Signals & Systems: Match Operation/System to Its Canonical Result List I A. Fourier transform of a Gaussian function B. Convolution of a rectangular pulse with itself C. Current through an inductor for a step input voltage List II Gaussian function Rectangular pulse Triangular pulse Ramp function Zero

Introduction / Context:
This item mixes three well-known results from transform theory, convolution, and basic circuit response. Recognizing these canonical outcomes helps quickly predict system behavior and verify computations without re-deriving details each time.

Given Data / Assumptions:

• Fourier transform properties of Gaussian functions.
• Convolution definition for two identical rectangular pulses.
• Ideal inductor with step DC voltage applied at t ≥ 0.

Concept / Approach:

1) The Fourier transform of a Gaussian is another Gaussian (self-reciprocal property). 2) Convolving a rectangle with itself produces a triangular pulse due to overlap length growing then shrinking linearly. 3) For an inductor, v = L * di/dt; applying a constant step voltage yields a current that increases linearly with time (a ramp), i(t) = (V/L) * t for t ≥ 0 (ignoring initial conditions and resistance).

Step-by-Step Solution:

Verification / Alternative check:

Use standard tables: Gaussian ↔ Gaussian; convolution of two width-T rectangles gives a triangle of base 2T and peak proportional to T; inductor step response measured on a scope shows a straight-line current ramp when series resistance is negligible.

Why Other Options Are Wrong:

Linking Gaussian to rectangle contradicts transform duality. A step across an inductor cannot give zero current (except at t = 0); instead, current ramps due to finite inductance.

Common Pitfalls:

Confusing convolution (time-domain overlap) with multiplication in the frequency domain; forgetting that ideal inductors resist instantaneous current change but do not hold current constant under constant applied voltage.

Final Answer:

A-1, B-3, C-4