Nonconvexity of the stability domain of digital filters

The stability of a causal digital filter with a rational transfer function H(z) is completely specified by the location of its poles with respect to the unit circle (UC). The set of points A corresponding to stable filters, i.e. to polynomials P with roots inside the UC, is called the stability domain Σ of digital filters. Results are established that concern the convexity of Σ. The denominator of a rational digital filter of nth order is a polynomial represented by a point A of the n-dimensional space of its coefficients. It is shown that for n⩾3, Σ is not convex and especially if point A belongs to Σ and a satisfies 0<a⩽1, then point aA does not necessarily belong to Σ