Auto-correlation function and frequency spectrum due to a super-position of uncorrelated exponential pulses

0.59 MB·

by  O. E. Garcia

Auto-correlation function and frequency spectrum due to a super-position of
uncorrelated exponential pulses
O. E. Garcia∗ and A. Theodorsen
Department of Physics and Technology,
UiT The Arctic University of Norway, N-9037 Tromsø, Norway
(Dated: February2,2017)
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Abstract
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The auto-correlation function and the frequency power spectral density due to a super-position of un-
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F correlated exponential pulses are considered. These are shown to be independent of the degree of pulse

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overlapandtherebytheintermittencyofthestochasticprocess. Forconstantpulsedurationandaone-sided

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h exponentialpulseshape,thepowerspectraldensityhasaLorentzianshapewhichisflatforlowfrequencies
p
- and apower law at high frequencies. The algebraic tail is demonstrated to result from the discontinuity in
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s the pulse function. For a strongly asymmetric two-sided exponential pulse shape, the frequency spectrum
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p isabroken powerlawwithtwoscaling regions. Inthecaseofasymmetricpulse shape, thepowerspectral
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c density is the square of a Lorentzian function. The steep algebraic tail at high frequencies in these cases
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y is demonstrated to follow from the discontinuity in the derivative of the pulse function. A random distri-
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p bution of pulse durations is shown to result in apparently longer correlation times but has no influence on
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theasymptotic powerlawtailofthefrequency spectrum. Theeffectofadditional randomnoise isalsodis-
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cussed,leadingtoaflatspectrumforhighfrequencies. Theprobability densityfunctionforthefluctuations
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1 isshowntobeindependentofthedistributionofpulsedurations. Thepredictionsofthismodeldescribethe
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inthescrape-off layerofmagnetically confinedplasmas.
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I. INTRODUCTION
Theboundaryregionofmagneticallyconfinedplasmasisgenerallyfoundtobeinaninherently
fluctuating state with order unity relative fluctuation levels of the plasma parameters.1–10 The
frequencypowerspectraofthesefluctuationsaregenerallyfoundtobecharacterizedbyaflatlow-
frequencyregionandasteephigh-frequencytail.11–27 Inmanypreviousworks,powerlawscaling
relationshipshavebeeninferredfromexperimentalmeasurementsandtheresultsinterpretedinthe
frameworks of scale-free inertial range cascade, self-similar processes and self-organized critical
behavior.11–21 By contrast, it will in this contribution be demonstrated that the auto-correlation
functions and frequency spectra reported from experimental measurements can be described by a
super-positionofuncorrelated exponentialpulses.
Large-amplitude fluctuations in the boundary region are attributed to the radial motion of
blob-like plasma filaments through the scrape-off layer, most clearly demonstrated by gas puff
imagingdiagnostics.22–35 Duringtheirradial propagation,theblob-likestructuresdevelopasteep
front and a trailing wake, which can also be observed in numerical simulations of isolated blob
structures36–50 and simulations of scrape-off layer turbulence.51–61 Conditional averaging of ex-
perimental measurement data have shown that large-amplitude fluctuations are well described by
an exponentialwave form.62–83 This universalobservationof front steepening in interchangemo-
tionsofplasmafilamentsmotivatesthepresent studyofexponentialpulses.
The statistical properties of large-amplitude fluctuations in the scrape-off layer of tokamak
plasmas have recently been elucidated by means of exceptionally long data time series under
stationaryplasmaconditions.78–84 From single-pointmeasurementsit has been demonstratedthat
there is an exponential distribution of both the peak amplitudes and the waiting times between
them. Moreover, the average duration time does not depend on the amplitudeand also appears to
berobustagainstchangesinplasmaparameters.75–84 Basedontheseproperties,astochasticmodel
for the plasma fluctuations has been developed by a super-position of uncorrelated exponential
pulses.85–90 The underlying assumptions and predictions of this model have recently been found
tocomparefavorably withexperimentalmeasurements.78–84
In this contribution, the auto-correlation function and frequency spectrum are derived and dis-
cussed in detail for one- and two-sided exponential pulses. These are shown to be independent
of the amplitude distribution of the pulses as well as the degree of pulse overlap and thereby the
intermittency of the stochastic process. For constant pulse duration and a one-sided exponential
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pulse shape, the power spectral density has a Lorentzian shape, which is flat for low frequencies
andapowerlawspectrumathighfrequencies. Foratwo-sidedexponentialpulseshape,thepower
spectral density is the product of two Lorentzian functions. The power law tails at high frequen-
cies are shown to result from discontinuitiesin the pulse function or its derivative. A distribution
of pulsedurationtimes is demonstratedto result in apparently longercorrelation timesbut has no
influenceon thepowerlawtailofthefrequencyspectrum.
