1. Offline and Real-time signal processing on fusion signals
R. Coelho, D. Alves
Associação EURATOM/IST, Instituto de Plasmas e Fusão Nuclear
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.

2. 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)

3. 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 !

4. 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 frequency resolution enhancement

5. 2. Empirical mode decomposition

6. 2. Empirical mode decomposition
Mirnov signal spectra, # using EMD 3 dominant IMF (signals + frequencies)

7. 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 

8. 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.

9. 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 

10. Background  With  Phase difference between signals :
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

11. 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

12. 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 matrix R.Coelho, D.Alves, RSI08

13. Kalman filter based spectrogram
Real-time replacement of spectrogram.
Amplitude, at a given time sample, estimated as
df=5kHz
s=2MHz

14. 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.

15. Synthetised results FFT algorithm Coherency (12 eq.spaced tor.coils)
s=100kHz
375 pt for averaging (3.75ms)
125pt/FFT
50pt overlap (0.5ms)

16. Synthetised results KCS algorithm Coherency (12 eq.spaced tor.coils)
s=100kHz
50 pt for averaging
=800Hz

17. 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

18. Experimental results KCS algorithm Coherency (first 5 tor.coils only)
s=1MHz
100 pt for averaging
=1000Hz

19. 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

20. 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)

21. 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 !