2 Wavelet transform

The wavelet transform has become a powerful tool for frequency analysis, in particular for non-stationary time series. Discussions of the wavelet transform and its applications can be found in a number of recent books and review articles [Daubechies1992,Chui et al.1994,Farge1992]. The wavelet transform of a function y(t) is defined as (here ^* denotes complex conjugate):

$$w(a,b) = a^{-\frac{1}{2}} \int_{-\infty}^{+\infty}y\left(t\right) g^*\left(\left(t-b\right)/a \right) {\rm d}t$$ (2)

where variable a is the scale dilation parameter and b the translation parameter. Both parameters are dimensionless. The real- or complex-valued function g(t) is called a mother (or analyzing) wavelet. Here a particular wavelet transform, the Morlet wavelet, will be used. The Morlet wavelet, being a locally periodic wave-train, is related to windowed Fourier analysis. It is obtained by taking a complex sine wave, and by localizing it with a Gaussian (bell-shaped) envelope. The Morlet wavelet is defined as:

$$g(t) = \exp\left(i\omega_{\mathrm{0}}t - \frac{t^\mathrm{2}}{2}\right)$$ (3)

and its Fourier transform:

$$G(\omega) = \sqrt{2\pi} \exp\left[-\frac{\left(\omega - \omega_{\mathrm{0}}\right)^2}{2}\right]\cdot$$ (4)

The Morlet wavelet gives the smallest time-bandwidth product [Lagoutte et al.1992]. $\omega_{\rm {0}}$ is a phase constant (in the present study $\omega_{\rm {0}}$ = 5). For large $\omega_{\rm {0}}$ the frequency resolution improves, though at the expense of decreased time resolution. The dilation parameter may be considered as equivalent to the frequency of the analyzed signal, while the translation parameter corresponds to the time elapsed along the analyzed sample. In practice, for analyzing a discrete-time signal y(t_i) we sample the continuous wavelet transform on a grid in the time-scale plane (b,a) by choosing a=j and b=k where j and k are integers. That is we compute wavelet coefficients

$$w_{j,k} = j^{-1/2} \int_{-\infty}^{\infty} \!\!\! y(t) g^*\left((t-k)/j\right) \, {\rm d}t$$ (5)

where $1 \leq j \leq J$ and $1 \leq k \leq N$. The integral in (5) is approximated using the discrete-time signal y(t_i).

Since the wavelet transform is an over-complete representation of the original signal (a one-dimensional signal is transformed to the two-dimensional plane) there are many possibilities for reconstructing the signal. One way is to use a discrete version of Morlet's formula

$$y(b) = c \int_{-\infty}^{\infty} \!\!\! w(a,b) \frac{{\rm d}a}{a^{3/2}}.$$ (6)

Note that the original signal's low-pass (DC) component is lost in the transform.

In the present study dilation number 1 corresponds to the highest frequency (a half of sampling rate). The highest dilation number corresponds to the lowest observable frequency.