This paper is organized as follows. In the following section the stochastic model describing
fluctuations due to a super-position of uncorrelated pulses is presented. In Sec. III the mean and
variance of the random variable are calculated and the intrinsic intermittency of the process is
discussed. General expressions for the auto-correlation function and the power spectral density
are derivedin Sec. IV. Thecases ofone- and two-sidedexponentialpulsefunctionswithconstant
pulse duration is considered in Sec. V. In Sec. VI the algebraic tail in the frequency spectra are
demonstratedto resultfromthediscontinuityin thepulsefunctionoritsderivative. A distribution
of pulse durations is in Sec. VII shown to result in apparently longer correlation times but has no
effect on the asymptoticpowerlaw tail in the frequency spectrum. The contribution of additional
random noiseis discussedin Sec. Cand finallya discussionof theresultsand theconclusionsare
giveninSec.VIII.TheprobabilitydensityfunctionsinthecaseofexponentialandLaplaceampli-
tudedistributionsarederivedin AppendixA. A discussionoftherelationbetween discontinuities
inthepulseshapeoritsderivatviesandpowerlawspectraisgiveninAppendixB.Finally,therole
ofadditionalrandom noiseisdiscussedinAppendixC.
II. STOCHASTICMODEL
Considerastochasticprocess givenby thesuper-positionofa randomsequence ofK pulsesin
atimeintervalofdurationT,85–94
K(T)
F (t)= (cid:229) A f t−tk , (1)
K k t
k=1 (cid:18) k (cid:19)
whereeachpulselabeledk ischaracterized byanamplitudeA ,arrivaltimet ,anddurationt ,all
k k k
assumed to be uncorrelated and each of them independent and identically distributed. The pulse
arrivaltimest areinthefollowingassumedtobeuniformlydistributedonthetimeintervalunder
k
consideration,thatis,theirprobabilitydensityfunctionis givenby1/T. Thepulsedurationtimes
t k are assumed to be randomly distributed with probability density Pt (t ), and the average pulse
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durationtimeisdefined by
¥
t d =ht i= dtt Pt (t ). (2)
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Z
Here and in the following, angular brackets denote the average of the argument over all random
variables. TheresultspresentedhereareindependentofthedistributionofpulseamplitudesP (A),
A
it is only assumed that the mean hAi and variance hA2i are finite. The role of the pulse amplitude
distributionis discussedfurtherin AppendixA.
Thepulseshapef (q )istakento bethesameforalleventsinEq. (1). Thisfunctionisnormal-
ized suchthat
¥
dq |f (q )|=1. (3)
−¥
Z
Thus,forconstantdurationeachpulsecontributesequallytothemeanvalueofF (t). Theintegral
K
ofthen’thpowerofthepulseshapeisdefined as
¥
I = dq [f (q )]n, (4)
n
−¥
Z
for positive integers n. It is assumed that T is large compared with the range of values of t for
which f (t/t ) is appreciably different from zero, thus allowing to neglect end effects in a given
realization of the process. Furthermore, the normalized auto-correlation function of the pulse
functionisdefined by90
1 ¥
r f (q )= dcf (c )f (c +q ), (5)
I2 −¥
Z
and theFouriertransformofthisgivesthefrequency spectrum,
¥ 1
̺f (J )= dqr f (q )exp(−iJq )= |j |2(J ), (6)
−¥ I2
Z
wherethetransformofthepulsefunctionisdefined by
¥
j (J )= dqf (q )exp(−iJq ). (7)
−¥
Z
In this contribution, the auto-correlation function and the power spectral density for the process
defined by Eq. (1) will be investigated for the case of exponential pulses. The influence of var-
ious distributions of the pulse duration times as well as additive random noise will be explored.
