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An energy signal has Fourier transform magnitude G(f) = 10 (constant over the band of interest). What is its energy spectral density S(f)?

Difficulty: Easy

Correct Answer: 100

Explanation:


Introduction / Context:
For energy signals, the energy spectral density (ESD) S(f) describes how the signal’s energy is distributed across frequencies. It relates directly to the magnitude of the Fourier transform through a simple squared-magnitude relationship.


Given Data / Assumptions:

  • G(f) denotes the Fourier transform magnitude of the energy signal (|G(f)|).
  • |G(f)| = 10 (constant) over the relevant frequency range.
  • Standard definition of ESD applies.


Concept / Approach:
By definition for energy signals, S(f) = |G(f)|^2. This ensures Parseval’s relation: total energy = ∫ S(f) df = ∫ |G(f)|^2 df. Therefore, when |G(f)| = 10, S(f) = 10^2 = 100 (with appropriate units).


Step-by-Step Solution:

Given |G(f)| = 10.Compute S(f): S(f) = |G(f)|^2.Therefore S(f) = 10^2 = 100.


Verification / Alternative check:

Parseval’s theorem: Energy E = ∫ |g(t)|^2 dt = ∫ |G(f)|^2 df = ∫ S(f) df, consistent.


Why Other Options Are Wrong:

10 or 20 or 50 or 5: these fail to square the magnitude, violating the definition S(f) = |G(f)|^2.


Common Pitfalls:

Confusing amplitude spectrum |G(f)| with energy spectral density S(f). The former is linear, the latter is quadratic.


Final Answer:

100
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