analytical signal as a tool for helioseismology yuzef d. zhugzhda izmiran, moscow region,...
TRANSCRIPT
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
,)(1
))(()(,)(
)(
,)()(sin)()(cos)()()()( )(
stdssu
tuHtvdttd
t
etAttAttAtivtutw ti
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.
0
2
0
2
2
2
|)(|
|)(|
)(
)()(
)(
dW
dW
dttA
dttAt
t
2
0
2
0
22
2
2
2
222
2
|)(|
|)(|
)(
)(
)(
)()()(
)(
dW
dW
dttA
dtdttdA
dttA
dttAt
dttdA
t
Mean instant frequency equals to average spectral frequency
)(t
Equality of second moments reads
To combine two equalities we arrive to
,)(
222
dttdA
where is average deviation of spectral frequency,
is mean deviation of instant frequency.
222
222 )()( tt
The deviation of spectral frequency defines half-width of spectral lines of p-modes. But it does not define stability of p-mode frequencies since fluctuations of amplitude widen spectral lines along with
fluctuations of frequency.
The stability of p-modes frequencies is defined by deviation of instant frequency. In contrast with the deviation of spectral
frequency the deviation of instant frequency does not depend on
fluctuations of amplitude. Thus, analytical signal makes possible to define frequency
stability of narrow-band signal. The definition of frequency stability is not possible in frames of spectral analysis.
Half-width of lines does not define frequency stability.
Thus, analytical signal makes possible in some cases to overcome time resolution of spectral analysis. Analytical signal can not be applied to quantum objects. This formula was obtained many years ago by brilliant Russian scientist Rytov (1940). It was obtained even before Gabor proposed to use Hilbert transformation to define analytical signal.
.))(())(())((,)())((
))(()(
))((
))(()(
),)cos((2)(
,)(/)))cos(()(()(
),cos()cos()(
22
21
222
22
12
21
21212122
21
2
2212121212
221
21
222111
tAtAtAttA
tAt
tA
tAt
tAAAAtA
tAtAAAAt
tAtAtu
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
Phase jumps and frequency spikes appear due to beats of two signals of distinct frequencies. Phase
jumps and frequency spikes occur at the nodes where amplitude reaches its minimum.
.))(())(())((,)())((
))(()(
))((
))(()( 2
22
12
22
22
12
21 tAtAtAt
tA
tAt
tA
tAt
P-modes are ideal object for application of analytical signal transformation since they are narrow-band oscillatory processes. To simplify treatment p-mode with l=0 (n=23) have been chosen. Two-months run of brightness observations by DIFOS photometer on the board of CORONAS-F satellite (channel 350nm) have been used. It is known that p-modes with l=0 are used to be splitted in few spectral components.
The surprising thing is that amplitudes of blue, red and even yellow components dominate
during two month. While amplitude of strongest green component is always less than these three
amplitudes.
Instant frequencies and phases show occurrence of frequency spikes and phase jumps. This is evidence of
unresolved quasi-harmonic components. I should focus your attention to location of phase jumps. One and all of them
appear at amplitude minima.
The variance of amplitude of 4 spectral components is 20 times more the variance of frequency which is a measure of frequency stability . Green spectral component is an exception of this rule. The increase of frequency variance arises due to multi-component nature of this spectral component which manifests itself by multiple phase jumps and frequency spikes
It turned out that two components plotted by cyan and magenta colors (=3303.9128, 3305.0794mkHz) have practically constant frequencies and phases during two months of observations. The variances of their frequencies are very small (=0.00031, 0.00086mkHz) that is 1/173 and 1/48 times variances of their amplitudes. Thus, line width of p-modes is defined only by amplitude fluctuations. The stability of frequency is defined by and is about 10^{-7} for cyan component.
The variance of frequency for the rest components are more because each of them consists of two unresolved components which is clear from the occurrence of spikes and phase jumps . It is instructive to point out that the amplitude of the most stable component (cyan curves) is affected by strong variations (about 4 times) which are not connected with beats. But no one of p-mode components disappears during two months of observations. Thus, so-called appearance and disappearance of p-modes sometimes is just manifestation of beats between components.
SummarySummary
1.1. Separate exploration of time evolution of Separate exploration of time evolution of amplitude and frequencyamplitude and frequency
2.2. Possibility to define frequency stabilityPossibility to define frequency stability
3.3. Separation of close spectral componentsSeparation of close spectral components
4.4. Exploration of the nature of phase jumps Exploration of the nature of phase jumps of p-modes (beats or stochastic of p-modes (beats or stochastic excitation)excitation)
5.5. Exploration of real life time of p-modes 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.