An inverse problem for Toeplitz matrices and the synthesis of discrete transmission lines

Let A_m be a positive definite, m x m, Toeplitz matrix. Let A_k be its k x k principal minor (for any k≤m), which is also positive definite and Toeplitz. Define the central mass sequence {ρ{variant}₁,...,ρ{variant}_m} by ρ{variant}_k = sup{ρ{variant}: A_k - ρ{variant}Π_k > 0}, in which Π_k is the k x k matrix of all 1's. We show how knowledge of the sequence {ρ{variant}_k} is equivalent to knowledge of the matrix A_m. This result has application to the direct and inverse problems for a transmission line which consists of piecewise constant components. Knowing the impulse response of the transmission line, we can calculate the capacitance taper of the line, and vice versa.