### Filter Diagonalization Method and Decimated Signal Diagonalization

```Aimé Lay-Ekuakille
University of Salento
Index:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Problem statement
Main motivation
FDM-Filter Diagonalization Method (mono)
DSD-Decimate Signal Diagonalization (mono)
Application for detection in pipeline
FDM-Multidimensional
DSD-Multidimensional
Application for EEG
Final outlook
Problem statement
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
Many engineering and physicals issues deal
with particles decaying problems, namely,
NMR, fMRI, new resisting vegetables,
industrial processes using radioactivity, light
and photonics, etc.
These issues can be modeled using special
transforms and particular descriptions
(Poisson)
Main motivation
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
Traditional methods FFT and Laplace transform are
not suitable for above problems.
Recently, new methods have been introduced,
termed Filter Diagonalization Method (FDM) and
Decimated Signal Diagonalization (DSD), for
obtaining the complete eigenspectra of arbitrarily
large matrices that are theoretically generated with
auto-correlation functions from time propagated
wave packets. Using the equivalence between the
auto-correlation functions and the exponentially
damped signals spectrum is obtained as sums of pure
Lorentzians
FDM-Filter Diagonalization Method (mono)
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
In FDM, we would like to fit the diagonalization
measured complex valued signal Cn as a sum of
damped sinusoids:

=
exp(−
=1
It can be represented as the time autocorrelation
function of a fictitious dynamical system with
non-Hermitian but symmetric Hamiltonian
= 0 | − 0 ≡ 0 |  0
so that the highly nonlinear fitting problem is
reduced to that of diagonalization , the
evolution operator over a single time step
FDM-Filter Diagonalization Method (mono)
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
Neither the explicit form of the Hamiltonian
nor the ‘initial state’ 0 need be known, as only a
matrix representation in specific basis is required
for a numerical solution. The primitive basis is
iteratively derived by letting  act on 0
=  0
0
so that the overlap matrix element  and the
1
matrix elements  of  are given by the measured
data
0
=  | = +
1
=  | = ++1
FDM-Filter Diagonalization Method (mono)
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
The extraction of the eigenvalues,  = exp(−  ,
that determine line position and width, and the
eigenvectors,  , that determine amplitude and phase,
results by solving a generalized eigenvalue problem of
the form

1
=
0

FDM-Filter Diagonalization Method (mono)
Frequency and amplitude are calculated from:
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
1
= −
∡
2
=
FDM-Filter Diagonalization Method (mono)
FDM Flowchart
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
DSD-Decimate Signal Diagonalization(mono)
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
The diagonalization procedure extracts all of relevant
peak parameters, the complex frequencies and
amplitudes,  ,  , from window min , max . The
DSD technique is restricted to signals that are given as
sums of damped exponentials. Therefore, we model
the band-limited decimated signal  as:

=
− , Im < 0
=1
where the condition   < 0 selects only those
physically
meaningful
harmonics
that
decay
exponentially with increasing time and k is the socalled local spectral rank which is equal to the
number of Lorentzians generated by the equation
above
DSD-Decimate Signal Diagonalization(mono)
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
Like FDM, DSD uses windows to reduce a large data
matrix to a number of simple ones before
diagonalization.
However, while FDM filters basis functions to create
its windows, DSD filters the time signal. A time
signal is processed to get a low-resolution spectrum
by DFT. This spectrum is divided into M windows
containing at most 200 data points to avoid an illposed problem.
DSD-Decimate Signal Diagonalization(mono)
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
A new signal is created for each window
setting to zero the content outside the window
and then recentering the window at zero. The
inverse DFT is performed to convert the
frequency data back into the time domain.
The decimation step occurs when this new
time signal is sampled at M times greater than
the original time step, creating a bandlimited
decimated signal, which is diagonalized to
extract the spectral parameters for the matrix
overlapping U0d and U1d.
The diagonalization procedure is realized for
each of the M signals in this way

=
− , Im < 0
=1
DSD-Decimate Signal Diagonalization(mono)
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
The ωk are extracted from the eigenvalues
u1k=e-jω1kτ of the operator  =  −
1
= − ∡

while the amplitude
calculated as
parameters
2
−1
= 0|
2
=
2

=
=0
are
DSD-Decimate Signal Diagonalization(mono)
We normalize the  as
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
0  = ′
And we build spectrum from

=
=1

+
2
DSD-Decimate Signal Diagonalization(mono)
DSD Flowchart
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
DSD-Decimate Signal Diagonalization(mono)
DSD vs FFT
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
DSD-Decimate Signal Diagonalization(mono)
DSD vs FFT
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
Application for detection in pipeline (mono)
Experimental setup
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
Application for detection in pipeline (mono)
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
Application for detection in pipeline (mono)
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
Application for detection in pipeline (mono)
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
Application for detection in pipeline (mono)
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
Application for detection in pipeline (multidimensional)
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
Application for detection in pipeline (multidimensional)
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
Application for detection in pipeline
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
Application for detection in pipeline
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
FDM-(Multidimensional)
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
To create the multi-dimensional FDM, in this case twodimensional, it has been constructed in addition a new
matrix of overlap  2 , as it has already been calculated
for the matrices  1 and  0 .
