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New signal processing technique for density prole
reconstruction using reectometry
F. Clairet, B. Ricaud, F. Briolle, S. Heuraux, C. Bottereau
To cite this version:
F. Clairet, B. Ricaud, F. Briolle, S. Heuraux, C. Bottereau. New signal processing technique for
density prole reconstruction using reectometry. Review of Scientic Instruments, American Institute
of Physics, 2011, 82, pp.3502. �10.1063/1.3622747�. �hal00961734�
REVIEW OF SCIENTIFIC INSTRUMENTS 82, 083502 (2011)
New signal processing technique for density proﬁle reconstruction
using reﬂectometry
F. C l a i r e t ,
1
B. Ricaud,
1,2
F. Briolle,
2,3
S. Heuraux,
4
and C. Bottereau
1
1
CEA, IRFM, F13108 SaintPaullezDurance, France
2
CPT UMR 6207, Campus de Luminy, case 907, F13288 Marseille, France
3
CReA, BA 701, F13306 Salon de Provence, France
4
IJLP2M, UMRCNRS 7198, Université Henri Poincaré, F54506 Vandoeuvre, France
(Received 11 March 2011; accepted 14 July 2011; published online 25 August 2011)
Reﬂectometry proﬁle measurement requires an accurate determination of the plasma reﬂected signal.
Along with a good resolution and a high signal to noise ratio of the phase measurement, adequate
data analysis is required. A new data processing based on timefrequency tomographic representation
is used. It provides a clearer separation between multiple components and improves isolation of the
relevant signals. In this paper, this data processing technique is applied to two sets of signals coming
from two different reﬂectometer devices used on the Tore Supra tokamak. For the standard density
proﬁle reﬂectometry, it improves the initialization process and its reliability, providing a more accu
rate proﬁle determination in the far scrapeoff layer with density measurements as low as 10
16
m
−1
.
For a second reﬂectometer, which provides measurements in front of a lower hybrid launcher, this
method improves the separation of the relevant plasma signal from multireﬂection processes due to
the proximity of the plasma. © 2011 American Institute of Physics. [doi:10.1063/1.3622747]
I. INTRODUCTION
Reﬂectometry relies on the fact that as an electromag
netic wave propagates through the plasma, its phase is shifted
due to the deviation of the local refractive index from the vac
uum value. At a certain critical density corresponding to the
cutoff layer, in a WKB description (or under WKB assump
tions), this refractive index goes to zero and the probing wave
is reﬂected. The In and Quadrature phase detection of the re
ﬂected wave provides a complex signal from which amplitude
and phase variation can be measured separately. The density
proﬁles are then reconstructed from the detected phase ac
cording to a recursive numerical algorithm.
1
Among the tradi
tional ﬁltering techniques and Fourier analysis to recover the
relevant plasma echo, many signal analysis procedures have
been extensively tested most of the time to compensate for the
lack of signal and phase s crambling due to the turbulence.
