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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 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 parameter**

τ : translation parameter

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**Practical use of wavelet transformation**

Decomposition and recomposition of a signal

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**PyWavelets pywt pywt.wavelist pywt.wavelet pywt.wavedec pywt.waverec**

['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()

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**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

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**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’)

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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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**PyWavelets decomposition reconstruction**

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

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**PyWavelets decomposition reconstruction**

pywt pywt.wavelist pywt.wavelet pywt.wavedec 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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Wavelet: Denoise in 2D

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