematical convenience. The values of c for the

three pickups in KEIRA are approximately 0 and ±0.038, and

change only slightly for different potential energy surfaces. If

?T ≈ 0, twremains constant and the amplitude of the Fourier

0.5

1.0

amplitude

0 0.51.01.5 2.02.53.03.5

0

0.5

0

1.0

0.3340.3360.3380.340

frequency (MHz)

0

0.5

1.0

3.3683.370 3.372

0

0.04

0.08

(a)

(b)

(c) (d)

FIG. 3. (Color online) FFT with a Welch window of the data in Fig. 2 for the

(a) central and (b) offset pickups, with closer inspection of (c) the 2nd and

(d) 20th harmonics for the central pickup only.

transform for positive frequencies ( f ) is

|S ( f )| ∝ e−π f tw

?????cosπf?2c −1

τ−2+ (2π ( f − hf0))2

2

?

f0

?????

×

∞

?

h=1

τ−1

(5)

(see Appendix A). Figure 3 shows the Fourier transforms of

the two pickup signals from Fig. 2. For harmonic frequencies

f ≈ hf0and a central pickup with c = 0, Eq. (5) reduces to

|S ( f )| ∝ τ?e−πhf0tw??1 + (−1)h?1/2.

Therefore, in Fig. 3(a) for the central pickup, the odd harmon-

ics are suppressed, while the even ones reduce exponentially

at a rate determined by tw. For the offset pickup [Fig. 3(b)],

odd harmonics are included and the magnitudes follow the

more complex formula given in Eq. (5). Note that the 13th

harmonic is absent since for c = −0.038, 2πf (1/2 − 2c)/f0

≈ 15π.

From Eq. (6), the FWHM of the harmonic peaks (?fτ

= 1/τπ) is independent of h, hence the resolution increases

linearly with the harmonic order

(6)

R =π

2hf0τ.

(7)

This has been exploited to achieve the very high resolutions

previously attained for a LEIT in self-bunching mode.12

B. Coherent bunch diffusion

Figures 3(c) and 3(d) show the 2nd and 20th harmonics

from the center pickup data. The second harmonic represents

Page 4

043103-4Greenwood et al. Rev. Sci. Instrum. 82, 043103 (2011)

a resolution of R ≈ 500. According to Eq. (3), ?f should be

the same for both harmonics, but due to bunch diffusion the

20th harmonic is broader. If the ion bunch diffuses coherently

[Eq. (1)], and we assume that the initial bunch width t0is

negligible, then tw= n?T in Eq. (4). The Fourier transform

for this signal is derived in Appendix B. For frequencies close

to the hth harmonic f = hf0+ δf and for c = 0, τ → ∞, h

even, we obtain

?

|S (δf )| ∝

1

?1 − e−π f ?T?2+ e−π f ?T(2πδf/f0)2

?1/2

,

(8)

which for low bunch spread per oscillation, i.e., f0?T ? 1,

yields

?

In this case, the contribution to the FWHM of the harmonic

due to coherent bunch dispersion is

√3 hf2

|S (δf )| ∝

1

(πhf0?T)2+(2πδf/f0)2

?1/2

.

(9)

?f?T=

0?T.

(10)

Therefore if the ion lifetime is long, the peak width increases

linearly with h and hence there is no improvement in resolu-

tion for higher harmonics. In Figs. 3(c) and 3(d) it can be seen

that the 20th harmonic is wider than that of the 2nd. It is also

evident from Fig. 2 that a proportion of the ions disperse more

slowly than the rest, resulting in the multiple component peak

in Fig. 3(d).

C. Windowing and discretization

In practice, the acquisition time TS is finite (5 ms in

Fig. 2) and the signal is multiplied by an apodization or win-

dowing function a (t/TS) to reduce spectral leakage in the

Fourier transform. Therefore S ( f ) is also convolved with

the Fourier transform of a (t/TS), which introduces additional

broadening to harmonic peaks so that

(?f )2≈ (?fτ)2+ (?f?T)2+ (?fA)2,

where ?fA= A/TS and A is a constant specific to the

apodization function, e.g., for a rectangle window A = 1.21;

Welch A = 1.59; Blackman A = 2.3. With a Welch window

it is only for TS> 5τ that ?fA< ?fτ. Therefore, the peak

width of the 2nd harmonic in Fig. 3(c) is dominated by the

apodization ?fA= 318 Hz.

As the data is also discrete, the Fast Fourier Transform

(FFT) calculates the frequency spectrum at T−1

(200 Hz), which is often similar to ?f [Fig. 3(c)]. This can be

mitigated by zero padding the data, but the discontinuity this

introduces at t = TScan result in additional spectral leakage.

