applied on reflectometry signals
9th International Reflectometry Workshop
Tomograms applied on reflectometry signals
Collaboration
F. Briolle, R. Lima, Centre de Physique Th´eorique, Marseille, France R. Vilela Mendes, IPFN - EURATOM/IST Association, Instituto Superior T´ecnico, Lisboa, Portugal
V.I. Man’ko and M. Man’ko, P. N. Lebedev Physical Institute, Moscow, Russia
F. Clairet, CEA, IRFM, F-13108 Saint-Paul-lez-Durance, France S. Heuraux, IJL, Nancy-University CNRS UMR 7198, BP 70239, F-54506 Vandoeuvre, France
F. Briolle () tomogram-data analysis 2 / 34
Aims
Reflectometry data : y (t) = A(t)e i Φ( t )
1
extract the components y k ˜ (t ) of the signal : reflections on the plasma, on the pothole, on the wall of the tokamak, ...
F. Briolle () tomogram-data analysis 3 / 34
1
extract the components y k ˜ (t ) of the signal : reflections on the plasma, on the pothole, on the wall of the tokamak, ...
2
estimate the phase derivative of the components, d dt φ ˜
k(t)
Aims
Reflectometry data : y (t) = A(t)e i Φ( t )
1
extract the components y k ˜ (t ) of the signal : reflections on the plasma, on the pothole, on the wall of the tokamak, ...
2
estimate the phase derivative of the components, d dt φ ˜
k(t) Tools
F. Briolle () tomogram-data analysis 3 / 34
1
extract the components y k ˜ (t ) of the signal : reflections on the plasma, on the pothole, on the wall of the tokamak, ...
2
estimate the phase derivative of the components, d dt φ ˜
k(t) Tools
1
Filtering, Fourier Transform
Aims
Reflectometry data : y (t) = A(t)e i Φ( t )
1
extract the components y k ˜ (t ) of the signal : reflections on the plasma, on the pothole, on the wall of the tokamak, ...
2
estimate the phase derivative of the components, d dt φ ˜
k(t) Tools
1
Filtering, Fourier Transform
2
Time-Frequency representation : Spectrogram, Wigner-Ville, Wavelet, Tomogram
F. Briolle () tomogram-data analysis 3 / 34
Tomogram
Some results on simulated data
Some results on reflectometry data
Tomogram
Time-Frequency operator ˆ
x = µ ˆ t + ν ω ˆ == µt + i ν d dt Eigen vectors
ψ X µ,ν
F. Briolle () tomogram-data analysis 5 / 34
Time-Frequency operator ˆ
x = µ ˆ t + ν ω ˆ == µt + i ν d dt Eigen vectors
ψ X µ,ν
Tomogram : Time-frequency distibution of the analytical signal s M s (X , µ, ν) =
< s, ψ X µ,ν >
2
- with normalisation R
M s (X , µ, ν)dX = 1
- µ = 1, ν = 0, distribution in the time domain M s (t, 1, 0) = | s(t ) | 2
- µ = 0, ν = 1, distribution in frequency domain M s (ω, 0, 1) = | ˜ s (ω) | 2
Tomogram Symplectic
single parameter θ
ˆ
x = cos θ ˆ t + sin θˆ ω M s (X , θ) =
Z
ψ X θ (t) s (t) dt
2
ψ X θ (t ) = 1
2π | sin θ | exp
i cos θ
2 sin θ t 2 − iX sin θ t
F. Briolle () tomogram-data analysis 6 / 34
Sinusoidal signal
y(t) = y 1 (t) + y 2 (t) + y 3 (t) + b(t) y 1 (t) = exp (i ω 0 t) , t ∈ [0, T ] y 2 (t) = exp (i ω 1 t) , t ∈
0, T
4
y 3 (t) = exp (i ω 1 t) , t ∈ T
2 , T
SD # 1 : Component decomposition
Sinusoidal signal
y(t) = y 1 (t) + y 2 (t) + y 3 (t) + b(t) y 1 (t) = exp (i ω 0 t) , t ∈ [0, T ] y 2 (t) = exp (i ω 1 t) , t ∈
0, T
4
y 3 (t) = exp (i ω 1 t) , t ∈ T
2 , T
Real part of the time signal (SNR = 10 dB)
0 5 10 15 20
−3
−2
−1 0 1 2 3
time
real(y)
F. Briolle () tomogram-data analysis 7 / 34
Tomogram (contour plot)
X
teta
0 pi/2
SD # 1 : Component decomposition
Projection, in the case of finite time t ∈ [0, T ] orthonormal basis
ψ T θ, X
< ψ T θ, X
nψ T θ, X
m>= δ m , n
F. Briolle () tomogram-data analysis 9 / 34
