Introduction to the Diagnosis of Magnetically Confined Thermonuclear - PowerPoint PPT Presentation

Fourier Analysis Tomographic reconstruction Introduction to the Diagnosis of Magnetically Confined Thermonuclear Plasma Selected Data Analysis Techniques J. Arturo Alonso Laboratorio Nacional de Fusión EURATOM-CIEMAT E6 P2.10 arturo.alonso@ciemat.es version 0.1 (February 13, 2012) Selected Data Analysis Techniques, A. Alonso, copyleft 2010 1 / 20

Fourier Analysis Tomographic reconstruction Outline 1 Fourier Analysis Fourier transform Some aplications Tomographic reconstruction 2 Pixel methods Singular Value Decomposition Selected Data Analysis Techniques, A. Alonso, copyleft 2010 2 / 20

Fourier Analysis Tomographic reconstruction Outline 1 Fourier Analysis Fourier transform Some aplications Tomographic reconstruction 2 Pixel methods Singular Value Decomposition Selected Data Analysis Techniques, A. Alonso, copyleft 2010 3 / 20

Fourier Analysis Tomographic reconstruction The Fourier Transform Given a function f ( x ) we define its Fourier Transform as Fourier Transform � ∞ f ( x ) e − 2 π ixs dx . F ( s ) = F f ≡ −∞ We assume this integral exist and can be inverted as Inverse Fourier Transform � ∞ f ( x ) = F − 1 F ≡ F ( s ) e 2 π ixs ds . −∞ Which follows from a definition of the Dirac’s delta (generalised) � ∞ function δ ( x ) = F − 1 { 1 } = −∞ exp ( 2 π ixs ) ds � ∞ � ∞ � ∞ dx ′ f ( x ′ ) e 2 π is ( x − x ′ ) = f ( x ) = F − 1 F f = dx ′ f ( x ′ ) δ ( x − x ′ ) ds −∞ −∞ −∞ Selected Data Analysis Techniques, A. Alonso, copyleft 2010 4 / 20

Fourier Analysis Tomographic reconstruction The convolution theorem The convolution of two functions f and g is a third function f ∗ g defined as � ∞ f ( x ′ ) g ( x − x ′ ) dx ′ , ( f ∗ g )( x ) ≡ −∞ then the convolution theorem states that F{ f ∗ g } = ( F f )( F g ) Proof: � ∞ �� ∞ � f ( x ′ ) g ( x − x ′ ) dx ′ e − 2 π isx dx F{ f ∗ g } = −∞ −∞ � ∞ �� ∞ � g ( x − x ′ ) e − 2 π isx dx f ( x ′ ) dx ′ = −∞ −∞ � ∞ G ( s ) e − 2 π isx ′ f ( x ′ ) dx ′ = G ( s ) F ( s ) = −∞ It also follows that F − 1 { F ∗ G } = fg . Selected Data Analysis Techniques, A. Alonso, copyleft 2010 5 / 20

Fourier Analysis Tomographic reconstruction Real functions For a general complex function f ( x ) we have � ∗ � �� F{ f ∗ ( x ) } = f ∗ ( x ) e − 2 π isx dx = f ( x ) e 2 π isx dx = F ∗ ( − s ) . For a real function f = f ∗ F ( s ) = F ( − s ) ∗ i.e., only half of the (complex) transform of a real function contains non-redundant information. Selected Data Analysis Techniques, A. Alonso, copyleft 2010 6 / 20

Fourier Analysis Tomographic reconstruction Energy conservation From the identity ( f ∗ h )( x ) = F − 1 { FH } evaluated at x = 0 we have � ∞ � ∞ f ( x ′ ) h ( − x ′ ) dx ′ = F ( s ) H ( s ) ds −∞ −∞ or, writting g ∗ ( x ′ ) = h ( − x ′ ) ⇒ G ∗ ( − s ) = H ( − s ) � ∞ � ∞ f ( x ) g ∗ ( x ) dx = F ( s ) G ∗ ( s ) ds −∞ −∞ know as the Power Theorem. If g = f then one obtains Rayleigh’s Theorem � ∞ � ∞ | f ( x ) | 2 dx = | F ( s ) | 2 ds −∞ −∞ Selected Data Analysis Techniques, A. Alonso, copyleft 2010 7 / 20

Fourier Analysis Tomographic reconstruction Periodic Functions: Fourier Series A periodic function f ( x ) : f ( x + nT ) = f ( x ) such as a physical quantity (say B ) defined over a periodic domanin (say the poloidal angle θ ) has a discrete transform 1 . � ∞ F ( s ) e 2 π is ( x + nT ) ds = f ( x ) f ( x + nT ) = −∞ ⇒ F ( s ) = 0 , except if e 2 π isnT = 1 → s k = k T In this case the function f ( x ) can be represented as a Fourier series � T / 2 ∞ ω k = 2 π c k e i ω k x ; � f ( x ) e − i ω k x dx f ( x ) = T k ; c k = − T / 2 k = −∞ 1 In reality the intergral is non-convergent, but it still illustrates that only some frequencies are present Selected Data Analysis Techniques, A. Alonso, copyleft 2010 8 / 20