2–4
This latter problem has been overcome over the past decade
by using fast sweep heterodyne techniques,
5
which provide
reliable and accurate phase measurements. However, reﬂec
tometry signals exhibit a multifrequency nature and the ex
traction of the relevant plasma echo can, in some circum
stances, appear to be delicate. As a matter of fact, the sig
nal analysis can become problematic when multireﬂection
processes occur simultaneously. Recently, a new tool
6
based
on a tomographic technique in the timefrequency plane has
been developed and it substantially improves the traditional
Fourier analysis in separating echoes with a similar frequency
but over different time domains with a great precision. While
Fourier transform is based on the projection onto a basis of
sine and cosine functions, the tomogram transform consists
in a decomposition of the signal onto an orthonormal basis of
linear chirps. In this paper we illustrate the performance of
the tomographic analysis applied to Tore Supra (TS) reﬂec
tometry signals through various examples. We ﬁrst present, in
Sec. II, the theoretical aspects of the tomogram technique to
motivate its application to improve the signal extraction corre
sponding to the ﬁrst plasma reﬂection and to provide a better
rejection of multireﬂections. In Secs. III and IV, these two
performances of the tomographic analysis are illustrated with
two different reﬂectometer conﬁgurations. In Sec. III, we ap
ply this analysis technique, to the standard density proﬁle re
ﬂectometer, for the ﬁrst Xmode cutoff detection. The prob
lem that we encounter on TS is to separate the plasma scrape
off layer echo from the back wall component. This problem
can prevent a proper density proﬁle initialization and modify
substantially the reconstruction at the far edge. The second ex
ample, treated in Sec. IV, corresponds to data obtained using
the same reﬂectometer but measuring density proﬁles in front
of a lower hybrid current drive (LHCD) launcher. Two waveg
uides bring the wave directly inside the vacuum chamber at a
few millimeters from the detected plasma edge. The multi
component nature of the signal is particularly signiﬁcant as
the reﬂectometer antennas are very close to the plasma, which
causes secondary multireﬂections. This reﬂectometer was in
stalled for a short period of time for test purposes. Such a
reﬂectometry conﬁguration will occur on ITER to address at
least two speciﬁc physic studies: the plasma positioning con
trol with a plasma position reﬂectometry system,
7, 8
and, the
power coupling to the plasma of additional powers such as ion
cyclotron resonance heating (ICRH) or LHCD systems where
measurements are foreseen in front of the launchers.
9
Then,
in Sec. V we discuss the advantages and limitations of this
new signal processing technique, and directions for further
research.
II. TOMOGRAM
A. Deﬁnition
The socalled tomogram transform was ﬁrst deﬁned in
the context of quantum mechanics and operator theory.
10, 11
00346748/2011/82(8)/083502/9/$30.00 © 2011 American Institute of Physics82, 0835021
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://rsi.aip.org/rsi/copyright.jsp
0835022 Clairet
et al.
Rev. Sci. Instrum. 82, 083502 (2011)
FIG. 1. (Color online) Short time window FFT analysis and instantaneous beat frequency (solid line) vs. probing frequency of a Vband reﬂectometer signal.
Both signals have been recorded during the same plasma at different time without (left) and with (right) additional ICRH.
A selfadjoint operator depending on a parameter θ is intro
duced together with its eigenfunctions: the tomogram trans
form performs the projection over the basis formed by these
eigenvectors. The properties of the operator ensure that the
eigenfunctions are θ dependent and orthogonal for all ﬁxed
θ. As a consequence, the signal can be projected, separated in
parts, and each part can be resynthesized in time in a fast and
efﬁcient way. Moreover, θ allows tuning the decomposition in
order to be adapted to the signal. It allows a better separation
of the different parts.
In the framework of Tore Supra reﬂectometry, the tomo
gram associated to the operator (θ)ofRef.12 is used. It
is the association of two parts, one related to time, the other
related to frequency and a parameter θ ∈ [−π/2,π/2]. The
parameter θ allows to pass from the time representation of the
signal (θ = 0) to the frequency one (θ = π/2) via intermedi
ate representations. The operator (θ)is
(θ) = cos θt + i sin θ
d
dt
,
where t and d/dt are, respectively, the tmultiplication oper
ator and the derivative operator. Its eigenfunctions ψ
θ
X
, i.e.,
deﬁned by (θ )ψ
θ
X
= X ψ
θ
X
,where X corresponds to an eigen
value of the operator (θ ), form an orthogonal basis. The re
ﬂectometry signals are of ﬁnite timelength T and the ψ
θ
X
on
the domain [0,T] have the following expressions:
ψ
θ
X
(t) =
1
√
T
exp
−i
1
2tanθ
t
2
−
X
sin θ
t
. (1)
The associated eigenvalue takes discrete values,
X =
2π sin θ
T
n, (2)
where n is an integer. The tomogram transform M of a signal
u is deﬁned by
M
u
(X,θ) =C
θ
(X)
2
, (3)
where C
θ
(X) is the scalar product:
C
θ
(X) =
1
√
T
T
0
u(t)ψ
θ
X
(t)dt. (4)
Remark that this transform represents an energy den
sity and for θ = 0 the tomogram is just

u(t)

2
, whereas for
θ = π/2, it is the density of the Fourier transform

U (ω)

2
.
