Introduction / Context:
Frequency response analysis in control and signal processing allows prediction of the output of an LTI system given a sinusoidal input. The output retains the same frequency as the input, but its amplitude and phase are modified according to the system's transfer function evaluated at that frequency.
Given Data / Assumptions:
- Input x(t) = A sin(ωt).
- System transfer function frequency response: G(jω).
- Assume linear, time-invariant, stable system with well-defined steady state.
Concept / Approach:
The output steady-state sinusoid is scaled by the magnitude of G(jω) and phase-shifted by its angle. Therefore, y(t) = |G(jω)| A sin(ωt + ∠G(jω)). Frequency does not change, only amplitude and phase.
Step-by-Step Solution:
Represent input as phasor: X(jω) = A ∠0°.Output phasor: Y(jω) = G(jω) × X(jω).Magnitude: |Y| = |G(jω)| A; Phase: ∠Y = ∠G(jω).Convert back to time domain: y(t) = |G(jω)| A sin(ωt + ∠G(jω)).
Verification / Alternative check:
Simulation of sinusoidal response in MATLAB/Scilab yields amplitude gain |G(jω)| and phase shift ∠G(jω), matching the formula.
Why Other Options Are Wrong:
Option (a): φ = tan⁻¹|G(jω)| is meaningless; phase is not tan⁻¹ of magnitude.Option (c): Doubles frequency (2ω), which does not occur in LTI systems.Option (d): A G(jω) sin[…] incorrectly multiplies sinusoid by a complex number directly.Option (e): Cosine form is valid but requires phase redefinition; the standard representation is sine with phase shift.
Common Pitfalls:
Forgetting that LTI systems preserve frequency; only amplitude and phase change.
Final Answer:
|G(jω)| A sin[ωt + ∠G(jω)]
Discussion & Comments