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Relation to Schur Functions

**Definition. **A *Schur function*
is defined as a complex function analytic and of modulus not exceeding
unity in .

**Property. **The function

is a Schur function if and only if is positive real.

**Proof. **

Suppose is positive real. Then for , rere is PR. Consequently, is minimum phase which implies all roots of lie in the unit circle. Thus is analytic in . Also,

Conversely, suppose is Schur. Solving Eq.(C.84) for and taking the real part on the unit circle yields

If is constant, then is PR. If is not constant, then by the maximum principle, for . By Rouche's theorem applied on a circle of radius , , on which , the function has the same number of zeros as the function in . Hence, is minimum phase which implies is analytic for . Thus is PR.

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Relation to functions positive real in the right-half plane

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Relation to Stochastic Processes