The general theory of selfadjoint operators gives the re
synthesis of the signal via the following relation:
u(t) =
X
C
θ
(X)ψ
θ
X
(t). (5)
In fact, each ψ
θ
X
is a linear chirp with phase derivative
φ
(t) =−
1
tan θ
t +
X
sin θ
= 2π (st + f
0
).
If plotted in the timefrequency domain, ψ
θ
X
is a ridge
of slope s = (−2π tan θ)
−1
and initial frequency (at t = 0)
f
0
= X/2π sin θ .
Thus, the tomogram transform can be seen as the projec
tion on a basis of linear chirps. In consequence, this transform
should be appropriate in the case of reﬂectometry data where
chirplike components are embedded in a complex signal, as
one can see, for instance, on the spectrogram of Fig. 1.
B. Practical processing steps and implementation
To extract the plasma reﬂection from the raw signal, the
tomogram analysis is done in several steps.
1. Choice of
θ
The selection of θ is not straightforward and depends on
the signal. As explained in Sec. II A, each ψ
θ
X
is a linear chirp
with phase derivative: φ
(t) = st + f
0
. Let us assume that the
signal to extract is a linear chirp with slope a and initial fre
quency b:ifs is taken equal to a, the tomogram decompo
sition will give the maximum of its efﬁciency since all the
chirp energy will be concentrated on one peak situated at f
0
= b. This would correspond to the orthogonal projection on
ψ
θ
X
with θ = arctan(−1/2aπ ) and X = 2π sin θ . The choice
of the best θ is usually made a posteriori with the help of the
tomogram plot: several projections are computed, associated
to different arbitrary chosen θ . Then a comparison of the re
sults is done by plotting the values of the function of two vari
ables M
u
. In the present study, for a more intuitive approach,
we have adopted the chirp point of view and we plot M
u
with
respect to θ and f
0
; an example is shown in Fig. 2. The 2D plot
highlights the different parts embedded in the signal which
can be extracted by the tomogram decomposition. The θ for
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://rsi.aip.org/rsi/copyright.jsp
0835023 Clairet
et al.
Rev. Sci. Instrum. 82, 083502 (2011)
FIG. 2. (Color online) Tomograms of the ohmic (left) and ICRH (right) signals. The dashed line represents the parameter chosen for the projection at θ
= π/5.
which the separation between the different parts is the great
est is chosen for the extraction (see Sec. III). It is important
to notice that the choice of the angle is robust, meaning that
a small change of θ will lead to a small change of the energy
pattern. This can be seen on the tomograms plotted in Fig. 2.
For identical reasons, the same robustness holds if the signal
to extract is not a perfectly linear chirp but possesses a small
bending.
2. Computation of the signal projections
C
θ
(X): The reﬂectometry signal is discrete and con
tains N = 2000 points per run of length T = 20 μs.
The sampling frequency is F
s
= N / T = 100 MHz.
The relation (4) is here the discrete sum: C
θ
(X)
=
N
n=1
u(
n
F
s
)
1
√
T
exp(i
1
2tanθ
n
2
F
2
s
)exp(−i
X
sin θ
n
F
s
). This quan
tity can be calculated using a fast Fourier transform (FFT)
algorithm (when N is a power of 2), as one can remark that
the above equation is the discrete Fourier transform of the
function s(
n
F
s
)exp(i
1
2tanθ
n
2
F
2
s
)
1
√
T
.
3. Choice of sets of C
θ
(
X
)
corresponding to the
plasma reﬂection
For a given θ, the energy density is calculated and plot
ted in Fig. 3. Several peaks of energy are present, which are
the consequence of the multireﬂections contained in the sig
nal. Using a threshold, regions with a high energy density
are isolated. The set of frequencies f
0
corresponding to the
plasma reﬂections is extracted. This gives the set S of ele
ments X = f
0
2π sin θ , which will be used for the signal re
construction.
