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

Question

Solve this multiple-choice question and choose the correct option.

Accepted Answer

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

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

More Questions from Signals and Systems

Discussion & Comments