Digital Filters Design for Signal and Image Processing

10.2. The Schur-Cohn criterion

In this section, we will present an algorithm that helps us learn, in a finite number of steps, if the following complex polynomial:

has all its zeros inside the unit disk; that is, in the subset:

For that, we introduce the polynomials and P*(z) represented as follows:

obtained by conjugating³ the coefficients of P(z) without conjugating the variable and

obtained by conjugating and inversing the order of the coefficients of P(z). The polynomial P*(z) is then called the reciprocal polynomial of P(z).

EXAMPLE 10.4.- the polynomial P(z) = (2 + j)z³ + 3z admits, for reciprocal polynomial P*(z) = 3z² + (2 − j).

COMMENT 10.1.- the reciprocal polynomials P*(z) and satisfy the following equality:

Now, from the polynomial P_N(z) = P(z), we construct a family of