L. PEROTTI ET AL. 243

becomes 12. The width is therefore

.. max

2

UC f

WWf

where

is the

dimensionless decay factor.

This means that, when using the Fourier Transform,

increasing numerical resolution (sampling rate) beyond

the natural linewidth of the signals involved is not much

use in separating neighbouring peaks.

When using the Padé Approximant approach, things

are very different: signals are represented by poles in the

complex plane and all poles are by definition singulari-

ties of Z(z) and therefore sharp. The basic point is that

poles corresponding to damped signals are off the unit

circle and are sharp only if we look at them in the

complex plane; if we only loo k on the unit circle, as it is

the case when using DFT, we do not see the singularity

itself but the profile of its tail as the intersection of Z(z)

with the unit circle.

This has two consequences:

1) as we are looking at the poles themselves, there are

no tails of strong wide peaks that can hide nearby peaks.

2) Since what counts is the distance of neighbouring

peaks in the comp lex plane, damped signals can be even

closer in frequency than reported above if their damping

constants (radial positions) are different.

2. Summary of the Method

Given a data series 012

,,, ,

ss s

N

N

, we build its generat-

ing function, or “Z-transform” Equation 1 and construct

its diagonal Padé Approximant, i.e. a rational function

with the numerator and denominator having the same

degree and whose Taylor expansion equals the Z-trans-

form up to order . The aim is to try and predict the

“Z-transform” for .

The choice of a diagonal rational approximation is the

best for both signal and noise because of the following

considerations.

For a finite ensemble of damped oscillators, the Z-

transform tends, when the number of data points goes to

infinity, to a

nn rational function in z, with 2nN

equal to twice the number of oscillators [5]. A diagonal

Padé Approximant therefore has the right structure for

the signal.

For pure noise, the organization of poles and zeros in

Froissart doublets [6-9] is again best approximated by a

nn rational function in

.

Most data analysts stopped using Padé Approximants

because of instabilities due to the fact that for a pure

signal, singularities appear when one tries to construct a

Padé Approximant of order higher than

nn. The prob-

lem is conveniently solved by the presence of noise who s e

Froissart doublets act as additional “signals”.

To numerically calculate poles and zeros of the Padé

Approximant of the Z-transform of a finite time series,

we build directly from the time series two tridiagonal

Hilbert space operators, called J-Matrices, one for the

numerator and one for the denominator. The eigenvalues

of these matrices readily provide the desired zeros and

poles. Details of our method can be found in [5]. Knowl-

edge of the positions of all poles and zeros also gives us

the residues for all poles and therefore the amplitudes

and phases of the signal oscillations.

3. Results and Sensitivity of the Method

When dealing with resolution, key parameters for both

Padé Approximant and Fourier Transform are:

1) the angular distance of the two signal poles, i.e. the

frequency difference scaled to the maximum detectable

frequency: this is the resolution itself.

2) The radial position

of the two signal poles, i.e.

the decay factor ln

of each of the signals.

3) The relative amplitude of the two signals.

4) The signal to noise ratio of the smaller of the two

signals, or equivalently , its precision in num ber of digits.

5) The number of data poin ts.

These are the factors that in practice can limit resolu-

tion, which assuming infinite data precision and arbitrar-

ily small noise has no limitatio n for Padé Approximants.

We now pass to look at a few cases that can help

clarify how parameters 2, 3, 4 and 5 affect resolution.

3.1. Two Equal Peaks on the Unit Circle

As a first example, let us consider two peaks on the unit

circle separated by along the circumference,

i.e. two constant amplitude waves whose frequency dif-

ference is

3

410

max

2

3

410

. Using the DFT to resolve them

we need a frequency step at least half the distance, i.e.

3

.. 410 2

fUC

3

10N

, which means a number of data poin ts

.

Figure 1 shows what can be seen with 256N

data

points. By increasing N to (Figure 2) a de-

cent resolution can be obtained. 8192N

Assuming the noise average amplitude to be 10–5 times

the amplitude of each of the two signals, and assuming

the data to be in double precision, the diagonal Padé

Approximant instead need s only 40 data points to resolve

the two signals with reasonable accuracy; a reduction by

two orders of magnitude. Figure 3 shows the positions

and residues of the reconstructed signal poles for 8

different realizations of the noise: there is some spread in

position and amplitude (pole residue) which completely

disappears by doublin g th e nu mber of d ata p oints, bu t the

16 poles are clearly grouped around the positions of the

two input poles marked by large red dots. 8 zeros fall

between the two groups of poles. The resolution transi-

tion between 1 and 2 signals takes place as follows. For

low only 8 poles with sizable residues are visible,

one for each noise realization, clustered halfway between

the positions of the two signal p oles; Figure 4 shows th e

N

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