(11)

S

intervals

D. Harmonic phase

Since the ion bunch has a well defined starting point in

time,inprinciplethere isaclear phase associated withFourier

components from ion trapping signal, which should allow dis-

crimination against uncorrelated electronic noise. If γ/f0is

the time taken by the ion bunch to reach the geometric center

of the trap after its initial formation (γ = 0.25 for KEIRA),

the phase of the Fourier frequencies is given by

φ ( f ) = −2πf

f0

?

γ +1

4

?

(12)

(see Appendix C). In principle, there is additional informa-

tion encoded in the phase which could be used to identify the

harmonic order and to distinguish signal from external noise

sources. However, this is very difficult to achieve in practice

as the phase changes rapidly across the finite width of a har-

monic peak.

E. Comparison with a time of flight mass

spectrometer

Some LEITs have been employed as multi-reflectron

time of flight mass spectrometers, where ions are trapped for

long enough to achieve mass separation and then extracted.18

By significantly increasing the effective time of flight length,

large enhancements in resolution capabilities can be achieved

compared to a standard linear or single-reflection device.

However, such a device is limited to studying similar mass

ions to ensure different ion bunches undergo the same num-

ber of oscillations.

If an ion bunch of width tw is extracted from a LEIT

after a time TS, the time of flight resolution is TS/2tw. To

compare with the resolution achievable using Fourier anal-

ysis, consider ions trapped for the same time with twf0

= 0.05 and τ → ∞. Using a Welch window, an equivalent

resolution would only be possible from the 32nd harmonic or

greater.

F. Drawbacks of Fourier analysis

The signal obtained from LEITs is nonsinusoidal un-

like those obtained from other mass spectrometers such as an

ICR or an Orbitrap. While higher harmonic components give

better resolution, there are a number of drawbacks in using

Fourier transforms to analyze this signal;

(1)The spectral amplitude of the signal is spread out among

many harmonics, reducing the signal-to-noise ratio, par-

ticularly for higher harmonics.

The presence of multiple harmonics makes conversion

of frequencies into a mass spectrum difficult, especially

if many different ions are present.

As it is possible to have many pickup detectors acquir-

ing data simultaneously, one would expect that these dif-

ferent channels could be combined to yield more infor-

mation and better signal-to-noise, but the spectra from

pickups offset from the trap center generate even more

complicated harmonic spectra.

There is rich phase information present in the signal

which could be used to discriminate against electronic

noise and improve resolution, but this is very difficult to

extract from the Fourier transform.

(2)

(3)

(4)

Page 5

043103-5Greenwood et al.Rev. Sci. Instrum. 82, 043103 (2011)

(5)Apodization of the data is necessary to reduce spectral

leakage, but this reduces the achievable resolution and

the signal-to-noise ratio.

IV. CHIMERA COMB-SAMPLING THEORY

A. Determining the frequency of a comb

Consider a time signal s (t) which consists of a series of

impulses at regular intervals,

s (t) =

N

?

n=0

δ

?

t −n

f0

?

.

(13)

In this case, we use δ to represent a narrow function of fi-

nite height. The frequency of such a distribution can be deter-

mined froma simple measurement, that is,to simply count the

time between one impulse and the next. For such a scheme,

the resolution increases as the number of impulses measured

is increased. However, executing this measurement becomes

nontrivial if the signal contains additional frequencies. Alter-

natively, using a comb function g (t), a spectral function S ( f )

may be generated, which gives the desired frequency f0,

g (t) =

m2

?

m=m1

δ

?

t −m

f

?

,

(14)

S ( f ) =

1

M

?∞

0

s (t)g (t)dt,

(15)

where M = m2− m1+ 1. However, S ( f ) would also con-

tain an infinite number of harmonics and “fractional

harmonics.”

Introducing an offset from t = 0, such that both s (t) and

g (t) are temporally offset by an amount γ/f0, gives

?∞

n=0

m2

?

In this case, more than one coincidence of the impulses only

occurs if

f

f0

Therefore if γ is chosen to be irrational, S ( f ) only has a sig-

nificant value when f = f0. LEITs are normally filled with

ions by injection of a bunch from outside the trap, where

bunches can be generated by, for example, chopping a contin-

uous beam, pulsing ions out of a pretrap, or ionization with a

short pulse laser. For ions which emanate from such a pulsed

ion source and subsequently execute trajectories dictated by

electrostatic optics, the value of γ will be constant, irrespec-

tive of the ion mass.

