Modulating Amplitude Affects FM Bandwidth

How Modulating Amplitude Affects FM Bandwidth (Mathematical Derivation)

1. FM Signal Definition

s(t) = A_c cos(2π f_c t + φ(t))

Where the phase is:

φ(t) = 2π k_f ∫ m(t) dt

• k_f = frequency sensitivity (Hz/Volt)
• m(t) = modulating signal

2. Sinusoidal Modulating Signal

m(t) = A_m cos(2π f_m t)

Integrating:

∫ m(t) dt = (A_m / 2π f_m) sin(2π f_m t)

Substitute into phase:

φ(t) = (k_f A_m / f_m) sin(2π f_m t)

3. Modulation Index

β = k_f A_m / f_m

FM signal becomes:

s(t) = A_c cos(2π f_c t + β sin(2π f_m t))

4. Frequency Deviation

Δf = k_f A_m

Δf ∝ A_m

5. FM Spectrum (Bessel Expansion)

s(t) = A_c Σ J_n(β) cos(2π(f_c + n f_m)t)

• J_n(β) = Bessel functions
• Number of significant sidebands depends on β

As β increases, more sidebands become significant → wider spectrum.

6. Carson’s Rule

B ≈ 2(Δf + f_m)

Substitute Δf:

B ≈ 2(k_f A_m + f_m)

7. Final Relationship

B ∝ A_m (for constant f_m and k_f)

Interpretation

• Increase A_m → increases Δf
• Increases modulation index β
• More sidebands appear
• Bandwidth increases

Conclusion

A_m ↑ → Δf ↑ → β ↑ → Bandwidth ↑