Page 1

Mon. Not. R. Astron. Soc. 000, 1–18 (2011)Printed 27 June 2011(MN LATEX style file v2.2)

Blazhko RRLyrae light curves as modulated signals

J. M. Benk˝ o?, R. Szab´ o, and M. Papar´ o

Konkoly Observatory, Konkoly Thege M. u. 15-17., H-1121 Budapest, Hungary

Accepted 2011 June 23. Received 2011 June 23; in original form 2011 April 28

ABSTRACT

We present an analytical formalism for the description of Blazhko RRLyrae light curves in

which employ a treatment for the amplitude and frequency modulations in a manner similar to

the theory of electronic signal transmitting. We assume monoperiodic RR Lyrae light curves

as carrier waves and modulate their amplitude (AM), frequency (FM), phase (PM), and as

a general case we discuss simultaneous AM and FM. The main advantages of this handling

are the following: (i) The mathematical formalism naturally explains numerous light curve

characteristics found in Blazhko RRLyrae stars such as mean brightness variations, compli-

cated envelope curves, non-sinusoidal frequency variations. (ii) Our description also explains

properties of the Fourier spectra such as apparent higher-order multiplets, amplitude distribu-

tion of the side peaks, the appearance of the modulation frequency itself and its harmonics. In

addition, comparing to the traditional method, our light curve solutions reduce the number of

necessary parameters. This formalism can be applied to any type of modulated light curves,

not just for Blazhko RRLyrae stars.

Key words: methods: analytical — methods: data analysis — stars: oscillations — stars:

variables: general — stars: variables: RR Lyrae

1INTRODUCTION

The Blazhko effect (Blazhko 1907) is a periodic amplitude and

phase variation of the RRLyrae variable stars’ light curve. The typ-

ical cycle lengths of these variations are about of 10-100 times

longer than the main pulsation periods (0.3−0.7 d). Almost half

of the RRLyrae stars pulsating in their fundamental mode (type

RRab) and a smaller but non-negligible fraction of the first over-

tone mode pulsating stars (type RRc) show the effect (Jurcsik et

al. 2009c; Chadid et al. 2009; Kolenberg et al. 2010; Benk˝ o et

al. 2010). It is usually interpreted as a modulation or a beating

phenomenon, but both hypotheses have their own problems. The

beating picture describes the main feature of the light curves and

Fourier spectra well (see Breger & Kolenberg 2006; Kolenberg et

al. 2006), but reproducing phase variations, multiplet structures

found in certain stars’ Fourier spectra (Jurcsik et al. 2008; Chadid

et al. 2010) in this framework is not possible. On the other side,

the stars showing doublet structures in their Fourier spectra (Al-

cock et al. 2000, 2003; Moskalik & Poretti 2003) seemed to be

contradicted with the modulation picture.

In this paper we describe the Blazhko effect as a modulation,

and derive the mathematical consequences of this assumption by

developing a consistent analytical framework. Using this frame-

work we demonstrate that many light curve characteristics are nat-

urally identified as mathematical consequences of the modulation

assumption. By disentangling these features we get closer to the

physics behind the Blazhko effect.

?E-mail: benko@konkoly.hu

The possibility of the modulation/Blazhko effect have been

raised for many types of pulsating stars from Cepheids to δ Scuti

stars (see e.g. Koen 2001; Henry, Fekel & Henry 2005; Moska-

lik & Kołaczkowski 2009; Breger 2010; Poretti et al. 2011). The

main motivation of this paper is the mathematical description of the

Blazhko RRLyrae stars’ light curves and investigate their proper-

ties. Most of our results can be applied directly to any other types

of variable stars, where modulation is suspected. Our deduced for-

mulae and the related phenomena may help to prove or reject the

modulation hypothesis.

The basic idea of this paper was raised in Benk˝ o et al. (2009).

Modulation is a technique that has been used in electronic commu-

nication for a long time, mostly for transmitting information signal

via a radio carrier wave. In those cases, the carrier wave is a sinu-

soidal electromagnetic (radio) wave that is modulated by a (gener-

ally non-periodic) information signal (e.g. speech, music). In this

paper the formalism developed by engineers for broadcasting radio

signals has been modified such a way that we assumed a monope-

riodic non-modulated RRLyrae light variation as a carrier wave.

While for communication usually only one type of modulation is

applied we allow both types of modulations (amplitude and angle).

In Section 2 we present a collection of classical formulae

that are well-known in physics of telecommunications (Carson

1922; van der Pol 1930; Roder 1931). Some of the more com-

plicated cases (multiple modulations, recursive or cascade modula-

tions) were investigated by mathematicians who developed the the-

ory of electric sound synthesizers in the years of 1960s and ’70s.

Theformulaearemodifiedtodescribethemodulatedlightcurvesin

Sec. 3 and investigated by a step-by-step process from the simplest

arXiv:1106.4914v1 [astro-ph.SR] 24 Jun 2011

Page 2

2Benk˝ o et al.

cases to the more complex ones. Section 4 compares the numerical

behaviour of the traditional method and this one. Section 5 summa-

rizes our results.

2BASIC FORMULAE

In this section we briefly review some classical definitions and for-

mulae (see e.g. Newkirk& Karlquist 2004;Schottstaedt 2003) that

will be used through the next sections. The simplest periodic signal

is a sinusoidal function. It has three basic parameters: amplitude,

frequency and phase and any of these can be modulated.

2.1Amplitude modulation

The amplitude modulation (AM) is the simplest of the three cases.

Let the carrier wave c(t) be a simple sinusoidal signal as

c(t) = Ucsin(2πfct + ϕc),

(1)

where Uc, fc and ϕc constant parameters are the amplitude, fre-

quency and initial phase of the carrier wave, respectively.

Let Um(t) represent an arbitrary waveform that is the mes-

sage to be transmitted. The transmitter uses the information signal

Um(t) to vary the amplitude of the carrier Uc to produce a modu-

lated signal:

UAM(t) = [Uc+ Um(t)]sin(2πfct + ϕc).

(2)

In the simplest case, where the modulation is also sinusoidal

Um(t) = UA

msin(2πfmt + ϕA

m).

(3)

Substituting Eq. (3) into (2) and using basic trigonometrical identi-

ties, expression (2) can be rewritten as

UAM(t) = Ucsin(2πfct + ϕc) +

UA

m

2

?sin?2π (fc− fm)t + ϕ−?+ sin?2π (fc+ fm)t + ϕ+??,

(4)

where ϕ−= ϕc−ϕA

shifts (±π/2) appear because we used (through all this paper) a

sinusoidal representation instead of sin and cos functions.

The exact analytical Fourier transformation of (4) is given in

Appendix A, however, the basic structure of the frequency spec-

trum can be easily read off from Eq. (4). Since the Fourier spectrum

of a single sinusoidal function shows a peak at the frequency of

the sinusoid, from the above expression (4) the well-known triplet

structure composed of the peaks fcand fc± fmcan be seen. The

amplitude of the side-peaks fc±fmare always equal. The Fourier

amplitude of the carrier wave (π√2πUc), that represents the energy

at the carrier frequency, is constant.

The ratio of the carrier wave amplitude A(fc) and the side

peaks A(fc ± fm) are connected to the modulation depth. We

rewrite Eq. (2) as

?

If Um(t) is a bounded function, let Umax

value of this modulation function, then modulation depth is de-

fined as h = Umax

m

/Uc. In the above discussed sinusoidal case

h = UA

amplitude of the central peak is twice of the side peak highs.

m+π/2 and ϕ+= ϕc+ϕA

m−π/2. The initial

UAM(t) =1 +Um(t)

Uc

?

represent the maximum

c(t).

(5)

m

m/Uc and A(fc ± fm)/A(fc) =

1

2h. In other words, the

2.2Angle modulations

The phase and frequency modulations together are called angle

modulations. Since, when we assume the sinusoidal carrier wave

Eq. (1) as c(t) = Ucsin[Θ(t)], the Θ(t) = 2πfct+ϕcdenotes the

angle part of the function.

Phase modulation (PM) changes the phase angle of the carrier

signal. Suppose that the modulating or message signal is Um(t),

then Θ(t) = 2πfct + [ϕc + Um(t)]. Let Um(t) be again a

bounded function. In this case we can define a constant as: kPM =

|Umax

UPM(t) = Ucsin2πfct + kPMUP

m

(t)|/2. This transforms the modulated signal

?

where |UP

signal is

m(t) + ϕc

?

,

(6)

m(t)| ? 1. The instantaneous frequency of the modulated

f(t) =dΘ

dt

= fc+ kPMdUP

m(t)

dt

.

(7)

Frequency modulation (FM) uses the modulation signal

Um(t) to vary the carrier frequency. Θ(t) = 2πf(t)t + ϕc and

here the instantaneous frequency f(t) is modulated by the signal

of kFMUF

m(t) as

f(t) = fc+ kFMUF

m(t).

(8)

In this equation kFM is the frequency deviation, which represents

the maximum shift from fc in one direction, assuming UF

limited to the range (−1,...,+1). Using the definitions of the in-

stantaneous frequency and phase, expression (8) can be rewritten as

Θ(t) = 2πfct+2πkFM

is

?

This definition of FM is the least intuitive of the three Eqs (2, 6

and 9). If we compare Eqs (6) and (9) we realize that the modu-

lation signals are in derivative-integral connection with each other.

In practice, we have modulating signals that can be represented by

analytical functions. Therefore, when we detect an FM or PM sig-

nal without any previous knowledge about them, it is impossible to

distinguish between FM and PM signals.

First, let the modulating signal be represented by a sinusoidal

wave with a frequency fm. The integral of such a signal is

m(t) is

?t

0UF

m(τ)dτ +ϕc. The modulated signal

UFM(t) = Ucsin2πfct + 2πkFM

?t

0

UF

m(τ)dτ + ϕc

?

. (9)

UF

m(t) =

UF

2πfmsin

m

?

2πfmt + ϕF

m

?

.

