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| In [[signal processing]], a '''polyphase matrix''' is a matrix whose elements are [[linear filter|filter mask]]s. It represents a [[filter bank]] as it is used in [[sub-band coder]]s alias [[discrete wavelet transform]]s.<ref name="strang1997filterbanks">
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| {{cite book
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| |first1=Gilbert|last1=Strang|author1-link=Gilbert Strang
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| |first2=Truong|last2=Nguyen
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| |title=Wavelets and Filter Banks
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| |publisher=Wellesley-Cambridge Press
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| |year=1997
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| |isbn=0-9614088-7-1
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| }}</ref>
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| If <math>\scriptstyle h,\, g</math> are two filters, then one level the traditional wavelet transform maps an input signal <math>\scriptstyle a_0</math> to two output signals <math>\scriptstyle a_1,\, d_1</math>, each of the half length:
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| :<math>\begin{align}
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| a_1 &= (h\cdot a_0) \downarrow 2 \\
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| d_1 &= (g\cdot a_0) \downarrow 2
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| \end{align}</math>
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| Note, that the dot means [[polynomial multiplication]]; i.e., [[convolution]] and <math>\scriptstyle\downarrow</math> means [[downsampling]].
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| 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.
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| :<math>\begin{array}{rclcrcl}
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| h_{\mbox{e}} &=& h \downarrow 2 &\qquad& a_{0,\mbox{e}} &=& a_0 \downarrow 2 \\
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| h_{\mbox{o}} &=& (h \leftarrow 1) \downarrow 2 && a_{0,\mbox{o}} &=& (a_0 \leftarrow 1) \downarrow 2
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| \end{array}</math>
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| The arrows <math>\scriptstyle\leftarrow</math> and <math>\scriptstyle\rightarrow</math> denote left and right shifting, respectively. They shall have the same [[operator precedence|precedence]] like convolution, because they are in fact convolutions with a shifted discrete [[Kronecker delta|delta impulse]].
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| :<math>\delta = (\dots, 0, 0, \underset{0-\mbox{th position}}{1}, 0, 0, \dots)</math>
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| The wavelet transformation reformulated to the split filters is:
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| :<math>\begin{align}
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| a_1 &= h_{\mbox{e}}\cdot a_{0,\mbox{e}} +
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| h_{\mbox{o}}\cdot a_{0,\mbox{o}} \rightarrow 1 \\
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| d_1 &= g_{\mbox{e}}\cdot a_{0,\mbox{e}} +
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| g_{\mbox{o}}\cdot a_{0,\mbox{o}} \rightarrow 1
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| \end{align}</math>
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| This can be written as [[matrix multiplication|matrix-vector-multiplication]]
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| :<math>\begin{align}
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| P &= \begin{pmatrix}
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| h_{\mbox{e}} & h_{\mbox{o}} \rightarrow 1 \\
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| g_{\mbox{e}} & g_{\mbox{o}} \rightarrow 1
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| \end{pmatrix} \\
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| \begin{pmatrix} a_1 \\ d_1 \end{pmatrix} &= P \cdot
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| \begin{pmatrix}
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| a_{0,\mbox{e}} \\
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| a_{0,\mbox{o}}
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| \end{pmatrix}
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| \end{align}</math>
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| This matrix <math>\scriptstyle P</math> is the polyphase matrix.
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| Of course, a polyphase matrix can have any size, it need not to have square shape. That is, the principle scales well to any [[filterbank]]s, [[multiwavelet]]s, wavelet transforms based on fractional [[refinable function|refinements]].
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| == Properties ==
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| 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 (mathematics)|matrix theory]] and [[module theory]]. The following properties are explained for a <math>\scriptstyle 2 \,\times\, 2</math> matrix, but they scale equally to higher dimensions.
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| === Invertibility/perfect reconstruction ===
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| 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 [[inverse matrix|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.
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| :<math>\begin{align}
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| \det P &= h_{\mbox{e}} \cdot g_{\mbox{o}} - h_{\mbox{o}} \cdot g_{\mbox{e}} \\
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| \exists A\ A\cdot P &= I \iff \exists c\ \exists k\ \det P = c\cdot \delta \rightarrow k
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| \end{align}</math>
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| <!--
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| (\dots,0,0,\underset{i-\mbox{th position}}{c},0,0,\dots),
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| defined above
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| -->
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| By [[Cramer's rule]] the inverse of <math>\scriptstyle P</math> can be given immediately.
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| :<math>P^{-1}\cdot\det P =
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| \begin{pmatrix}
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| g_{\mbox{o}} \rightarrow 1 & - h_{\mbox{o}} \rightarrow 1 \\
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| -g_{\mbox{e}} & h_{\mbox{e}}
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| \end{pmatrix}
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| </math>
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| === Orthogonality ===
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| Orthogonality means that the [[adjoint matrix]] <math>\scriptstyle P^*</math> is also the inverse matrix of <math>\scriptstyle P</math>. The adjoint matrix is the [[transposed matrix]] with [[adjoint filter]]s.