Projection, in the case of finite time t ∈ [0, T ] orthonormal basis
ψ T θ, X
< ψ T θ, X
nψ T θ, X
m>= δ m , n
Projections
c n =< s , ψ T θ, X
n>
SD # 1 : Component decomposition
Projection, in the case of finite time t ∈ [0, T ] orthonormal basis
ψ T θ, X
< ψ T θ, X
nψ T θ, X
m>= δ m , n
Projections
c n =< s , ψ T θ, X
n>
Representation of the projections c n of the signal y(t) for θ = π 5
−20 0 20 40 60 80 100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
X
abs(cn)
F. Briolle () tomogram-data analysis 9 / 34
Component ˜ y 2 (t)
0 5 10 15 20
−2
−1 0 1 2 3 4 5
time
real(y2)
SD # 1 : Component decomposition
Component ˜ y 2 (t)
0 5 10 15 20
−2
−1 0 1 2 3 4 5
time
real(y2)
Component ˜ y 3 (t) components
0 5 10 15 20
−2
−1 0 1 2 3 4 5
time
real(y3)
F. Briolle () tomogram-data analysis 10 / 34
Chirps : y (t) = e i Φ
1( t ) + e i Φ
2( t ) + b(t ) SNR=10dB
SD # 2 : Component decomposition and phase derivative
Chirps : y (t) = e i Φ
1( t ) + e i Φ
2( t ) + b(t ) SNR=10dB with ∂ ∂ t Φ 1 ( t ) = a 1 t 2 + b 1 t + c 1
and ∂ ∂ t Φ 2 (t) = b 2 t + d 2 √ t + c 2 .
F. Briolle () tomogram-data analysis 11 / 34
Chirps : y (t) = e i Φ
1( t ) + e i Φ
2( t ) + b(t ) SNR=10dB with ∂ ∂ t Φ 1 ( t ) = a 1 t 2 + b 1 t + c 1
and ∂ ∂ t Φ 2 (t) = b 2 t + d 2 √ t + c 2 .
Representation of dt d Φ 1 (t) and dt d Φ 2 (t) as a function of time.
0 5 10 15 20
20 30 40 50 60 70 80 90
time
Phase derivative
SD # 2 : Component decomposition and phase derivative
Frequency representation
−500 0 50
5 10 15
Frequency
abs(fft(y))
F. Briolle () tomogram-data analysis 12 / 34
Tomogram of the chirps signal
X
teta
0 pi/2
SD # 2 : Component decomposition and phase derivative
Projection θ = atan( ∆ ∆ T F ) ≈ π 8
0 10 20 30 40 50
0 0.05 0.1 0.15 0.2 0.25
X
abs(cn)
y1
y2
F. Briolle () tomogram-data analysis 14 / 34
Phase derivative
The phase derivative can be estimated from the projections { c n } n
∂
∂t φ(t ) = Im Γ(t)
˜ y(t)
(1)
SD # 2 : Component decomposition and phase derivative
Phase derivative
The phase derivative can be estimated from the projections { c n } n
∂
∂t φ(t ) = Im Γ(t)
˜ y(t)
(1) with
Γ(t) = X
n
c n
∂
∂t ψ θ, x
nT (t) (2)
F. Briolle () tomogram-data analysis 15 / 34
Phase derivative
The phase derivative can be estimated from the projections { c n } n
∂
∂t φ(t ) = Im Γ(t)
˜ y(t)
(1) with
Γ(t) = X
n
c n
∂
∂t ψ θ, x
nT (t) (2) and
˜
y ( t ) = X
n
c n ψ θ, x
nT ( t ) (3)
SD # 2 : Component decomposition and phase derivative
Phase derivative θ ≈ π 8
0 5 10 15 20
0 10 20 30 40 50 60 70 80 90
time
phase derivative
F. Briolle () tomogram-data analysis 16 / 34
Time representation of the signal
Fast-sweep X-mode reflectometer on Tore Supra, V band
0 20
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 0.6 0.8
time
real(y)
Reflectometry data : choc 42824
Frequency representation of the signal
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018
Frequency (MHz)
Power Spectrum
choc 42824 (prof 1003/2086) tps : 9.9996 s
Plasma
First wall Porthole
F. Briolle () tomogram-data analysis 18 / 34
Spectrogram of the signal
Frequency (GHz)
Phase derivative
choc 42824 (prof 1003/2086) tps : 9.9996 s
55 60 65 70 75
50 40 30 20 10 0
−10
−20
−30
−40
−50
First wall Plasma
Porthole
Reflectometry data : choc 42824
Tomogram of the signal
X
teta
choc 42824 (prof 1003/2086) tps:9.9996s
0 pi/2
Porthole
First wall Plasma
F. Briolle () tomogram-data analysis 20 / 34
Spectrum c x θ
nof the reflectometry signal y(t) for θ = π − π 5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X
abs(cn).