(10)

Thus, in this case Eq. (9) gives

UFM(t) = Ucsin

?

2πfct + η sin

?

2πfmt + ϕF

m

?

+ ϕc

?

, (11)

where the modulation index is defined as η = (kFMUF

Eq. (11) can be deduced from Eq. (6) as well. The only difference

is the value of η = kPMUP

lation frequency fm. Let us transcribe Eq. (11) using relations for

trigonometricalandBesselfunctions(Abramowitz&Stegun 1972)

as:

?

where Jk(η) is the Bessel function of first kind with integer order

k for the value of η (Fig. 1); ϕm denotes either ϕF

formula is known as the Chowning relation (Chowning 1973). Al-

though it had been deduced by many different authors formerly,

m)/fm.

m, which is independent of the modu-

UFM(t) = Uc

∞

k=−∞

Jk(η)sin[2π (fc+ kfm)t + kϕm+ ϕc],

(12)

mor ϕP

m. This

Page 3

Blazhko RRLyrae light curves as modulated signals3

-0.6-0.6 -0.6-0.6

-0.4 -0.4-0.4-0.4

-0.2-0.2 -0.2-0.2

0 0 0 0

0.2 0.2 0.2 0.2

0.4 0.4 0.4 0.4

0.6 0.6 0.6 0.6

0.8 0.8 0.8 0.8

1 1 1 1

0 0 0 0 5 5 5 5 10 10 10 10 15

xxxx

20 20 20 20 25 25 25 25 30 30 30 30

f(x) f(x)f(x)f(x)

15 15 15

Figure 1. The graph of the first three and the 10th Bessel functions of first

kind with integer order k for the value of x. J0(x) – (red) continuous line,

J1(x) – (green) dashed line, J2(x) – (blue) dotted line, J10(x) – (purple)

dash-dotted line.

Chowning recognized its key role in the electronic sound creation

method called FM synthesis.

Similarly to Eq. (4) expression (12) also helps to imagine the

spectrum. It is made up of a carrier at fcand symmetrically placed

side peaks separated by fm. The amplitudes follow the Bessel func-

tions. The behavior of the Bessel functions is well known: except

for small arguments (x < k), they behave like damped sine func-

tions (see also, Fig. 1). For higher indices the higher order side

peaks gradually become more and more important. As a conse-

quence, the amplitude of the central peak gets reduced. The fre-

quency spectrum of an actual FM signal has an infinite number of

side peak components, although they become negligibly small be-

yond a point.

If |η| ? 1, we find that J0(η) ≈ 1, |J±1| = η/2 and Jk ≈ 0

for k > 1. That is, the spectrum can be approximated with an

equidistant triplet similarly to AM, but the character of the sig-

nal differs from AM: the total amplitude of the modulated wave

remains constant. When η increases the amplitude of side peaks

also increases, but the Fourier amplitude of the carrier decreases.

In other words, the side peaks could be larger than the central peak,

on the other hand higher order side frequencies could also be of

larger amplitude than the lower order ones.

A more general case is formulated by Schottstaedt (1977)

UFM(t) = Ucsin

?

2πfct +

q

?

q

?

?

p=1

U(p)

m sin

?

2πf(p)

m t + ϕ(p)

m

?

+ ϕc

?

= Uc

∞

?

kp=−∞

···

∞

?

?

k1=−∞

?

p=1

Jkp(U(p)

m )

?

sin

?

2πfct +

q

p=1

kp

2πf(p)

m t + ϕ(p)

m

?

+ ϕc

?

.

(13)

Here the modulating signal is assumed to be a linear combination

of a finite number of sinusoidal functions with arbitrary frequen-

cies f(p)

spectrum contains equidistant frequencies on both sides of the car-

rier frequency. The amplitudes of the side peaks fc ± kpf(p)

determined by products of Bessel functions.

m , amplitudes U(p)

m and phases ϕ(p)

m , p = 1,2,...,q. The

m

are

2.3Combined modulation

In practice, the electronic circuits that generate modulated signals

generally produce a mixture of amplitude and angle modulations.

This combined modulation is disliked in radio techniques but wel-

comed in sound synthesis and as we will see they appear in the case

of Blazhko RRLyrae stars, as well. Let us overview the basic phe-

nomena of combined modulations following Cartianu (1966). We

start with the simplest case: both the AM and FM are sinusoidal

and their frequencies are the same.

UComb(t) = Uc(1 + hsin2πfmt)sin[2πfct+

η sin(2πfmt + φm) + ϕc].

(14)

By suitable choice of the starting epoch, without any restriction

of the general validity we can set ϕc = 0. Here φm is the rela-

tive phase difference between the modulating FM and AM signals.

Other designations are the same as before. The third term of the

product (14) is the same as in Eq. (11), therefore, after applying the

Chowning relation (12)

UComb(t) = Uc(1 + hsin2πfmt)·

∞

?

This expression results in an infinite number of amplitude modu-

lated waves. After trigonometrical transformations we get:

?

h

2Jk−1(η)sin

h

2Jk+1(η)sin

It can be seen that each terms consists of three sinusoidal functions

with different phases. On the basis of expression (16), the spectrum

of combined modulation (14) is comprehensible as a combination

of three FM spectra. The peaks are at the same places as the fre-

quencies of the spectrum of (12), but the amplitudes of a pair of

side peaks are generally asymmetrical. Using some trigonometrical

identities, the rules of summation of parallel harmonic oscillations

and relations for Bessel functions we arrive to the expression for

the Fourier amplitudes of a certain frequency:

?

k=−∞

Jk(η)sin[2π (fc+ kfm)t + kφm].

(15)

UComb(t) = Uc

∞

?

?

2π (fc+ kfm)t + (k + 1)φm+π

k=−∞

2π (fc+ kfm)t + (k − 1)φm−π

Jk(η)sin[2π (fc+ kfm)t + kφm]+

2

?

+

?

2

??

.

(16)

A(fc+ kfm) ∼ Uc

J2

k(η)

?

1 −hk

η

sinφm

?2

+

h2

4cos2φm[Jk+1(η) − Jk−1(η)]2

?1

2

,

(17)

(k = 0,±1,±2,...). Introducing the power difference of the side

peaks as it was done by Szeidl & Jurcsik (2009) ∆l := A2(fc+

lfm)−A2(fc−lfm), where l = 1,2,3... and taking into account

formula (17), we get

∆l= −4hl

ηU2

cJ2

l(η)sinφm.

(18)

This formula is a direct generalisation of the formulae given by

Szeidl & Jurcsik (2009) for l = 1 and l = 2. It is evident, that

this asymmetry parameter depends only on φm, the relative phase

of AM and FM. The left hand side peaks are higher than the right

hand side ones (∆l< 0), if 0 < φm < π, otherwise the situation is

opposite: π < φm < 2π and ∆l > 0. In those very special cases,

Page 4

4Benk˝ o et al.

where φm = 0 or φm = π the side peaks’ amplitudes are equal.

We have to note that if one of the modulations from AM and FM

dominates in the combined case (h ? η or η ? 1), the amplitude

of the side peaks are almost the same.

3 BLAZHKO MODULATION

RRLyrae light curves traditionally are described by a Fourier series

of a limited number of terms. In Blazhko modulated RRLyrae stars

the sum builds up from terms of harmonics of the main pulsation

frequency, side peaks due to the modulation and the modulation

frequency and even its harmonics:

m(t) = A0+

N

?

i=1

Aisin[2πFit + Φi],

(19)

where either Fi = jf0, (j = 1,2,...,n); or Fi = kfm,

(k = 1,2,...,m); or Fi = j?f0 + k?fm, (j?= 1,2,...,n?,

k?= 1,2,...,m?); Fi = j??f0−k??fm, (j??= 1,2,...,n??, k??=

1,2,...,m??); f0and fmare the main pulsation frequency and the

modulation one, respectively. The amplitudes Ai and phases Φi

are considered as independent quantities and determined by a non-

linear fit. The necessary number of parameters for a complete light

curve solution is 2N + 3 (amplitudes and phases and two frequen-

cies and the zero point A0). The number of parameters can be as

high as 500-600 for a long time series of good quality (see e.g.

Chadid et al. 2010).

Inthenextsubsectionsweshowhowthemodulationparadigm

can be applied and what advantages it has compared to this tradi-

tional handling Eq. (19).

3.1Blazhko stars with AM

To start with, we discuss Blazhko stars’ light curves with pure AM

effect, although the recent space-born data suggest that all Blazhko

RRLyrae stars show amplitude modulation and simultaneous pe-

riod changes (Chadid et al. 2010; Benk˝ o et al. 2010; Poretti et al.

2010). We follow a step-by-step generalization process that allows

us to separate effects more clearly. We note that the most striking

feature of a Blazhko RRLyrae light curve is the amplitude varia-

tion, which is generally easy to find and in many cases the only

detectable modulation (see Stothers 2010 and references therein).

To apply the framework described in Sec. 2.1 to an RRLyrae

light curve the course-book formulae need some extensions.

We choose a continuous, infinite, periodic function with a non-

modulated RRLyrae shape as a “carrier wave”. This function is

described by the frequency f0 and its harmonics, that is c∗(t) :=

m(t) if Fi = jf0in Eq. (19).

Although, the exact analytical Fourier spectra of any of the

modulated signals discussed in this paper can be calculated with-

out any problems, at least in theory (see also Appendix A), to illus-

trate the different formulae synthetic light curves and their Fourier

spectra were also generated and plotted. An artificial light curve

was constructed as a carrier wave with typical RRLyrae parame-

ters (f0 = 2 d−1and its 9 harmonics) on a 100 day long time span

sampled by 5 min (insert in Fig. 2). The Fourier transform of such

a signal is well-known (Fig. 2): it consists of the transformed si-

nusoidal components given in Eq. A2. (More precisely, due to the

finite length of the data set and its sampling, the Fourier transfor-

mations should always be multiplied by the Fourier transform of

the appropriate window function.)