However,first thetwolowestordermomentsoftheprocess willbederived.
III. MEAN,VARIANCEANDINTERMITTENCY
Thetwo lowestordermomentsofarealizationofthestochasticprocessdefined by Eq.(1)can
becalculated inastraightforward mannerby averagingoverallrandom variables.85–93
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A. Meanvalue
Starting with thecase of exactly K eventsin a timeintervalwith duration T, themean valueis
givenby
¥ ¥ T dt
hF Ki= dA1PA(A1) dt 1Pt (t 1) 1
−¥ 0 0 T
Z Z Z
···Z−¥¥ dAKPA(AK)Z0¥ dt KPt (t K)Z0T dTtK k(cid:229)=K1Akf (cid:18)t−t ktk(cid:19), (8)
using that the pulse arrival times are uniformly distributed. Neglecting end effects by taking the
integrationlimitsfor thearrivaltimest inEq. (8)to infinity,themeanvalueofthesignalfollows
k
directly,
K
hF i=t I hAi . (9)
K d 1
T
Herea changeofintegrationvariabledefined by q =(t−t )/t has been made,giving
k k
¥ ¥ t−t ¥ ¥
dt kPt (t k) dtkf t k = dt kt kPt (t k) dqf (q )=t dI1. (10)
Z0 Z−¥ (cid:18) k (cid:19) Z0 Z−¥
Takingintoaccount thatthenumberofpulsesK isalsoarandom variableandaveraging overthis
as wellgivesthemean valueforthestationaryprocess,
t
hF i= d I hAi, (11)
t 1
w
where t =T/hKi is the average pulse waiting time. For a non-negative pulse function, I = 1,
w 1
themean valueoftheprocess isgivenby theaverage pulseamplitudeand theratiooftheaverage
pulsedurationand waitingtimes. Themean valuevanishesboth foranti-symmetricpulseshapes,
I =0,andforpulseamplitudedistributionswithvanishingmean,hAi=0. Itshouldalsobenoted
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thatthemean valuesdoesnotdepend on thedistributionofpulsedurationtimes.
B. Variance
The variance can similarly be calculated by averaging the square of the random varible. From
thedefinitionofF (t)itfollowsthat
K
F 2(t)= (cid:229)K (cid:229)K A A f t−tk f t−tℓ . (12)
K k ℓ t t
k=1ℓ=1 (cid:18) k (cid:19) (cid:18) ℓ (cid:19)
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Averagingthisoverpulseamplitudesas wellas durationand arrivaltimesfor fixedt and K gives
¥ ¥ T dt
hF 2Ki= −¥ dA1PA(A1) 0 dt 1Pt (t 1) 0 T1
Z Z Z
···Z−¥¥ dAKPA(AK)Z0¥ dt KPt (t K)Z0T dTtK k(cid:229)=K1ℓ(cid:229)=K1AkAℓf (cid:18)t−t ktk(cid:19)f (cid:18)t−t ℓtℓ(cid:19). (13)
There are two contributionsto the variance from thedouble sum. When k6=ℓ there are K(K−1)
termswiththevalue
¥ ¥ T dt t−t
Z−¥ dAkPA(Ak)Z0 dt kPt (t k)Z0 Tk Akf (cid:18) t k k(cid:19)
¥ ¥ T dt t−t
×Z−¥ dAℓPA(Aℓ)Z0 dt ℓPt (t ℓ)Z0 TℓAℓf (cid:18) t ℓ ℓ(cid:19), (14)
whilefork=ℓthereare K termsgivenby theintegral
¥ ¥ T dt t−t
Z−¥ dAkPA(Ak)Z0 dt kPt (t k)Z0 TkA2kf 2(cid:18) t k k(cid:19). (15)
Neglecting end effects due to the finite duration of individual pulses by extending the integration
limitsfort andt toinfinity,thevarianceforlargeT in thecase ofexactlyK pulsesisgivenby
k ℓ
K K(K−1)
hF 2i=t I hA2i +t 2I2hAi2 . (16)
K d 2 T d 1 T2
By averagingoverall realizationswherethenumberofpulsesK isstatisticallydistributedgives
t
hF 2i= d I hA2i+hF i2, (17)
t 2
w
wherehK(K−1)i=hKi2 hasbeenassumed. ThisisanexactrelationforaPoissondistributionof
the number of pulses, and approximatewhen there is a large number of pulses in each realization
oftheprocess. Itfollowsthatthestandarddeviationorrootmeansquare(rms)valueoftherandom
variableis givenby
t
F 2 = d I hA2i. (18)
rms t 2
w
Thus, the absolute fluctuation level is large when there is significant overlap of pulse events, that
is, for long pulsedurations and short pulse waitingtimes. As for the mean valuegiven above, the
variancedoesnot dependon thedistributionofthepulsedurationtimes.
