It represents a filter bank as it is used in sub-band coders alias discrete wavelet transforms.
are two filters, then one level the traditional wavelet transform maps an input signal
, each of the half length: Note, that the dot means polynomial multiplication; i.e., convolution and
If the above formula is implemented directly, you will compute values that are subsequently flushed by the down-sampling.
You can avoid their computation by splitting the filters and the signal into even and odd indexed values before the wavelet transformation: The arrows
The wavelet transformation reformulated to the split filters is: This can be written as matrix-vector-multiplication This matrix
Of course, a polyphase matrix can have any size, it need not to have square shape.
That is, the principle scales well to any filterbanks, multiwavelets, wavelet transforms based on fractional refinements.
The representation of sub-band coding by the polyphase matrix is more than about write simplification.
matrix, but they scale equally to higher dimensions.
It implies, that the Euclidean norm of the input signals is preserved.
The orthogonality condition can be written out For non-orthogonal polyphase matrices the question arises what Euclidean norms the output can assume.
A signal, where these bounds are assumed can be derived from the eigenvector corresponding to the maximizing and minimizing eigenvalue.
For instance the decomposition into addition matrices leads to the lifting scheme.
[3] However, classical matrix decompositions like LU and QR decomposition cannot be applied immediately, because the filters form a ring with respect to convolution, not a field.