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 h , g {\displaystyle \scriptstyle h,\,g} are two filters, then one level the traditional wavelet transform maps an input signal a 0 {\displaystyle \scriptstyle a_{0}} to two output signals a 1 , d 1 {\displaystyle \scriptstyle a_{1},\,d_{1}} , each of the half length:
a 1 = ( h ⋅ a 0 ) ↓ 2 d 1 = ( g ⋅ a 0 ) ↓ 2 {\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:
h e = h ↓ 2 a 0 , e = a 0 ↓ 2 h o = ( h ← 1 ) ↓ 2 a 0 , o = ( a 0 ← 1 ) ↓ 2 {\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.
δ = ( … , 0 , 0 , 1 0 − th position , 0 , 0 , … ) {\displaystyle \delta =(\dots ,0,0,{\underset {0-{\mbox{th position}}}{1}},0,0,\dots )}
The wavelet transformation reformulated to the split filters is:
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