VLSI Digital Signal Processing Systems: Design and Implementation

D.2    ORTHONORMALITY OF SCHUR POLYNOMIALS

In the previous section, we proved the orthogonality of the Schur polynomials using the inner product formulation. Therefore, to prove the orthonormality of Schur polynomials, we only need to show

For convenience, an i-th order Schur polynomial is expressed as

Using this notation, (12.37) can be rewritten as

Then, from (12.4) and (D.13), the coefficient of z^N−1 of Φ_N−1(z) is

By the same way, the coefficient of zⁱ of Φ_i(z) can be expressed as

By repeated application of (D.15),

To evaluate the inner product in (D.11), Φ_i(z) can be expressed using (D.3) and (D.6) as

The reverse Schur polynomials in P(z) do not have any effect on the above inner product since the constant of zΦ_i(z) is zero. The only term in P(z) that produces a nonzero result is ...