Construction and Properties

There are equally spaced large half width wavelets of low mean frequency. They enter into the representation of the low resolution, slowly varying features of the signal. There are also equally spaced small half width wavelets of corresponding higher spread in frequency. They enter into the representation of the high resolution, abruptly changing features of the signal.

Instead of the equally spaced frequency windows of the wave packets, the wavelets are synthesized over frequency windows whose width increases exponentially.

Figure 2.16: Unequal frequency windows of the set of o.n. wavelets. Each window is an octave.

Wave packets have variable frequency window $\varepsilon =2^{-k} \varepsilon_0$ . Inserting this into Eq. (2.59), we obtain the Fourier transform of a wavelet as the following windowed phase factor:

$$F_{2^{-k}\ell} (\omega ) = \left\{\begin{array}{cc} \displaystyle... ...silon_0, 2^{1-k}\varepsilon_0]\\ 0 &\textrm{otherwise} \end{array}\right. ~~.$$

Here $\varepsilon =2^{-k} \varepsilon_0$ is the variable frequency window. Such a wavelet is a wave packet $P_{j\ell}^\varepsilon(t)$ (see Section 2.4.1) for which $j=1$ and $\varepsilon =2^{-k} \varepsilon_0$ . The integer $k=0,\pm 1,\pm 2,\dots$ is the octave number of the wavelet. Let us designate this wavelet by $W^+_{k\ell}$ . Its key properties are as follows:

1. Its explicit form is

   $$W^+_{k\ell} (t) =\sqrt\frac{1}{2^{-k}\varepsilon_0 }\int\limits_{2^{-k} \varepsil... ...\ell\omega /2^{-k}\varepsilon_0} \frac{e^{i\omega t}}{\sqrt{2\pi}} d\omega\, ~,$$ (265)
   and its Fourier transform is

   $$\hat W^+_{k\ell}(\omega) =F_{2^{-k}\ell} (\omega )$$ (266)

   
   

2. Its mean frequency is
   $$\overline{\omega}=\frac{1}{2} (2^{-k}+2^{1-k})\varepsilon_0~.$$

3. Its mean position along the time axis is
   $$\overline{t}=2\pi\ell / 2^{-k}\varepsilon_0~.$$

4. The wavelet has half width

   $$\Delta t =2\pi / 2^{-k}\varepsilon_0~,$$ (267)
   its frequency spread is

   $$\Delta\omega = 2^{-k}\varepsilon_0~,$$ (268)
   and its phase space area is

   $$\Delta\omega\Delta t =2\pi~,$$ (269)

   
   
   like that of any o.n. wave packet.
5. The wavelets, as well as their Fourier transforms, are orthonormal:

   $$\int^\infty_{-\infty}\overline{W^+}_{k\ell} (t)W^+_{k'\ell '}(t)dt = \delta_{\ell\ell '}\delta_{kk'}\,,$$

   $$\int^\infty_{-\infty}\overline{\hat W^+}_{k\ell} (\omega)\hat W^+_{k'\ell '}(\omega)d\omega = \delta_{\ell\ell '}\delta_{kk'}\,~.$$

   
   

6. They form a complete set in the given domain,

   $$\sum^\infty_{k=-\infty}\sum^\infty_{\ell =-\infty} W^+_{k\ell}(t) \overline{W^+}_{k\ell}(t')+W^-_{k\ell}(t)\overline{W^-}_{k\ell}(t') = \delta (t-t')\,,$$
   as well as in the Fourier domain

   $$\sum^\infty_{k=-\infty}\sum^\infty_{\ell =-\infty} \hat W^+_{k\el... ..._{k\ell}(\omega')+ \hat W^-_{k\ell}(\omega)\overline{\hat W^-}_{k\ell}(\omega') = \delta (\omega-\omega') ~.$$

   
   

Note that the negative frequency wavelets,

$$W^-_{k\ell} (t)=\sqrt\frac{1}{2^{-k}\varepsilon_0 }\int^{-2^{k} \... ...\ell\omega /2^{-k}\varepsilon_0} \frac{e^{i\omega t}}{\sqrt{2\pi}} d\omega\, ~,$$ (270)

must be included in order to form a complete set. These completeness relations imply that these wavelets as well as their Fourier transforms form bases for the vector space of square integrable functions $L^2(-\infty ,\infty )$ .

Figure: Partitioning of phase space by o.n. wavelets into cells of equal area ( $\triangle t~\triangle \omega =2\pi$ ), but unequal shape ( $\Delta t=2\pi/2^{-k} \varepsilon _0,~\Delta \omega =2^{-k} \varepsilon _0$ ). For a given mean frequency the successive wavelets have equal width.

