Fourier / Wavelet Analysis ASTR 3010 Lecture 19 Textbook : N/A.
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Transcript of Fourier / Wavelet Analysis ASTR 3010 Lecture 19 Textbook : N/A.
![Page 1: Fourier / Wavelet Analysis ASTR 3010 Lecture 19 Textbook : N/A.](https://reader035.fdocuments.us/reader035/viewer/2022062511/551c58ce550346a66a8b4fd6/html5/thumbnails/1.jpg)
Fourier / Wavelet Analysis
ASTR 3010
Lecture 19
Textbook : N/A
![Page 2: Fourier / Wavelet Analysis ASTR 3010 Lecture 19 Textbook : N/A.](https://reader035.fdocuments.us/reader035/viewer/2022062511/551c58ce550346a66a8b4fd6/html5/thumbnails/2.jpg)
Fourier Transform
in signal processing, (time and frequency)
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Add bunch of zeros in your data!
Number of input data points number of frequency sampling in FT!
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Example of FFT in astronomy : defringing a spectrum
heavily fringed raw spectrum
power spectrum of the input
defringed spectrum
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Limits on Fourier Transform
it can only “see” one variable (period or time) at a time at sufficient precision!
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Short-Time Fourier Transform
• Using a window function in time
• Limited by the Uncertainty Principle : t*ω = constant
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STFT resolution problem
• Four different Gaussian windows
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Wavelet Transform
• Wavelet transform can get two different information (i.e., time and frequency) simultaneously!
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Wavelet Transform
where basis function is
s : scale parameterτ : translation parameter
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Practical use of wavelet transformation
• Decomposition and recomposition of a signal
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PyWaveletshttp://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']
• pywto pywt.wavelisto pywt.waveleto pywt.wavedeco pywt.waverec
import pywtpywt.wavelist()
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PyWaveletshttp://www.pybytes.com/pywavelets
• pywto pywt.wavelisto pywt.waveleto pywt.wavedeco pywt.waverec
import pywtmyw=pywt.wavelet(‘db4’)phi,psi,wx = myw.wavefun()plot(wx,phi,’r’)plot(wx,psi,’b’)
Daubechies Wavelet : order 4
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PyWaveletshttp://www.pybytes.com/pywavelets
• pywto pywt.wavelisto pywt.waveleto pywt.wavedeco pywt.waverec
import pywtmyw=pywt.wavelet(‘sym20’)phi,psi,wx = myw.wavefun()plot(wx,phi,’r’)plot(wx,psi,’b’)
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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.
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PyWaveletsdecompositionreconstruction
• pywto pywt.wavelisto pywt.waveleto pywt.wavedeco pywt.waverec
import pywtmyw=pywt.wavelet(‘db4’)dec =
myw.wavedec(data,’db4’,’zpd’,5)
![Page 16: Fourier / Wavelet Analysis ASTR 3010 Lecture 19 Textbook : N/A.](https://reader035.fdocuments.us/reader035/viewer/2022062511/551c58ce550346a66a8b4fd6/html5/thumbnails/16.jpg)
PyWaveletsdecompositionreconstruction
• pywto pywt.wavelisto pywt.waveleto pywt.wavedeco pywt.waverec
import pywtmyw=pywt.wavelet(‘sym20’)dec =
myw.wavedec(data,’sym20’,’zpd’,5)
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pywt : Denoising
import pywt… set high order “difference” coeffs to zero.… among “diff” coeffs, clip small coeffs < 0.2*sigma… then, reconstructdec = myw.wavedec(data,’db4’,’zpd’,5)
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Wavelet: Denoisinghttp://www.toolsmiths.com/docs/CT199809.pdf
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Wavelet: Denoise in 2D
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Wavelet: Denoise in 2D
http://www.pixinsight.com/doc/legacy/LE/21_noise_reduction/example_1/04.html