# Fourier / Wavelet Analysis

## Presentation on theme: "Fourier / Wavelet Analysis"— Presentation transcript:

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

Fourier Transform in signal processing, (time and frequency)

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 Transform
it 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 parameter
τ : translation parameter

Practical use of wavelet transformation
Decomposition and recomposition of a signal

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()

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

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’)

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.

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)

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)

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)

Wavelet: Denoising

Wavelet: Denoise in 2D

Wavelet: Denoise in 2D