02468 1012141618 20

0

.1

.2

.3

.4

Frequency [d−1]

Amplitude [mag]

0 .2.4.6.811.21.4

.4

.2

0

−.2

−.4

−.6

−.8

−1

Time [days]

Brightness [mag]

Figure 2. Fourier amplitude spectrum of the artificial RRLyrae light curve

acting as a carrier wave of all modulated light curves constructed in this

paper (main panel), and a part of the light curve itself (insert).

Substituting c∗(t) carrier wave into the definition of AM in

Eq. (5):

?

?

j=1

m∗

AM(t) =1 +U∗

m(t)

U∗

c

?

c∗(t) =

[1 + m∗

m(t)]a0+

n

?

ajsin(2πjf0t + ϕj)

?

.

(20)

Expression (20) describes a general amplitude modulated RRLyrae

light curve, U∗

of the non-modulated light curve. On the one hand, the non-zero

constant term of a0is obligatory from mathematical point of view,

otherwise the Fourier sum does not compose a complete set of

functions. On the other hand, this value represents the difference

between the magnitude and intensity means. More precisely, either

we use physical quantities (viz. positive definite fluxes) or we trans-

form normalized fluxes into magnitude scale. In this latter case the

average of the transformed light curve differs from zero. In this

paper, for traditional purposes we use the second approach. For

RRLyrae stars the typical value of this difference is about some

hundredths of a magnitude (a0 ? 1). It is evident that this con-

stant differs from the zero point of the light curve A0 given in the

apparent magnitude scale.

m(t) is the modulation signal, U∗

c is the amplitude

3.1.1Sinusoidal amplitude modulation

In the simplest case the modulation is sinusoidal:

U∗

m(t) = amsin(2πfmt + ϕm).

(21)

Sample light curves obtained with this assumption from Eq. (20)

are shown in Fig. 3. Introducing the modulation depth as h =

am/U∗

resulting in modulations symmetrical to an averaged value viz. a

horizontal line (left panels). Right panels show cases with higher

modulation depths (am > a0), where this symmetry is broken.

A common feature of these light curves is that the maxima and

minima of the envelope curves coincide in time. Furthermore, the

average brightness of all light curves vary with fm. It can be seen

directly from Eq. (20): the m∗

haviour. That is, the found mean brightness (V ) variations during

c the parameters were chosen as a0 ? U∗

c and am ? a0,

m(t)a0term is responsible for this be-

Page 5

Blazhko RRLyrae light curves as modulated signals5

Time [days]

Brightness [mag]

.5

0

−.5

am <= a0

.5

0

−.5

0 20406080100

.5

0

−.5

.5

0

−.5

−1

am > a0

.5

0

−.5

−1

0204060 80100

.5

0

−.5

−1

Figure 3. Artificial light curves with a sinusoidal AM computed with the

formula Eq. (20). Left panels show symmetrical modulation (am ? a0;

a0 = 0.2), the right ones are asymmetrical (am > a0; a0 = 0.005).

The modulation depth h is increasing from the top to bottom as h =

0.1,0.2,0.4; fm= 0.05 d−1and ϕm= 270 deg are fixed.

Time [d]

Brightness [mag]

020406080100

1

0

−1

99.51010.511

1

.5

0

−.5

−1

−1.5

1919.520 20.521

.5

0

−.5

Figure 4. Bottom panel: Artificial light curve with a sinusoidal AM com-

puted with the formula (20). The modulation depth is h = 1.2. Other pa-

rameters are a0= 0.01, fm= 0.05 d−1and ϕm= 270 deg. Top panels:

Two-day zooms around a maximum (top left) and a minimum (top right) of

the modulation cycle.

the Blazhko cycle (Jurcsik et al. 2005) is a natural consequence of

the AM.

There is a fascinating case, when the modulation is very strong

i.e. when the modulation depth is h > 1. Beside the strong light

curve changes (Fig. 4) in some Blazhko phases the shape of the

light curve looks very unfamiliar (see top right panel in Fig. 4). The

relevance of this mathematical case is corroborated by the Kepler

observation of V445Lyr that shows similar characteristics (fig 2. in

Benk˝ o et al. 2010).

Using some trigonometrical relations, Eq. (20) with (21) can

be converted to a handy sinusoidal decomposition form from where

the Fourier spectrum is easily seen:

m∗

AM(t) = a0+ ha0sin(2πfmt + ϕm) +

Frequency [d−1]

Amplitude [mag]

02468101214161820

0

.02

.04

.06

.08

0.02.04.06.08.1.12.14.16.18.2

0

.001

.002

.003

1.51.61.71.81.922.12.22.32.42.5

0

.02

.04

.06

.08

Figure 5. Fourier amplitude spectrum of the artificial sinusoidal AM light

curve in bottom right panel of Fig. 3 after the data are prewhitened with the

main frequency and its harmonics. Inserts are zooms around the positions

of the main frequency f0 = 2 d−1(top), and the modulation frequency

fm= 0.05 d−1(bottom), respectively.

n

?

j=1

ajsin(2πjf0t + ϕj) +

h

2

n

?

j=1

aj

?sin?2π (jf0− fm)t + ϕ−

sin?2π (jf0+ fm)t + ϕ+

= ϕj − ϕm + π/2, ϕ+

Fourier spectrum of such an AM signal is familiar for RRLyrae

stars’ experts (see also Fig. 5). It consists of the spectrum of the

non-modulated star (third term) as in Fig. 2 and two equidistant

side peaks around each harmonic (last term). The amplitudes of the

pairs of side peaks are always equal: A(jf0± fm) ∼ ajh/2. The

second term in Eq. (22), that causes the average brightness varia-

tion, produces the frequency fmitself in the spectrum. The Blazhko

modulation frequency is always found in observed data sets ex-

tended enough (see e.g. Kov´ acs 1995; Nagy 1998; Jurcsik et al.

2005, 2008; Chadid et al. 2010; Poretti et al. 2010; Kolenberg et

al. 2011).

It is a long-standing question whether there is any Blazhko

phase where the modulated light curve is identical to the unmod-

ulated one (see Jurcsik, Benk˝ o & Szeidl 2002 and further refer-

ences therein). In this simplest case the answer is easy. It happens

if the second and fourth terms in (22) disappear simultaneously,

namely, in the zero points of the modulated sinusoidal function at

t = (kπ − ϕm)/(2πfm), k is an arbitrary integer.

The number of used parameters for solving such a light curve

(Fig. 3) in the traditional way (according to Eq. 19) is 6n + 5,

where n denote the number of detected harmonics including the

main frequency. The necessary number of parameters in our han-

dling is 2n + 5. The modulation is described by 3 parameters (fm,

am, ϕm) as opposed to the traditional framework where this num-

ber is 4n + 3.

j

?+

j

??,

(22)

where ϕ−

jj

= ϕj + ϕm − π/2. The

Page 6

6Benk˝ o et al.

3.1.2Non-sinusoidal AM

As a next step, we assume the modulation function m∗

an arbitrary periodic signal represented by a Fourier sum with a

constant frequency fm. Substituting it into Eq. (20) we get

?

p=1

m(t) to be

m∗

AM(t) =aA

0+

q

?

aA

psin

?

0 = 1 + (am

2πpfmt + ϕA

p

??

c∗(t),

(23)

where constants are defined by aA

am

ters. Some typical light curves are shown is Fig. 6. It is evident that

their envelope curves are non-sinusoidal and their shapes depend

on the actual values of aA

these envelope curves occur, however, at the same Blazhko phase

as in the previous sinusoidal cases. Rewriting of (23) similarly to

Eq. (22) but in a more compact form yields

0/U∗

c), and aA

p =

p/U∗

c. From now on, the upper index A denotes the AM parame-

p and ϕA

p. The maxima and minima of

m∗

AM(t) =

q

?

p=0

n

?

j=0

aA

p

2ajsin?2π (jf0± pfm)t + ϕ±

jp

?,

(24)

where the two sinusoidal terms appearing analogously to Eq. (22)

are formally unified into one formula and denoted by ± signs;

ϕ+

constants are chosen to be ϕA

By investigating the Fourier amplitude of the side peaks we

found that A(jf0±pfm)/A(jf0) ∼ aA

the side peaks of a given order and the central peak is constant. (ii)

ThecommonlyusedamplituderatioA(jf0±pfm)/A(f0±pfm) ∼

aj/a1vs. harmonic order is the same as the amplitude ratio of the

main frequency A(jf0)/A(f0) ∼ aj/a1 vs. harmonic order. (iii)

Since the same coefficient aA

the amplitudes of left-hand-side and right-hand-side peaks are the

same. According to this, the generated Fourier spectrum (Fig 7)

now shows symmetrical multiplet structure of peaks around the

main frequency and its harmonics (jf0 ± pfm). Each multiplet

structure is the same at each harmonic order, that is the number

of the side peaks, their frequency differences and amplitude ratios

to their central peaks are the same. It is important to note that the

number of side peaks (in one side) is equal to p. In addition, the har-

monic components of the modulation frequency pfm also appear.

(This can be obtained from Eq. (24) if j = 0.)

Such a phenomenon was undetected in the observed data of

Blazhko stars until recently. Hurta et al. (2008) found equidistant

quintuplets in the spectrum of RVUMa for the first time. Besides

triplets and quintuplets, sextuplet structures were also found by Ju-

rcsik et al. (2008) in the spectrum of MWLyr, while Chadid et

al. (2010) detected even 8th order (sepdecaplet) multiplet frequen-

cies in the spectrum of CoRoT data of V1127Aql. According to

Sec.2.2,theanglemodulationscauseinfinitenumbersofsidepeaks

around each harmonic, therefore, the origin of the observed multi-

plets as a non-sinusoidal amplitude modulation is certain for those

cited cases (e.g. MWLyr and V1127Aql), where the harmonics of

the modulation frequency are also detected.

