Page 277 - Introduction to Information Optics

262 5. Transformation with Optics

The time-frequency joint representation has an intrinsic limitation; the
product of the resolutions in time and frequency is limited by the uncertainty
principle:

AfAto ^ 1/2.

This is also referred to as the Heisenberg inequality. A signal cannot be
represented as a point of infinitely small size in time-frequency space. The
position of a signal in time-frequency space can be determined only within a
rectangle of At Act).

5.4.3. PROPERTIES OF WAVELETS

The wavelet transform is of particular interest for analysis of nonstationary
and fast-transient signals, because of its property of localization in both time
and frequency domains. In the definition of the wavelet transform in Eq. (5.9),
the kernel wavelet functions are not specified. This is the difference between the
wavelet transform and many other mathematical transforms, such as the
Fourier transform. Therefore, when talking about the wavelet transform one
must specify what wavelet is used in the transform.
In fact, any square integrable function can be a wavelet, if it satisfies the
admissibility and regularity conditions. The admissible condition is obtained
as

(5.11,

where H(co) is the Fourier transform of the mother wavelet h(f). If the
condition in Eq. (5.11) is satisfied, the original signal can be completely
recovered by the inverse wavelet transform. No information is lost during the
wavelet transform. The admissible condition implies that the Fourier transform
of a wavelet must be zero at the zero frequency

and, equivalently, in the time domain the wavelet must be oscillatory, like a
wave, to have a zero mean:

k(t)dt = Q. (5.1.2)