Wavelet Toolbox 
Details and Approximations
The equations for continuous and discrete synthesis are of considerable interest and can be read in order to define the detail at level j:
The details have just been defined. Take a reference level called J. There are two sorts of details. Those associated with indices correspond to the scales which are the fine details. The others, which correspond to j > J, are the coarser details.
We group these latter details into
which defines what is called an approximation of the signal s. We have just created the details and an approximation. They are connected. The equality
signifies that s is the sum of its approximation A_{J} and of its fine details. From the previous formula, it is obvious that the approximations are related to one another by
For an orthogonal analysis, in which the _{j,k} is an orthonormal family,
The following table contains definitions of details and approximations.
Definition of the detail at level j 

The signal is the sum of its details 

The approximation at level J 

Link between A_{J}_{1} and A_{J} 
A_{J}_{1} = A_{J} + D_{J} 
Several decompositions 
From a graphical point of view, when analyzing a signal, it is always valuable to represent the different signals (s, A_{j}, D_{j}) and coefficients (C(j,k)).
Let us consider the following figure. On the left side, s is the signal; a5, a4, a3, a2, and a1 are the approximations at levels 5, 4, 3, 2, and 1. The best approximation is a1; the next one is a2, and so on. Noise oscillations are exhibited in a1, whereas a5 is smoother.
On the right side, cfs represents the coefficients (for more information, see Wavelet Transforms: Continuous and Discrete), s is the signal, and d5, d4, d3, d2, and d1 are the details at levels 5, 4, 3, 2, and 1.
The different signals that are presented exist in the same time grid. We can consider that the t index of detail D_{4}(t) identifies the same temporal instant as that of the approximation A_{5}(t) and that of the signal s(t). This identity is of considerable practical interest in understanding the composition of the signal, even if the wavelet sometimes introduces dephasing.
Figure 67: Approximations, Details, and Coefficients
Synthesis: An Inverse Transform  The Fast Wavelet Transform (FWT) Algorithm 
© 19942005 The MathWorks, Inc.