Transcript Слайд 1 - IZMIRAN

ANALYTICAL SIGNAL AS A
TOOL FOR
HELIOSEISMOLOGY
Yuzef D. Zhugzhda
IZMIRAN, Moscow Region, Troitsk-city
The definition of an analytical signal was given by Gabor (1946).
Quasi-harmonic physical signal u(t) can be replaced by compex signal
w(t )  u (t )  iv(t )  A(t ) cos  (t )  A(t ) sin  (t )  A(t )e i ( t ) ,
d (t )
1
 (t ) 
, v (t )  H (u (t )) 
dt


u ( s ) ds
,

ts

where H(u(t)) is the Hilbert transformation of physical signal, A(t),
(t) and (t) are instant amplitude, phase and frequency of signal.
It was different ideas how is to define analytical signal.
It takes more than 30 years to realize finally that only Hilbert transformation
provides the correct way from the real physical signal to complex analytical
signal (Fink 1978).
The analytical signal allows to define uniquely the instant amplitude,
phase and frequency of narrow-band signal.
Advantage taken from analytical signal use is that time evolution
of frequency, phase and amplitude can be explored separately.
The
deviation
ofofspectral
frequency
defines
half-width
Mean
instant
frequency
to average
spectral
frequency
 (t ) equals
The
stability
p-modes
frequencies
is defined
by 
 But it does not define

of
spectraloflines
of p-modes.
deviation
instant
frequency.
In
contrast
with
the
2
2
 | W ( ) | d
 (t ) A(t ) dt


stability
of
p-mode
frequencies
since
fluctuations
of
deviation
of
spectral
frequency
0 the deviation of instant

 (t ) 
 


amplitude
widen
spectral
lines along
with
fluctuations
2
2
frequency
does
not
depend
on
fluctuations
of
A
(
t
)
dt
|
W
(

)
|
d




0
Thus,
analytical
signal
makes
possible
inmakes
some cases
to
of
frequency.
amplitude.
Thus,
analytical
signal
possible
to
Equality
of second
reads Analytical
overcome time
resolution
of spectral
analysis.
define
frequency
stability
ofmoments
narrow-band
signal. The
signal can not be applied to quantum
This formula
t) 
 dA(objects.
 | W ( )in
| d

(t ) A(t ) dt stability

 dt
definition
of
frequency
is
not
possible



was
obtained
many
years
ago
by
brilliant
Russian
scientist
dt


 dA(t ) 

(t )  





frames
of
spectral
analysis.
Half-width
of
lines
does
not
dt  It was obtained even before Gabor proposed to
Rytov (1940).
A(t ) dt
A(t ) dt
| W ( ) | d



define frequency
use Hilbert transformation
to definestability.
analytical signal.

2

2

2
2
2
2
2



0

2

2

2
2

0
To combine two equalities we arrive to
( 2  ( 2
(
( 2  (t )2  ((t )2
2
 dA(t ) 

 ,
 dt 
where ( 2   2   2 is average deviation of spectral frequency,
is mean deviation of instant frequency.
Theory of analytical signal (Wakman&Veistein 1973, 1985, Fink 1978) restricts its
application only for quasi-harmonic signals. Zhugzhda (2006) lifted this
restriction. It appeared that analytical signal can be applied to signals which
consist of two or more quasi-harmonic signals of close frequencies. In the
simplest case of two-component signal
u (t )  A1 cos(1t  1 )  A2 cos(2t   2 ),
 (t )  ( A121  A222  A1 A2 (1  2 ) cos((1  2 )t  1   2 )) / A(t ) 2 ,
A(t ) 2  A12  A22  2 A1 A2 cos((1  2 )t  1   2 ),
( A1 (t ))2
( A2 (t ))2
2
2
2
 (t ) 

(
t
)


(
t
)
,
(
A
(
t
))

(
A
(
t
))

(
A
(
t
))
.
1
2
1
2
2
2
( A(t ))
( A(t ))
2 and frequency spikes
2
Phase( Ajumps
appear due to beats of two
(
t
))
(
A
(
t
))
2
2
2
2
signals
(t )  of1 distinct

(
t
)


(
t
)
,
(
A
(
t
))

(
A
(
t
))

(
A
(
t
))
.
spikes2
1 frequencies. 2Phase
2 jumps and frequency
1
2
( A(t ))
( A(t ))
occur at the nodes where amplitude reaches its minimum.
The
P-modes
variance
are
ideal
of amplitude
object
forofphases
application
4 is
spectral
components
of occurrence
analytical is
signal
20
transformation
more
variance
since
they
of
Instant
frequencies
and
show
of
frequency
spikes
and
phase
The
surprising
thing
that
amplitudes
oftimes
blue,
redthe
and
even
arejumps.
frequency
narrow-band
which
oscillatory
is evidence
a measure
processes.
of unresolved
frequency
To simplify
stability
treatment
. Green spectral
p-mode
component
with l=0 (n=23)
is an
Thiscomponents
is
of
quasi-harmonic
components.
I should
yellow
dominate
during
two
month.
While
have
exception
been
of
chosen.
this
rule.
Two-months
The
run of
brightness
variance
observations
arises
bydue
DIFOS
to multiphotometer
focus
your
attention
to increase
location
offrequency
phase jumps.
One
and
all
of
them
appear at
amplitude
of
strongest
green
component
is
always
less
than
on the boardnature
component
of CORONAS-F
of this spectral
satellite
component
(channelwhich
350nm)
manifests
have been
itself
used.
by multiple
It is known
phase
that
amplitude
minima.
p-modes
jumps
and
with
frequency
l=0 are spikes
used to
be splitted
few spectral components.
these
threein amplitudes.
The variance of frequency for the rest components are more because each of them
It turned out that two components plotted by cyan and magenta colors (n=3303.9128,
consists of two unresolved components which is clear from the occurrence of spikes
3305.0794mkHz) have practically constant frequencies and phases during two months of
and phase jumps . It is instructive to point out that the amplitude of the most stable
observations. The variances of their frequencies are very small (n=0.00031,
component (cyan curves) is affected by strong variations (about 4 times) which are not
0.00086mkHz) that is 1/173 and 1/48 times variances of their amplitudes. Thus, line
connected with beats. But no one of p-mode components disappears during two
width of p-modes is defined only by amplitude fluctuations. The stability of frequency
months of observations. Thus, so-called appearance and disappearance of p-modes
is defined by n/n and is about 10^{-7} for cyan component.
sometimes is just manifestation of beats between components.
Summary
1.
2.
3.
4.
5.
Separate exploration of time evolution of
amplitude and frequency
Possibility to define frequency stability
Separation of close spectral components
Exploration of the nature of phase jumps of
p-modes (beats or stochastic excitation)
Exploration of real life time of p-modes
APPLICATION OF ANALYTICA SIGNAL TO EXPLORATION OF SUNSPOT OSCILLATIONS
Observations of sunspot oscillations by Centeno et al (2006) in He I 10830 A
multiplet have been used. In addition to analysis of Centeno et al. passband
frequencies and their variance have been obtained at 26 points along the sleet.