1 / 20

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.

astro
Télécharger la présentation

Fourier / Wavelet Analysis

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related