In practice, the measured impulses from an ion bunch

passing through the pickup are finite in time, so that there

will still be contributions from fractional harmonics even if γ

is an irrational number. If a second pickup is used to acquire

data simultaneously, it will have a different value of γ as the

S ( f ) =

1

M

0

?

N

?

?

δ

?

t −(n + γ)

f0

??

?

×

m=m1

δ

t −(m + γ)

f

dt.

(16)

=m + γ

n + γ.

(17)

distance between ion source/pulsar and pickup has changed.

Analysis of this data for the new γ will also have fractional

harmonics but at different values to the signal from the first

pickup. Therefore, by multiplying the two spectra and taking

the square root, the fractional harmonics will be suppressed,

producing a frequency spectrum with the main contribution

being from the fundamental frequencies only.

B. Resolution

As a comb function is a better match to the pickup signal

than a sinusoid, one would expect the frequency resolution to

be significantly improved. To simulate a more realistic sig-

nal, we convolve the comb function with a pickup response

functionq (t).Formathematicalconvenience,wewillapprox-

imate q (t) by a triangle function of full width in time 2twand

amplitude t−1

?

q (t) = 0 , |t| > tw.

w, such that

q (t) =

1

tw

1 −|t|

tw

?

, |t| ≤ tw,

(18)

Convolving this with the comb function in Eq. (13), we

may express the signal as

s (t) =

N

?

n=0

q

?

t −n

f0

?

(19)

(where we have set γ = 0). The corresponding frequency

function [cf. Eq. (16)] is thus

S ( f ) =

1

M

?∞

0

N

?

m2

?

n=0

q

?

?m

t −n

f0

?

m2

?

?

m=m1

δ

?

t −m

f

?

dt,

=

1

M

N

?

n=0

m=m1

q

f

−n

f0

.

(20)

Now, with twas the FWHM of q (t), where tw? 1/f0, and if

we let f = f0+ δf whereδf

to S ( f ) only when m = n, giving

1

M

m=m1

To model the effect of coherent diffusion, we let

f0< 1/m, there are contributions

S (δf ) =

m2

?

q

?mδf

f2

0

?

.

(21)

tw= (k + mp)/f0,

(22)

where k/f0is the initial FWHM, and the increase in width per

oscillation is ?T = p/f0, giving

f0

M

m=m1

For no bunch diffusion, p = 0, and for small values of δf this

yields

?

S (δf ) =

m2

?

?k + pm − m|δf |

(k + pm)2

f0

?

.

(23)

S (δf ) =

f0

k

1 −|δf |

2 f0k(m2+ m1)

?

.

(24)

Page 6

043103-6Greenwood et al.Rev. Sci. Instrum. 82, 043103 (2011)

0

0.2

0.4

0.6

0.8

1.0

0

0.2

0.4

0.6

0.8

1.0

frequency (kHz)

amplitude

100.0

99.9

(a)

(b)

(c) (d)

99.9

100.1

100.0

100.1

FIG. 4. (Color online) Comb-sampling frequency analysis for simulated data

with a frequency of 100 kHz and FWHM of 200 ns for analysis windows now

extending from (a) 0–5, (b) 0.5–5, (c) 2.5–5, and (d) 4.5–5 ms.

(see Appendix D). This is also a triangle function, giving res-

olution R,

R =

f0

2?f

=m2+ m1

4k

.

(25)

It is worth noting that as the sampling window nar-

rows toward the end of the time data, i.e., m2= N, m1

→ m2, where N = f0TSis the number of oscillations within

the sample time, the resolution for a time of flight instrument

is recovered,

R →m2

2k=

TS

2tw.

(26)

At the other extreme, if the sampling window spans the whole

data set (m1= 0, m2= N), the resolution is a factor of two

lower. In Fig. 4, synthetic data with parameters f0= 100 kHz,

TS= 5 ms, N = 1000, tw= 200 ns, k = 0.04, and p = 0 has

been analyzed using comb-sampling for four sample windows

with different starting points (m1/f0= 0, 0.5, 2.5, 4.5 ms) and

the same end point (m2/f0= 5 ms). Note that the mean value

of the whole synthetic data set has been subtracted from each

data point prior to analysis.

As predicted from Eq. (25), delaying the start of the sam-

pling increases the resolution, so that in Fig. 4(d) the reso-

lution is close to that expected for a conventional TOF spec-

trometer with a 5 ms flight time. Figure 5 shows the same data

but focusing on amplitudes close to zero and over a greater

range of frequencies. On either side of the main peak the

frequency function dips below zero. This corresponds to fre-

quencies where the comb “misses” the pickup impulses and

samples only the baseline (= −mean). There is also a ring-

ing artifact which begins when the sampling frequency is far

enough away from f0to start sampling the next impulse at the

end of the time window when

N f0= m2f = (N ± 1) f,

(27)

−0.1

0

0.1

0.2

0.3

amplitude

99100 101

−0.1

0

0.1

0.2

0.3

frequency (kHz)

99100 101

(c)

(a) (b)

(d)

FIG. 5. (Color online) As for Fig. 4 but with the frequency range increased

and a reduced amplitude range.

f − f0= ±f

N= ±100 Hz.