Fourier Analysis Tomographic reconstruction The Discrete-time Fourier Transform (DFT) A physical measurement always consist of a vector with finite number N of discrete samples f τ = f ( τ ∆ t ) of an ideal continuos function f ( t ) . We define the Discrete Fourier Transform N − 1 N − 1 f τ e − 2 π i ( ν/ N ) τ ; F ν = N − 1 � � F ν e 2 π i ( ν/ N ) τ f τ = τ = 0 ν = 0 The consistency of these transformation follows from the fact that for any two intergers τ, τ ′ � 1 N − 1 if τ = τ ′ e − 2 π i ( ν/ N )( τ − τ ′ ) = � N − 1 0 otherwise . ν = 0 There are DFT versions of the FT theorems (convolution, power, etc.) Selected Data Analysis Techniques, A. Alonso, copyleft 2010 9 / 20

Fourier Analysis Tomographic reconstruction Frequencies in the DFT The following relations hold for real f τ F ν = F ν + nN F ν = F ∗ − ν = F ∗ N − ν 2 + ν = F ∗ F N N 2 − ν The discrete Fourier component F ν is associated with the physical ν frequency ν p = N ∆ t . Frequency restolution is then ∆ ν p = 1 / N ∆ t = 1 / T Maximum frequency is that of ν = N / 2 or ν Ny = 1 / 2 ∆ t p Selected Data Analysis Techniques, A. Alonso, copyleft 2010 10 / 20

Fourier Analysis Tomographic reconstruction Average over realizations Estimating real data spectral properties. Experimental data often consist of � 2 20 ≈ 10 6 samples. Rather than using all the points to get a single spectrum can be split into realizations and average the spectrum over them to gain some statistical significance. Floating potential (2 MHz sampling) 50 0 −50 0 10 20 30 40 50 60 0 10 Power Spectrum (a.u.) FFT of all 2 17 points −2 Periodogram 2 10 points x 2 7 windows 10 Welch 2 10 points 50% ovelap windows −4 10 −6 10 −8 10 −10 10 0 100 200 300 400 500 600 700 800 900 1000 f (kHz) Selected Data Analysis Techniques, A. Alonso, copyleft 2010 11 / 20

Fourier Analysis Tomographic reconstruction Spectrum evolution TJII#18998 V float 1000 800 Frequency (kHz) 600 400 200 0 50 100 150 200 250 Time (ms) Selected Data Analysis Techniques, A. Alonso, copyleft 2010 12 / 20

Fourier Analysis Tomographic reconstruction Cross power spectrum and coherence Just as the power spectrum is of a signal x ( t ) estimated as P x ( ω ) = � X ω X ∗ ω � where �·� is the average over ‘realisations’, the cross-power spectrum of x ( t ) and y ( t ) is defined as P xy = � X ω Y ∗ ω � The cross-coherence is 50 defined as x(t), y(t) 0 | P xy ( ω ) | 2 −50 C xy ( ω ) = 1040 1050 1060 1070 1080 P x ( ω ) P y ( ω ) 0.8 TJII#18998 V float 1 and 2 coherence for stationary spectra 0.6 C xy ( ν ) 0.4 C xy ( ω ) ∼ |� e i ( φ x ( ω ) − φ y ( ω )) �| 2 0.2 so it is high for frequencies 0 3 4 5 6 10 10 10 10 ν (Hz) with approx. constant phase difference. Selected Data Analysis Techniques, A. Alonso, copyleft 2010 13 / 20

Fourier Analysis Tomographic reconstruction The Continuous Wavelet Transform Wavelets were originally intended to analyse highly non-stationary and inhomogeneous signals (as human speech). FT provides all the frequency information about the signal but no temporal localization of the frequencies. WT give certain degree of localization in both frequency and time ( local frequency analyser). It is defined as � WT ψ { x } ( a , b ) = x ( t ) ψ a , b ( t ) dt R with � t − b � 1 ψ a , b ( t ) = √ a ψ a Selected Data Analysis Techniques, A. Alonso, copyleft 2010 14 / 20

Fourier Analysis Tomographic reconstruction Outline 1 Fourier Analysis Fourier transform Some aplications Tomographic reconstruction 2 Pixel methods Singular Value Decomposition Selected Data Analysis Techniques, A. Alonso, copyleft 2010 15 / 20

Fourier Analysis Tomographic reconstruction Tomography: posing the problem To recover the spatial distribution of, say, the emissivity g ( r ) from line-integrated measurements � f l = d sg ( r ) , l = 1 . . . n l S l Ill-posed problem: never sufficient information in a finite number of chords to determine g ( r ) Needs to reduce the function search space 1 Expansion in a finite set of ortogonal basis functions and fit coefficients to data (Cormak) 2 Pixel methods (or expasion in ‘pixel’ basis functions) Selected Data Analysis Techniques, A. Alonso, copyleft 2010 16 / 20

Fourier Analysis Tomographic reconstruction Tomography: posing the problem To recover the spatial distribution of, say, the emissivity g ( r ) from line-integrated measurements � f l = d sg ( r ) , l = 1 . . . n l S l Selected Data Analysis Techniques, A. Alonso, copyleft 2010 16 / 20

Fourier Analysis Tomographic reconstruction Pixel methods (I) Space is divided into (possibly non-square) pixels of approximatelly constant g Arrange the n l f ’s and n pixels = n x × n y g ij ’s in two column vectors f and g Then compute the ‘contribution’ n l × n pixel matrix T f = Tg Direct inversion is usually inpractical. Instead one tries to minimise the deviation from the data 1 2 ξ ⊺ ξ , ξ ( g ) = Tg − f . min g Selected Data Analysis Techniques, A. Alonso, copyleft 2010 17 / 20