Egerton University — Faculty of Engineering

Virtual Laboratory: Frequency Modulation (FM)

EEEN 462 — Analog Communication  |  4th Year B.Sc. Electrical Engineering
fc = 38 kHz, Ac = 1.5 V  |  fm: 0–2 kHz  |  Modulation index β: 0–10

1. Laboratory Objectives

By the end of this virtual laboratory, the student should be able to:

  1. Explain the principle of frequency modulation and derive the time-domain expression of an FM wave.
  2. Define and distinguish between the frequency deviation Δf and the modulation index β = Δf/fm.
  3. Generate FM waveforms on the simulator and observe the effect of varying β (0–10) and fm (0–2 kHz) on the modulated signal.
  4. Interpret the FM spectrum in terms of Bessel functions Jn(β), including the carrier J0(β) and sidebands at fc ± n fm.
  5. Estimate the transmission bandwidth using Carson's rule, BT ≈ 2(Δf + fm), and verify it against the number of significant sidebands.
  6. Distinguish narrowband FM (NBFM) from wideband FM (WBFM) and compare FM with AM in bandwidth, power and noise performance.

2. Theory

2.1 Concept of Frequency Modulation

In frequency modulation, the instantaneous frequency of the carrier is varied in proportion to the message signal, while the carrier amplitude remains constant. For a sinusoidal message m(t) = Amcos(2πfmt):

fi(t) = fc + Δf·cos(2πfmt)

where Δf = kfAm/(2π) is the peak frequency deviation from the carrier. The modulated wave is:

s(t) = Ac cos[2πfct + β·sin(2πfmt)]
β = Δf / fm   (modulation index)

2.2 Narrowband vs Wideband FM

TypeConditionCharacteristics
Narrowband FM (NBFM)β << 1 (practically β < 0.5)Bandwidth ≈ 2fm (like AM); low fidelity; used in mobile radio.
Wideband FM (WBFM)β > 0.5Bandwidth much greater than 2fm; high fidelity and noise immunity; used in FM broadcasting.

2.3 Frequency-Domain Representation (Bessel Functions)

Expanding the FM wave as a Fourier series with fundamental fm:

s(t) = Ac Σn=−∞+∞ Jn(β) · cos[2π(fc + n fm)t]

The spectrum consists of the carrier at fc (amplitude AcJ0(β)) and an infinite set of sidebands at fc ± n fm, each with amplitude Ac|Jn(β)|. Key properties of Bessel functions:

βJ0J1J2J3J4Sig. pairs (≈β+1)
0.50.9380.2420.031≈0≈01
1.00.7650.4400.1150.0200.0032
2.00.2240.5770.3530.1290.0343
3.0−0.2600.3390.4860.3090.1324
4.0−0.397−0.0660.3640.4300.2815
5.0−0.178−0.3280.0470.3650.3916
6.00.151−0.277−0.2430.1150.3587
8.00.1720.235−0.113−0.291−0.1059
10.0−0.2460.0430.2550.058−0.22011

Note how J0 falls and the spectrum widens as β increases; at β ≈ 2.4 the carrier component disappears entirely (J0 = 0).

2.4 Transmission Bandwidth — Carson's Rule

BT ≈ 2(Δf + fm) = 2fm(β + 1)

Carson's rule captures about 98% of the transmitted power. For comparison, the significant-spectrum estimate is BT ≈ 2n fm with n ≈ β + 1.

2.5 Power in FM

PT = Ac2/2 = (1.5)2/2 = 1.125 W
Key contrast with AM: The FM amplitude is constant, so the transmitted power does not depend on β or fm — increasing β redistributes the same power over more, weaker sidebands. In AM, more modulation means more total power.

2.6 Practical Context

FM sound broadcasting (88–108 MHz) uses Δf = 75 kHz with fm(max) = 15 kHz, giving β = 5 and BT ≈ 2(75+15) = 180 kHz — hence 200 kHz channel spacing. The wide bandwidth buys excellent immunity to amplitude noise (static), which is why FM audio sounds much cleaner than AM.

Display note: A 38 kHz carrier is far too fast to draw on screen, so the time-domain plot shows a fixed 4 ms window with a display carrier of 20 kHz (y-axis fixed at −2 V to +2 V); the message and deviation behaviour are exactly those of true FM. The spectrum plot uses the true 38 kHz carrier with a fixed ±30 kHz window, with exact Bessel-computed amplitudes.

3. Procedure

3.1 Pre-Lab

  1. Review Sections 2.1–2.5 and write the expressions for s(t), β, Δf and Carson's bandwidth in your notebook.
  2. For fm = 1 kHz, pre-calculate Δf and BT (Carson) for β = 0.5, 1, 2, 4, 6, 8, 10.

