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 "$\downarrow$":

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

See also

• Hurwitz determinant

References

• Template:Cite thesis (contains proofs of the above properties)