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Tomograms and data analysis applied on reflectometry signals

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applied on reflectometry signals

9th International Reflectometry Workshop

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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

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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

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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)

(6)

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

(7)

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

(8)

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

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Tomogram

Some results on simulated data

Some results on reflectometry data

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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

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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

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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

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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

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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

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Tomogram (contour plot)

X

teta

0 pi/2

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SD # 1 : Component decomposition

Projection, in the case of finite time t ∈ [0, T ] orthonormal basis

ψ T θ, X

< ψ T θ, X

n

ψ T θ, X

m

>= δ m , n

F. Briolle () tomogram-data analysis 9 / 34

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Projection, in the case of finite time t ∈ [0, T ] orthonormal basis

ψ T θ, X

< ψ T θ, X

n

ψ T θ, X

m

>= δ m , n

Projections

c n =< s , ψ T θ, X

n

>

(18)

SD # 1 : Component decomposition

Projection, in the case of finite time t ∈ [0, T ] orthonormal basis

ψ T θ, X

< ψ T θ, X

n

ψ 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

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Component ˜ y 2 (t)

0 5 10 15 20

−2

−1 0 1 2 3 4 5

time

real(y2)

(20)

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

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Chirps : y (t) = e i Φ

1

( t ) + e i Φ

2

( t ) + b(t ) SNR=10dB

(22)

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

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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

(24)

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

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Tomogram of the chirps signal

X

teta

0 pi/2

(26)

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

(27)

Phase derivative

The phase derivative can be estimated from the projections { c n } n

∂t φ(t ) = Im Γ(t)

˜ y(t)

(1)

(28)

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

n

T (t) (2)

F. Briolle () tomogram-data analysis 15 / 34

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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

n

T (t) (2) and

˜

y ( t ) = X

n

c n ψ θ, x

n

T ( t ) (3)

(30)

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

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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)

(32)

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

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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

(34)

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

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Spectrum c x θ

n

of 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).

2

choc 42824 (prof 1003/2086) tps : 9.9996 s

Porthole

First wall Plasma

(36)

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

(37)

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)

(38)

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

(39)

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

(40)

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

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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

F. Briolle () tomogram-data analysis 28 / 34

(43)

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

(44)

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

(45)

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=10

Antenna coupling

X2 X1

Antenna

position

(46)

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

(47)

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)

(48)

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

F. Briolle () tomogram-data analysis 34 / 34

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