In searching for a Blazhko phase where the modulated and

carrier waves are identical we concluded that the modulation

terms can only be entirely disappearing from the formula (24) if

aA

ists. This necessary condition is complemented by an additional

one: the modulation virtually disappears in the moments when

?q

the parameters aA

jp= ϕj + ϕA

p− π/2; ϕ−

jp= ϕj − ϕA

0:= ϕ0 := π/2.

p+ π/2. The arbitrary

p. (i) The amplitude ratio of

pbelongs to both side peaks (at ±p),

0 = 1 (am

0 = 0) is true, otherwise no such Blazhko phase ex-

p=1ajapsin[2π(jf0± pfm)t + ϕ∓

zero or infinite numbers of zero points depending on the values of

p, ϕA

jp] = 0. The sum has either

p. That is, generally there are no such phases

Brightness [mag]

.5

0

−.5

−1

.5

0

−.5

−1

.5

0

−.5

−1

.5

0

−.5

−1

020406080 100

.5

0

−.5

−1

Time [days]

Figure 6. Synthetic light curves of non-sinusoidal AM signals computed

by the formula (23). A two-term sum of modulation signal was assumed:

aA

the second modulation term varies from top to bottom as: ϕA

220, 270, 360, respectively.

1= 0.01, aA

2= 0.2 mag; ϕA

1= 270 deg are fixed and the phase of

2= 110, 140,

Frequency [d−1]

Amplitude [mag]

0246810 1214 1618 20

0

.02

.04

.06

.08

0 .02.04 .06.08.1.12 .14.16 .18.2

0

.001

.002

.003

1.51.6 1.71.8 1.92 2.12.2 2.32.4 2.5

0

.02

.04

.06

.08

Figure 7.

AM light curves in Fig. 6 after the data were prewhitened with the main

frequency and its harmonics. Inserts are zooms around the positions of

the main frequency f0 = 2 d−1(top), and the modulation frequency

fm= 0.05 d−1(bottom), respectively.

Fourier amplitude spectrum of the synthetic non-sinusoidal

Page 7

Blazhko RRLyrae light curves as modulated signals7

where a non-sinusoidal AM light curve and its carrier wave are

identical.

The necessary number of the parameters for a light curve fit of

(19) and (23) is (2q+1)2n+2q+3 and 2n+2q+3, respectively.

Here n denotes the total number of used harmonics including the

main frequency and q is the order of side peak structures as above,

(i.e. q = 1 means triplets, q = 2 quintuplets, etc). In the traditional

description each additional side peak order increased the number

of terms by 4n+2 as opposed to our method, where this increment

is only 2.

3.1.3Parallel AM modulation

Multiperiodic modulation was suspected in XZCyg (LaCluyz´ e et

al. 2004), UZUMa (S´ odor et al. 2006), SUCol (Szczygieł & Fab-

rycky 2007), and LSHer (Wils, Kleidis & Broens 2008). The

Blazhko RR Lyrae stars of the MACHO and OGLE surveys (Al-

cock et al. 2000; Moskalik & Poretti 2003) that have unequally

spaced triplet structures in their Fourier spectra are possibly also

multiperiodically modulated variables. CZLac (S´ odor et al. 2011)

is the first Blazhko star with multiperiodic modulation where both

modulation periods are identified. Not only modulation side peaks

but linear combinations of the modulation frequencies also appear.

Signs of multiple modulation were discovered in Kepler data of

V445Lyr (Benk˝ o et al. 2010). There are numerous possibilities for

creating a multiply modulated light curve. Let us review some of

them.

The most simple case is a natural generalization of Eq. (23)

when the modulation signal is assumed to be a sum of signals with

differentˆfr

signals be independent, i.e., the modulation signal consists of lin-

early superimposed waves. In this case, Eq. (23) reads as:

?

r=1p=1

m, where r = 1,2,... signs constant frequencies. Let

m∗

AM(t) =ˆ a0+

s

?

qr

?

ˆ aprsin

?

2πpˆfr

mt + ˆ ϕpr

??

c∗(t),

(25)

where ˆ a0 = 1 +?s

are taken into account and the only varied parameter is the fre-

quency of the second modulation isˆf2

frequencies are comparable (ˆf1

panel a the envelope shape of the light curve shows the well-known

beatingphenomenon.Herethebeatingperiodis200days,albeitthe

modulation periods are close to the shortest known ones. It is easy

to understand that the observations taken on a moderate time span

often detect only the gradual increase or decrease of the amplitude

of the Blazhko cycles. In panel b of Fig. 8ˆf2

that is the ratio of the modulation frequencies is 4:3, similarly to

the case of CZLac during its second observed season in S´ odor et

al. (2011). The amplitude changes of the consecutive Blazhko cy-

cles need well-covered long-term time series observations, other-

wise the interpretation becomes difficult. Panel c in Fig. 8 shows

the case where the frequency of the second modulation is half of

the first one (ˆf2

cause alternating higher and lower Blazhko cycles. The exact 2:1

ratio between the two modulation frequencies leads to the same re-

sult as a two-term non-sinusoidal modulation in Eq. (23) (see also

top panel in Fig. 6). The bottom panel in Fig. 8 shows a case where

the second modulation has a much longer period than the primary

Blazhko cycle. In a first inspection the top and bottom panels are

very similar apart from a phase shift.

To reveal the real situation we need to compare their Fourier

r=1am

0r/U∗

c, and ˆ apr = am

pr/U∗

c. This formula

is demonstrated in Fig. 8. In this figure, only two modulation waves

m. When the two modulation

m= 0.1 d−1andˆf2

m= 0.09 d−1in

mwas set to 0.075 d−1,

m = 0.05 d−1). These specially selected values

Brightness [mag]

Time [days]

.5

0

−.5

−1

a

.5

0

−.5

−1

b

.5

0

−.5

−1

c

0 204060 80100

.5

0

−.5

−1

d

Figure 8. Artificial light curves calculated with two independent sinusoidal

AM modulations according to the formula of (25). The fixed parameters

were a0= 0.01, ˆ a11= 0.5, ˆ a12= 0.2 mag,ˆf1

ˆ ϕ12= 120 deg, whereˆf2

and 0.01 d−1, respectively.

m= 0.1 d−1, ˆ ϕ11= 270,

mchanges from top to bottom as 0.09, 0.075, 0.05

spectra starting with

m∗

AM(t) =

s

?

r=1

qr

?

p=0

n

?

j=0

ˆ apr

2

ajsin

?

2π

?

jf0± pˆfr

m

?

t + ˆ ϕ±

jpr

?

.

(26)

Where the constants are chosen similarly to (24): ˆ ϕ−

ˆ ϕpr+ π/2; ˆ ϕ+

toseethattheFourierspectrumof expression (26)containsssetsof

side peaks shown in Fig. 7. The qualitative structure of these sets

is the same. It consists of the carrier’s spectrum (jf0), the peaks

of the different modulation frequencies and their harmonics (pˆfr

and the side peaks around the main frequency and its harmonics:

jf0± pˆfr

the independence of the modulation waves no further side peaks

appear.

jpr= ϕj −

jpr= ϕj+ ˆ ϕpr− π/2; ϕ0 = ˆ ϕ0r = π/2. It is easy

m),

m, where p = 1,2, ..., qr, and r = 1,2,...,s. Due to

3.1.4Modulated modulation – the AM cascade

It is hard to imagine, however, that in a real stars’s case, the dif-

ferent modulating waves are independently superimposed without

any interactions. Let us investigate the possibility of the modulated

modulation: the cascade. In other words, the modulation signal is

composed of recursively modulated waves as

Page 8

8Benk˝ o et al.

f0−f1

m

f0+f1

m

f0−f2

m

f0+f2

m

f0−(f1

m+f2

m)

f0+(f1

m+f2

m)

f0−f1

m+f2

m

f0+f1

m−f2

m

1.71.81.92 2.1 2.22.3

0

.02

.04

.06

.08

.1

Frequency [d−1]

Amplitude [mag]

Figure9.FourieramplitudespectraoftheartificialcascadeAMlightcurves

with parameters of panel b in Fig. 8 – (black) dotted line – and its cascade

equivalent – (red) continuous line. The spectra show the interval around the

main pulsation frequency after the data are prewhitened with it. The (blue)

dashedlineshowsthespectrumofcascadecaseafterthesidepeaksf0±f1

f0± f2

are shifted with +0.01 and −0.01 mag, respectively).

m,

mare also removed (for better visibility the top and bottom spectra

c(1)(t) := c∗(t), m(1)

m(t) = m∗

m(t),

c(2)(t) := m(1)

AM(t) = [1 + m(1)

m(2)

m(t)]c(1)(t),

AM(t) = [1 + m(2)

m(s)

m(t)]c(2)(t),...,

AM(t) = [1 + m(s)

m(t)]c(s)(t).

(27)

m∗

AM(t) =

s

?

r=1

?

˜ a0r+

qr

?

p=1

˜ aprsin

?

2πp˜fr

mt + ˜ ϕpr

??

c∗(t),

(28)

where ˜ a0r = 1 + am

plitude of the rth carrier wave c(r)(t). On the basis of a visual in-

spection,thereareimperceptibledifferencesamongthelightcurves

produced by this expression (28) and those that can be seen in

Fig. 8. The Fourier spectrum, however, contains additional peaks

at the linear combinations of f0and˜fr

understand this spectrum we generate Eq. (28) in a form similar to

Eq. (26).

sin2π

0r/U∗

cr, ˜ apr = am

pr/U∗

cr. U∗

crdenotes the am-

mas it is shown in Fig 9. To

m∗

AM(t) =

s

?

r=1

qr

?

p=0

n

?

j=0

1

2

s

?

r??=r

?

˜ a0r?

r?=1

˜ a0r?

˜ apraj·

m

??

??

?

?

?

jf0± p˜fr

?

t + ˜ ϕ±

?

?

jpr

?

+

s

?

?

k=2

?