2
=  | 2  = ++2
With the new matrix, the new eigenvalues and
eigenvectors will be calculated overlap, and from their
processing we will derive new amplitudes 2 and pulse
2 of the second spectrum similar to the onedimensional case.

2
2 = 2
0
2
FDM-(Multidimensional)
For the construction of bispectrum it will be
calculated the value of:
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
8.
9.
CROSS AMPLITUDE

= 1
0 2 1 2
BISPECTRUM
DSD(multidimensional)
Re 1,2
Application for EEG
Final outlook
1,2
1
1
Im
Im
1 − 21
2 − 22
FDM-(Multidimensional)
FDM-2D Flowchart
START
1.
2.
Problem statement
Main motivation
Acquire N samples cn of the analog signal with sampling time
τ and sampling frequency fs= 1/ τ
Build square matrices U0,U1 e U2
3.
M=N/2
FDM-(mono)
Rank(U0)<M
4.
DSD-(mono)
ρ(U0)→+∞
5.
6.
7.
8.
Application for
detection in pipeline
FDM(multidimensional)
DSD(multidimensional)
Application for EEG
Solve the generalized eigenvalue problem
with DLSP (damped least squares pseudoinverse)
Make normalization of eigenvector BK
such that BKTU0BK=δkk
Make normalization of eigenvector BK
such that BKTU0BK=δkk
Fix a smally and retain only eigenvalues
u1k such that 1-ϒ<|u1k|<1-ϒ
Fix a smally and retain only eigenvalues u1k
such that 1-ϒ<|u2k|<1-ϒ
Calculate ω1k=-1/τ
Calculate ω2k=-1/τ
Calculate d1k=(CTBK)
Calculate d1k=(CTBK)
Calculate cross amplitude
9.
Final outlook
Build bi-spectrum as
END
FDM-(Multidimensional)
The figures below show an application of the FDM
bispectrum
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
3D-VIEW
CONTOURVIEW
DSD-(Multidimensional)
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
As for the FDM method and decimation step, a
diagonalization
procedure
for
each
of
the M signals is realized

1
=
1  −1  → 1 1 = 1 0 1
=1

2
=
2  −2  → 2 2 = 2 0 2
=1
DSD-(Multidimensional)
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
9.
The ω1k and ω2k are extracted from the
eigenvalues
u1k=e-jω1kτ and u2k=e-jω2kτ of the
operator  =  −
1
1
= − ∡ 1

2
while the amplitude parameters are calculated as
2
−1
1 = 0|1
2
= 1 1
2

1 1
=
Application for EEG
Final outlook
1
= − ∡ 2

=0
2
−1
2 = 0|2
2
= 2 2
2

2 2
=
=0
DSD-(Multidimensional)
Then we calculate the cross amplitude as
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook

= 1
0 2 1 2
And we build bi-spectrum as
′1,2
1
Im
Im
1 − 21
2 − 22
(1 , 2 =
1,2
DSD-(Multidimensional)
DSD-2D Flowchart
1.
START
Problem statement
Acquire N samples of the analog signal with sampling time τ
2.
Main motivation
Compute FFT of the acquired samples obtaing the spectrum X
Create a moving window W of size ND=N/M
3.
FDM-(mono)
K=1
Multiply X by W obtaing a small Nd-point spectrum XK
4.
DSD-(mono)
Center Xk to zero frequency (shift)
IFFT-transform Xk obtaing a Nd-point signal Xk
5.
6.
7.
Application for
detection in pipeline
FDM(multidimensional)
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
Build matrix overlapp U0,U1 e U2
Compute FDM analysis to obtain amplitudes
and frequencies {(d1,ω1)}k of xk spectrum
Compute FDM analysis to obtain amplitudes
and frequencies {(d2,ω2)}k of xk spectrum
Both compress frequencies and resize
amplitudes by a factor M
Both compress frequencies and resize
amplitudes by a factor M
Cut frequencies at right and left spectrum
borders due to overlapping
Cut frequencies at right and left spectrum
borders due to overlapping
Inverse shift xk spectrum
Inverse shift xk spectrum
Move forward W by (Nd-overlap)-point
Move forward W by (Nd-overlap)-point
k=k+1
yes
k<=M ?
no
Calculate cross-amplitude Dkk
Build spectrum as
END
DSD-(Multidimensional)
The figures below show an application of the FDM
bispectrum
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
3D-VIEW
CONTOURVIEW
Application for EEG
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
Normal
EEG
Epileptic
EEG
Final outlook
1.
Problem statement
2.
Main motivation
3.
FDM-(mono)
4.
DSD-(mono)
5.
Application for
detection in pipeline
6.
FDM(multidimensional)
7.
DSD(multidimensional)
8.
Application for EEG
9.
Final outlook
The FDM and DSD are parameter estimators which
exhibit a 2-fold advantage over the most frequently
applied spectral estimator, the Fast Fourier Transform
(FFT).
1) FDM and DSD determine all the peak parameters
(positions, magnitudes, relaxation times,phases,
etc.) and then construct a spectrum in any
desired mode. This includes absorption, which has
a better resolving power than the corresponding
magnitude spectrum. The absorption spectra are
easily obtained without any additional
experimental effort, as no phase problems exist.
2) When a spectrum is not too densely packed with
spectral or noise features, remarkably good
results can be achieved with shorter computation
time than FFT.
```