Question

For a system to be physically realizable, the degree of numerator polynomial M and the degree of denominator polynomial N should be

• A. \( M \ge N \)
• B. \( M \le N \)
• C. \( M = N \)
• D. No constraint on M and N

Hint:

Physical realizability for LTI systems usually implies causality.
For a rational transfer function \(H(s)\), causality requires deg(Numerator) \(\le\) deg(Denominator).
This ensures the system does not respond before an input is applied.

Answer 1

The Correct Option is B

For a continuous-time LTI system with a rational transfer function \(H(s) = \frac{\text{Numerator}(s)}{\text{Denominator}(s)}\), where M is the degree of the numerator polynomial and N is the degree of the denominator polynomial, physical realizability is linked to causality. For the system to be causal (and thus physically realizable in most practical contexts), the degree of the numerator M must be less than or equal to the degree of the denominator N. \[ M \le N \] If \(M>N\), the system's impulse response would contain impulses or derivatives of impulses at \(t=0\), implying non-causal behavior (response before or at the instant of input in a way that requires future knowledge). Such a transfer function is called a proper rational function. If \(M<N\), it is strictly proper. \[ \boxed{M \le N} \]