These six wavelet properties are summarized geometrically in terms of their phase space representatives. The set of o.n. wavelets induces a partitioning of phase space into cells of equal area

$$\Delta\omega\Delta t =2\pi / 2^{-k}\varepsilon_0~ 2^{-k}\varepsilon_0=2\pi~,$$

but unequal shape, Eq.(2.67)-(2.68). The orthogonality in the time and in the frequency domains implies that the areas of these cells should be pictured as nonoverlapping. The completeness relations imply that these cells cover the whole phase space without any gaps between them. In brief, the phase space is partitioned by the wavelets into mutally exclusive and jointly exhaustive cells of equal area, but different shapes as in Figure 2.17. This is different from Figure 2.13, which depicts the partitioning by the o.n. wave packets into cells. They also are mutually exclusive and jointly exhaustive and have equal area. But they have identical shape.

The variable $\varepsilon_0$ is a positive parameter which effects all wavelets at once. It therefore controls the way they partition phase space. What happens when one increases $\varepsilon_0$ ? Reference to propert 4. indicates that such an increase produces a global distortion which dilates all phase space cells along the vertical (frequency) direction while compressing them along the horizontal (time) direction.

The distortion corresponds to that suffered by an incompressible fluid.

Once the parameter has doubled in value, the new partitioning is congruent to the old one. However, the integer octave label $k$ gets shifted by one unit in the process: $k\rightarrow k+1$ . More precicely, reference to Eq.(2.65) shows that one has

$$\left.W^\pm_{k\ell}(t)\right\vert _{\varepsilon_0=\varepsilon_1} = \left.W^\pm_{k+1\,\ell}(t)\right\vert _{\varepsilon_0=2\varepsilon_1}~.$$ (271)

The set of o.n. wavelets is characterized by a seventh fundamental property:

7.

All wavelets are derivable from a single standard wave packet. By translating and compressing this standard ``mother wavelet'', as it is known informally, one recovers any one of the o.n. wavelets. This recovery holds separately for the positive and negative frequency wavelets and follows directly from their defining equations, Eq.(2.65) and (2.70): By changing the Fourier integration variable from $\omega$ to $\Omega =\frac{\omega}{2^{-k}}$ , one obtains, for the typical wavelet, the alternate integral expression

$$W^\pm_{k\ell}(t) = \sqrt{\frac{1}{ 2^{-k}\varepsilon_0}} (\pm) \int^{\pm 2^{1-k}\var... ...pi i\ell\omega / 2^{-k}\varepsilon_0} \frac{e^{i\omega t}}{\sqrt{2\pi}} d\omega$$

$$= \sqrt{\frac{2^{-k}}{\varepsilon_0}} (\pm)\int^{\pm2\varepsilon_0}... ..._0} e^{i\Omega (2^{-k} t-2\pi \ell /\varepsilon_0 )}\frac{d\Omega}{\sqrt{2\pi}}$$

$$\equiv \sqrt{2^{-k}}\psi^\pm(2^{-k} t- \frac{2\pi}{\varepsilon_0}\ell )~~ , \quad k,\ell=0,\pm 1,\pm 2,\cdots~~~~.$$ (272)

The simplicity of this expression is striking. To obtain it, all one needs to do is apply a translation, compression and amplification to a single universal wave packet. Start with the mother wavelet (standard wave packet function),

$$W^\pm_{00}(t)\equiv\psi^\pm (t) = \frac{1}{\sqrt{2\pi\varepsilon_0}} (\pm)\int^{\pm2\varepsilon_0}_{\pm\varepsilon_0} e^{i\Omega t}d\Omega$$

$$= \sqrt{\frac{\varepsilon_0}{2\pi}} e^{\pm3i t\varepsilon_0/2}~ \frac{\sin t\varepsilon_0/2}{t\varepsilon_0/2} \quad ,$$

and translate it along the $t$ -axis by an amount $\frac{2\pi}{\varepsilon_0} \ell$ to obtain

$$\psi^\pm(t-\frac{2\pi}{\varepsilon_0}\ell)~.$$

Next compress it uniformly along the $t$ -axis by the compression factor $2^{-k}$ to obtain the compressed wave packet

$$\psi^\pm(2^{-k} t-\frac{2\pi}{\varepsilon_0}\ell )~.$$

To preserve normalization amplify its amplitude by $\sqrt{2^{-k}}$ to obtain

$$\sqrt{2^{-k}}\psi^\pm(2^{-k} t-\frac{2\pi}{\varepsilon_0}\ell )~~.$$

This three step process is sufficient to yields the generic wavelet, Eq.(2.72).

Note that the resulting set of orthonormal wavelets decomposes into different classes. Those wavelets belonging to the same class ($k$ fixed) have the same mean frequency and the same temporal width, but are time translated relative to each other ( $\ell =0,\pm 1,\pm 2, \cdots$ ). By contrast, different classes are distinguished by different mean frequencies and hence different widths.