Solution:

Following are the characteristics for rational function $F(S)$ to be a P.R.F:

(1)

i) Degree difference between polynomial equation in numerator \& denominator allowed is maximum 1 (one)

ii) S terms should be present from the Mar. degree to a minimum degree in numerator & denominator.

iii) If some terms of S are missing then it should be either an even polynomial or an odd polynomial equation.

(2)

A polynomial equation of S in numerator denominator should be a Hurwitz polynomial.

(3)

If poles for a given rational function F(S) are in ju form (Imagery), the residuals for a given function should be positive.

(4)

For a given rational function F(S)

$A(w) \geqslant 0$ for all values of $w^{\circ}$ where $A(w)=m_1 m_2-n_1 n_2$

$m \rightarrow$ Even terms,$n \rightarrow$ ODD terms

$1 \rightarrow$ Numeratov, $2 \rightarrow$ Denominator