EEEN 462 – Introduction to FM Modulation

1. Learning Objectives

By the end of this study unit, the student should be able to:

Define frequency modulation (FM) and distinguish it from amplitude modulation.
Identify and sketch the modulating (message), carrier, and frequency-modulated signals in the time domain, labelling the key features of each.
Explain the concepts of instantaneous frequency, frequency deviation, and deviation ratio.
Derive the single-tone FM wave equation and define the modulation index β.
Describe the FM spectrum in terms of Bessel coefficients and determine the number of significant sidebands.
Apply Carson's rule to compute the transmission bandwidth of an FM signal.
State the constant-power property of FM and explain its noise advantage over AM.
Describe methods of FM generation (direct and indirect/Armstrong) and demodulation (slope detector, Foster–Seeley, PLL).
Solve quantitative problems on FM parameters, bandwidth, and system comparison.

Contents at a Glance

1. Learning Objectives
2. Introduction
3. The Three Key Signals
4. Mathematical Model of FM
5. Modulation Index & Deviation
6. Spectrum & Bandwidth
7. Generation of FM
8. FM Demodulation
9. FM versus AM
10. Worked Examples
11. Summary
12. Review Questions

2. Introduction

In angle modulation, it is the angle (phase) of the carrier that is varied in accordance with the message, while the carrier amplitude stays constant. Frequency modulation (FM) is the most important form of angle modulation: the instantaneous frequency of the carrier is made to deviate from its resting value f_c in proportion to the instantaneous amplitude of the message.

Contrast with AM: In AM the information sits in the amplitude of the carrier, so any noise that corrupts amplitude corrupts the message. In FM the information sits in the frequency; the constant-amplitude carrier can be passed through limiters that strip amplitude noise before demodulation. This gives FM its famous noise immunity and high audio quality — the reason Edwin Armstrong invented wideband FM for broadcasting in 1933.

Narrowband FM (NBFM) has β ≪ 1 and occupies about the same bandwidth as AM; wideband FM (WBFM), used in FM broadcasting (88–108 MHz) and two-way radio, uses large deviation to trade bandwidth for noise immunity.

Property	Narrowband FM (NBFM)	Wideband FM (WBFM)
Modulation index β = Δf/f_m	β ≪ 1 (typically < 0.5)	β ≫ 1 (broadcast: up to 5)
Bandwidth	≈ 2f_m (like AM)	≈ 2(Δf + f_m) (much wider)
Noise immunity	Poor (similar to AM)	Excellent (capture effect)
Typical use	Two-way mobile radio	FM broadcasting, TV sound

3. The Three Key Signals

3.1 Modulating (Message) Signal — m(t)

A single-tone message is m(t) = A_m cos(2πf_mt). In FM it is the amplitude of m(t) that determines how far the carrier frequency swings away from f_c (the frequency deviation), while the frequency f_m determines how fast the carrier swings.

Figure 1 — Modulating signal m(t). Its amplitude sets the carrier frequency deviation; its frequency sets the rate of deviation.

3.2 Carrier Signal — c(t)

The carrier is a high-frequency sinusoid c(t) = A_c cos(2πf_ct) with constant amplitude A_c and constant resting (unmodulated) frequency f_c, where f_c ≫ f_m. In FM its amplitude never changes — only its instantaneous frequency does.

Figure 2 — Unmodulated carrier c(t) = A_ccos(2πf_ct), a sinusoid of constant amplitude and resting frequency. FM keeps A_c fixed and varies the instantaneous frequency around f_c.

3.3 Frequency-Modulated Signal — s(t)

The FM wave has constant amplitude everywhere, but its instantaneous frequency varies between f_c + Δf (when m(t) = +A_m) and f_c − Δf (when m(t) = −A_m). The result: the waveform is compressed (crowded cycles) at message peaks and stretched (spread cycles) at message troughs.

Figure 3 — FM wave: the amplitude A_c is strictly constant at every instant, while the instantaneous frequency f_i(t) = f_c + k_fm(t) reaches its maximum f_c+Δf exactly where the message m(t) peaks (crowded cycles) and its minimum f_c−Δf at message troughs (stretched cycles). The dashed curve traces f_i(t)._{c, but the spacing between zero-crossings varies with the message. Cycle spacing is minimum (frequency highest, f_c+Δf) at message peaks and maximum (frequency lowest, f_c−Δf) at message troughs. The dashed blue curve traces the instantaneous frequency.}

Reading the waveform: Unlike AM, there is no envelope to trace — every peak reaches the same height A_c. All the information is hidden in the density of the cycles. This is why FM receivers can use a limiter (clipper) to remove amplitude noise without touching the message.