2choc 42824 (prof 1003/2086) tps : 9.9996 s
Porthole
First wall Plasma
Reflectometry data : choc 42824
Components of the reflectometry signal : θ = π − π 5
56 58 60 62 64 66 68 70 72 74 76
−0.5 0 0.5 1 1.5 2 2.5
Frequency (GHz)
Amplitude (real part)
choc 42824 (prof 1003/2086) tps : 9.9996 s
First wall
Plasma
Porthole
F. Briolle () tomogram-data analysis 22 / 34
Separation of the components in the reflectometry signal
55 60 65 70 75
−60
−50
−40
−30
−20
−10 0
Frequency (GHz)
Amplitude (dB)
choc 42824 (prof 1003/2086) tps : 9.9996 s
First wall
Plasma Full signal
Figure: Full signal and reflexions on the wall and on the plasma (dB)
Reflectometry data : choc 42824
Components of the reflectometry signal : θ = π 2
56 58 60 62 64 66 68 70 72 74 76
−0.5 0 0.5 1 1.5 2 2.5
Frequency (GHz)
Amplitude (real part)
choc 42824 (prof 1003/2086) tps : 9.9996 s
Plasma First wall
Porthole
F. Briolle () tomogram-data analysis 24 / 34
Components in the reflectometry signal : θ = π 2
56 58 60 62 64 66 68 70 72 74 76
−60
−50
−40
−30
−20
−10 0
choc 42824 (prof 1003/2086) tps : 9.9996 s
Frequency (GHz)
Ampitude (dB)
First wall Full signal Plasma
Reflectometry data : choc 42824
Phase derivative of the three components (θ = π − π 5 )
55 60 65 70 75
−5 0 5 10 15 20 25
Frequency (GHz)
Phase derivative
choc 42824 (prof 1003/2086) tps : 9.9996 s
First wall
Plasma
Porthole
F. Briolle () tomogram-data analysis 26 / 34
Reflectometry data : LH antenna choc 29286 tps: 1.43536
Time representation of the data (red line filtered data)
0 20
−3
−2.5
−2
−1.5
−1
−0.5 0
time
Amplitude (dB)
TS 29296 (prof 2) tps=1.43536 sweep time : 20 micros npts=2000
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Frequency representation of the data
−50 0 0 50
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
TS 29296 (prof 2) tps=1.43536 sweep time : 20 micros npts=2000
Power Spectrum
Reflectometry data : LH antenna choc 29286 tps: 1.43536
Extraction of the two components
60 62 64 66 68 70
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Frequency
abs(signal)
Y2(t)
Y1(t)
signal(t)
F. Briolle () tomogram-data analysis 30 / 34
Extraction of the two components
Beat frequency (MHz)
55 60 65 70 75
0 5 10 15 20 25 30 35 40 45
50
TS 29296 (prof 2) tps = 1.4353s sweep time: 20us npt=2000 FFT : win=40 step=10Antenna coupling
X2 X1
Antenna
position
Reflectometry data : LH antenna choc 29286 tps: 1.43536
Extraction of the two components
2.8 2.85 2.9 2.95 3 3.05 3.1 3.15 3.2
0 1 2 3 4 5 6 7 8 9x 1018
radius (m)
density (m-3)
TS 29296 t= 1.4353s (#2)separatrix LH antenna
non filtered filtered
F. Briolle () tomogram-data analysis 32 / 34
Tomogram gives promissing results for reflectometry data separation of components
phase derivative estimation Applications
First frequency cutoff determination
Separation of close multicomponents (reflectometers in Heating
Antennas ICRH, LH)
Tomograms associated to the conformal group
Time-frequency tomogram
x 1 = µ ˆ t + ν ω ˆ = µt + iν d dt Time-scale
x 2 = µt + i ν
t d dt + 1
2
Frequency-scale
x 3 = iµ d dt + i ν
t d
dt + 1 2
Time-conformal
x 4 = µt + i ν
t 2 d dt + t
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