# Transfer matrix

In applied mathematics, the transfer matrix is a formulation in terms of a block-Toeplitz matrix of the two-scale equation, which characterizes refinable functions. Refinable functions play an important role in wavelet theory and finite element theory.

For the mask ${\displaystyle h}$, which is a vector with component indexes from ${\displaystyle a}$ to ${\displaystyle b}$, the transfer matrix of ${\displaystyle h}$, we call it ${\displaystyle T_{h}}$ here, is defined as

${\displaystyle (T_{h})_{j,k}=h_{2\cdot j-k}.}$

More verbosely

${\displaystyle T_{h}={\begin{pmatrix}h_{a}&&&&&\\h_{a+2}&h_{a+1}&h_{a}&&&\\h_{a+4}&h_{a+3}&h_{a+2}&h_{a+1}&h_{a}&\\\ddots &\ddots &\ddots &\ddots &\ddots &\ddots \\&h_{b}&h_{b-1}&h_{b-2}&h_{b-3}&h_{b-4}\\&&&h_{b}&h_{b-1}&h_{b-2}\\&&&&&h_{b}\end{pmatrix}}.}$

The effect of ${\displaystyle T_{h}}$ can be expressed in terms of the downsampling operator "${\displaystyle \downarrow }$":

${\displaystyle T_{h}\cdot x=(h*x)\downarrow 2.}$

## Properties

• ${\displaystyle T_{h}\cdot x=T_{x}\cdot h}$ .
• 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.
More precisely:
Let ${\displaystyle h_{\mathrm {e} }}$  be the even-indexed coefficients of ${\displaystyle h}$  (${\displaystyle (h_{\mathrm {e} })_{k}=h_{2k}}$ ) and let ${\displaystyle h_{\mathrm {o} }}$  be the odd-indexed coefficients of ${\displaystyle h}$  (${\displaystyle (h_{\mathrm {o} })_{k}=h_{2k+1}}$ ).
Then ${\displaystyle \det T_{h}=(-1)^{\lfloor {\frac {b-a+1}{4}}\rfloor }\cdot h_{a}\cdot h_{b}\cdot \mathrm {res} (h_{\mathrm {e} },h_{\mathrm {o} })}$ , where ${\displaystyle \mathrm {res} }$  is the resultant.
This connection allows for fast computation using the Euclidean algorithm.
${\displaystyle \mathrm {tr} ~T_{g*h}=\mathrm {tr} ~T_{g}\cdot \mathrm {tr} ~T_{h}}$
• For the determinant of the transfer matrix of convolved mask holds
${\displaystyle \det T_{g*h}=\det T_{g}\cdot \det T_{h}\cdot \mathrm {res} (g_{-},h)}$
where ${\displaystyle g_{-}}$  denotes the mask with alternating signs, i.e. ${\displaystyle (g_{-})_{k}=(-1)^{k}\cdot g_{k}}$ .
• If ${\displaystyle T_{h}\cdot x=0}$ , then ${\displaystyle T_{g*h}\cdot (g_{-}*x)=0}$ .
This is a concretion of the determinant property above. From the determinant property one knows that ${\displaystyle T_{g*h}}$  is singular whenever ${\displaystyle T_{h}}$  is singular. This property also tells, how vectors from the null space of ${\displaystyle T_{h}}$  can be converted to null space vectors of ${\displaystyle T_{g*h}}$ .
• If ${\displaystyle x}$  is an eigenvector of ${\displaystyle T_{h}}$  with respect to the eigenvalue ${\displaystyle \lambda }$ , i.e.
${\displaystyle T_{h}\cdot x=\lambda \cdot x}$ ,
then ${\displaystyle x*(1,-1)}$  is an eigenvector of ${\displaystyle T_{h*(1,1)}}$  with respect to the same eigenvalue, i.e.
${\displaystyle T_{h*(1,1)}\cdot (x*(1,-1))=\lambda \cdot (x*(1,-1))}$ .
• Let ${\displaystyle \lambda _{a},\dots ,\lambda _{b}}$  be the eigenvalues of ${\displaystyle T_{h}}$ , which implies ${\displaystyle \lambda _{a}+\dots +\lambda _{b}=\mathrm {tr} ~T_{h}}$  and more generally ${\displaystyle \lambda _{a}^{n}+\dots +\lambda _{b}^{n}=\mathrm {tr} (T_{h}^{n})}$ . This sum is useful for estimating the spectral radius of ${\displaystyle T_{h}}$ . There is an alternative possibility for computing the sum of eigenvalue powers, which is faster for small ${\displaystyle n}$ .
Let ${\displaystyle C_{k}h}$  be the periodization of ${\displaystyle h}$  with respect to period ${\displaystyle 2^{k}-1}$ . That is ${\displaystyle C_{k}h}$  is a circular filter, which means that the component indexes are residue classes with respect to the modulus ${\displaystyle 2^{k}-1}$ . Then with the upsampling operator ${\displaystyle \uparrow }$  it holds
${\displaystyle \mathrm {tr} (T_{h}^{n})=\left(C_{k}h*(C_{k}h\uparrow 2)*(C_{k}h\uparrow 2^{2})*\cdots *(C_{k}h\uparrow 2^{n-1})\right)_{[0]_{2^{n}-1}}}$
Actually not ${\displaystyle n-2}$  convolutions are necessary, but only ${\displaystyle 2\cdot \log _{2}n}$  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 ${\displaystyle \varrho (T_{h})}$ . It holds
${\displaystyle \varrho (T_{h})\geq {\frac {a}{\sqrt {\#h}}}\geq {\frac {1}{\sqrt {3\cdot \#h}}}}$
where ${\displaystyle \#h}$  is the size of the filter and if all eigenvalues are real, it is also true that
${\displaystyle \varrho (T_{h})\leq a}$ ,
where ${\displaystyle a=\Vert C_{2}h\Vert _{2}}$ .

## References

• Strang, Gilbert (1996). "Eigenvalues of ${\displaystyle (\downarrow 2){H}}$  and convergence of the cascade algorithm". IEEE Transactions on Signal Processing. 44. pp. 233–238.
• Thielemann, Henning (2006). Optimally matched wavelets (PhD thesis). (contains proofs of the above properties)