(28)

In Fig. 5(d) the ringing artifact is unacceptably high since

a number of different frequencies are capable of sampling

most of the impulses in this small sample window. The best

outcome can be found for Fig. 5(b) where the start time of the

window is 10% of the end time. Here, nearly all of the ring-

ing is confined to negative values which can be eliminated if

values less than zero are rejected. Delaying the start of the

sampling also ensures that for real data, no ions on unstable

trajectories contribute to the frequency spectrum.

Alternatively, coherent bunch diffusion with k = 0,

yields

?

S (δf ) =

f0

Mp

1 −|δf |

pf0

?

ln

?m2

m1

?

,

if m1,m2? 1

(29)

(see Appendix D), giving a resolution of R = 1/2p. In this

case, the resolution is solely determined by the rate at which

the bunch diffuses and not upon the length of the acquisition

time.

C. Influence of the ion lifetime and windowing

To simulate the influence of the ion lifetime τ, the data

is given a weighting which linearly decreases in time with a

gradient of τ−1, so that q (t) now becomes

1

tw

tw

effectively creating a trapezoidal window for the data. Tak-

ing the case of zero bunch diffusion (p = 0) and m1= 0, the

resolution becomes

R =m2(3τ f0− 2m2)

6k (2τ f0− m2)

q (t) =

?

1 −|t|

??

1 −

m

τ f0

?

, |t| ≤ tw,

(30)

(31)

Page 7

043103-7Greenwood et al. Rev. Sci. Instrum. 82, 043103 (2011)

(see Appendix E). This reduces to Eq. (25) as τ → ∞ and

if the lifetime is equal to the width of the analysis window (τ

= m2/f0),theresolutionisafactorof1.5lower(R = m2/6k).

Another advantage of this technique, compared to FFT, is

that any shape of window can be employed. Since the analysis

is more sensitive to impulse peaks near the end of the time

spectrum, their relative weight can be enhanced. For instance,

a simple linear weighting increases the resolution by a factor

of 4/3 (R = m2/3k, Appendix E).

V. IMPLEMENTATION OF CHIMERA ALGORITHM

In this section, we extend the analysis to the general case

and apply it for the setup currently employed in KEIRA. If

we include the pickup offset, c, then Eqs. (13) and (14) now

become

?

?

s (t) =

N

?

n=0

δ

?

t −(n + γ + c)

?n + γ +1

f0

?

+δ

t −

2− c?

f0

??

,

(32)

g (t) =

m2

?

m=m1

?

δ

?

t −(m + γ + c)

f

?

??

+δ

?

t −

?m + γ +1

2− c?

f

.

(33)

For a proportion of the teeth in these two combs to coincide

periodically, i.e., contribute to S ( f ) more than once, one of

the following conditions must be met:

f

f0

=m + γ + c

n + γ + c,

=m + γ +1

n + γ +1

f

f0

=m + γ +1

n + γ + c

m + γ + c

n + γ +1

2− c

,

f

f0

2− c

2− c,

f

f0

=

2− c,

(34)

where m,n are integers in the specified ranges. The mag-

nitude of S ( f ) is determined by the fraction of the sample

comb data which yields coincidences normalized to the value

at f = f0. Theoretical spectra for a pure comb signal with

similar conditions to those used in KEIRA (γ = 1/4,c = 0

and γ = 1/4,c = −1/28) are shown in Fig. 6. For the cen-

ter pickup (c = 0), it can be seen that the relative contribu-

tion of some fractional harmonics is high, particularly for

f/f0= 1/h where h is odd. The even harmonics are absent

while odd harmonics are reduced in amplitude by a factor of

h. In contrast, for the offset pickup fractional harmonics are

significantly reduced in magnitude, with no contribution at in-

teger values.

As the frequency peaks in Figs. 6(a) and 6(b) only coin-

cide for f = f0, when the two spectra are multiplied together

only the fundamental remains. For real data, the finite width

of the image charge impulses does result in some contribu-

tions at other frequencies giving an incomplete suppression

of fractional harmonics. In this case, acquiring an additional

0

0.5

1.0

01.02.0

relative frequency

3.04.05.0

0

0.5

1.0

amplitude

(a)

(b)

FIG. 6. (Color online) Spectrum from analysis of a pure data comb at fre-

quency f0for (a) central pickup ring with γ = 1/4,c = 0, and (b) offset

pickup ring with γ = 1/4,c = −1/28 (approximately equal to the experi-

mental value of −0.038).

spectrum from a third pickup enables fractional harmonics to

be further suppressed by multiplying all three together and

taking the cube root.

