4. (Extra Problem to those who enjoy challenges!!!) In class we say that there a

Question

4. (Extra Problem to those who enjoy challenges!!!) In class we say that there are equal number 1/1 of zeros and poles without full explanation as below. Prove (or, at least explain) under what condition it is true and why Rf: Poles and Zeros From the fundamental theorem of algebra, the numerator has M roots zeros of H) and the denominator has N roots (poles). Re-wrting in poles & zeros: In addition, tere may also exist zeros and poles at z= 0 and at infinity. · H(e) ·| lf we take the poles and zeros at z=0 and at infinity as well as multiple-order zeros and poles, the number of zeros and poles are always equal . Zeros: 21, 2., zM . Poles: p1, p2 pN zeros of multiplicity (N-M) at 0: or (if N

Explanation / Answer

usually in transfer function the degree of denominator is always greater than the numerator. only then the partial fractions are possible and the system can be realised.

When the system as equal number of solutions for both numerator and denominator, partial fractions may not be possible. and the system may not be practically realizable

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