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 $h$, which is a vector with component indexes from $a$ to $b$, the transfer matrix of $h$, we call it $T_h$ here, is defined as

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

More verbosely

$T_h=\left(\begin{array}{cccccc}{h}_a& & & & & \\ h_{a+2}& h_{a+1}& h_a& & & \\ h_{a+4}& h_{a+3}& h_{a+2}& h_{a+1}& h_a& \\ \ddots & \ddots & \ddots & \ddots & \ddots & \ddots \\ & h_b& h_{b-1}& h_{b-2}& h_{b-3}& h_{b-4}\\ & & & h_b& h_{b-1}& h_{b-2}\\ & & & & & h_b\end{array}\right).$

The effect of $T_h$ can be expressed in terms of the downsampling operator "$\downarrow$":

$T_h\cdot x=(h*x)\downarrow 2.$

Properties

• $T_h\cdot x=T_x\cdot h$.
• 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 $h_{\mathrm{e}}$ be the even-indexed coefficients of $h$ ($(h_{\mathrm{e}}{)}_k=h_{2k}$) and let $h_{\mathrm{o}}$ be the odd-indexed coefficients of $h$ ($(h_{\mathrm{o}}{)}_k=h_{2k+1}$).

  Then $\det T_h=(-1{)}^{\lfloor \frac{b-a+1}{4}\rfloor }\cdot h_a\cdot h_b\cdot \mathrm{r}\mathrm{e}\mathrm{s}(h_{\mathrm{e}},h_{\mathrm{o}})$, where $\mathrm{r}\mathrm{e}\mathrm{s}$ is the resultant.

  This connection allows for fast computation using the Euclidean algorithm.
• For the trace of the transfer matrix of convolved masks holds
  $\mathrm{t}\mathrm{r}{T}_{g*h}=\mathrm{t}\mathrm{r}{T}_g\cdot \mathrm{t}\mathrm{r}{T}_h$
• For the determinant of the transfer matrix of convolved mask holds

  $\det T_{g*h}=\det T_g\cdot \det T_h\cdot \mathrm{r}\mathrm{e}\mathrm{s}(g_{-},h)$

  where $g_{-}$ denotes the mask with alternating signs, i.e. $(g_{-}{)}_k=(-1{)}^k\cdot g_k$.
• If $T_h\cdot x=0$, then $T_{g*h}\cdot (g_{-}*x)=0$.
  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 $\lambda$, i.e.

  $T_h\cdot x=\lambda \cdot x$,

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

  $T_{h*(1,1)}\cdot (x*(1,-1))=\lambda \cdot (x*(1,-1))$.
• Let ${\lambda }_a,\dots ,{\lambda }_b$ be the eigenvalues of $T_h$, which implies ${\lambda }_a+\dots +{\lambda }_b=\mathrm{t}\mathrm{r}{T}_h$ and more generally ${\lambda }_a^n+\dots +{\lambda }_b^n=\mathrm{t}\mathrm{r}(T_h^n)$. 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 $n$.

  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 $\uparrow$ it holds

  $\mathrm{t}\mathrm{r}(T_h^n)={(C_{k}h*(C_{k}h\uparrow 2)*(C_{k}h\uparrow 2^2)*\cdots *(C_{k}h\uparrow 2^{n-1}))}_{[0{]}_{2^n-1}}$

  Actually not $n-2$ convolutions are necessary, but only $2\cdot {\log }_{2}n$ 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 $\varrho (T_h)$. It holds

  $\varrho (T_h)\ge \frac{a}{\sqrt{\mathrm{\#}h}}\ge \frac{1}{\sqrt{3\cdot \mathrm{\#}h}}$

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

  $\varrho (T_h)\le a$,

  where $a=\Vert C_{2}h{\Vert }_2$.

See also

• Hurwitz determinant

References

• Strang, Gilbert (1996). "Eigenvalues of $(\downarrow 2)H$ and convergence of the cascade algorithm". IEEE Transactions on Signal Processing 44: 233–238. doi:10.1109/78.485920. 
• Thielemann, Henning (2006). Optimally matched wavelets (PhD thesis). (contains proofs of the above properties)