then its discrete wavelet transform consists of the signals 
and 
.
The signals are computed by the iteration
where G is a lowpass and H a highpass filter and 
denotes a convolution operation.
We say that a signal with index i belongs to octave i. Because
G is a lowpass filter the a-signals are called approximations. The
d-signals are called detail signals because H is a highpass filter. In this
terminology the approximation and details at octave i+1 are obtained by
applying a filter to the approximation at octave i and subsampling the
results with a factor of 2.
If 
is the mother wavelet and 
the scaling
function of the corresponding multiresolution analysis, we can interpret the discrete
wavelet transform in the following manner:
Let f be the square integrable function defined by
Then we have
For a more detailed discussion of the a- and d-signal in the context
of the wavelet transform consult the chapter on multiresolution
analysis in the book of Daubechies [6].
Since the input signal x has finite length,
each signal at octave i+1 has half the length of the approximation at
octave i. If we do not restrict the number of octaves it may happen
that the filters G and H become longer than the signals to which we
want to apply G and H. Therefore the maximum number of octaves is such
that the last approximation at octave p will have at least the same
length as G and H.
The result of "Decompose" is a signal of the DWT type.
At the top of the display window the detail at octave 1 is displayed and at the bottom you find the last approximation. Between these channels the detail signals are in the order of increasing octaves.
Warning: Note that the x-axis offset (shift) of the input signal is
not taken into account in the resulting DWT signal.