In practice, the teeth of the comb are given a finite width

so that more than one data point is sampled per peak. The

average of all these data points is then normalized by the

number of teeth in the comb M. Increasing the width of the

teeth reduces the achievable resolution, but the change is neg-

ligible provided that the width remains less than tw. If ions

of different mass have the same bunch length in space, then

tw∝ m1/2

cies f , the algorithm changes the teeth width in proportion

to f−1.

Figure 7 shows CHIMERA sampling analysis for sim-

ulated data generated using the same parameters as those

used in Fig. 4, for the frequency range 0.1 f0–5.1 f0. Data

for three pickups (c = 0,−0.038, +0.06) has been gener-

ated [Figs. 7(a), 7(b), and 7(c), respectively]. Also shown are

the results of combining two of the spectra [c = 0,−0.038,

Fig. 7(d)] and all three [Fig. 7(e)] together. It can be seen that

there is a steady improvement in the purity of the spectrum

as the number of combined data sets increases. In principle

this will improve further if data from additional pickups is

included.

Toachieveastillpurerspectrum,theFFTcanbeusedasa

discriminator.Comparingacomb-samplingspectrumwiththe

FFT for c = 0 (at double the frequency since odd harmonics

aresuppressed),iftheFFTliesbelowausersetdiscrimination

level, the contribution to the comb-sampling spectra may be

set to zero at that frequency.

0

∝ f−1

0. Therefore, for different sampling frequen-

VI. ANALYSIS OF KEIRA RESULTS

So far we have applied the CHIMERA algorithm only to

synthetic data. Figure 8 shows the analysis of real data ac-

quired for TS= 8 ms from ionization of Xe gas by 40 fs laser

pulses with a maximum intensity of 2.5 × 1013W cm−2, av-

eraged over 4000 shots. In Fig. 8(a), the FFT of data from

the center pickup shows 14 even harmonics above the gen-

eral noise level. In Fig. 8(b), the comb-sampling method has

Page 8

043103-8 Greenwood et al.Rev. Sci. Instrum. 82, 043103 (2011)

0 0.1 0.2 0.3 0.4 0.5

0

0.5

1.0

amplitude

0 0.1 0.20.30.4 0.5

0 0.1 0.2 0.3 0.40.5

00.10.2 0.3 0.40.5

0

0.5

1.0

frequency (MHz)

00.10.20.3 0.40.5

(a)

(d)

(b) (c)

(e)

FIG. 7. (Color online) Spectra from analysis of simulated data for a single ion species oscillating at a frequency of 100 kHz with γ = 1/4 and (a) c = 0, (b)

c = −0.038, (c) c = +0.06, (d) c = 0,−0.038 combined, (e) all combined.

been applied using a window from 0.8 – 8 ms for both center

and offset pickup data, with a discriminator set just above the

general FFT noise in Fig. 8(a). Apart from a few minor contri-

butions, harmonics are absent from the comb analyzed spec-

trum. The frequency spectrum for isotopes of Xe+is shown in

Fig. 8(c), where the second harmonic of the FFT is compared

with CHIMERA analysis, which gives a resolving power of

5000, a factor 15 greater than the second harmonic in the FFT.

Whilst an equivalent FFT resolution can be obtained from the

30th harmonic as predicted by Eq. (3), this peak is barely dis-

cernable above the noise level.

It can be seen that the relative amplitudes of the xenon

isotopes obtained from FFT and CHIMERA are different.

While the results of the CHIMERA analysis are in good

agreement with natural abundances of these isotopes, the

agreement for the FFT is poor. This is probably due to the

limitation on the FFT frequency step size as discussed in

Sec. III C.

As CHIMERA removes harmonics, it is also straightfor-

ward to generate a mass spectrum. To demonstrate this, an

experiment was carried out on a gaseous target of α-Cyano-

4-hydroxycinnamic acid (CHCA), which has a formula of

C10H7NO3and a molecular mass of 189.17. The neutral tar-

get of this molecule was produced by depositing a sample of

CHCA onto the surface of the final electrode, which was then

desorbed off the surface using a 4 ns, 355 nm laser pulse.

