A qualitative descriptive presentation of the decomposition of a signal
using wavelet systems or wavelet transforms consists of partitioning the
time–scale plane into *tiles* according to the indices

The energy in a signal is given in terms of the DWT by Parseval's relation
in (Reference) or (Reference). This shows the energy is a function of the
translation index

The wavelet transform allows analysis of a signal or parameterization of a signal that can locate energy in both the time and scale (or frequency) domain within the constraints of the uncertainty principle. The spectrogram used in speech analysis is an example of using the short-time Fourier transform to describe speech simultaneously in the time and frequency domains.

This graphical or visual description of the partitioning of energy in a
signal using tiling depends on the *structure* of the system, not the
*parameters* of the system. In other words, the tiling partitioning
will depend on whether one uses

**Nonstationary Signal Analysis**

In many applications, one studies the decomposition of a signal in
terms of basis functions. For example, stationary signals are decomposed
into the Fourier basis using the Fourier transform. For nonstationary
signals (i.e., signals whose frequency characteristics are time-varying
like music, speech, images, etc.) the Fourier basis is ill-suited because
of the poor time-localization. The classical solution to this problem is
to use the short-time (or windowed) Fourier transform (STFT). However, the
STFT has several problems, the most severe being the fixed time-frequency
resolution of the basis functions. Wavelet techniques give a new class of
(potentially signal dependent) bases that have desired time-frequency
resolution properties. The “optimal” decomposition depends on the signal
(or class of signals) studied. All classical time-frequency
decompositions like the Discrete STFT (DSTFT), however, are signal
independent. Each function in a basis can be considered *schematically* as a tile in the time-frequency plane, where most of its
energy is concentrated. Orthonormality of the basis functions can be
schematically captured by nonoverlapping tiles. With this assumption, the
time-frequency tiles for the standard basis (i.e., delta basis) and the
Fourier basis (i.e., sinusoidal basis) are shown in
(Reference).

**Tiling with the Discrete-Time Short-Time Fourier Transform**

The DSTFT basis functions are of the form

where *all* DSTFT
bases with the particular time-frequency resolution
characteristic).

**Tiling with the Discrete Two-Band Wavelet Transform**

The discrete wavelet transform (DWT) is another signal-independent tiling
of the time-frequency plane suited for signals where high frequency signal components
have shorter duration than low frequency signal components. Time-frequency
atoms for the DWT,

The tiling of the time-frequency plane is a powerful graphical method for understanding the properties of the DWT and for analyzing signals. For example, if the signal being analyzed were a single wavelet itself, of the form

the DWT would have only one nonzero coefficient,

**General Tiling**

Notice that for general, nonstationary signal analysis, one desires
methods for controlling the tiling of the time-frequency plane, not just
using the two special cases above (their importance notwithstanding). An
alternative way to obtain orthonormal wavelets

Remember that the tiles represent the relative size of the translations and scale change. They do not literally mean the partitioned energy is confined to the tiles. Representations with similar tilings can have very different characteristics.