Wavelets

Abstract

In the previous chapter, we showed that a key issue with the short-time Fourier transform (STFT) is the choice of the window length, given the basic limitation imposed by the uncertainty principle of signal analysis. Short windows give good time (but bad frequency) resolution, and conversely, long windows give good frequency (but bad time) resolution (see Sect. 3.3). In the late 1970s, Jean Morlet, a geophysicist working for a French oil company, realized that the STFT was not suitable for the study of his seismic data. He observed that a good compromise between time and frequency resolution was not possible because high-frequency patterns had a shorter duration compared to the low-frequency ones. So, a single window for all frequencies would not do. His solution was quite straightforward: he just took different window sizes for different frequencies, or more accurately, he took a cosine function tapered with a Gaussian (a Gabor function, see Sect. 3.2) and compressed it or stretched it in time to get the different frequencies (see Fig. 4.1). Then, instead of always having the same window size, he had the same wave shape at different scales, that is, with a variable size. With this simple trick, he created the basis of wavelets!

Appendices

Appendices

4.1.1 Continuous and Dyadic Wavelet Transforms

Having provided an intuitive introduction to wavelets and their main advantages over other decompositions, let us now formalize these ideas. The wavelet transform gives a time-frequency (or more accurately time-scale) representation that is defined as the correlation between the signal x(t) and the wavelet functions ${y}_{a,b}(t)$.

$$ {W}_{y}X\left(a,b\right)=\langle x(t),{y}_{ab}\rangle \rm\rm\equiv \rm\rm{\left|a\right|}^{-1/2}{\displaystyle {\int }_{-\infty }^{+\infty }x(t)\rm{{0.05em}}}{y}^{*}\left(\frac{t-b}{a}\right)\rm{{0.05em}}dt $$

(4.1)

where ^* denotes complex conjugation and ${y}_{a,b}(t)$are scaled and shifted versions of a unique mother wavelet $y(t)$(see Fig. 4.7):

Fig. 4.7

A wavelet function ${y}_{a,b}(t)$at different scales a and times b

$$ {y}_{a,b}(t)={\left|a\right|}^{-\frac{1}{2}}y\left(\frac{t-b}{a}\right)$$

(4.2)

where $a,b\rm\in$ ℜ are the scale and translation parameters, respectively. The wavelet transform gives a decomposition that is maximum at those scales and times where the wavelet best matches the signal x(t). Moreover, Eq. 4.1 can be inverted, thus giving the reconstruction of x(t) from the wavelet coefficients (Grossmann and Morlet 1984).

The wavelet transform maps a signal of one independent variable t onto a function of two independent variables a, b. This representation is overly redundant, and without losing any information, it is sometimes more practical to define the wavelet transform only at discrete scales a and discrete times b by choosing the dyadic set of parameters $\left\{{a}_{j}={2}^{j},\text{b}_{j,k}={2}^{j}k\text{{1em}}j,k\in \mathbb{Z}\right\}$, as shown in Fig. 4.8. This dyadic sampling gives a nonredundant transform that has many samples for the high frequencies – where we actually want to have high time resolution-and less and more spaced samples for the lower frequencies – where high time resolution is not that crucial given that precise times are not well defined for low frequencies.

Fig. 4.8

Lattice showing the points at which the dyadic wavelet transform is calculated

Figure 4.9 shows the continuous and the dyadic wavelet transform of an average evoked potential. In the average evoked potential, we observe two main components: the P100-N200 (a positive peak at about 100 ms followed by a negative peak at about 200 ms) and the P300 response (see Sect. 1.5 for details). Note that both the continuous and dyadic transforms give essentially the same information: an increased activity reflecting the P100-N200 complex in the low (i.e., high frequency) scales and an increased activity correlated with the P300 response in the higher (low frequency) scales. The continuous plot may look smoother but the main advantage of the dyadic transform is computational speed, especially considering that this transform can be implemented with a very fast algorithm, as we will see in the next section.

Fig. 4.9

The continuous and dyadic wavelet transform of the average evoked potential shown in Fig. 4.4

4.1.2 Multiresolution Decomposition

The correlation of the signal x with contracted versions of the dyadic wavelets of Eq. 4.2 gives the high-frequency components, and the correlation with the dilated versions gives the low-frequency ones. These correlations can be arranged in a recursive algorithm called multiresolution decomposition (Mallat 1989). The multiresolution decomposition separates the signal into details (high-frequency components) and approximations (coarser representations of the signal) at different scales. Each detail (D _j ) and approximation (A _j ) at a given scale j is obtained from the previous approximation (A ₁ ) (see Fig. 4.10). This pyramidal scheme makes the multiresolution decomposition very fast, even faster than the fast Fourier transform: the time required for the computation of the multiresolution decomposition is of the order of N (with N the number of data points), whereas for the fast Fourier transform is N * log N (Mallat 1989).

Fig. 4.10

Implementation of the multiresolution decomposition algorithm

Let us now see the steps for the decomposition and reconstruction of the signal following the scheme of Fig. 4.10. First, the signal x is high-pass and low-pass filtered using the filters G and H, respectively. Both sets of coefficients obtained after filtering are decimated by two (one every two data points is deleted), thus giving the first level detail D ₁ , containing the activity in the upper half of the frequency spectrum (i.e., from f_s/4 to f_s/2),³ and the first approximation A ₁ , containing the lower half (from 0 to f_s/4). After decimation, the number of data points of D ₁ plus the ones of A ₁ is equal to the number of data points of x, thus obtaining a nonredundant representation. Then, the approximation is further decomposed, and the whole procedure is repeated j times, where j is the number of chosen levels. As a result we obtain the signal x decomposed into D ₁ to D _j details and one final approximation A _j . Note that the rescaling of the mother function is given by the decimation of the coefficients.

From this set of coefficients (the details and approximations), the reconstruction of the signal x is done in a similar way using the inverse filters G’ and H’ and upsampling the data (i.e., inserting zeros between samples), as shown in the right side of Fig. 4.10.