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C. Intermittency
Theratio oftheaverage pulsedurationand waitingtimes,
t
g = d, (19)
t
w
determinesthedegreeofpulseoverlapinthestochasticprocess. Wheng issmall,pulsesgenerally
appear isolated in realizations of the process, resulting in a strongly intermittent signal where
most of the time is spent at small values. When g is large, there is significant overlap of pulses
and realizations of the process resemble random noise. Indeed, it can be demonstrated that the
probabilitydensityfunctionfortherandomvariableF (t)approachesanormaldistributioninthe
K
limitofinfinitelylargeg , independentlyofthepulseshapeand amplitudedistribution.85,86
Foranon-zero mean valueoftheprocess,therelativefluctuationlevelisgivenby
F 2 1I hA2i
rms = 2 . (20)
hF i2 g I2hAi2
1
This is large for long pulse waiting times and short pulse durations. In Appendix A it is shown
that alsotheskewnessand flatness momentsbecomelarge forsmallvaluesof g . In thefollowing,
therescaled variablewithzero meanand unitstandarddeviationwillbefrequentlyconsidered,
F −hF i
F (t)= . (21)
F
rms
e
Some realizations of this process are presented in Fig. 1 for one-sided exponential pulses with
constant duration and exponentially distributed amplitudes. It is clearly seen that the signal is
strongly intermittent for small values of g , while pulse overlap for large values of g makes the
signals resemble random noise. Further description of the intermittency effects in this process is
givenin AppendixAand are discussedin Refs. 85–90.
IV. CORRELATIONSANDSPECTRA
In the same way as the variance was calculated above, the auto-correlation function for F (t)
K
can be calculated by a straight forward average over all random variables and the power spectral
densityisthen givenbya transformationtothefrequency domain.
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g =102 g =10 g =1 g =10-1
5 0
) 0 5
t
(
~
F
5 0
0
00 2200 4400 6600 8800 110000
t/t
d
FIG. 1. Realizations of the stochastic process for a one-sided exponential pulse shape with constant dura-
tion t and exponentially distributed pulse amplitudes. The degree of pulse overlap is determined by the
d
intermittency parameterg =t /t .
d w
A. Auto-correlation function
Considering first the signal F (t) defined by Eq. (1), the auto-correlation function for a time
K
lag r isgivenby adoublesum,
¥ ¥ T dt
hF K(t)F K(t+r)i= dA1PA(A1) dt 1Pt (t 1) 1···
−¥ 0 0 T
Z Z Z
Z−¥¥ dAKPA(AK)Z0¥ dt KPt (t K)Z0T dTtK k(cid:229)=K1ℓ(cid:229)=K1Akf (cid:18)t−t ktk(cid:19)Aℓf (cid:18)t−ttℓℓ+r(cid:19). (22)
Thereareagaintwocontributionstothedoublesum,comprisingK(K−1)termswhenk6=ℓgiven
by
hAi2 (cid:229)K ¥ dt kPt (t k) T dTtk f t−t tk ¥ dt ℓPt (t ℓ) T dTtℓf t−ttℓ+r , (23)
k,ℓ=1Z0 Z0 (cid:18) k (cid:19)Z0 Z0 (cid:18) ℓ (cid:19)
k6=ℓ
and K termswhen k=ℓ,
hA2i (cid:229)K ¥ dt kPt (t k) T dTtkf t−t tk f t−ttk+r . (24)
k=1Z0 Z0 (cid:18) k (cid:19) (cid:18) k (cid:19)
Neglecting end effects due to thefinite pulseduration by taking theintegrationlimitsfort andt
k ℓ
toinfinityforthecaseofexactlyK pulseeventsresultsin
K(K−1) K ¥
hF K(t)F K(t+r)i=t d2I12hAi2 T2 +I2hA2iT dtt Pt (t )r f (r/t ). (25)
0
Z
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Averaging over the number of pulses occurring in the interval of duration T, it follows that the
auto-correlationfunctionforthestationaryprocessis givenby90
1 ¥
RF (r)=hF i2+F 2rmst dtt Pt (t )r f (r/t ), (26)
d Z0
wherer f (q )isthenormalizedauto-correlationfunctionforthepulsefunctiondefinedbyEq.(5).
