The Continuos Wavelet Transform

The Continuous Wavelet Transform (CWT) was formalized in 1984 [7,15] and has since been used as a powerful tool to analyze hidden patterns in complex time series in both the time and frequency domains [21,9,10].

Consider two signals composed of sinusoids with frequency 1 Hz and 1.001 Hz, respectively. It may be difficult to distinguish between these two signals in the presence of background noise unless many cycles are observed, implying the need of many-seconds of observation. Now consider two signals with pure frequencies of 1000 Hz and 1001 Hz-again, a 0.1% difference. Here it should be possible to distinguish the two signals in an interval of much less than one second. In other words, good frequency resolution requires longer observation times as frequency decreases. Thus, it might be more convenient to construct a basis whose elements have larger temporal width at low frequencies.

The previous example motivates a multi-resolution time-frequency tiling of the form:

Figure 2: Time-Frequency map of wavelet transform. Note that $\Delta \omega \Delta t$, area of the boxes remain the same.

The Continuous Wavelet Transform (CWT) accomplishes the above multi-resolution tiling by time-scaling and time-shifting a prototype function $\Psi(t)$ , often called the analyzing function or wavelet. The a-scaled and b-shifted basis elements is given by

$$\Psi_{a,b}(t) = \frac{1}{a}\Psi\left(\frac{t-b}{a}\right)\,,$$ (1)

where $\Psi_{a,b}(t)$ is a copy of $\Psi(t)$ rescaled by $a$ and centered at $b$ and

$$\int_{-\infty}^{+\infty}\Psi(t)dt = 0\,\,.$$ (2)

The CWT $F(a,b)$ of a function $f(x)$ is defined by:

$$F(b,a)=\int f(x)\Psi_{a,b}(x)dx\,\,.$$ (3)

We applied the CWT to both the experimental data and the model generated time series, using an analyzing function called the Mexican Hat wavelet (see Fig.3),

$$\Psi(x)=(x^2-1)\exp(-x^2/2)\,\,.$$ (4)

Figure 3: (left) First derivative of a Gaussian function for 2 scales with respective FFT; (rigth) Second derivative of a Gaussian function, or Mexican Hat for 2 scales with respective FFT.