?

r∈Rk

qr

?

n

?

p=1

a0

r?∈RC

k

r

˜ apr

Sk(α)+

s

?

k=2

r∈Rk

qr

p=1

j=1

aj

r?∈RC

k

˜ a0r?

r

˜ apr

Sk+1(β),

(29)

˜ ϕ−

An index set Rsis defined so that it contains all r indices from r =

1,2,...,s. Index sets Rkmeans all subsets of Rswhat contains k

elements. Therefore, the total number of Rksets is?s

jpr= ϕj − ˜ ϕpr+ π/2; ˜ ϕ+

jpr= ϕj + ˜ ϕpr− π/2; ˜ ϕ0r = π/2.

k

?. The sums

over r ∈ Rkmean a sum over the all possible combinations of the

k number of different indices r. Similarly r?always runs over the

complement of a set RC

up sums of sinusoidal functions (see for the definition and further

details in Appendix B) of the linear combinations of k angles. Here

the components of the α and β vectors are αr = 2πp˜fr

r ∈ Rk, βk+1= 2πjf0t + ϕj and βr = αr.

For better comparison we present the formula in the form

of Eq. (29) instead of the most possible compact one. Although

Eq. (29) seems to be complicated, the meaning of each term is

simple: the first term can be directly compared to the linearly su-

perimposed case Eq. (26). It produces all the peaks in the Fourier

spectrum appearing in that case: the main pulsation frequency and

its harmonics jf0, modulation frequencies and its harmonics p˜fr

the side frequencies around the main frequency and its harmonics

(jf0± p˜fr

all the possible linear combinations of p˜fr

is responsible for the many linear combination frequencies around

the main frequency and its harmonics (Fig. 9). The latter two types

of combination terms were detected by S´ odor et al. (2011) in the

Fourier spectrum of CZLac, the only well-studied multiply modu-

lated RRLyrae star.

Long-term secondary changes in Blazhko cycles can be ex-

plained by the variable strength of the modulation. To formulate

this assumption we arrived at

AM(t) =?1 +?1 + m?

The formula (30) can be considered as a special case of (28) when

s = 2 and ˜ a01 = 0.

kof the actual Rk. The functions Skbuild

mt + ˜ ϕpr,

m,

m). The second sum in Eq. (29) belongs to the peaks of

m, whilst the last term

m∗

m(t)?m??

m(t)?c∗(t)

(30)

3.2Blazhko stars with FM

We remind the reader of the possible absence of real Blazhko stars

with pure AM that was mentioned in the introductory paragraphs

of Sec. 3.1. The only difference between pure AM and FM cases is

that RRLyrae stars showing pure PM/FM are much more rarely re-

ported than pure AM ones, but there are some examples (e.g. Kurtz

et al. 2000; Derekas et al. 2004).

How can the formalism discussed in Sec. 2.2 be applied to

RRLyrae stars? Let us assume the same carrier wave as in the case

ofAM,butheretheinstantaneousfrequencyf(t)isdenotedasf0+

m∗

m(t), and m∗

m(t) is an arbitrary (bounded) modulation signal.

m∗

FM(t) = a0+

n

?

j=1

ajsin{2πj [f0+ m∗

m(t)]t + ϕj}.

(31)

Expression (31) describes a general frequency modulated RRLyrae

light curve.

3.2.1The sinusoidal FM

When the modulating function is sinusoidal and expressed in the

same form as (21), Eq. (31) becomes

m∗

FM(t) = a0+

?

where aF= am/fm, ϕF= ϕm+π/2 and the upper index F marks

theparametersofFM.Theamplitudeofthissignalisdeterminedby

the Fourier amplitudes aj of the carrier signal, hence no amplitude

changes are present. In the bottom panel of Fig. 10 a simulated light

n

j=1

ajsin

?

2πjf0t + jaFsin

?

2πfmt + ϕF?

+ ϕj

?

,

(32)

Page 9

Blazhko RRLyrae light curves as modulated signals9

Time [d]

Brightness [mag]

0 20406080 100

.5

0

−.5

−1

10.610.8 1111.2 11.4

.5

0

−.5

20.6 20.82121.221.4

Figure 10. Bottom: Artificial FM light curve produced with sinusoidal

modulation according to Eq. (32). Fourier parameters of the “carrier light

curve” are the same as before, aF= 0.279;ϕF= 0. Boxes show the lo-

cations of the top panels. Top panels: One-day zooms from two different

phases of the modulation cycle. The non-modulated “carrier” light curve

is shown by a (red) continuous lines while (blue) dotted lines denotes FM

signal. The periodic phase shift (viz. PM) caused by FM can be clearly

identified.

Frequency [d−1]

Amplitude [mag]

02468101214161820

0

.02

.04

.06

15.615.81616.216.4

0

.002

.004

.006

.008

.01

1.61.822.22.4

0

.02

.04

.05 .1.15.2.25

0

.0002

.0004

Figure 11. Bottom: Fourier amplitude spectrum of the artificial sinusoidal

FM light curve in Fig. 10 after the data are prewhitened with the main fre-

quency and its harmonics. Top panels are zooms around the positions of the

mainfrequencyf0= 2d−1(topleft),andits7thharmonics8f0= 16d−1

(top right), respectively. The modulation frequency fmis missing from the

spectrum (insert).

curve is shown. It is evident that there is no amplitude change. Two-

day zooms from two different phases of the modulation cycle are

shown in the top panels. The periodic phase shift caused by FM can

be identified well: in the left-hand-side panel the non-modulated

light curve is to the left from the FM light curve whilst in the right-

hand side panel the situation is opposite.

Using Chowning’s relation (12) we get from (32):

m∗

FM(t) = a0+

n

?

j=1

∞

?

k=−∞

ajJk

?

jaF?

sin

?

2π (jf0+ kfm)t + kϕF+ ϕj

?

(33)

.

This equation shows the main characteristics of the Fourier spec-

trum (Fig. 11). It consists of peaks at f0 and at its harmonics

jf0 and each of them is surrounded by side peaks at jf0 ± kfm

with symmetrical amplitudes at the two sides. This symmetry of

the amplitudes can be seen from the expression of amplitude ra-

tio A(jf0 ± kfm)/A(jf0) ∼ |J±k(jaF)|, and it is known that

J−k(z) = (−1)kJk(z). It is worth to compare the AM spectra

in Figs. 5 and 7 to this FM spectrum. The Fourier amplitude of

the side peaks are proportional to the Bessel function, and an im-

mediate consequence can be seen in the figure: the amplitude of

the triplet peaks are higher at 3f0 than at 2f0. (Although it is not

shown in the figure, the higher order j > 5 harmonics have also

smaller amplitudes than their side frequencies’ ones.) Since the ar-

gument of the Bessel functions depends on the order of harmonics

j, higher order harmonics “feel” larger modulation index, which

results in more side peaks around the higher order harmonics (see

inserts in Fig. 11). This effect was found for V1127Aql from its

CoRoT data by Chadid et al. (2010). A further remarkable flavour

of this Fourier spectrum is, that it does not include fmas opposed

to any of the AM spectra (insert in Fig. 11).

Let us return to the question whether there is any phase in the

modulation cycle where the light curve is identical to the monope-

riodic light curve (carrier signal). As it was shown in the case of

sinusoidal AM modulation, there are some such possible phases.

Looking at the formula (32) it can be realised that the modulation

disappears at the moments of time if t = (lπ−ϕF)/(2πfm), where

l is an arbitrary integer.

Estimating the number of the necessary parameters for a light

curve fit the traditional description Eq. (19) needs ≈ 2n + 3 +

4?n

well. At the same time, Eq. (32) requires only 2n+5 parameters, no

more than in the sinusoidal AM case. For a typical case plotted in

Fig. 10 (n = 10 and aF= 0.27), the difference is 143 parameters

versus 25.

j=1[int(jaF)+1], where “int” means the integer function, and

n is the number of all harmonics including the main frequency as

3.2.2The case of non-sinusoidal FM

Assuming an arbitrary periodic modulation with a fixed frequency

wesubstituteaFouriersumrepresentingthismodulationsignalinto

Eq. (31) and get

m∗

FM(t) = a0+

n

?

j=1

ajsin

?

2πjf0t +

j

q

?

p=1

aF

psin

?

2πpfmt + ϕF

p

?

+ φj

?

,

(34)

where the constant terms are contracted as φj = jaF

the previous sinusoidal case the equation can be rewritten as

0+ ϕj. As in

m∗

FM(t) = a0+

n

?

j=1

∞

?

k1,k2,...,kp=−∞

q

?

aj

?

q

?

p=1

Jkp(jaF

p)

?

·

sin

?

2π

?

jf0+

p=1

kppfm

?

t +

q

?

p=1

kpϕF

p+ φj

?

.

(35)

Page 10

10Benk˝ o et al.

Frequency [d−1]

Amplitude [mag]

02468 101214161820

0

.02

.04

.06

15.615.816 16.216.4

0

.002

.004

.006

.008

.01

1.61.82 2.22.4

0

.02

.04

Figure 12. Bottom: Fourier amplitude spectrum of an artificial non-

sinusoidal FM light curve calculated by the formula of (34) after the data

are prewhitened with the main frequency and its harmonics. Parameters of

the generated light curve were the same as for the light curve in Fig. 10, and

p = 2, aF

positions of the main frequency f0= 2 d−1(top left), and its 7th harmonic

8f0= 16 d−1(top right), respectively.

2= −0.1 mag, ϕF

2= π/4. Top panels are zooms around the

In one sense, this formula is a generalisation of (13) to the case

for a non-sinusoidal carrier wave, in the other sense however, the

modulation frequencies are chosen specially as fp

Comparing equation (35) with (33) it can be realised that the

structure of both Fourier amplitude spectra should be similar (cf.

also Fig. 11 and Fig. 12), although there are significant differences,

as well. First of all, besides the same values of common Fourier pa-

rameters of a sinusoidal and a non-sinusoidal case, the detectable

side peaks are more numerous than for the non-sinusoidal case. The

reason is simple: the higher order terms in the modulation signals’

sum increase the “effective modulation index”. The most notewor-

thy difference is the disappearance of the symmetry between the

amplitude of the side peaks from the lower and higher frequency

parts.

