Digital Filters Design for Signal and Image Processing

11.5. Appendix A: demonstration of the Schur-Cohn criterion

Let us consider the polynomial:

of degree n ≥ I and its reciprocal polynomial:

We write:

so this is a complex polynomial with two variables x and y.

We observe that . This means that for all , and so on, by writing a_h,k = A_h,n−k−1, the application:

is a hermitian quadratic form (HQF).¹³

EXAMPLE 11.2.– if P = α₀(x + α), then .

It seems that for P = a₀(x + α), the signature¹⁴ of H(P; u₁) gives the position of the roots of P in relation to the unit circle. This result is not limited to the polynomials of degree 1, but extends to all the polynomials P(z) as we will see.

We assume that P = P₁P₂ with p₁(x) = b₀x^r +… +b_r, and P₂(x) =