Page 280 - Introduction to Information Optics

5.5. Physical Wavelet Transform 265

where £ —- (4 t') is a 4D translation of the wavelet in the 4D space time, s is
the scalar scaling factor with a dimension of [M], 8(so)/c) is the step function
defined as

if fl <0,

and exp( — ipc,) is the linear phase related to the translation £ of the wavelet.
The electromagnetic wavelet defined in Eq. (5.13) is localized in the temporal
frequency axis by an one-sided window

There is no window in the spatial-frequencies domain in the definition of the
electromagnetic wavelet. However, the spatial and temporal frequencies are
related by the light cone constraint a)/c = \p\.
Expression of the electromagnetic wavelet in the space-time domain is
obtained with the inverse Fourier transform of Eq. (5.13).

01
" ' exp[ip-(r - £)], (5.14)

+
where C is the light cone with the positive frequencies with CD > 0 and
3
3
sw/c > 0, and d p/(l6n o)/c) is the Lorentz-invariant measure. To compute the
integral in Eq. (5.14), one needs to first put the translation £ = 0, that results
in the reference wavelet, or mother wavelet, h 0>s(r), which is not translated but
only scaled by the scale factor s. Then, put r = 0 to consider the wavelet h 0 S(Q)
and compute the integral on the right-hand side of Eq. (5.14), which is an
integral over a light cone in the 3D spatial-frequency space. For computing this
integral an invariance under the Lorentz transform should be used. Finally,
/i 0, s(r) can be obtained from fc 0iS(0) as [16]

and the h^ s(r) is obtained from the reference wavelet h 0s(r) by introducing
translations in space-time as

According to Eq. (5.15) the electromagnetic wavelet is localized in space
time. At the initial time f = 0 the reference wavelet with no translation = 0