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## 12.12    PROBLEMS

1. Consider the orthonormal Schur polynomial basis {Φi(z)}. Starting with definitions, prove the following:

(a)  ΦN−1(z), ΦN−1(z) = 1,

(b)  ΦN(z), ΦN−1(z) = 0,

(c)  ΦN(z), ΦN−2(z) = 0.

2. Prove the following inner product identities:

3. Consider the following transfer function

(a)   Obtain orthonormal basis functions from the denominator using the Schur algorithm.

(b)   Expand the numerator using the orthonormal polynomials obtained in (a).

(c)  Calculate the output signal power when the input is a white wide-sense stationary random process with unit power.

(d)  Determine a scaling constant a such that the scaled transfer function αH(z) has unit output signal power.

4. Consider the ...

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