Minimum function in network synthesis: For a minimum function, what is the correct relationship between the degrees of the numerator and denominator polynomials?

Introduction / Context:
In network synthesis, a minimum function (a special positive-real function) is a proper rational function with simple, interlacing poles and zeros on the imaginary axis (or in specific symmetric locations), often used when deriving canonical ladder forms. Understanding its polynomial degree relationship is essential for recognizing realizable driving-point impedances or admittances with minimal reactive elements.

Given Data / Assumptions:

• We are discussing proper rational functions used as driving-point functions.
• Minimum functions are “proper,” meaning the function tends to zero at infinite frequency for impedance-type forms.
• No non-proper (improper) behavior is allowed for passive realizations without ideal transformers when targeting standard ladder forms.

Concept / Approach:

A proper rational function has numerator degree strictly less than denominator degree for impedance or admittance forms that vanish at infinite frequency (or behave appropriately for passivity). For minimum functions, this property holds: the degree of the numerator is exactly one less than that of the denominator, ensuring realizability with a minimal count of energy storage elements and compliance with PR conditions.

Step-by-Step Solution:

Verification / Alternative check:

Classical synthesis derivations (Cauer forms) show continued-fraction expansions where each division step reduces the degree by one, consistent with numerator being exactly one less than denominator for the driving-point impedance form under discussion.

Why Other Options Are Wrong:

• Equal degrees: Borderline proper/improper; not characteristic of the minimum function targeted in ladder synthesis.
• Numerator higher by one: Improper; violates passivity for basic realizations.
• Unequal (vague): Not specific enough and can include improper cases.

Common Pitfalls:

• Confusing “minimum function” with “minimum phase” (control theory). Here, the term is a synthesis-specific construct.

Final Answer:

the degree of numerator is one less than degree of denominator