Transfer matrix explained

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 ⋅} . More verboselyT _{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 T _{h} can be expressed in terms of the downsampling operator "\downarrow ":T _{h ⋅} x = (h*x)\downarrow 2 .

See also

• Hurwitz determinant

References

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