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
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AA-1, B-2, C-3
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BA-2, B-4, C-5
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CA-1, B-3, C-4
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DA-2, B-3, C-5
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EA-1, B-4, C-2
Answer
Correct Answer: A-1, B-3, C-4
Explanation
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:
A → FT{Gaussian} = scaled Gaussian ⇒ 1.B → rect(t) * rect(t) = tri(t) by geometric area overlap ⇒ 3.C → v_L = L di/dt; with v_L = V (constant), integrate: di/dt = V/L ⇒ i(t) = (V/L) t (ramp) ⇒ 4.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