An FM signal with modulation index M_f passes through an ideal frequency doubler. What is the modulation index at the output of the doubler (assume the modulating frequency remains the same)?
Correct Answer: 2 M_f
Introduction / Context:Frequency multipliers are common in FM transmitters and test setups. They multiply the carrier frequency by an integer n and also scale the peak frequency deviation. Knowing how modulation index transforms is crucial for meeting deviation and spectral mask limits after multiplication stages.
Given Data / Assumptions:
- Input FM has modulation index Mf = Δf / fm.
- A frequency doubler multiplies instantaneous frequency deviation and carrier frequency by 2.
- Modulating frequency fm is unchanged by the multiplier.
Concept / Approach:
For an n-times frequency multiplier, Δfout = n * Δfin, and fm,out = fm,in. Therefore, Mf,out = Δfout / fm,out = n * Δfin / fm,in = n * Mf,in. With n = 2, the output modulation index doubles.
Step-by-Step Solution:
Write Mf = Δf / fm.For a doubler, Δf → 2Δf; fm unchanged.Hence Mf,out = 2 * Mf,in.Verification / Alternative check:
Spectral lines (Bessel components) shift and scale consistently with the multiplier, reflecting increased deviation (wider spectrum) proportional to n.
Why Other Options Are Wrong:
- Mf or Mf/2: contradicts the standard transformation.
- Mf^2: not supported by FM theory.
- “Unchanged with limiter”: a limiter clips amplitude, not frequency deviation.
Common Pitfalls:
Assuming the modulating frequency doubles—only the carrier and deviation are multiplied by the frequency multiplier.
Final Answer:
2 M_f