Windowed Fourier Transform

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Presentation transcript:

Windowed Fourier Transform Complex demodulation Appropriate for burst data Questionable for continuous data (may miss important frequencies)

Complex demodulation

Wavelet Analysis Seasonal SST averaged over Central Pacific (Torrence and Compo, 1998) “… to analyze time series that contain nonstationary power at different frequencies (Daubechies 1990)” (Derivative of a Gaussian)

WAVELET TRANSFORM  t is sampling interval s is scale (frequency) Continuous Wavelet Transform: Convolution of time series xn [N is the # of observations] with a scaled and translated version of a wavelet 0() [ is non-dimensional time] n’ s The wavelet 0() must have zero mean and be localized in time and frequency and have a complex conjugate  *

plane wave modulated by a Gaussian (in time domain) plane wave modulated by a Gaussian Considerably faster to do the wavelet transform calculation in frequency (or Fourier space) H() is the Heaviside step function: H() = 1 @  > 0 H() = 0, otherwise  - scaling factor Real Imaginary Fourier transform of Complex Morlet

WAVELET FUNCTIONS Real Imaginary Fourier transform m = 6

(derivative of Gauss order 2) m = 2

m = 2

WAVELET FUNCTIONS Fourier transform m = 2

WAVELET FUNCTIONS Fourier transform m = 4

WAVELET FUNCTIONS Fourier transform m = 6

k is the frequency index WAVELET TRANSFORM Torrence and Compo (1998) Convolution of time series xn’ with a scaled and translated version of a base function: a wavelet 0 () – continuous function in time and frequency – “mother wavelet” Convolution needs to be effected N (# of points in time series) times for each scale s; n is a translational value Much faster to do the calculation in Fourier space Convolution theorem allows N convolutions to be done simultaneously with the Discrete Fourier Transform: k is the frequency index Convolution theorem: Fourier transform of convolution is the same as the pointwise product of Fourier transforms

Inverse Fourier transform of is Wn (s) WAVELET TRANSFORM Fourier transform of Inverse Fourier transform of is Wn (s) where the angular frequency: With this relationship and a FFT routine, can calculate the continuous wavelet transform (for each s) at all n simultaneously

EXAMPLE OF WAVELET APPLICATION

(Torrence and Compo, 1998)