Introducción a Wavelets (ondeletas) ANALISIS MULTIRESOLUCION     http://www.jhu.edu/~signals/phasorlecture2/indexphasorlect2.htm

Transformadas espectrales: Transformada de Fourier: u = 0,1,2, ..., N-1

PARTE REAL PARTE IMAGINARIA

Transformada Hartley: Nucleo o Kernel

Transformada Discreta Coseno (DCT)

=

=

Espectro de Fourier f(x) F(u)

STFT (Short time Fourier transform) Or windowed Fourier transform g(t-r) : Window

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

Ventanas: Ventana de Hamming: Ventana rectangular:

A segment of a vowel extracted with a rectangular window The amplitude spectrum using a rectangular window Calculated using Matlab: abs(fft(sig))

      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

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).

Ejemplos de espectrogramas (Ver MATLAB)

Ejemplos de espectrogramas Here is the sum of two parallel linear chirps with its spectrogram.

Here is the sum of two hyperbolic chirps and its spectrogram.

Introducción a Wavelets

four frequency components at different times 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)

Wavelet de Morlet

The mexican hat wavelet

                                                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)

Escala

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.

En un espectrograma:

En un escalograma:

Suma de dos señales CHIRP hiperbólicas Windowed fourier transform (Espectrograma) Continuous Wavelet Transform CWT (Escalograma)

Aplicaciones

Análisis de señales Oxímetro de Pulso

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

DWT (Discrete Wavelet Transform)

A friendly guide to wavelets: http://perso.orange.fr/polyvalens/clemens/wavelets/wavelets.html#section7

Ahora dejamos fija la Ondeleta y lo que vamos comprimiendo por etapas es la señal

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).

Sub-band coding

Sub-band coding algorithm

Transformada inversa

2-D Discrete Wavelet Transform

Wavelet Packet

En resumen: CWT DWT 2D - DWT

... En resumen: CWT DWT 2D - DWT

http://www.gisdevelopment.net/technology/ic/techip0003a.htm COMPRESION DE LA DCT A WAVELETS http://www.acm.org/crossroads/xrds6-3/sahaimgcoding.html#Fig6

Espectrograma