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. Offline and Real-time signal processing on fusion signals.

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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.

Offline and Real-time signal processing on fusion signals

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

CoherencyPhase


  • 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

CoherencyPhase


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  NNw 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=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

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 !


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