If are two filters, then one level the traditional wavelet transform maps an input signal to two output signals , each of the half length:
If the above formula is implemented directly, you will compute values that are subsequently flushed by the down-sampling. You can avoid that by splitting the filters and the signal into even and odd indexed values before the transformation.
The wavelet transformation reformulated to the split filters is:
This can be written as matrix-vector-multiplication
This matrix is the polyphase 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. It allows the adaptation of many results from matrix theory and module theory. The following properties are explained for a matrix, but they scale equally to higher dimensions.
The case that a polyphase matrix allows reconstruction of a processed signal from the filtered data, is called perfect reconstruction property. Mathematically this is equivalent to invertibility. According to the theorem of invertibility of a matrix over a ring, the polyphase matrix is invertible if and only if the determinant of the polyphase matrix is a Kronecker delta, which is zero everywhere except for one value.
By Cramer's rule the inverse of can be given immediately.
The orthogonality condition
can be written out
For non-orthogonal polyphase matrices the question arises what Euclidean norms the output can assume. This can be bounded by the help of the operator norm.
A signal, where these bounds are assumed can be derived from the eigenvector corresponding to the maximizing and minimizing eigenvalue.
The concept of the polyphase matrix allows matrix decomposition. For instance the decomposition into addition matrices leads to the lifting scheme. 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.
- Strang, Gilbert; Nguyen, Truong (1997). Wavelets and Filter Banks. Wellesley-Cambridge Press. ISBN 0-9614088-7-1.
- Thielemann, Henning (2001). Adaptive construction of wavelets for image compression (Diploma thesis). Martin-Luther-Universität Halle-Wittenberg, Fachbereich Mathematik/Informatik.
- Daubechies, Ingrid; Sweldens, Wim (1998). "Factoring wavelet transforms into lifting steps". J. Fourier Anal. Appl. 4 (3). pp. 245–267.