Subsequent ionization/fragmentation was enforced by a 40 fs

laser pulse of intensity 1013W cm−2. Data was acquired for

3 ms using the central pickup and one offset pickup detec-

tor. Figure 9 compares mass spectra produced from a FFT of

the central pickup signal and from the comb-sampling anal-

ysis using data from both pickups. As well as the improved

0

0.5

1.0

amplitude

0 0.2 0.4 0.60.81.01.21.4

0

0.5

1.0

0.10150.10200.10250.1030

frequency (MHz)

0.10350.1040 0.10450.1050

0

0.5

1.0

136

132

129

128

130

131

134

(b)

(c)

(a)

FIG. 8. (Color online) Frequency analysis of Xe+isotopes trapped for 8 ms using (a) FFT with a Welch window and (b) comb-sampling with a window from

0.8–8 ms. Plot (c) shows the fundamental frequency from comb-sampling (narrow peaks) compared to the second harmonic of the FFT. Corresponding masses

of Xe isotopes are indicated.

Page 9

043103-9Greenwood et al.Rev. Sci. Instrum. 82, 043103 (2011)

0

0.5

1.0

amplitude

020 4060 80 100120140160180 200

0

0.5

1.0

mass (atomic units)

C2Hn

+

C3Hn

+

C4Hn

+

C5Hn

+

C6Hn

+

C7Hn

+

–COOHCN

–COOH

–OH

M+(b)

(a)

FIG. 9. (Color online) Frequency analysis of ions generated from ionization of CHCA trapped for 3 ms from (a) FFT of the center pickup data with a Welch

window, and (b) comb-sampling of data from the center pickup and the offset pickup closest to the femtosecond laser focus. M+is the parent ion, with other

fragments corresponding to the hydrocarbons shown, or to the loss (−) of a particular chemical group.

resolution of the comb-sampling method, it can be seen that

multiple harmonics in Fig. 9(a) make analysis of masses less

than 30 very difficult with FFT. For low masses in Fig. 9(b)

there are some fractional harmonics present, but these are at

a low level and could be further suppressed if data from addi-

tional pickups was available.

VII. SUMMARY

In this paper, a new algorithm (CHIMERA) for analyz-

ing the oscillation frequencies of ions in a linear electrostatic

trap has been described. By using a comb function to sam-

ple the data at different frequencies, it has been shown that

more complete utilization of the information contained in the

ion signal is possible. The key to extracting only the funda-

mental frequency is for the initial ion bunch generation to be

temporally and spatially well defined. By generating multiple

spectra using data acquired from more than one pickup de-

tector, integer and fractional harmonics arising from the finite

width of the pickup impulses can be suppressed.

We have tested CHIMERA on simulated data and real

data acquired from a linear electrostatic ion trap using in situ

ion generation with a femtosecond laser. As the sampling of

the narrow pickup peaks is very sensitive to the frequency

of the comb, mass resolutions obtained from relatively short

trapping periods (<10 ms) are shown to approach those which

could be obtained for a linear time of flight device of equiv-

alent length (about 100 m). For instance, a mass resolution

of 5000 was obtained for Xe+isotopes for a trapping time

of 8 ms, which was a 15 fold improvement over the second

harmonic obtained from a Fourier transform. CHIMERA also

benefits in that it does not suffer from spectral leakage, which

is a feature of the windowing process in a Fourier transform.

And unlike FFT algorithms, there are no limitations on the

frequency steps used.

The value of the initial offset (γ) of the comb data and

that of the pickup electrodes from the trap center (c), in the

present setup have been fixed by the existing geometry and

operational mode (γ=1/4, c=0, −0.038). While ideally these

values should be irrational, this is not possible in practice due

to the finite width of the recorded impulses. However, if γ and

c are chosen to avoid values close to factorized fractions with

small denominators, significant contributions from fractional

harmonics can be avoided.

As such our present value of γ=1/4 is not ideal, but could

be changed by pulsing the potential of the end plate R7 once

all the ions have first left the ion generation region. For in-

stance, if R7 = 3.2 kV when the ions are created and is raised

to 3.55 kV once the ions reach the field free region, a new γ

value of 0.283 would be generated. With appropriate choices

of γ and c, CHIMERA could also be easily applied to other

electrostatictrapsandringsforwhichionbunchesareinjected

from an external source. For instance, ion bunches from a

matrix-assisted laser desorption and ionization source19and a

pulsed beam20have previously been injected into this type of

trap using electrostatic optics. Another possible implementa-

tion would be for the case where ions are confined and cooled

in a radiofrequency trap prior to injection into a storage ring,

provided the ions are injected directly rather than being mass

selected by a magnet.21

Application to other mass spectrometers, such as the Or-

bitrap and ICR may be possible if highly anharmonic signals

are acquired. The potential for generating higher mass resolu-

tion from anharmonic signals in an ICR has previously been

recognised.22,23For some pickup electrode arrangements and

ion excitation schemes, signals from multiple pickups or the

differentialsignalfrompairscouldexhibitasignalresembling

the pulse trains analyzed in this paper.24

In conclusion, the ability to extract high resolution mass

spectra from a linear electrostatic ion trap has been enhanced

by employing a new CHIMERA analysis method, which sam-

ples ion oscillation data with a comb function. Compared with

Fourier analysis, the CHIMERA algorithm makes better use

of the phase and frequency information present in the data.