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| :<math>P^* = \begin{pmatrix}
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| h_{\mbox{e}}^* & g_{\mbox{e}}^* \\
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| h_{\mbox{o}}^* \leftarrow 1 & g_{\mbox{o}}^* \leftarrow 1
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| \end{pmatrix}
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| </math>
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| It implies, that the [[Euclidean norm]] of the input signals is preserved. That is, the according wavelet transform is an [[isometry]].
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| :<math>||a_1||_2^2 + ||d_1||_2^2 = ||a_0||_2^2</math>
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| The orthogonality condition
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| :<math>P \cdot P^* = I</math>
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| can be written out
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| :<math>\begin{align}
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| h_{\mbox{e}}^* \cdot h_{\mbox{e}} + h_{\mbox{o}}^* \cdot h_{\mbox{o}} &= \delta \\
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| g_{\mbox{e}}^* \cdot g_{\mbox{e}} + g_{\mbox{o}}^* \cdot g_{\mbox{o}} &= \delta \\
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| h_{\mbox{e}}^* \cdot g_{\mbox{e}} + h_{\mbox{o}}^* \cdot g_{\mbox{o}} &= 0
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| \end{align}</math>
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| === Operator norm ===
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| 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]].
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| :<math>\forall x\ \|P\cdot x\|_2 \in \left[\|P^{-1}\|_2^{-1}\cdot\|x\|_2, \|P\|_2\cdot\|x\|_2\right]</math>
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| For the <math>\scriptstyle 2 \,\times\, 2</math> polyphase matrix the Euclidean operator norm can be given explicitly using the [[Frobenius norm]] <math>\scriptstyle\|\cdot\|_F</math> and the [[z transform]] <math>\scriptstyle Z</math>:<ref name="thielemann2001adaptivewavelet">
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| {{cite thesis
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| |first=Henning|last=Thielemann
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| |title=Adaptive construction of wavelets for image compression
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| |type=Diploma thesis
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| |publisher=Martin-Luther-Universität Halle-Wittenberg, Fachbereich Mathematik/Informatik
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| |year=2001
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| |url=http://edoc.bibliothek.uni-halle.de/servlets/DocumentServlet?id=2134
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| }}</ref>
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| :<math>\begin{align}
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| p(z) &= \frac{1}{2}\cdot \|Z P(z)\|_F^2 \\
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| q(z) &= \left|\det [Z P(z)]\right|^2 \\
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| \|P\|_2 &= \max\left\{\sqrt{p(z) + \sqrt{p(z)^2-q(z)}} : z\in\mathbb{C}\ \land\ |z| = 1\right\} \\
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| \|P^{-1}\|_2^{-1} &= \min\left\{\sqrt{p(z) - \sqrt{p(z)^2-q(z)}} : z\in\mathbb{C}\ \land\ |z| = 1\right\}
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| \end{align}</math>
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| This is a special case of the <math>n\times n</math> matrix where the operator norm can be obtained via [[z transform]] and the [[spectral radius]] of a matrix or the according [[spectral norm]].
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| :<math>\begin{align}
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| \|P\|_2
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| &= \sqrt{\max\left\{\lambda_{\mbox{max}}\left[Z P^*(z)\cdot Z P(z)\right] : z\in\mathbb{C}\ \land\ |z| = 1\right\}} \\
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| &= \max\left\{\|Z P(z)\|_2 : z\in\mathbb{C}\ \land\ |z| = 1\right\} \\
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| \|P^{-1}\|_2^{-1}
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| &= \sqrt{\min\left\{\lambda_{\mbox{min}}\left[Z P^*(z)\cdot Z P(z)\right] : z\in\mathbb{C}\ \land\ |z| = 1\right\}}
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| \end{align}</math>
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| A signal, where these bounds are assumed can be derived from the eigenvector corresponding to the maximizing and minimizing eigenvalue.
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| === Lifting scheme ===
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| The concept of the polyphase matrix allows [[matrix decomposition]]. For instance the decomposition into [[triangular matrix|addition matrices]] leads to the [[lifting scheme]].<ref name="sweldens1998liftingfactor">
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| {{cite article
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| |first1=Ingrid|last1=Daubechies|author1-link=Ingrid Daubechies
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| |first2=Wim|last2=Sweldens|author2-link=Wim Sweldens
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| |title=Factoring wavelet transforms into lifting steps
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| |journal=J. Fourier Anal. Appl.
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| |volume=4
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| |issue=3
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| |pages=245-267
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| |year=1998
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| |url=http://cm.bell-labs.com/who/wim/papers/factor/index.html
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| }}</ref> However, classical matrix decompositions like [[LU decomposition|LU]] and [[QR decomposition]] cannot be applied immediately, because the filters form a [[ring (algebra)|ring]] with respect to convolution, not a [[field (algebra)|field]].
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| == References ==
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| <references/>
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| [[Category:Wavelets]] <!-- it is central part of wavelet theory -->
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| [[Category:Digital signal processing]] <!-- but also of interest elsewhere -->
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