3.2 In-Lab Steps

  1. Open the Simulation. Note the defaults: fm = 1 kHz, β = 1, Ac = 1.5 V.
  2. Step A — Waveform observation: Set fm = 1 kHz and sweep β = 0, 0.5, 1, 2, 4, 6, 8, 10. Observe that the envelope amplitude stays constant (contrast with AM) while the instantaneous frequency swings more widely. Sketch the waveform at β = 2.
  3. Step B — Frequency deviation: For each β in Step A, record Δf = β·fm from the readout. Confirm Δf is independent of fm for fixed β by repeating at fm = 500 Hz.
  4. Step C — Spectrum versus β: With fm = 1 kHz, record the carrier amplitude J0(β) and count the significant sideband pairs for β = 0.5, 1, 2, 4, 6, 8, 10. Verify that the carrier vanishes near β = 2.4 and note the 180° phase reversals (negative Jn) reported by the simulator.
  5. Step D — Bandwidth verification: For each setting in Step C, record Carson's bandwidth 2(Δf+fm) and the measured bandwidth 2n fm from the significant sideband count n. Compare the two.
  6. Step E — Effect of message frequency: Fix β = 2. Vary fm = 0.2, 0.5, 1, 1.5, 2 kHz. Observe how the sideband spacing changes, record Δf, and verify that BT scales with fm at fixed β.
  7. Step F — NBFM limit: Set β = 0.2 and compare the spectrum with an AM spectrum (only carrier + one strong pair). Comment.
  8. Answer the discussion questions (Section 5) and write your report.

4. Interactive Simulation

Fixed parameters: carrier fc = 38 kHz, amplitude Ac = 1.5 V. Vary the message frequency fm (0–2 kHz) and the modulation index β (0–10). Time plot: fixed 4 ms window, y-axis −2 V to +2 V, display carrier 20 kHz. Spectrum: true 38 kHz carrier over a fixed ±30 kHz window, Bessel amplitudes Ac|Jn(β)|.

FM type
WBFM
Frequency deviation Δf
—
Carson bandwidth 2(Δf+fm)
—
Significant sideband pairs n
—
Spectrum width 2n·fm
—
Total power Ac²/2
1.125 W
Carrier J0(β)
—

4.1 Waveforms — message (red dashed) and FM signal (blue); constant envelope, varying instantaneous frequency

4.2 Frequency Spectrum (true carrier at 38 kHz, Bessel amplitudes)

5. Guidelines for Report Writing

Your report should be a formal, individually written, typed A4 document with the following structure:

  1. Title Page: Egerton University; Faculty of Engineering; EEEN 462 — Analog Communication; experiment title (Frequency Modulation Virtual Laboratory); your name and registration number; date; lecturer's name.
  2. Abstract: 5–8 lines summarising the experiment and key findings (bandwidth growth with β, constant power, Carson verification).
  3. Objectives: As listed in Section 1.
  4. Theory: In your own words: the FM equation, β and Δf, the Bessel-series spectrum, Carson's rule, and the NBFM/WBFM distinction.
  5. Procedure: Concise description of the steps performed, with the simulator settings used.
  6. Results:
    • Table 1: β sweep at fm = 1 kHz (β, Δf, J0(β), pairs n, 2n fm, Carson BT).
    • Table 2: fm sweep at β = 2 (Δf, spacing, bandwidths).
    • Waveform screenshots at β = 0.5, 2 and 4; spectrum screenshots at β = 2, 4, 8.
    • Graph of BT versus β (0–10) showing both Carson's rule and the 2n fm estimate.
  7. Analysis / Discussion: Answer:
    1. Verify Carson's rule against your measured significant-spectrum width. What percentage of cases agree within ±10%?
    2. Show from your Step A observations that the FM envelope is constant. What does this imply about the transmitted power as β changes?
    3. Explain, using Bessel functions, why the carrier component can vanish even though the carrier "frequency" is always present in the waveform.
    4. At β = 0.2, why does the FM spectrum resemble an AM spectrum? What distinguishes them?
    5. Compare FM and conventional AM in terms of bandwidth, power efficiency and noise immunity. Why is FM preferred for high-fidelity broadcast despite its larger bandwidth?
    6. An FM broadcast transmitter has Δf = 75 kHz and fm(max) = 15 kHz. Compute β and BT and justify the 200 kHz channel spacing.
  8. Conclusion: Relate findings to the objectives.
  9. References: Cite the course textbook and this virtual laboratory (IEEE style).
Presentation: Typically 8–12 pages including figures. All figures must be numbered, captioned and referenced in the text. Indicative marking: Theory 15%, Procedure 10%, Results 35%, Discussion 30%, Presentation 10%.

6. References

  1. L. W. Couch II, Digital and Analog Communication Systems, 8th ed., Pearson, 2013.
  2. S. Haykin and M. Moher, Communication Systems, 5th ed., Wiley, 2009.
  3. B. P. Lathi and Z. Ding, Modern Digital and Analog Communication Systems, 4th ed., Oxford University Press, 2009.
  4. EEEN 462 Course Notes, Department of Electrical and Electronic Engineering, Egerton University.