Nonconvexity of the stability domain of digital filters

The denominator of a rational digital filter of nth order is a polynomial represented by a point A of the n-dimensional space of stability [ 6 ] , we can, for example, compute the reflection coefficients from vector a . which requires O ( n ’ / 2 ) numerical operations at each time. In addition, these coefficients can be updated from time sample to time sample in an adaptive filtering context [7]. But it is clear that this procedure is slow compared to some sufficient stability conditions on a given in [8] and directly related to the geometry of E. In the design of linear predictive coding digital filters, the constraints involved and C are represented [ 5 ] by sets in the same n-dimensional space of the filter coefficients and topological properties of these sets are useful. Unfortunately, the structure of C is complicated for 17 2 3, even in the real case. An approach to its study is given for I I = 3 in [3] and some general results are discussed in [5] and [8]. This correspondence establishes results that concern the convexity of the stability domain C. A region CR of an n-dimensional space is said to be convex if it satisfies v ( a , b ) E C R 2 , then aa + ( I a ) b E C R , va E [0, I ] . ( 2 ) In the real case and for n = 2, it is well known that C is the interior of a triangle [ I ] , [21. As a consequence, C is convex and, in particular, if A belongs to E, then all the points a A , 0 s a s 1, also belong to C. Unfortunately, we will prove in the following section that this property is not valid for n > 2 and, as a consequence of this result, the nonconvexity of the stability domain for n > 2. Comment: Let a polynomial P ( z ) be represented by a point R of the n-dimensional space of its roots and be the stability domain in this space. If R is the point associated with the polynomials P ( z ) having r , , 1 5 i s n , as roots, the point a R is associated with Q ( z ) = P ( z / a ) having as roots ari, 1 s i I n. It is obvious that if R belongs to , then a R also belongs to 6 for 0 5 a 5 1 and n 2 1. This property is to compared to the result above. its coefficients. The set of all points A corresponding to stable filters defines a region Z in this space called the stability domain. It is shown that for n 2 3, E is not convex and especially if point A belongs to Z and a satisfies 0 < a I 1, then point aA does not necessarily belong to E. 11. FUNDAMENTAL RESULT If the vector or the point is associated with the polynomial ( 3 ) PI(:) = ( z I ) ” the vector a s or the point a S is associated with the polynomial