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

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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 Using a window function in time Limited by the Uncertainty Principle : t*ω = constant Limited by the Uncertainty Principle : t*ω = constant

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STFT resolution problem Four different Gaussian windows Four different Gaussian windows

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Wavelet Transform Wavelet transform can get two different information (i.e., time and frequency) simultaneously! 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 parameter τ : translation parameter

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Practical use of wavelet transformation Decomposition and recomposition of a signal Decomposition and recomposition of a signal

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PyWavelets ['bior1.1', 'bior1.3', 'bior1.3', 'bior1.5', 'bior1.5', 'bior2.2', 'bior2.2', 'bior2.4', 'bior2.4',… 'coif1', 'coif1', 'coif2', 'coif2',… 'db1', 'db1', 'db2', 'db2', 'db3', 'db3',… 'sym15', 'sym15', 'sym16', 'sym16', 'sym17', 'sym17', 'sym18', 'sym18', 'sym19', 'sym19', 'sym20'] 'sym20'] pywt pywt o pywt.wavelist o pywt.wavelet o pywt.wavedec o pywt.waverec import pywt pywt.wavelist()

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PyWaveletshttp://www.pybytes.com/pywavelets pywt pywt o pywt.wavelist o pywt.wavelet o pywt.wavedec o 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

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PyWaveletshttp://www.pybytes.com/pywavelets pywt pywt o pywt.wavelist o pywt.wavelet o pywt.wavedec o pywt.waverec import pywt myw=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. 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; D 1 ) and low- frequency (approximation coefficients; A 1 ) components (bands). For example, the low- pass filter will remove all half- band highest frequencies. Information from only the low frequency band (A 1 ), with a half number of points, will be filtered in the second decomposition level. The A 2 outcome will be filtered again for further decomposition. 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; D 1 ) and low- frequency (approximation coefficients; A 1 ) components (bands). For example, the low- pass filter will remove all half- band highest frequencies. Information from only the low frequency band (A 1 ), with a half number of points, will be filtered in the second decomposition level. The A 2 outcome will be filtered again for further decomposition.

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PyWavelets decomposition reconstruction pywt pywt o pywt.wavelist o pywt.wavelet o pywt.wavedec o pywt.waverec import pywt myw=pywt.wavelet(‘db4’) dec = myw.wavedec(data,’db4’,’zpd’,5)

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PyWavelets decomposition reconstruction pywt pywt o pywt.wavelist o pywt.wavelet o pywt.wavedec o pywt.waverec import pywt myw=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, reconstruct dec = myw.wavedec(data,’db4’,’zpd’,5)

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Wavelet: Denoising

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Wavelet: Denoise in 2D

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