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Digital Filters Design for Signal and Image Processing by Mohamed Najim

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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:

images

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

images

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

images

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

images

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 images satisfy the following equality:

images

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

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