4. Mathematical Model of FM

Start with an angle-modulated carrier of constant amplitude:

General angle-modulated waves(t) = A_c cos[θ(t)] = A_c cos[2π f_c t + φ(t)]

The instantaneous frequency (in Hz) is the rate of change of the total phase:

Instantaneous frequencyf_i(t) = (1/2π) · dθ/dt = f_c + (1/2π) · dφ/dt

Definition of FM: make the instantaneous frequency deviation from f_c proportional to the message:

FM defining lawf_i(t) = f_c + k_f m(t)

where k_f (Hz/V) is the frequency sensitivity of the modulator. For a single-tone message m(t) = A_m cos(2πf_mt), the maximum deviation (peak deviation) is:

Peak frequency deviationΔf = k_f A_m

Integrating the frequency law to get the phase, the FM wave becomes:

Single-tone FM wave (time domain)s(t) = A_c cos[2π f_c t + β sin(2π f_m t)]

Modulation indexβ = Δf / f_m = k_fA_m / f_m

Compare with AM: the AM modulation index m_a can never exceed 1 without distortion. The FM index β has no such limit — it may be far greater than 1. Larger β means more sidebands and more bandwidth, but better noise immunity (wideband FM).

Using Bessel functions of the first kind, the FM wave expands into a carrier plus an infinite set of sidebands at f_c ± n f_m:

Bessel expansions(t) = A_c Σ_n=−∞^∞ J_n(β) cos[2π(f_c + n f_m) t]

Component	Frequency	Amplitude
Carrier	f_c	A_c J₀(β)
1st sidebands (n = ±1)	f_c ± f_m	A_c J₁(β)
2nd sidebands (n = ±2)	f_c ± 2f_m	A_c J₂(β)
n-th sidebands	f_c ± n f_m	A_c J_n(β)

5. Modulation Index, Deviation, and Deviation Ratio

Modulation index (single tone)β = Δf / f_m

Deviation ratio (general message, band-limited to W)D = Δf_max / W

Parameter	Symbol	Meaning
Frequency sensitivity	k_f	Hz of deviation per volt of message
Peak (maximum) deviation	Δf	Largest swing of instantaneous frequency from f_c
Modulation index	β	Δf relative to the modulating tone frequency
Deviation ratio	D	Worst-case β for a general message band W

Broadcast example (Kenya/EU band): FM broadcast uses Δf = 75 kHz maximum deviation and message bandwidth W = 15 kHz, giving D = 5 — the origin of the "modulation index up to 5" seen in broadcast practice.

Significant Sidebands — the 1% Rule

The number of sideband pairs that carry significant power is finite. By convention, sidebands with amplitude |J_n(β)| ≥ 0.01 (1% of the unmodulated carrier) are retained. This defines significant sideband pairs ≈ β + 1.

6. FM Spectrum and Bandwidth

The FM spectrum has a carrier line of amplitude A_cJ₀(β) plus sideband lines at f_c ± nf_m with amplitudes A_cJ_n(β). Unlike AM, the carrier component can vanish for certain β (J₀(β) = 0 at β ≈ 2.405, 5.520, 8.654, …), yet the signal is still perfectly modulated — power has simply moved into the sidebands.

Figure 4 — Single-tone FM spectrum (drawn for β = 2): carrier A_cJ₀(β) at f_c plus sideband pairs at f_c ± nf_m. Sidebands beyond n ≈ β + 1 are negligible (< 1%).

Carson's rule — approximate bandwidthBW ≈ 2(Δf + f_m) = 2(β + 1) f_m (single tone)

General band-limited message (bandwidth W)BW ≈ 2(Δf_max + W) = 2(D + 1) W

Sanity checks: NBFM (β ≪ 1) → BW ≈ 2f_m, same as AM. Broadcast WBFM: 2(75 + 15) kHz = 180 kHz, which is why FM stations are spaced 200 kHz apart.