This enables the fundamental ion oscillation frequencies to

be extracted with much higher resolution, while also sup-

pressing uncorrelated electronic noise. When this algorithm is

combined with the mass independent trapping and ion detec-

tion characteristics of the electrostatic trap, very complex ion

Page 10

043103-10Greenwood et al. Rev. Sci. Instrum. 82, 043103 (2011)

mixtures with an enormous mass range can be simultaneously

analyzed with very high resolution.

ACKNOWLEDGMENTS

The authors wish to note with appreciation, insightful

discussions with Gleb Gribakin who provided valuable input

to this paper. This work was supported by Leverhulme Trust

and utilized the Artemis Laser at the STFC Central Laser Fa-

cility. C.R.C. and O.K. acknowledge funding from the Lev-

erhulme Trust. L.B., M.J.D., and R.B.K. acknowledge fund-

ing from Department of Employment and Learning (Northern

Ireland) and J.D.A. acknowledges funding from the European

Social Fund.

APPENDIX A: FOURIER TRANSFORM FOR AN ION

BUNCH OF CONSTANT LENGTH

s (t) ∝ e−|t|/τ

?

tw

t2

w/4 + t2⊗

2− c?

?

∞

?

n=−∞

δ

?

t −(n + c)

f0

?

+δ

t −

?n +1

f0

??

,

(A1)

for this signal with Lorentzian peaks and constant width tw,

the Fourier transform is

τ−1

τ−2+ (2πf )2

?

n=−∞

S ( f ) ∝

⊗

e−π| f |tw

?

∞

?

e−

2πif (n+c)

f0

+e−

2πif(n+1

2−c)

f0

??

,

(A2)

S ( f ) ∝

τ−1

τ−2+ (2πf )2

?

⊗

e−π| f |tw

?

1+e

2πif (2c−1

f0

2)

?

e−

2πif c

f0

∞

?

n=−∞

e−

2πif n

f0

?

,

(A3)

S ( f ) ∝

τ−1

τ−2+ (2πf )2

?

⊗

e−π| f |tw

?

1+e

2πif(2c−1

2)

f0

?

e−

2πif c

f0

∞

?

h=−∞

δ ( f −h f0)

?

.

(A4)

If the width of the Lorentzian is much less than the fun-

damental frequency (i.e, τ−1? f0), only one delta impulse in

the comb makes a significant contribution to the convolution

at each frequency, yielding

?

S ( f ) ∝ e−π| f |tw

1 + e

2πif(2c−1

2)

f0

?

e−2πif c

f0

×

∞

?

h=−∞

τ−1

τ−2+ (2π ( f − hf0))2,

(A5)

S ( f ) ∝ e−π| f |twcosπf?2c −1

∞

?

and the amplitude for positive frequencies is

2

?

f0

e−πif

2 f0

×

h=−∞

τ−1

τ−2+ (2π ( f − hf0))2,

(A6)

|S ( f )| ∝ e−π f tw

?????cosπf?2c −1

τ−2+ (2π ( f − hf0))2.

2

?

f0

?????

×

∞

?

h=1

τ−1

(A7)

APPENDIX B: FOURIER TRANSFORM FOR A

COHERENTLY DIFFUSING ION BUNCH

s (t) ∝ e−|t|/τ

∞

?

t −(n + c)

n=0

n?T

(n?T)2

4

+ t2

⊗

?

δ

?

f0

?

+ δ

?

t −

?n +1

2− c?

f0

??

.

(B1)

With the ion bunch lengthening linearly in time, the

Fourier transform for positive frequencies is

S ( f ) ∝

τ−1

τ−2+ (2πf )2

∞

?

⊗

n=0

e−π f n?T

?

e−

2πif (n+c)

f0

+ e−

2πif(n+1

2−c)

f0

?

.(B2)

Using the identity?∞

S ( f ) ∝

n=0pneinx=

τ−1

τ−2+ (2πf )2

e−2πif c/f0

1 − e−π f ( f0?T+2i)/f0

1

1−peix,

⊗

?

