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 .
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.