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

heavily fringed raw spectrum

power spectrum of the input

defringed spectrum

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

Short-Time Fourier Transform• Using a window function in time

• Limited by the Uncertainty Principle : t*ω = constant

STFT resolution problem

• Four different Gaussian windows

Wavelet Transform • Wavelet transform can get two different information (i.e., time and

frequency) simultaneously!

Wavelet Transform

where basis function is

s : scale parameterτ : translation parameter

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

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()

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

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’)

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.

PyWaveletsdecompositionreconstruction

• pywto pywt.wavelisto pywt.waveleto pywt.wavedeco pywt.waverec

import pywtmyw=pywt.wavelet(‘db4’)dec =

myw.wavedec(data,’db4’,’zpd’,5)

PyWaveletsdecompositionreconstruction

• pywto pywt.wavelisto pywt.waveleto pywt.wavedeco pywt.waverec

import pywtmyw=pywt.wavelet(‘sym20’)dec =

myw.wavedec(data,’sym20’,’zpd’,5)

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)

Wavelet: Denoisinghttp://www.toolsmiths.com/docs/CT199809.pdf

Wavelet: Denoise in 2D

Wavelet: Denoise in 2D

http://www.pixinsight.com/doc/legacy/LE/21_noise_reduction/example_1/04.html