Lesson plan
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 \scriptstyle h,\, g are two filters, then one level the traditional wavelet transform maps an input signal \scriptstyle a_0 to two output signals \scriptstyle a_1,\, d_1 , each of the half length: : \begin{align} a_1 &= (h \cdot a_0) \downarrow 2 \\ d_1 &= (g \cdot a_0) \downarrow 2 \end{align} Note, that the dot means polynomial multiplication; i.e., convolution and \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: : \begin{align} 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{align} The arrows \scriptstyle\leftarrow and \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.…
Article summary
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 \scriptstyle h,\, g are two filters, then one level the traditional wavelet transform maps an input signal \scriptstyle a_0 to two output signals \scriptstyle a_1,\, d_1 , each of the half length: : \begin{align} a_1 &= (h \cdot a_0) \downarrow 2 \\ d_1 &= (g \cdot a_0) \downarrow 2 \end{align} Note, that the dot means polynomial multiplication; i.e., convolution and \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: : \begin{align} 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{align} The arrows \scriptstyle\leftarrow and \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.…
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