The expression of the auto-correlation function is simplified by considering the rescaled variable
defined by Eq.(21),
¥
1
RF (r)= t dtt Pt (t )r f (r/t ). (27)
d 0
Z
It is emphasized that this expresseion for the auto-correlation function does not rely on a Poisson
distribution of the number of pulse events. The auto-correlation function is determined by the
pulseshapethroughr f (q ) andtheprobabilitydensityfunctionforthepulsedurationtimes.
Equation(26) emphasizes theroleof thepulseshape in determiningthe auto-correlationfunc-
tion for the process. However, the time lag r in the integrand can be transfered to the distribution
function for thepulsedurations timesby thechange of variables q =|r|/t , which givesthealter-
nativeformulation
1 ¥ dq r
RF (r)= t q 2 q |r|Pt (|r|/q )r f (q ), (28)
d 0
Z
wheretheprobabilitydensityfeunctionforthepulsedurationtimesis normalizedsuchthat
¥ ¥ dq
dt Pt (t )= |r|Pt (|r|/q )=1. (29)
q 2
0 0
Z Z
In the case of constant pulse duration, the latter is given by the degenerate distribution Pt (t ) =
d (t −t ), where d is the delta distribution. The auto-correlation function is then R (r) =
d F
r f (r/t d), that is, it is simply given by the normalized auto-correlation function of the individ-
e
ual pulsesintheproecss.
B. Powerspectraldensity
From the auto-correlation function given above, thepower spectral density follows directly by
aFouriertransformationto thefrequency domain,90
¥ 1 ¥
W F (w )= drRF (r)exp(−iw r)=2p hF i2d (w )+F 2 dtt 2Pt (t )̺f (tw ), (30)
rmst
Z−¥ dZ0
where w is the angular frequency and ̺f (J ) is the Fourier transform of the normalized auto-
correlation function defined by Eq. (6). Here the first term on the right hand side of the second
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equality is a zero-frequency contribution from the mean value of the random variable. The ex-
pression for the power spectral density is simplified by considering the rescaled random variable
defined by Eq.(21),
1 ¥
W F (w )= t dtt 2Pt (t )̺f (tw ). (31)
dZ0
e
It should be noted that this frequency spectrum is independent of the amplitude distribution P
A
and does not depend on the intermittency parameter g . The latter property is evidently due to the
assumption of uncorrelated pulses. Moreover, the above expression is not restricted to a Poisson
distributionfor the number of pulses K(T). The only assumptionsmade are that the pulse arrival
timeshaveauniformdistributionandthatthetwolowestordermomentsfortheprocessarefinite.
Inthespecialcaseofconstantpulseduration,theexpressionforthepowerspectraldensitybecome
W F (w ) = t d ̺f (t dw ), that is, the spectrum is simply determined by that of the pulse function
f (q ).
e
By the simple change of variables, J = t |w |, Eq. (31) for the power spectral density can be
writtenin thealternativeform
1 ¥ dJ J 2
W F (w )= t |w |w 2 Pt (J /|w |)̺f (J ), (32)
dZ0
e
wheretheprobabilitydensityfunctionforthepulsedurationtimesis normalizedsuchthat
¥ ¥ dJ
dt Pt (t )= Pt (J /|w |)=1. (33)
|w |
0 0
Z Z
Equation (32) is suggestive of a power law spectrum for sufficiently broad distributions of pulse
durations. In general, the power spectral density is determined by both the pusle shape and the
distributionfunctionforpulsedurations.90
V. EXPONENTIALPULSES
The above expressions for the auto-correlation function and the power spectral density were
derived for an arbitrary pulse function under the assumption that the two lowest order moments
exist. In this section, the properties of the stochastic process will be investigated for the case of
one-and two-sidedexponentialpulseswithconstantduration.
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