To understand this let us investigate the simplest non-

sinusoidal case, if q = 2 and concentrate only on the side peaks

around the main pulsation frequency (j = 1). Then the above ex-

pression (35) is simplified to

m:= pfm.

∞

?

?

k1=−∞

sin

∞

?

k2=−∞

a1Jk1(aF

1)Jk2(aF

2)·

2π [f0+ (k1+ 2k2)fm]t + k1ϕF

1+ k2ϕF

2+ φ1

?

.

(36)

For calculating the amplitude of the triplet’s side peaks A(f0±fm)

we have to sort out the corresponding terms from the above infinite

sum such as k1 = 1 − 2k2 and k1 = −(2k2+ 1), for the right-

hand-side and left-hand-side peaks, respectively (k2is an arbitrary

integer). It can be seen, that both sums include the same elements,

because J−3(aF

J5(aF

ences have the same values with an opposite sign. The only dif-

fering terms contain J1(aF

J−1(aF

sponsible for the asymmetry of the side peaks. Introducing power

1)J1(aF

2),... for each pair and the relative phase differ-

2) = J3(aF

1)J−1(aF

2), J−5(aF

1)J2(aF

2) =

1)J−2(aF

1)J0(aF

2) in the sum of A(f0+ fm) and

1)J0(aF

2) in the sum of A(f0 − fm). These terms are re-

Harmonic order

Logarithm of the amplitude ratio

−3

−2

−1

0

1

2

0123456789

−2

−1

0

1

Figure 13. Amplitude ratios of the harmonic components of the main pul-

sation (A(jf0)/A(f0)) plotted on decimal logarithmic scale compared to

theamplituderatiosofthemodulationcomponents:A(jf0+pfm)/A(f0+

pfm). Top: sinusoidal FM case; p = 0, 1, 2 and 3 (black) asterisks, (red)

circles, (blue) triangles and (green) squares, respectively. Bottom: non-

sinusoidal FM; shape of the symbols denotes p as in the top panel, but

filled symbols mark positive ps (higher frequency side peaks) while the

open symbols show negative ps (lower frequency side peaks).

difference of the side peaks as in Sec. 2.3 we get

∆1 = 4ˆ A1J1(aF

1)J0(aF

2)cos(ˆΦ1− ϕF

1).

(37)

Hereˆ A1 andˆΦ1 indicate the amplitude and phase of a sinusoidal

oscillation obtained by summing all the terms in (36) except the

different ones. The asymmetry of the higher order side peaks (|k1+

2k2| > 1) can be verified in a similar manner.

This asymmetry has a further consequence. The functions of

amplitude ratio vs. harmonic orders belonging to a given pair of

side peaks are diverge from each other (Fig. 13). This behaviour

is well-known from the similar diagrams of observed Blazhko

RRLyrae stars (Jurcsik et al. 2009b; Chadid et al. 2010; Kolen-

berg et al. 2011). It can also be seen that the actual character of the

asymmetry can change with the harmonic order j or even within a

given order with the different p. For example, in Fig. 13 if p = 1

(triplets), the right-hand-side peaks are always higher than the left-

hand-side ones and the difference between the pairs are increas-

ing with harmonic orders. Meanwhile, if p = 3 (septuplets) the

situation is the opposite. In the case of p = 2 (quintuplets) the

lower frequency peaks have higher amplitude around the lower or-

Page 11

Blazhko RRLyrae light curves as modulated signals 11

Epoch

O−C [d]

0 20 4060 80100120140160180200

−.1

−.08

−.06

−.04

−.02

0

.02

.04

Figure 14. O-C diagram of the maxima of FM light curves with sinusoidal

– (blue) filled circles – and non-sinusoidal – (red) empty circles – modu-

lations, respectively. The input light curves are generated from the formu-

lae of (32) and (34) with the same parameters as the light curve shown in

Fig. 10 (sinusoidal case) and Fourier plot in Fig. 12 (non-sinusoidal case).

(For better visibility the non-sinusoidal curve is shifted by −0.05.)

der (j < 5) harmonics, for higher harmonics (j > 7) the amplitude

ratios of the pairs of side peaks reversed.

As it was discussed in the introductory Section 2.3 simulta-

neous and sinusoidal amplitude and phase modulations result in

an asymmetrical spectrum, therefore, the asymmetry of the ampli-

tude spectrum alone is not a good criterion for detecting a non-

sinusoidal FM. The classical O−C diagram is an ideal tool for this

purpose (see e.g. Sterken 2005). Fig. 14 illustrates the O−C dia-

grams of the maxima for two artificial FM light curves: a sinusoidal

and a simple non-sinusoidal one.

At the end of this section we compare the necessary param-

eters of a potential fit based on the classical description (19) and

the present (31) one. In the latter case this value is 2n + 2q + 3,

where n and q are defined in (31). The expression is the same as

in the case of a non-sinusoidal AM. The traditional formula needs

≈ 2n + 3 + (4?n

values are 27 and 163, respectively.

j=1[int(j?q

p=1aF

p) + 1]) parameters. For the

1 = 0.27, aF

case showed in Fig. 12 (n = 10, aF

2 = 0.1) these

3.2.3Parallel FM

We continue the discussion as in the case of AM. The next step

is the multiply modulated FM with independently superimposed

modulation signals (parallel modulation). As has already been

noted, the chance of such a scenario is very low for stars, but this

case shows a new phenomenon, which is why it is worth to have a

look at it.

?

?

r=1p=1

ˆ m∗

FM(t) = a0+

n

?

qr

?

j=1

ajsin2πjf0t +

j

s

?

0r+ϕj. The formula (38) can easily be transformed

with the help of Eqs. (13) and (35) to

ˆ aF

prsin

?

2πpˆfr

mt + ˆ ϕF

pr

??

+ˆφj

?

.

(38)

Hereˆφj := jˆ aF

f0−2f1

m

f0+2f1

m

f0−f1

m−f2

m

f0+f1

m+f2

m

f0−f1

m+f2

m

f0+f1

m−f2

m

f0−2f2

m

f0+2f2

m

1.71.8 1.92 2.12.2 2.3

0

.005

.01

.015

.02

Frequency [d−1]

Amplitude [mag]

Figure 15. Fourier amplitude spectra of the artificial light curves containing

two parallel FM modulations in formula (38). The figure shows a zoom

around the main pulsation frequency after the data are prewhitened with it

and the triplet components f0±ˆf1

quintuplet frequencies (f0±2ˆf1

frequencies. (The used parameters are:ˆf1

ˆ aF

m, f0±ˆf2

m, f0±2ˆf2

m. The highest peaks are at the

m) and at the linear combination

m= 0.1 d−1,ˆf2

m= 0.01 d−1,

11= 0.5 mag, ˆ aF

12= 0.2 mag.)

ˆ m∗

FM(t) = a0+

n

?

s

?

j=1

∞

?

k11,k12,...,kqss=−∞

qr

?

aj

?

s

?

r=1

qr

?

p=1

Jkpr(jˆ aF

pr)

?

·

sin

?

2π

?

jf0+

r=1p=1

kprpˆfr

m

?

t +

s

?

r=1

qr

?

p=1

kprˆ ϕF

pr+ˆφj

?

.

(39)

There is a fundamental difference between the construction of

FourierspectraofparallelAMandFMsignals.WhiletheAMspec-

tra build up from a simple sum of the component spectra belonging

to a given modulation frequencyˆfr

ble linear combinations ofˆfr

sation frequency jf0. This is illustrated in Fig. 15. In practice, this

effect complicates distinguishing Fourier spectra from the cascade

AM and parallel FM.

m, FM spectra contain all possi-

mand the harmonics of the main pul-

3.2.4 The FM cascade

Although a parallel FM modulation results in a more complex

Fourier spectrum than either a parallel or even a cascade AM, our

former statement is still true. There is a very low chance for in-

dependently superimposed modulation signals in real stars. Let us

turn to the FM cascade (viz. the modulated modulation) case now!

?

where

˜ m∗

FM(t) = a0+

n

j=1

ajsin

?

2πjf0t + j˜CFM(t) +˜φj

?

,

(40)

˜CFM(t) := m(1)

FM(t) =

q1

?

?

p=1

˜ aF

p1sin

?

2πp˜f1

mt + pm(2)

FM(t) + ˜ ϕF

p1

?

,

m(2)

FM(t) =

q2

?

p=1

˜ aF

p2sin2πp˜f2

mt + pm(3)

FM(t) + ˜ ϕF

p2

?

,...,

m(s)

FM(t) =

qs

?

p=1

˜ aF

pssin

?

2πp˜fs

mt + ˜ ϕF

ps

?

,

(41)

Page 12

12Benk˝ o et al.

and˜φj = j˜ aF

of a modulation cascade with s elements, where all elemental mod-

ulation functions m(r)

That is, they are assumed to be independent periodic signals with

the frequencies˜fr

infinite series of sinusoidal functions (see Chowning relation), it is

not a surprise that the sinusoidal decomposition of the expression

(40) is very similar to the parallel case (39). Namely

01+ ϕj. As we can see, the function˜CFM(t) consists

FM(t) are represented by finite Fourier sums.

m. Since an FM modulation can be reproduced by

˜ m∗

FM(t) = a0+

n

?

?

?

j=1

∞

?

qr

?

?

k11,k12,...,kqss=−∞

?

aj·

?q1

p=1

?

?

Jkp1

j˜ aF

p1

s

?

r=2p=1

Jkpr

?

s

kp−1,r˜ aF

pr

??

·

sin

?

2πjf0+

s

?

r=1

qr

p=1

kprp˜fr

m

t +

?

r=1

qr

?

p=1

kpr˜ ϕF

pr+˜φj

?

.

(42)

The frequency content is exactly the same as in the parallel case,

only the values of amplitudes and phases are different.

3.3The case of PM

Here we discuss the phase modulation. As we stated in the Sec. 2

there is no chance to distinguish between FM and PM phenomena

onthebasisoftheirmeasuredsignals(inverseproblem),ifthemod-

ulation function m∗

same time, if the basic physical parameters such as effective tem-

perature, radius and logg are changing during the Blazhko cycle as

was found recently (S´ odor, Jurcsik & Szeidl 2009; Jurcsik et al.

