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

2. Complex demodulation

3. 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)

4. 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  *

5. 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() =  > 0
H() = 0, otherwise
 - scaling factor
Real
Imaginary
Fourier transform of Complex Morlet

6. WAVELET FUNCTIONS
Real
Imaginary
Fourier transform
m = 6

7. (derivative of Gauss order 2)
m = 2

8. m = 2

9. WAVELET FUNCTIONS
Fourier transform
m = 2

10. WAVELET FUNCTIONS
Fourier transform
m = 4

11. WAVELET FUNCTIONS
Fourier transform
m = 6

12. 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

13. 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

14. EXAMPLE OF WAVELET APPLICATION

15. (Torrence and Compo, 1998)