Introducci

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Introducción a Wavelets (ondeletas)
ANALISIS MULTIRESOLUCION
 
   
http//www.jhu.edu/signals/phasorlecture2/indexph
asorlect2.htm
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Transformadas espectrales
Transformada de Fourier
u 0,1,2, ..., N-1
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PARTE REAL
PARTE IMAGINARIA
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Transformada Hartley
Nucleo o Kernel
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Transformada Discreta Coseno (DCT)
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Espectro de Fourier
f(x) F(u)
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STFT (Short time Fourier transform)
Or windowed Fourier transform
g(t-r) Window
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Spectrogram

• The square modulus of the windowed Fourier
  transform is the spectrogram of a signal

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Ventanas
Ventana de Hamming
Ventana rectangular
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A segment of a vowel extracted with a
rectangular window
The amplitude spectrum using a rectangular
window Calculated using Matlab abs(fft(sig))
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  The amplitude spectrum using a hamming
window. Calculated using Matlab
abs(fft(hamming(512) . sig))
A segment of a vowel extracted with a hamming
window. Calculated using Matlab hamming(512) .
sig
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This is the basis for most computer generated
spectrograms (display pixel intensity on a log
scale by limiting the dynamic range to about
60-80 dB).
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Ejemplos de espectrogramas

• (Ver MATLAB)

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Ejemplos de espectrogramas
Here is the sum of two parallel linear chirps
with its spectrogram.
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Here is the sum of two hyperbolic chirps and its
spectrogram.
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• Introducción a Wavelets

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four frequency components at different times.
The interval 0 to 250 ms is a sinusoid of 300 Hz,
and the other 250 ms intervals are sinusoids of
200 Hz, 100 Hz, and 50 Hz
w(t)exp(-a(t2)/2)
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Wavelet de Morlet
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The mexican hat wavelet
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                                                Gráficos de varios tipos distintos de wavelets. (a) Wavelet de Haar, (b) Wavelet de Daubechies, (c) Wavelet de Morlet. (Cortesía de Ofer Levi, Universidad de Stanford)
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Escala
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Ejemplos de escalogramas (CWT)Continuous Wavelet
Transform
These signals are drawn from a database signals
that includes event related potentials of normal
people, and patients with Alzheimer's disease.
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En un espectrograma
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En un escalograma
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Suma de dos señales CHIRP hiperbólicas
Windowed fourier transform (Espectrograma)
Continuous Wavelet Transform CWT (Escalograma)
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Aplicaciones
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Análisis de señales

• Oxímetro de Pulso

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• Representación frecuencia-tiempo para
• Datos muestreados
• (b) FT
• (c) WFT
• (d) CWT

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DWT (Discrete Wavelet Transform)
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A friendly guide to wavelets

• http//perso.orange.fr/polyvalens/clemens/wavelets
  /wavelets.htmlsection7

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Ahora dejamos fija la Ondeleta y lo que vamos
comprimiendo por etapas es la señal
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• El análisis multiresolución se consigue a través
  de filtrado y submuestreo de la señal original.
• La exploración en tiempo se consigue a través de
  operaciones de convolución (filtrado digital).

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Sub-band coding
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Sub-band coding algorithm
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Transformada inversa
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2-D Discrete Wavelet Transform
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Wavelet Packet
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En resumen
CWT
DWT
2D - DWT
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... En resumen
CWT
DWT
2D - DWT
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• http//www.gisdevelopment.net/technology/ic/techip
  0003a.htm

COMPRESION DE LA DCT A WAVELETS
http//www.acm.org/crossroads/xrds6-3/sahaimgcodin
g.htmlFig6
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Espectrograma