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

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

More verbosely

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 can be expressed in terms of the downsampling operator "":

Read more about Transfer Matrix:  Properties, See Also

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