Constant Power Property

Because the amplitude A_c never changes, the total power of an FM wave is constant and equals the unmodulated carrier power, independent of β:

Total FM power (R = 1 Ω)P_T = A_c²/2 = constant

Modulation only redistributes this fixed power among the carrier and sidebands, as guaranteed by the Bessel identity Σ J_n²(β) = 1.

FM vs AM on power: AM power rises with m_a and wastes up to 67% in the carrier. FM keeps power constant and delivers more of it into information-bearing sidebands as β grows — one reason FM achieves better SNR at the same peak transmitter power.

7. Generation of FM

7.1 Direct Method (Voltage-Controlled Oscillator)

Vary a reactive element of an LC oscillator (varactor diode whose capacitance depends on the message voltage, or a reactance transistor/FET) so that the oscillator frequency follows m(t) directly.

Simple; easily produces large deviation.
Frequency stability is poor (drifts with temperature, supply) → must be stabilized with a frequency-locked loop or AFC.

7.2 Indirect Method (Armstrong, 1936)

Integrate the message, use it to phase-modulate a crystal oscillator at low deviation (NBFM, β small), then multiply the frequency with a chain of frequency multipliers (×n) and mix/translate to the final carrier.

Excellent carrier stability from the crystal.
More complex: multipliers + mixers needed to reach wideband deviation.

Why the integrator? Feeding the message integral to a phase modulator produces FM: φ(t) = k_p∫m(t)dt ⇒ f_i = (1/2π)dφ/dt = (k_p/2π) m(t). A PM modulator preceded by an integrator is an FM modulator.

8. FM Demodulation

Demodulating FM means converting frequency variations back into amplitude variations. A standard chain: limiter → discriminator → envelope detector → LPF.

8.1 Slope (Differentiator) Detector

Differentiate the FM wave: ds/dt is an AM signal whose envelope is proportional to the instantaneous frequency; a diode envelope detector then recovers the message. In practice the differentiation is approximated by a tuned circuit operated on its linear slope of the |H(f)| curve.

8.2 Foster–Seeley Discriminator and Ratio Detector

Balanced discriminator circuits convert frequency deviations above/below f_c into positive/negative output voltages with good linearity; the ratio detector additionally provides amplitude-limiting action.

8.3 Phase-Locked Loop (PLL) Demodulator

The PLL is the modern standard: a VCO tracks the incoming instantaneous frequency; the control voltage that steers the VCO is, after the loop filter, a replica of the message. PLL demodulators offer excellent linearity and are easily integrated.

Standard receiver chain: antenna → RF amp → limiter (removes AM noise — possible only because FM information is not in amplitude) → mixer/IF (10.7 MHz in broadcast FM) → discriminator/PLL → de-emphasis → audio amplifier.

9. FM versus AM — System Comparison

Attribute	Amplitude Modulation (AM)	Frequency Modulation (FM)
Carrier property varied	Amplitude	Frequency (constant amplitude)
Information location	Amplitude / envelope	Density of zero crossings
Modulation index	m_a = A_m/A_c ≤ 1	β = Δf/f_m — unlimited
Bandwidth (single tone)	2f_m (fixed)	2(β+1)f_m (varies with β)
Total transmitted power	Increases with modulation; ≤ 33% efficient	Constant; all power usable
Noise immunity	Poor (noise adds to amplitude)	Excellent (limiter removes AM noise)
Demodulation	Envelope detector (very simple)	Discriminator / PLL (more complex)
Capture effect	None	Stronger station suppresses weaker
Constant transmitted power	No	Yes
Typical services	MW/SW broadcasting, air band	VHF sound broadcasting, PMR, TV audio, telemetry

Trade-off summary: AM buys simplicity with poor noise performance and power waste; FM buys noise immunity and constant power at the price of wider bandwidth and more complex transmitters/receivers. This is the classic bandwidth–noise trade at the heart of analog communication design.

10. Worked Examples

Example 10.1 — Deviation and modulation index

An FM modulator has k_f = 10 kHz/V. A message of amplitude 4 V and frequency 5 kHz modulates a 100 MHz carrier. Find Δf, β, and the instantaneous frequency range.

Δf = k_fA_m = 10 × 4 = 40 kHz

β = Δf/f_m = 40/5 = 8

f_i swings from 100 MHz − 40 kHz = 99.96 MHz to 100 MHz + 40 kHz = 100.04 MHz

Example 10.2 — Bandwidth by Carson's rule

For the signal of Example 10.1, find the transmission bandwidth.

