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Offline and Real-time signal processing on fusion signals

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Outline

1 – The Fourier space methods

2 – Empirical mode decomposition

3 – (k,ω) space methods - Coherency spectrum and SVD

4 – Beyond the Fourier paradigm Real-time based techniques.

– Motional Stark Effect data processing.

R. Coelho, D. Alves

Associação EURATOM/IST, Instituto de Plasmas e Fusão Nuclear

- Fourier space methods (time dual)
- Eigenmode decomposition providing signal support (even for discontinuous signals)
continuous

discrete

Some Useful Properties

If h(ω)=f(ω)g(ω)

If h(x)=f(x)g(x) then h(ω)=f(ω)*g(ω)

- Fourier space methods (time dual)
Some Useful Properties

If h(ω)=f(ω)g(ω)

- FILTERING in time !
If h(x)=f(x)g(x) then h(ω)=f(ω)*g(ω)

- FILTERING in frequency !

- Fourier space methods
Time-frequency analysis

- Sliding FFT method : S(t,ω) where midpoint of time window corresponds to a FFT.
- Windowed spectrogram : same as above but with window function to reduce noise and enhance time localization
- Spectrogram with zero padding : same as above but zero padding to each time window shadow frequencyresolution enhancement

2. Empirical mode decomposition

2. Empirical mode decomposition

Mirnov signal spectra, # 11672 using EMD 3 dominant IMF (signals + frequencies)

3. (k,ω) space methods - Coherency spectrum and SVD

Coherency-Spectrum – standard tool for mode number analysis of

fluctuation spectra

Formal definition

, - auto-spectrums

- cross-spectrum densities of two signals

CoherencyPhase

- Singular value decomposition (SVD)
- SVD is a decomposition of an array in time and space, finding the most significant time and space characteristics.
- The SVD of an NxM matrix A is A=UWVT
- W - MxM diagonal matrix with the singular values
- Columns of matrix V give the principal spatial modes and the product UW the principal time components.

Mode number analysis by coherence spectrum

Cross-Spectrum – standard tool for mode number analysis of

MHD fluctuation spectra

Formal definition

, - auto-spectrums

- cross-spectrum densities of two signals

CoherencyPhase

Background

With

m is the mode number and the frequency

Phase difference between signals :

Generalisation of full coil array naturally leads to a linear fit of entire coil set

Time/frequency constraints

Ensemble averaging is in practice replaced by time averaging

Spectral estimation done usually with FFT

…FFT Coherency spectrum drawbacks…

Each FFT (N-samples) gives ONE estimate for AMPLITUDE and

PHASE for each frequency component.

Average over Nw windows NNw samples to ONE Coherency

spectrum

Trade-off Time/frequency resolution

Beyond FFT paradigm...

State variable recursive estimation according to linear model + measurements

F – process matrix

K – filter gain

z – measurements

R,Q – noise covariances

The process matrixR.Coelho, D.Alves, RSI08

Kalman filter based spectrogram

Real-time replacement of spectrogram.

Amplitude, at a given time sample, estimated as

- df=5kHz
- s=2MHz

Kalman coherence spectrum

Real-time estimation of in-phase and quadratures of each -component allows for cross-spectrum estimation :

Two coil signals (labelled a and b)

in-phase ( )

quadrature ( )

ADVANTAGE

Streaming estimation of phase difference.

Much less “sample consuming” than FFT.

Effective filtering of estimates “sharpens” coherency.

Synthetised results

FFT algorithm

Coherency (12 eq.spaced tor.coils)

n=-3,4

s=100kHz

375 pt for averaging (3.75ms)

125pt/FFT

50pt overlap (0.5ms)

Synthetised results

KCS algorithm

Coherency (12 eq.spaced tor.coils)

n=-3,4

s=100kHz

50 pt for averaging

=800Hz

Experimental results #68202 (n=1 ST precursor)

FFT algorithm

Coherency (first 5 tor.coils only)

n=1

s=1MHz

1500 pt for averaging (1.5ms)

1000pt/FFT

100pt overlap

Experimental results

KCS algorithm

Coherency (first 5 tor.coils only)

s=1MHz

100 pt for averaging

=1000Hz

Experimental results #72689 (m=3,n=2 NTM)

FFT algorithm

Coherency (first 5 tor.coils only)

n=1s=1MHz

1500 pt for averaging (1.5ms)

1000pt/FFT

100pt overlap

Experimental results

KCS algorithm

Coherency (first 5 tor.coils only)

s=1MHz

100 pt for averaging

=1000Hz

n=3, IDL “fake contouring”

Earlier detection in coherency (threshold effect)

Conclusions

A novel method for space-frequency MHD analysis using Mirnov data was developed.

A Kalman filter lock-in amplifier implementation is used to replace the FFT in the coherence function calculation.

Particularly suited technique for real-time analysis with limited number of streaming data

Saving in data samples arises from the streaming estimation of in-phase and quadrature components of any given frequency mode existent in the data, not possible in a FFT based algorithm.

Ongoing work…better candidates will be targeted !