2009a,b) the cyclic variation of the fundamental pulsation period

(vis. frequency) that results in FM would be a plausible explana-

tion for observed effects. There is an additional possible argument

against the existence of PM in RRLyrae stars.

If we assume that the modulating function m∗

plicit time variation – as in the usual definition for PM modulation

in electronics – Eq. (31) reads as

m(t) in Eq. (31) is allowed to be arbitrary. At the

mcontains no ex-

m∗

PM(t) = a0+

n

?

j=1

ajsin[2πjf0t + m∗

m(t) + ϕj].

(43)

When this formula is expressed as Eq. (33) or Eq. (35) according

to a sinusoidal or an arbitrary periodic modulating function, respec-

tively, the arguments of Bessel functions are independent from the

harmonic order j as opposed to the case of FM. It causes a sys-

tematic difference between Fourier spectra of FM and PM. While

the number of detectable side peaks in FM increases with the order

of harmonics, for PM the number of side peaks is the same for all

harmonics.

There are two Blazhko RRLyrae stars that show both strong

phase variations and their data are precise enough, these are the

CoRoT targets V1127Aql and CoRoT 105288363 (Chadid et al.

2010; Guggenberger et al. 2011). The spectrum of V1127Aql

clearly shows the existence of FM: 3rd order side frequencies

are detected around the main pulsation frequency while order of

8th around the 19th harmonic. The Fourier analysis of separate

Blazhko cycles of CoRoT 105288363 showed that with the increas-

ing strength of phase variation, the number of detected side peaks

around higher order harmonics are also increased (Guggenberger

et al. 2011). It is an evidence of (changing) FM.

3.4 Real Blazhko stars with simultaneous AM and FM

In this section we discuss the general combined case, when both

types of modulations occur simultaneously. As it was mentioned

before both AM and FM type modulations were detected for all

observed Blazhko RRLyrae stars if the observed data sets were

precise and long enough. This is the situation for ground-based

(Jurcsik et al. 2009c; and references therein) and space-born ob-

servations of CoRoT and Kepler (Chadid et al. 2010; Poretti et al.

2010; Benk˝ o et al. 2010; Kolenberg et al. 2011) as well.

Generalising the sinusoidal case of combined modulation

Eq. (14) discussed in Sec. 2.3 we get

m∗

Comb(t) = [1 + m∗

m(t)]m∗

FM(t),

(44)

where m∗

Eq. (31). Since all observed Blazhko stars show AM and FM with

the same frequency, we have investigated only those cases where

this assumption is fulfilled.

FM(t) is the general modulated FM function defined by

3.4.1Combined modulations with sinusoidal functions

The simplest case similarly to the pure AM and FM cases is the

simultaneous but sinusoidal modulations.

m∗

Comb(t) = (1 + hsin2πfmt)·

n

?

?

a0+

j=1

ajsin

?

2πjf0t + jaFsin(2πfmt + φm) + ϕj

??

,

(45)

where the notations are the same or directly analogous with the

previously defined ones: h = am/U∗

of the second term (the FM modulated “carrier wave”). The relative

phase between AM and FM signals is φm = ϕF− ϕm.

According to the schema of (16) expression (45) can be refor-

mulated into

FMand U∗

FMis the amplitude

m∗

?

h

2Jk−1(jaF)sin?2π (f0+ kfm)t + (k − 1)φm+ ϕ−

h

2Jk+1(jaF)sin?2π (f0+ kfm)t + (k + 1)φm+ ϕ+

Comb(t) = a0+ a0hsin2πfmt+

?

n

j=1

∞

k=−∞

aj

?

Jk(jaF)sin[2π (f0+ kfm)t + kφm+ ϕj]+

j

?+

j

??

,

(46)

where ϕ±

in Fig. 16 can be interpreted as a sum of the combined modulation

with sinusoidal carrier wave (16) with an additional term describ-

ing the modulation frequency itself (insert in bottom panel). Each

harmonic is surrounded by a multiplet structure of peaks just like

the main frequency. The number of side peaks increases with the

harmonic order j similarly for FM (Sec. 3.2.1).

The asymmetrical amplitudes of pairs of side frequencies be-

longing to a given harmonic can be characterised similarly to the

sinusoidal carrier wave case (18) as

j= ϕj ± π/2. Based on Eq. (45) the Fourier spectrum

∆jl= −4hl

jaFa2

jJ2

l(jaF)sinφm.

(47)

Here ∆jl = A2(jf0+ lfm) − A2(jf0− lfm) is the power dif-

ference of the lth side peaks at the jth harmonics (l = 1,2,...).

Similarly to the course book case discussed in Sec. 2.3 the asym-

metry depends on the actual value of h and aF, (viz. the relative

Page 13

Blazhko RRLyrae light curves as modulated signals 13

Frequency [d−1]

Amplitude [mag]

0246810 1214 1618 20

0

.02

.04

.06

.08

.1

15.615.81616.2 16.4

0

.002

.004

.006

.008

.01

1.61.822.2 2.4

0

.02

.04

.06

.08

.1

.05.1.15 .2 .25

0

.001

.002

.003

Figure 16. Bottom: Fourier amplitude spectrum of an artificial com-

bined (AM & FM) light curve computed from Eq. (45) after the data are

prewhitened with the main frequency and its harmonics. Top panels are

zooms around the positions of the main frequency f0 = 2 d−1(top left),

and its 7th harmonics 8f0= 16 d−1(top right), respectively. The relative

phase between AM and FM is set to φm= 270 deg.

strengths of AM and FM) and the relative initial phase angle φm.

The most extreme possibility is when one of the side peaks com-

pletely disappears. The necessary conditions are φm = ±π/2 and

jaF= hl. The asymmetry decreases with the increasing harmonic

order j (see also top panels in Fig. 16), because all the Bessel func-

tions quickly converge to zero with increasing arguments, therefore

dominate the right hand side of expression (47).

Non-equidistant sampling and large gaps in the observed time

series can cause significant differences between side-peak ampli-

tudes (see Jurcsik et al. 2005). Such sampling effects, however,

can not explain huge differences, such as when side peaks com-

pletely disappear in one side and the spectra show doublets, though

numerous examples were found by large surveys as MACHO and

OGLE (Alcock et al. 2000, 2003; Moskalik & Poretti 2003). But

as illustrated by Fig. 16, highly asymmetrical side peaks can eas-

ily be generated by (45). This asymmetry effect can be a possible

explanation for the observed doublets (RR-ν1 stars) and even for

triplets (RR-ν2 stars). In the latter case the two side frequencies

can originate from a quintuplet structure (equidistant triplet on one

side) or from a multifrequency modulation (non-equidistant triplet

on one side).

Searching for phases where the modulated and non-modulated

light curves are identical we conclude that such phases exist only

if φm = (k1− k2)π; (k1, k2 are integers) and then the moments

of the coincidences are t = k2/2fm. Assuming all Blazhko stars

showing both AM and FM this conclusion supports the finding of

Jurcsik, Benk˝ o & Szeidl (2002), who studied light and radial ve-

locity curves of Blazhko RRLyrae stars,

The amplitude ratio vs. harmonic order diagrams show sim-

ilar shapes and relative positions as were discussed in Sec. 3.2.2

with the connection of Fig. 13. Let us look at the maximum bright-

ness vs. maximum phase diagrams, a classical tool for analysing

Blazhko RRLyrae stars. Such diagrams are plotted in Fig. 17 for

synthetic light curves generated from the formula of (45). All

diagrams have a simple round shape. They reflect the relative

strength of AM and FM components. In panels A and B the rel-

Maximum phase

Maximum brightness [mag]

φm=90

A

−.6

−.7

−.8

−.9

φm=120

C

−.6

−.7

−.8

−.9

φm=60

E

−.020 .02

−.6

−.7

−.8

−.9

φm=90

B

φm=240

D

φm=300

F

−.020 .02

Figure17. Maximumbrightnessvs.maximumphasediagramsforsomear-

tificial light curves with combined sinusoidal modulations. Between panels

A and B the relative strengths of AM and FM are changed as (A) h = 0.1

and aF= 0.2 and (B) h = 0.2 and aF= 0.1, respectively. From panel

B to F the amplitudes are fixed and only the relative phase φmis changed

as shown at the upper left corner in each panel. The arrows in panels C-F

indicate the direction of motion.

ative strengths are opposite 2h = aFand h = 2aF, respec-

tively. As a consequence, the loop is deformed vertically or hor-

izontally. When the angle φm differs from the special values of

lπ/2, (l = 0,1,2,3,4), the axes of the loops are inclined to the ver-

tical horizontal position. This angle also determines the direction of

motion. If 0 < φm < π it is clockwise, whilst if π < φm < 2π

it is anti-clockwise. (These conditions are the same as it was found

by Szeidl & Jurcsik 2009 for sinusoidal carrier waves.) It is note-

worthy, that the same ranges of φmalso determine the character of

power difference of the side peaks: if the right hand side peaks are

higher than the left hand side ones then the direction of motion is

anti-clockwise and vice versa.

3.4.2 Non-sinusoidal combined modulation

On the basis of the previous sections it is easy to define the light

curves which are modulated by general periodic signals simultane-

ously both in their amplitudes and phases:

Comb(t) =m∗

c∗(t)

where the functions m∗

and (34), respectively. Since the two modulations are described

m∗

AM(t)

m∗

FM(t),

(48)

AM(t) and m∗

FM(t) are defined by Eqs. (23)

Page 14

14Benk˝ o et al.

by different functions (they are represented by different order of

Fourier sums), no such simple relative phase can be defined as φm

for the sinusoidal case in Sec.3.4.1. Therefore, we obtain for the

mathematical form of such a generally modulated light curve:

p?=1

?

j=1

?

where the notations are the same as in Eqs. (23) and (34). This

expression describes all the discussed phenomena of a light curve

modulated regularly with a single frequency fm. The envelopes

of these light curves are very similar to the envelopes of non-

sinusoidal AM light curves shown in Fig. 6. The light curves show

non-sinusoidal phase variation as well (see also Figs. 10 and 14).

