Frequency Modulation (FM)

In Weeks 2–4, every AM variant — DSB-SC, DSB-TC, SSB — changed the carrier's amplitude. The modulated signal's envelope carried the message.

Higher message amplitude → bigger envelope. Demodulation recovered \(m(t)\) from the amplitude variations — whether by coherent detection or envelope detection.

An unmodulated carrier has three parameters:

If we keep \(A_c\) fixed and instead vary the frequency or phase, we enter the domain of angle modulation.

Angle modulation takes two major forms:

This week focuses on FM. We will give PM just enough attention to see how the two relate.

Part 1

Instantaneous Phase and Frequency

Sections 02–03 — From the carrier to the FM equation.

The argument of the cosine — \(2\pi f_c\,t + \varphi\) — is the total phase. For an unmodulated carrier, this phase increases linearly with time at a constant rate determined by \(f_c\).

When the phase argument can be any function of time, we write:

where \(\theta_i(t)\) is the instantaneous phase — the total accumulated phase at time \(t\).

• Unmodulated carrier: \(\theta_i(t) = 2\pi f_c\,t + \varphi\) — linear in \(t\), constant frequency.
• Modulated signal: \(\theta_i(t)\) is a more general function — the frequency can change with time.

The instantaneous frequency is defined as:

Check: for an unmodulated carrier \(\theta_i(t)=2\pi f_c\,t + \varphi\):

The defining idea of frequency modulation:

• When \(m(t) > 0\): \(f_i > f_c\) — frequency increases.
• When \(m(t) < 0\): \(f_i < f_c\) — frequency decreases.
• When \(m(t) = 0\): \(f_i = f_c\) — returns to the carrier.

Since \(f_i(t) = \tfrac{1}{2\pi}\,\tfrac{d\theta_i}{dt}\), integrate to recover the instantaneous phase:

Substitute into \(s(t) = A_c\cos[\theta_i(t)]\):

Part 2

Single-Tone FM

Sections 04–06 — Frequency deviation, waveform, and modulation index.

Let \(m(t) = A_m \cos(2\pi f_m\,t)\). Integrate:

The instantaneous phase becomes:

Define two new quantities:

Substituting \(\beta = \Delta f / f_m\) into the general FM equation:

Verify — differentiate \(\theta_i(t)\) to recover the instantaneous frequency:

\(\Delta f = k_f A_m\) is the maximum deviation of the instantaneous frequency from \(f_c\).

• For fixed \(f_m\): larger \(\Delta f\) → larger \(\beta\) → stronger frequency excursion → more significant spectral sidebands.
• For fixed \(\Delta f\): higher \(f_m\) → smaller \(\beta\) — the message reverses before the frequency swings very far.

\(\beta\) is also the peak phase deviation in radians — it tells you the maximum phase swing beyond the unmodulated carrier.

Examples A and B have the same \(\Delta f\) but \(\beta\) differs by a factor of 10, because the message frequency changed. Example C matches FM broadcast parameters.

Part 3

FM Spectrum and Bandwidth

Sections 07–08 — Sidebands, bandwidth, and Carson's rule.

The practical bandwidth containing approximately 98% of FM signal power:

where \(B_m\) is the highest frequency in the message. For single-tone modulation (\(B_m = f_m\)):

Check limiting cases:

• \(\beta \ll 1\) (NBFM): \(B_{\text{FM}} \approx 2f_m\) — same as AM. ✓
• \(\beta \gg 1\) (WBFM): \(B_{\text{FM}} \approx 2\Delta f\) — dominated by the frequency deviation. ✓

FM broadcast uses roughly 18× more bandwidth than AM broadcast for the same type of audio content — but FM gets much better noise performance in return.

Part 4

Engineering Perspective

Sections 09–10 — Comparison, noise advantage, and the angle-modulation family.

In ideal FM, the carrier amplitude is always \(A_c\) — the message is entirely in the phase argument. This has a powerful engineering consequence:

1. Channel noise, interference, and fading primarily affect signal amplitude.
2. An FM receiver can hard-limit (clip) the received signal to remove all amplitude variations before detecting the frequency.
3. This discards most noise without losing the message, because the message was never in the amplitude.

Since frequency is the derivative of phase:

• FM of \(m(t)\) is equivalent to PM of \(\displaystyle\int m(t)\,dt\).
• PM of \(m(t)\) is equivalent to FM of \(\dfrac{dm(t)}{dt}\).

We will not develop PM further this week — but the relationship explains why FM and PM share the same spectral structure.

1. If \(\Delta f\) doubles and \(f_m\) stays fixed, what happens to \(\beta\)?
2. If \(f_m\) doubles while \(\Delta f\) stays fixed, what happens to \(\beta\)?
3. When \(m(t)\) reaches its positive peak, where is \(f_i(t)\)?
4. Does the envelope amplitude change in ideal FM?
5. Which occupies more bandwidth for the same message: DSB-AM or wideband FM?

EE 351 · Week 6

Next: from FM mathematics to the practical FM spectrum and receiver

This week you learned how FM encodes information in the carrier's instantaneous frequency rather than its amplitude. You derived the single-tone FM equation, understood \(\Delta f\) and \(\beta\), and estimated bandwidth using Carson's rule.