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Transcript of Introducción a Wavelets (ondeletas) signals/phasorlecture2/indexphasorlect2.htm ANALISIS...
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Introducción a Wavelets (ondeletas)Introducción a Wavelets (ondeletas)
http://www.jhu.edu/~signals/phasorlecture2/indexphasorlect2.htm
ANALISIS MULTIRESOLUCION
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Transformadas espectrales:Transformadas espectrales:
F uN
f x ejux
N
x
N
( ) ( )
12
0
1
u = 0,1,2, ..., N-1
nucleo
xfuxKuF
)(),()(
Transformada de Fourier:
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xuNj
euxK2
),(
PARTE REAL PARTE IMAGINARIA
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Transformada Hartley:Transformada Hartley:
Nucleo o KernelNucleo o Kernel
)(22
cos1
)(1
0
xfxuN
senxuNN
uFN
x
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Transformada Discreta Coseno Transformada Discreta Coseno (DCT)(DCT)
)(2
)12(cos
1)(
1
0
xfN
ux
NuF
N
x
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nucleo
xfuxKuF
)(),()(
=
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nucleo
xfuxKuF
)(),()(
=
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Espectro de FourierEspectro de Fourier
f(x)
F(u)
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STFT (Short time Fourier transform)STFT (Short time Fourier transform)
Or windowed Fourier Or windowed Fourier transformtransform
dtertgtfwrS jwtf
)()(),(
g(t-r) : Window
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SpectrogramSpectrogram
The square modulus of the windowed Fourier The square modulus of the windowed Fourier transform is the transform is the spectrogramspectrogram of a signal: of a signal:
22
)()(),(),( dtertgtfwrSwrSpec jwtff
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Ventanas: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 espectrogramasEjemplos de espectrogramas
(Ver MATLAB)(Ver MATLAB)
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Ejemplos de espectrogramasEjemplos 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 WaveletsIntroducción 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*(t^2)/2)
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Wavelet de MorletWavelet de Morlet
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The mexican hat The mexican hat waveletwavelet
224
1 2
)1(3
2)(
x
exx
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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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EscalaEscala
fs
1
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waveletmothert)(
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Ejemplos de escalogramas (CWT)Ejemplos de escalogramas (CWT)Continuous Wavelet TransformContinuous 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:En un espectrograma:
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En un escalograma: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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AplicacionesAplicaciones
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AnAnálisis de señalesálisis de señales
OxOxímetro de Pulsoímetro de Pulso
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Representación frecuencia-tiempo para:
(a)Datos muestreados
(b) FT
(c) WFT
(d) CWT
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DWT (Discrete Wavelet DWT (Discrete Wavelet Transform)Transform)
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A friendly guide to wavelets:A friendly guide to wavelets:
http://perso.orange.fr/polyvalens/clemens/wavelets/wavelets.html#section7
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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 El análisis multiresolución se consigue a través de filtrado y consigue a través de filtrado y submuestreo de la señal original.submuestreo de la señal original.
La exploración en tiempo se La exploración en tiempo se consigue a través de operaciones consigue a través de operaciones de convolución (filtrado digital).de convolución (filtrado digital).
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)(nxW
)(nx
Sub-band coding
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Sub-band coding algorithmSub-band coding algorithm
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Transformada inversaTransformada inversa
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2-D Discrete Wavelet Transform2-D Discrete Wavelet Transform
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Wavelet PacketWavelet Packet
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En resumen:En resumen:
CWT
DWT
2D - DWT
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... En resumen:... En resumen:
CWT
DWT
2D - DWT
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http://www.gisdevelopment.net/technology/ic/techip0003a.htm
http://www.acm.org/crossroads/xrds6-3/sahaimgcoding.html#Fig6
COMPRESION DE LA DCT A WAVELETS
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EspectrogramaEspectrograma