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Fourier / Wavelet Analysis

Fourier / Wavelet Analysis. ASTR 3010 Lecture 19 Textbook : N/A. Fourier Transform. in signal processing, (time and frequency). Add bunch of zeros in your data!. Number of input data points  number of frequency sampling in FT!. Example of FFT in astronomy : defringing a spectrum.

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Fourier / Wavelet Analysis

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  1. Fourier / Wavelet Analysis ASTR 3010 Lecture 19 Textbook : N/A

  2. Fourier Transform in signal processing, (time and frequency)

  3. Add bunch of zeros in your data! Number of input data points  number of frequency sampling in FT!

  4. Example of FFT in astronomy : defringing a spectrum heavily fringed raw spectrum power spectrum of the input defringed spectrum

  5. Limits on Fourier Transform it can only “see” one variable (period or time) at a time at sufficient precision!

  6. Short-Time Fourier Transform • Using a window function in time • Limited by the Uncertainty Principle : t*ω = constant

  7. STFT resolution problem • Four different Gaussian windows

  8. Wavelet Transform • Wavelet transform can get two different information (i.e., time and frequency) simultaneously!

  9. Wavelet Transform where basis function is s : scale parameter τ : translation parameter

  10. Practical use of wavelet transformation • Decomposition and recomposition of a signal

  11. PyWavelets http://www.pybytes.com/pywavelets ['bior1.1', 'bior1.3', 'bior1.5', 'bior2.2', 'bior2.4', … 'coif1', 'coif2', … 'db1', 'db2', 'db3', … 'sym15', 'sym16', 'sym17', 'sym18', 'sym19', 'sym20'] • pywt • pywt.wavelist • pywt.wavelet • pywt.wavedec • pywt.waverec import pywt pywt.wavelist()

  12. PyWavelets http://www.pybytes.com/pywavelets • pywt • pywt.wavelist • pywt.wavelet • pywt.wavedec • pywt.waverec import pywt myw=pywt.wavelet(‘db4’) phi,psi,wx = myw.wavefun() plot(wx,phi,’r’) plot(wx,psi,’b’) Daubechies Wavelet : order 4

  13. PyWavelets http://www.pybytes.com/pywavelets • pywt • pywt.wavelist • pywt.wavelet • pywt.wavedec • pywt.waverec import pywt myw=pywt.wavelet(‘sym20’) phi,psi,wx = myw.wavefun() plot(wx,phi,’r’) plot(wx,psi,’b’)

  14. Wavelets Decomposition Tree • decomposition of a signal into several resolution levels. • First, the original signal is decomposed by two complementary half-band filters (high-pass and low-pass filters) that divide a spectrum into high-frequency (detail coefficients; D1) and low-frequency (approximation coefficients; A1) components (bands). For example, the low-pass filter will remove all half-band highest frequencies. Information from only the low frequency band (A1), with a half number of points, will be filtered in the second decomposition level. The A2 outcome will be filtered again for further decomposition.

  15. PyWaveletsdecompositionreconstruction • pywt • pywt.wavelist • pywt.wavelet • pywt.wavedec • pywt.waverec import pywt myw=pywt.wavelet(‘db4’) dec = myw.wavedec(data,’db4’,’zpd’,5)

  16. PyWaveletsdecompositionreconstruction • pywt • pywt.wavelist • pywt.wavelet • pywt.wavedec • pywt.waverec import pywt myw=pywt.wavelet(‘sym20’) dec = myw.wavedec(data,’sym20’,’zpd’,5)

  17. pywt : Denoising import pywt … set high order “difference” coeffs to zero. … among “diff” coeffs, clip small coeffs < 0.2*sigma … then, reconstruct dec = myw.wavedec(data,’db4’,’zpd’,5)

  18. Wavelet: Denoising http://www.toolsmiths.com/docs/CT199809.pdf

  19. Wavelet: Denoise in 2D

  20. Wavelet: Denoise in 2D http://www.pixinsight.com/doc/legacy/LE/21_noise_reduction/example_1/04.html

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