Laplace transform basics for standard test signals What is the Laplace transform of an impulse input of magnitude X applied at t = 0?

Introduction / Context:
Standard test signals—impulse, step, and ramp—are fundamental in control theory and system identification. Knowing their Laplace transforms allows quick derivations of transfer functions and responses. The impulse is particularly important because it is the identity input for linear time-invariant systems in the convolution sense.

Given Data / Assumptions:

• Impulse input δ(t) scaled by magnitude X: u(t) = X δ(t).
• Laplace transform is defined for Re(s) greater than the region of convergence pertinent to the system.
• Transform is unilateral (from 0^- or 0^+), standard in control applications.

Concept / Approach:
The Laplace transform of δ(t) is 1. By linearity, the transform of X δ(t) is X * 1 = X. Physically, an impulse instantaneously transfers an area X to the system input, exciting all modes according to initial-value theorems. This property underpins impulse responses and Green’s functions in LTI system analysis.

Step-by-Step Solution:

Verification / Alternative check:
Impulse–step relationship: step is integral of impulse → L{step of height X} = X/s; differentiating 1/s gives δ(t), consistent with L{δ(t)} = 1.

Why Other Options Are Wrong:

• X^2, 1/X, 1: Do not follow from linearity and the unit impulse transform (only 1 is for unit impulse without scaling; with magnitude X the result is X).

Common Pitfalls:
Confusing the impulse (area X at t = 0) with a step of height X (transform X/s); mixing bilateral and unilateral Laplace conventions does not change the impulse result.

Final Answer:
X