Multiscale Analysis as a Method of Measurement

Multiscale analysis introduces a breakthrough in the measurement of signals. It quantifies not only the location of characterisic features within a given signal (see Figure 2.19 on page ), but, like a telescope with variable and calibrated zoom, it also quantifies their amplitudes in an optimally efficient way. Measuring rods capable of this dual capability are depicted schematically in Figure 2.24 on page .

Such an application of a multiscale analysis to any given signal always requires two steps:

1. specifying the scaling function, the standard of measurement and
2. measuring, and hence representing, the signal relative to the basis elements generated from that scaling function.

Let us consign the task of specifying the scaling function to the next subsection. Thus we assume that a choice of a scaling function has been made, and we endeavor to measure the given signal, say $f(t)$ . This means that we find the coefficients which represent the $k$ th approximation of $f$ , i.e. the least squares approximation of $f$ in the subspace $\mathbf{V}_k$ , Eq.(2.101). This array of coefficients, the array of inner products

$$\left[ P_{\mathbf{V}_k}f\right] = \{ \langle\sqrt{2^{-k}} \phi(2^{-k} u-\ell),f(u)\rangle :~\ell=0,\pm 1,\cdots \} ~,$$

is called the discrete approximation of $f$ at resolution $2^{-k}$ , and it constitutes the result of the measuring process. It consists of the inner products

$$\langle \phi(2^{-k} u-\ell),f(u)\rangle = \int\limits_{-\infty}^\infty \overline\phi \left(-2^{-k}(2^{k}\ell-u)\right) f(u) du$$

$$=\left[f(u)\ast \overline\phi (-2^{-k} u)\right](2^{k}\ell)~,$$

which is the convolution integral evaluated at the equally spaced points $2^k\ell$ . These values of the convolution integral are the output resulting from the signal $f(t)$ being passed through the filter $\overline\phi (-2^{-k} u)$ . This is because in the Fourier domain the convolution integral is the product of two Fourier transforms. Thus one finds that the discrete approximation consists of the set of sampled values of the given signal after it has passed through a filter which is expressed by the Fourier integral

$$\int\limits_{-\infty}^\infty \overline\phi (-2^{-k}u) \frac{e^{-i\omega u}}{\sqrt{2\pi}}du~.$$