Transfer matrix

{{unordered list

| $T\_h\backslash cdot\; x\; =\; T\_x\backslash cdot\; h$.

| If you drop the first and the last column and move the odd-indexed columns to the left and the even-indexed columns to the right, then you obtain a transposed Sylvester matrix.

| The determinant of a transfer matrix is essentially a resultant.

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More precisely:

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Let $h\_\{\backslash mathrm\{e\}\}$ be the even-indexed coefficients of $h$ ($(h\_\{\backslash mathrm\{e\}\})\_k\; =\; h\_\{2k\}$) and let $h\_\{\backslash mathrm\{o\}\}$ be the odd-indexed coefficients of $h$ ($(h\_\{\backslash mathrm\{o\}\})\_k\; =\; h\_\{2k+1\}$).

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Then $\backslash det\; T\_h\; =\; (-1)^\{\backslash lfloor\backslash frac\{b-a+1\}\{4\}\backslash rfloor\}\backslash cdot\; h\_a\backslash cdot\; h\_b\backslash cdot\backslash mathrm\{res\}(h\_\{\backslash mathrm\{e\}\},h\_\{\backslash mathrm\{o\}\})$, where $\backslash mathrm\{res\}$ is the resultant.

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This connection allows for fast computation using the Euclidean algorithm.

| For the trace of the transfer matrix of convolved masks holds

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$\backslash mathrm\{tr\}~T\_\{g*h\}\; =\; \backslash mathrm\{tr\}~T\_g\; \backslash cdot\; \backslash mathrm\{tr\}~T\_h$

| For the determinant of the transfer matrix of convolved mask holds

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$\backslash det\; T\_\{g*h\}\; =\; \backslash det\; T\_g\; \backslash cdot\; \backslash det\; T\_h\; \backslash cdot\; \backslash mathrm\{res\}(g\_-,h)$

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where $g\_-$ denotes the mask with alternating signs, i.e. $(g\_-)\_k\; =\; (-1)^k\; \backslash cdot\; g\_k$.

| If $T\_\{h\}\backslash cdot\; x\; =\; 0$, then $T\_\{g*h\}\backslash cdot\; (g\_-*x)\; =\; 0$.

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This is a concretion of the determinant property above. From the determinant property one knows that $T\_\{g*h\}$ is singular whenever $T\_\{h\}$ is singular. This property also tells, how vectors from the null space of $T\_\{h\}$ can be converted to null space vectors of $T\_\{g*h\}$.

| If $x$ is an eigenvector of $T\_\{h\}$ with respect to the eigenvalue $\backslash lambda$, i.e.

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$T\_\{h\}\backslash cdot\; x\; =\; \backslash lambda\backslash cdot\; x$,

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then $x*(1,-1)$ is an eigenvector of $T\_\{h*(1,1)\}$ with respect to the same eigenvalue, i.e.

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$T\_\{h*(1,1)\}\backslash cdot\; (x*(1,-1))\; =\; \backslash lambda\backslash cdot\; (x*(1,-1))$.

| Let $\backslash lambda\_a,\backslash dots,\backslash lambda\_b$ be the eigenvalues of $T\_h$, which implies $\backslash lambda\_a+\backslash dots+\backslash lambda\_b\; =\; \backslash mathrm\{tr\}~T\_h$ and more generally $\backslash lambda\_a^n+\backslash dots+\backslash lambda\_b^n\; =\; \backslash mathrm\{tr\}(T\_h^n)$. This sum is useful for estimating the spectral radius of $T\_h$. There is an alternative possibility for computing the sum of eigenvalue powers, which is faster for small $n$.

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Let $C\_k\; h$ be the periodization of $h$ with respect to period $2^k-1$. That is $C\_k\; h$ is a circular filter, which means that the component indexes are residue classes with respect to the modulus $2^k-1$. Then with the upsampling operator $\backslash uparrow$ it holds

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$\backslash mathrm\{tr\}(T\_h^n)\; =\; \backslash left(C\_k\; h\; *\; (C\_k\; h\backslash uparrow\; 2)\; *\; (C\_k\; h\backslash uparrow\; 2^2)\; *\; \backslash cdots\; *\; (C\_k\; h\backslash uparrow\; 2^\{n-1\})\backslash right)\_\{[0]\_\{2^n-1\}\}$

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Actually not $n-2$ convolutions are necessary, but only $2\backslash cdot\; \backslash log\_2\; n$ ones, when applying the strategy of efficient computation of powers. Even more the approach can be further sped up using the Fast Fourier transform.

| From the previous statement we can derive an estimate of the spectral radius of $\backslash varrho(T\_h)$. It holds

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$\backslash varrho(T\_h)\; \backslash ge\; \backslash frac\{a\}\{\backslash sqrt\{\backslash \#\; h\}\}\; \backslash ge\; \backslash frac\{1\}\{\backslash sqrt\{3\backslash cdot\; \backslash \#\; h\}\}$

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where $\backslash \#\; h$ is the size of the filter and if all eigenvalues are real, it is also true that

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$\backslash varrho(T\_h)\; \backslash le\; a$,

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where $a\; =\; \backslash Vert\; C\_2\; h\; \backslash Vert\_2$.

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• Hurwitz determinant

• {{cite journal

|first=Gilbert|last=Strang

|author-link=Gilbert Strang

|title=Eigenvalues of $(\backslash downarrow\; 2)\{H\}$ and convergence of the cascade algorithm

|journal=IEEE Transactions on Signal Processing

|volume=44

|pages=233–238

|year=1996

|doi=10.1109/78.485920

}}

• {{cite thesis

|first=Henning

|last=Thielemann

|url=http://nbn-resolving.de/urn:nbn:de:gbv:46-diss000103131

|title=Optimally matched wavelets

|type=PhD thesis

|year=2006

}} (contains proofs of the above properties)

Category:Wavelets

Category:Numerical analysis