Polyphase matrix

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:

${\displaystyle {\begin{aligned}a_{1}&=(h\cdot a_{0})\downarrow 2\\d_{1}&=(g\cdot a_{0})\downarrow 2\end{aligned}}}$

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:

${\displaystyle {\begin{aligned}h_{\mbox{e}}&=h\downarrow 2&a_{0,{\mbox{e}}}&=a_{0}\downarrow 2\\h_{\mbox{o}}&=(h\leftarrow 1)\downarrow 2&a_{0,{\mbox{o}}}&=(a_{0}\leftarrow 1)\downarrow 2\end{aligned}}}$

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.

${\displaystyle \delta =(\dots ,0,0,{\underset {0-{\mbox{th position}}}{1}},0,0,\dots )}$

The wavelet transformation reformulated to the split filters is:

${\displaystyle {\begin{aligned}a_{1}&=h_{\mbox{e}}\cdot a_{0,{\mbox{e}}}+h_{\mbox{o}}\cdot a_{0,{\mbox{o}}}\rightarrow 1\\d_{1}&=g_{\mbox{e}}\cdot a_{0,{\mbox{e}}}+g_{\mbox{o}}\cdot a_{0,{\mbox{o}}}\rightarrow 1\end{aligned}}}$

This can be written as matrix-vector-multiplication

${\displaystyle {\begin{aligned}P&={\begin{pmatrix}h_{\mbox{e}}&h_{\mbox{o}}\rightarrow 1\\g_{\mbox{e}}&g_{\mbox{o}}\rightarrow 1\end{pmatrix}}\\{\begin{pmatrix}a_{1}\\d_{1}\end{pmatrix}}&=P\cdot {\begin{pmatrix}a_{0,{\mbox{e}}}\\a_{0,{\mbox{o}}}\end{pmatrix}}\end{aligned}}}$

This matrix ${\displaystyle \scriptstyle P}$ 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.