The tree as a filter-generating algorithm

We have seen that the output at any stage is produced by a sequence of identical filtering and decimating operations. An alternative algorithm is to calculate the single long filter and the single multiple decimation that would produce exactly the same result. Here we are not operating on data but on the wavelet filters themselves. At each stage, the new low-pass filter is the old low-pass filter convolved with the basic low-pass filter decimated by a factor of 2ⁿ, where n is the stage. Similarly the new high-pass filter is the old low-pass filter convolved with the multiply-decimated basic high-pass filter.