4. Extracted signals
The reconstruction of the extracted signals is then per
formed according to the relation (5). For the plasma reﬂec
tions, the sum is hence over the values of X belonging to S.
The action of step 3 is to clear out the projection from the
multireﬂection components of the signal. Then, the output of
part 4 gives a temporal signal containing only the plasma re
ﬂections. The phase of t his timedomain signal can then be
used as the input of an algorithm for reconstructing the den
sity proﬁle.
III. STANDARD DENSITY PROFILE REFLECTOMETRY
A complete set of reﬂectometers, covering the fre
quency range from 33 to 150 GHz, provides density proﬁle
measurements on TS plasma from the edge to the centre
and beyond.
13, 14
For normal operation, the reﬂectometer
antennas are located in a dedicated porthole outside the
θ
θ
θ
θ
FIG. 3. (Color online) Slices of the tomogram at θ = π/5 for the ohmic signal (left) and the ICRH signal (right). Three components are present in the signal
and correspond, respectively, to the quartz window, the plasma, and the back wall echoes.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://rsi.aip.org/rsi/copyright.jsp
0835024 Clairet
et al.
Rev. Sci. Instrum. 82, 083502 (2011)
FIG. 4. (Color online) Spectrograms of the plasma signals separated by the tomographic analysis for ohmic and ICRH conditions (the solid lines correspond to
a calculation from the instantaneous phase derivative).
vacuum vessel behind a quartz window. Xmode polarisation
can provide an accurate initialization of the density proﬁle as
long as the magnetic ﬁeld proﬁle is known. The detection of
the ﬁrst cutoff frequency, which equals the electron cyclotron
frequency at zero density, provides the starting position of
the radial density proﬁle. According the TS speciﬁcities
(plasma size, magnetic ﬁeld aspect ratio, etc.) during routine
operation, the proﬁle initialization is mainly performed with
the Vband reﬂectometer signal, which is treated here. The
reﬂectometer signal frequency, also called the beat frequency
(F
b
), is related to the time derivative of the phase, such
that
F
b
=
1
2π
∂φ
∂t
.
The beat frequency can be determined either at a given
point using the phase derivative at this point (the instanta
neous beat frequency used to calculate the density proﬁles)
or from sliding FFT which, in this case, realises an average
over the time window analysis. Despite this averaging, the
advantage of the FFT analysis is to conveniently provide a
full picture of every signal components. In the following,
several spectrograms are shown and are all constructed from
the 2000 points signal by using shortterm Fourier transform
with windows of 100 points and a step of 10 points. Zero
padding has been done using 1900 points to increase the
smoothness of Fig. 1. Moreover, to make the different com
ponents visible even when the signal amplitude is low, the
signal energy has been normalized to one for each window.
The scrapeoff layer (SOL) region is characterized by
outboard plasma with very low density (∼10
17
m
−3
). Due to
the low reﬂection efﬁciency of this very low edge density
plasma, part of the probing wave crosses the cutoff layer and
reﬂects onto the back wall (Fig. 1). Unfortunately, the fre
quency discrimination of these two echoes with traditional ﬁl
tering techniques appears to be somewhat difﬁcult due to their
closeness in frequency. Figure 1 shows two different plasma
edge behaviours according the experimental scenarios: under
ohmic condition and with additional heating during the same
plasma discharge.
Evidences of different plasma boundary conditions be
tween the two plasma phases are observed. During the ohmic
phase, the ﬁrst plasma cutoff frequency occurs around 70
GHz, while during the additional heating phase the cutoff
appears above 60 GHz. In this latter case, additional plasma
is created in the periphery probably due to some extra power
deposition at the edge and/or because of the enhanced particle
transport. When this edge plasma starts reﬂecting the wave, it
does not screen totally the back wall because of its low den
sity. The beat frequency increases continuously from 10 to
20 MHz with the probing wave frequency. A change in the
reﬂection behaviour appears above 68 GHz where the beat
FIG. 5. (Color online) Reﬂected signal amplitude comparison using tomography analysis (dashed line) and bandpass ﬁlter 519 MHz (solid line), for ohmic
(left), and ICRH (right).
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