First-order LTI realizations: Which responses cannot be realized by a first-order linear time-invariant system?

Introduction / Context:
Filter order dictates the shapes a system can realize. Understanding the limitations of first-order LTI systems helps in selecting the right architecture for desired frequency responses.

Given Data / Assumptions:

• Continuous-time or discrete-time first-order LTI systems.
• \u201cRealization\u201d means a proper transfer function of order one.
• Standard definitions: low-pass, high-pass, band-pass, band-stop (notch), and all-pass.

Concept / Approach:

A first-order transfer function has one pole and possibly one zero. It can realize monotonic magnitude responses such as low-pass or high-pass, and even a first-order all-pass (with a pole and a mirrored zero). To obtain a band-pass (peaked passband between two cutoffs) or a band-stop (attenuated notch around a center), you need at least two reactive elements → a second-order (or higher) system.

Step-by-Step Reasoning:

Verification / Alternative check:

Bode plots of first-order systems are monotonic (±20 dB/decade slopes) and cannot form a passband flanked by two roll-offs or a notch between two rises.

Why Other Options Are Wrong:

• (a) or (b) alone are too narrow; both responses are unrealisable with first order.
• (d) All-pass can be implemented first-order.
• (e) Low-pass/high-pass are the canonical first-order forms.

Common Pitfalls:

• Assuming any desired magnitude shape can be approximated with first order—band-selective responses need higher order.
• Confusing all-pass (flat magnitude, phase shaping) with band-pass.

Final Answer:

Both band-pass and band-stop responses