Let H(z) be a rational transfer function, with associated nonnegative impulse response sequence. The paper considers the question: When does there exist a triple A is an element of R(NXN), b is an element of R(N), c is an element of R(N) with all nonnegative entries and H(z) = c'(zI - A)(-1)b? An essentially complete characterization is given of the H(z) allowing such a realization, in terms of the location of the pole or poles of H(z) with maximum modulus

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