Signal processing tools Lisbon 18/02/09R Coelho 1/29 Offline and Real-time signal processing on...

Signal processing tools Lisbon 18/02/09 R Coelho 1/29

Offline and Real-time signal processing on fusion signals

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


1. 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)


1. 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 !


1. 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


2. Empirical mode decomposition

N

jjj

N

jj ttAtIMFtS

11

))(cos().()()(


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

2/121

1212

)(S)(S

)(S)(C

)(S1 )(S2

)(S12

212 )(C )(CArg 12


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

2/121

1212

)(S)(S

)(S)(C

)(S1 )(S2 )(S12

212 )(C )(CArg 12


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

t2mcos)r(BB

121212 mmm

2/121

1212

)(S)(S

)(S)(C


Time/frequency constraints

• Ensemble averaging is in practice replaced by time averaging

• Spectral estimation done usually with FFT

……FFT Coherency spectrum drawbacks…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 – measurementsR,Q – noise covariances

The process matrix R.Coelho, D.Alves, RSI08

1kk x̂Fx̂

kkkkk x̂HzKx̂x̂

N

2

1

N

F

F

F

Fh

)cos()sin(

)sin()cos(F

ii

iii sii /2


Kalman filter based spectrogram

Real-time replacement of spectrogram.

Amplitude, at a given time sample, estimated as

2i2

21i2i x̂x̂A

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.

cosx̂

sinx̂

€

S12(ω) = ˆ x cos_ bˆ x cos_ a + ˆ x sin_ b

ˆ x sin_ a( ) + i ˆ x sin_ bˆ x cos_ a − ˆ x cos_ b

ˆ x sin_ a( )


Synthetised results

FFT algorithm Coherency (12 eq.spaced

tor.coils)

n=-3,4s=100kHz375 pt for averaging (3.75ms)125pt/FFT50pt overlap (0.5ms)


Synthetised results

KCS algorithm Coherency (12 eq.spaced

tor.coils)

n=-3,4s=100kHz50 pt for averaging=800Hz


Experimental results #68202 (n=1 ST precursor)

FFT algorithm Coherency (first 5 tor.coils only)

n=1s=1MHz1500 pt for averaging (1.5ms)1000pt/FFT100pt overlap


Experimental results

KCS algorithm Coherency (first 5 tor.coils

only)

s=1MHz100 pt for averaging=1000Hz


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

FFT algorithm Coherency (first 5 tor.coils only)

n=1s=1MHz1500 pt for averaging (1.5ms)1000pt/FFT100pt overlap


Experimental results

KCS algorithm Coherency (first 5 tor.coils

only)

s=1MHz100 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 !

Signal processing tools Lisbon 18/02/09R Coelho 1/29 Offline and Real-time signal processing on...

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