In signal processing, a polyphase matrix is a matrix whose elements are filter masks. It represents a filter bank as it is used in sub-band coders alias discrete wavelet transforms.

If ${\displaystyle \scriptstyle h,\,g}$ are two filters, then one level the traditional wavelet transform maps an input signal ${\displaystyle \scriptstyle a_{0}}$ to two output signals ${\displaystyle \scriptstyle a_{1},\,d_{1}}$, each of the half length:

Note, that the dot means polynomial multiplication; i.e., convolution and ${\displaystyle \scriptstyle \downarrow }$ means downsampling.

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 ${\displaystyle \scriptstyle \leftarrow }$ and ${\displaystyle \scriptstyle \rightarrow }$ denote left and right shifting, respectively. They shall have the same precedence like convolution, because they are in fact convolutions with a shifted discrete delta impulse.

The wavelet transformation reformulated to the split filters is:

This can be written as matrix-vector-multiplication

This matrix ${\displaystyle \scriptstyle P}$ is the polyphase matrix.