BW ≈ 2(Δf + f_m) = 2(40 + 5) = 90 kHz

Significant sideband pairs ≈ β + 1 = 9, i.e. sidebands out to f_c ± 9f_m = f_c ± 45 kHz (98% power rule)

Example 10.3 — Broadcast FM parameters

A broadcast FM station uses maximum deviation 75 kHz and audio bandwidth 15 kHz. Find the deviation ratio and bandwidth.

D = Δf_max/W = 75/15 = 5

BW ≈ 2(75 + 15) = 180 kHz (hence 200 kHz channel spacing)

Example 10.4 — Power of the FM wave

An FM transmitter has A_c = 100 V across R = 50 Ω. Find the total transmitted power.

P_T = A_c²/(2R) = 100²/(2 × 50) = 100 W — and it stays 100 W no matter how strongly it is modulated.

Example 10.5 — Instantaneous frequency

An FM wave has f_c = 100 MHz and β = 2 with f_m = 1 kHz. The phase is θ(t) = 2πf_ct + β sin(2πf_mt). Show that f_i(t) = f_c + Δf cos(2πf_mt) and evaluate Δf.

f_i = (1/2π)dθ/dt = f_c + βf_m cos(2πf_mt) ⇒ Δf = βf_m = 2 × 1 = 2 kHz

11. Summary of Key Results

Quantity	Formula
FM defining law	f_i(t) = f_c + k_f m(t)
Instantaneous frequency	f_i(t) = (1/2π) dθ(t)/dt
Peak deviation	Δf = k_f A_m
FM wave (single tone)	s(t) = A_c cos[2πf_ct + β sin(2πf_mt)]
Modulation index	β = Δf/f_m (general: D = Δf_max/W)
Spectrum	carrier A_cJ₀(β) at f_c; sidebands A_cJ_n(β) at f_c ± nf_m
Significant sidebands	pairs ≈ β + 1 (1% amplitude rule)
Bandwidth (Carson)	BW ≈ 2(Δf + f_m) = 2(β+1)f_m
Total power	P_T = A_c²/(2R) — constant, independent of β
NBFM vs WBFM	β ≪ 1 (BW ≈ 2f_m) vs β ≫ 1 (wide BW, high noise immunity)
Generation	Direct (VCO/varactor) or indirect (Armstrong: integrate → NBFM PM → multiply)
Demodulation	Limiter → discriminator (slope/Foster–Seeley) or PLL

12. Review Questions

Define frequency modulation and state how it differs fundamentally from amplitude modulation in terms of where the message information resides.
Sketch and label the modulating signal, the carrier, and the FM wave for a single-tone message, indicating the points of maximum and minimum instantaneous frequency.
Define instantaneous frequency and use it to derive the single-tone FM wave s(t) = A_ccos[2πf_ct + β sin(2πf_mt)].
An FM modulator has k_f = 5 kHz/V. The message is 3 V peak at 2 kHz. Find Δf, β, the range of instantaneous frequencies, and the bandwidth by Carson's rule.
Explain why the FM carrier amplitude can fall to zero at certain values of β (e.g. β = 2.405) yet the signal still carries full information.
State Carson's rule and verify it numerically for a broadcast FM signal (Δf = 75 kHz, W = 15 kHz). Why are FM broadcast channels 200 kHz apart?
Prove that the total power of an FM signal is independent of the modulation index, and comment on the engineering significance of this result.
Describe the direct and indirect (Armstrong) methods of FM generation, highlighting the stability trade-offs.
Draw the block diagram of an FM receiver front end and explain why a limiter can be used before demodulation — and why the same trick fails for AM.
Compare AM and FM in a table under the headings: bandwidth, power efficiency, noise immunity, transmitter complexity, receiver complexity, and typical applications.

Selected answers: Q4 — Δf = 15 kHz, β = 7.5, f_i ∈ [f_c−15 kHz, f_c+15 kHz], BW ≈ 2(15+2) = 34 kHz. Q5 — J₀(2.405) ≈ 0, so the carrier line vanishes while sidebands carry the power (ΣJ_n² = 1).

Introduction to Frequency Modulation (FM)

EEEN 462 — Analog Communications | 4th Year Study Guide