1 + e

2πif(2c−1

2)

f0

?

.

(B3)

For f = hf0+ δf , where δf ? f0, c = 0, τ → ∞ and h

even

S (δf ) ∝ (1 − e−π f ?Te−2πiδf/f0)−1,

|S (δf )| ∝ ((1 − e−π f ?T)2+ e−π f ?T(2πδf/f0)2)−1

(B4)

2.

(B5)

APPENDIX C: FREQUENCY DEPENDENCE OF THE

FOURIER PHASE

If we include additional time delay γ/f0, and consider

only imaginary multiplicative terms in Eq. (A3) we obtain

?

S ( f ) ∝ e−

2πif (c+γ)

f0

1+e

2πif(2c−1

2)

f0

?

,

(C1)

Page 11

043103-11Greenwood et al. Rev. Sci. Instrum. 82, 043103 (2011)

S ( f ) ∝

?

?

cos2πf (c + γ)

f0

− i sin2πf (c + γ)

+ isin2πf?2c −1

f0

?

1+cos2πf?2c −1

2

?

f0

2

?

f0

?

, (C2)

S ( f ) ∝ cos2πf?c −1

?

4

?

f0

×

cos2πf?γ +1

4

?

f0

− i sin2πf?γ +1

?

4

?

f0

?

, (C3)

φ ( f ) = −2πf

f0

γ +1

4

?

.

(C4)

APPENDIX D: COMB-SAMPLING THEORETICAL

RESOLUTION

S (δf ) =

f0

M

m2

?

m=m1

?k + pm − m|δf |

(k + pm)2

f0

?

.

(D1)

For no bunch diffusion (p = 0),

f0

Mk

m=m1

S (δf ) =

m2

?

?

1 − m|δf |

f0k

?

=

f0

k

−|δf |

Mk2

m2

?

m=m1

m ,

(D2)

S (δf ) =

f0

k

−

|δf |

2Mk2(m2− m1+ 1)(m2+ m1),

(D3)

S (δf ) =

f0

k

?

1 −|δf |

2 f0k(m2+ m1)

?

.

(D4)

This expression [in Eq. (D4)] is only valid for

m2|δf |

f0k

≤ 1.

(D5)

For coherent bunch diffusion (k = 0)

S (δf ) =

f0

Mp2

?

p −|δf |

f0

?

m2

?

m=m1

1

m.

(D6)

Using the following identity for an infinite harmonic series:

lim

m2→∞

m2

?

?

m=1

1

m= lnm2 + 0.5772 ,

(D7)

S (δf ) =

f0

Mp

1 −|δf |

pf0

?

ln

?m2

m1

?

,

if m1,m2? 1,

(D8)

S (δf ) =

if m1= 1,m2? 1 .

f0

Mp

?

1 −|δf |

pf0

?

(lnm2+ 0.5772 ),

(D9)

APPENDIX E: RESOLUTION OBTAINED WITH A

LINEAR TIME WEIGHTING ON THE DATA

For a signal decreasing linearly with a gradient of τ−1,

m1=0, and p = 0,

f0

Mk

m=0

S (δf ) =

m2

?

?

1 −m

k

|δf |

f0

??

1 −

m

τ f0

?

,

(E1)

S (δf ) ∝

m2

?

m=0

1 −

1

τ f0

m2

?

m=0

m−|δf |

kf0

?m2

m=0

?

m −

1

τ f0

m2

?

m=0

m2

?

,

(E2)

S (δf ) ∝ m2+ 1 −

?

1

2τ f0m2(m2+ 1)

1

3τ f0m2(m2+ 1)(2m2+ 1)

−|δf |

2kf0

m2(m2+ 1)−

?

(E3)

,

S (δf ) ∝ 1 −

m2

2τ f0

−|δf |m2

2kf0

?

1 −(2m2+ 1)

3τ f0

?

,

(E4)

S (δf ) ∝ 1 −|δf |m2(3τ f0− 2m2− 1)

3kf0(2τ f0− m2)

Since m2? 1, this gives a resolution of

R =m2(3τ f0− 2m2)

With a linearly increasing weighting ,

.

(E5)

6k (2τ f0− m2).

(E6)

S (δf ) ∝

m2

?

m=0

m −|δf |

kf0

m2

?

m=0

m2,

(E7)

S (δf ) ∝ m2(m2+ 1) −|δf |

3kf0

m2(m2+ 1)(2m2+ 1),

(E8)

S (δf ) ∝ 1 −|δf |

3kf0

(2m2+ 1).

(E9)

Giving a resolution of R = m2/3k if m2? 1.

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