As in the former simpler cases, the Fourier spectrum can also

be constructed analytically with the help of the sinusoidal decom-

position of (49):

m∗

Comb(t) =

aA

0+

q?

?

?

aA

p? sin

?

?

2πp?fmt + ϕA

p?

?

·

a0+

n

ajsin2πjf0t+

j

q

?

p=1

aF

psin2πpfmt + ϕF

p

?

+ φj

??

,

(49)

m∗

Comb(t) = a0aA

0+

q?

?

∞

?

?

p=1

p?=1

a0aA

p? sin

?

?

?

2πp?fmt + ϕA

p?

?

+

n

?

j=1

q?

?

p?=0

k1,k2,...,kq=−∞

aA

p?

2aj

q

?

p=1

Jkp(jaF

p)

?

?

·

sin

?

2π

?

jf0+

q

?

kpp ± p?

fm

?

t + ψ±

pp?j

.

(50)

Here the ψ±

constant is chosen as ϕA

spectrum is simple and well-understandable on the basis of the pre-

viously discussed cases. The second term is responsible for the

appearance of the modulation frequency and its higher harmon-

ics (see insert in Fig. 7). The next (infinite number of) terms de-

scribe a spectrum which is similar to the non-sinusoidal FM spec-

trum (Fig. 12) but it also shows the AM splitting which is present

in the sinusoidal combined case. These effects make the calcula-

tion of the peaks’ amplitude complicated. The asymmetry between

each pair from a multiplet around a given harmonic is determined

by two factors: one of them is the the non-sinusoidal nature of the

FM (Sec. 3.2.2) and the other one is the combination of AM and

FM (Sec. 3.4.1).

The maximum brightness vs. maximum phase diagrams gen-

erally show complicated shapes. They could form knots, loops and

other non-trivial features. A collection of such diagrams is plotted

in Fig. 18. The direction of the motion can arbitrarily change by

tuning the initial phases.

pp?j:=?q

p=1kpϕF

0 := π/2. The qualitative structure of this

p± ϕA

p? + φj ∓ π/2, the arbitrary

3.4.3Combined multifrequency modulations

A combined modulation with multiperiodic AM or FM or both can

be handled analogously to the simpler presented cases. We can

substitute m∗

fined by Eq. (25) (parallel AM) or Eq. (28) (AM cascade). Writ-

ing m∗

m(t) into the general expression (44) as it was de-

FM(t) as Eq. (38) (parallel FM) or Eq. (40) (FM cascade) in

Maximum phase

Maximum brightness [mag]

G

−.020.02

−.6

−.7

−.8

−.9

H

−.020 .02

I

−.020 .02

D

−.6

−.7

−.8

−.9

EF

A

−.6

−.7

−.8

−.9

BC

Figure 18. Some typical maximum brightness vs. maximum phase dia-

gramsforsyntheticlightcurveswithcombinednon-sinusoidalmodulations.

The relative strength, initial phases and number of used harmonics of AM

and FM has been varied.

principle is straightforward. In practice, however, calculating co-

efficients (amplitudes, phases) is more complicated. The resulting

light curves and Fourier spectra can be interpreted on the basis of

their constituents. They do not show new features except their max-

imum brightness vs. maximum phase diagrams which show time-

dependent and generally non-closed curves as opposed to those in

Fig 18. If the ratio of modulating frequencies are commensurable,

the curve is closed, otherwise it has a non-repetitive behaviour. The

reason is that if the modulation is described by N independent fre-

quencies the proper diagram would be 2N-dimensional and the

classical one is only a 2-D projection of it.

4 PRACTICAL APPLICATION – A CASE STUDY

To demonstrate how our formalism works in practice, we gener-

ated two artificial light curves with simultaneous non-sinusoidal

AM (with two harmonics, p?= 2) and FM (with three harmon-

ics, p = 3). The light curves are 100-days long and sampled by

5-minutes in the same manner as the all synthetic light curves in

the paper. We added Gaussian noise to the light curves either with

rms=0.01 mag (model A) or 10−4mag (model B), respectively.

The model A is similar to a good quality ground-based observation,

while the model B simulates a typical space-borne data set. These

two artificial light curves (top panels in Fig. 19) were analysed with

a blind test (i.e. without knowledge about frequencies, amplitudes

and phases) both by the traditional way and by our method.

4.1Classical light curve analysis

Constructing the mathematical model Eq. (19) of the light curves

in the traditional analysis is a successive prewhitening process. It

consists of Fourier spectra building, fitting the data with the param-

eters of the highest peak(s) in the spectrum by a non-linear algo-

rithm, and subtracting the fitted function from the data and so on.

The process continues as long as significant peaks are detected. At

the end of this analysis the noise of the residual data reaches the

observational scatter.

Page 15

Blazhko RRLyrae light curves as modulated signals15

Brightness [mag]

Amplitude [mag]

.5

0

−.5

−1

Model A

0

.1

.2

.3

.4

(σ=0.01)

−.06

.04

−.04

−.02

0

.02

.04

0

.0005

.001

020 40 60 80 100

Time [days]

−.06

−.04

−.02

0

.02

05101520

0

.0005

.001

Frequency [d−1]

.5

0

−.5

−1

Model B

0

.1

.2

.3

.4

(σ=0.0001)

−.004

.004

−.002

0

.002

.004

0

10−4

.0002

0 20 40 60 80 100

Time [days]

−.004

−.002

0

.002

05101520

0

10−4

.0002

Frequency [d−1]

Figure 19. Artificial light curves with Gaussian noise and their Fourier spectra. Model A (σ = 10−2) is plotted in the left, model B (σ = 10−4) in the right.

The residual light curves and their spectra after the traditional fitting process (middle panels) and the present one (bottom panels). (Residuals and their spectra

are in different scales for the two models.)

In the case of model A the highest peaks belong to the main

frequency f0 and its harmonics. In the further prewhitening steps

the triplets (f0±fm), quintuplets (f0±2fm), septuplets (f0±3fm),

nonuplets (f0 ± 4fm) and the modulation frequency (fm) were

found and fitted. The significance level was chosen at S/N = 4,

where signal-to-noise ratio (S/N) is estimated as Breger et al.

(1993). To remove all significant peaks from the spectrum five from

the undecaplet peaks (f0+ 5fm) and two aliases have to be fitted

andsubtracted.(Wenotethatthefrequency2fmwasdetectable,but

under the significance level, therefore it was not fitted.) The resid-

ual light curve of this process and its Fourier spectrum are plotted

in the middle panels of Fig. 19. The rms of the residual light curve

is 10−2, so we got back the input noise value. The number of fit-

ted frequencies are 103 and used parameters in this successive fit is

201. If the frequency of the side peaks are also fitted independently

(“let it free approach”) this value increases to 286.

The process works similarly in the case of model B as well.

Naturally, many more significant peaks are detectable. From the

highest peaks to the lowest: f0, its harmonics, the side peaks up

their orders of six (f0 ± 6fm), the modulation frequency and its

harmonic (fm, 2fm) are significant. Many alias peaks originating

from the finite data length are also detectable: 46 such frequencies

were removed up to the significance level of the sixth order of side

peaks. We stopped the analysis here at S/N ≈ 40, because we

already found 168 frequencies and used 368 parameters (or 478 if

we fit each frequency independently). The resulted light curve and

its Fourier spectrum are shown in the middle panels of Fig. 19. The

rms of this light curve is 10−3an order of magnitude higher than

the input noise parameter is.

4.2Light curve analysis in our framework

When we apply the approach of this work we need to calculate only

oneFourierspectrumandasinglenon-linearfitforeachlightcurve.

The Fourier spectrum and the characteristics of the light curve help

us to choose the proper fitting formula and to determine the initial

values of the fit.

The Fourier spectrum for any of the light curves A and B pro-

vide us with the necessary parameters (f0, j, a0, aj, ϕj) of the

carrier wave. The amplitude modulation and its non-sinusoidal na-

ture in both light curves are apparent. Searching for peaks in the

low frequency range of the Fourier spectra results in good initial

values for fm, p?= 1,2, aA

detectable around the harmonics of the main frequency than the

number of harmonics of fm and the side peaks are asymmetrical,

an FM has to be assumed. To check its non-sinusoidal nature we

may prepare e.g. an O−C or maximum brightness vs. maximum

phase diagram. At the end of this preparatory work we can choose

the fitting formula (49). We note that to determine the correct initial

values of aF

quency variations. In the worst case, they can be estimated by some

numerical trials with the non-linear fit. Using initial values that are

good enough the non-linear fit converged fast for models A and B.

The algorithm reached the noise levels 10−2and 10−4within few

(less than 10) iterations automatically. The residual light curves and

their Fourier spectra are plotted in the bottom panels of Fig 19. The

number of fitted parameters for both models is only 33. Due to the

finite numerical accuracy the residual spectra always show a struc-

ture reflecting the original spectra at very low levels.

In conclusion our method fits the light curves in a single step

with much less parameters than the traditional one. In addition we

avoid the time-consuming alias fitting and subtracting processes.

The difference in the number of used parameters increases with the

increasing accuracy of the observed data sets. In our example the

number of regressed parameters is reduced from 6 times (model A)

to more than 10 times (model B). Our description has advantages in

the numerical fit of the ground-based observations as well, but its

advantages are outstanding in the analysis of the space-born time

series.

p? and ϕA

p?. While more side peaks are

pand ϕF

pdepends on the tool used for finding the fre-

5DISCUSSION AND SUMMARY

In this paper we have investigated mathematical representation of

artificial light curves. These light curves are defined as modulated

signals where their carrier wave is a monoperiodic RRLyrae light

curve defined by its finite Fourier sum. Different types of periodic

functions are taken into account as modulation functions from the