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Analog Modulation

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Description

A sinusoidal message \(m(t) = A_m \sin(2\pi f_m t)\) is placed on a sinusoidal carrier of amplitude \(A_c\) and frequency \(f_c\) in one of two ways:

\[\text{AM:}\quad s(t) = \left[A_c + m(t)\right]\sin(2\pi f_c t)\]
\[\text{FM:}\quad s(t) = A_c \sin\!\left(2\pi f_c t - \beta\cos(2\pi f_m t)\right), \qquad \beta = \frac{\Delta f}{f_m},\quad \Delta f = k_f A_m\]

so the FM signal's instantaneous frequency is \(f_i(t) = f_c + k_f\,m(t)\). The sensitivity is fixed at \(k_f = 0.05\) Hz per unit of message amplitude.

  • Modulation selects AM (red), FM (blue), or AM + FM to stack both signals on the same scale.
  • Carrier amplitude Ac (0–200) and Message amplitude Am (0–200) are in arbitrary amplitude units.
  • Carrier frequency fc and Message frequency fm (1–30 Hz) set the frequencies. The plots span 1 s, so a frequency in hertz equals the number of cycles you see.

In AM mode, the dashed envelope \(\pm[A_c + m(t)]\) follows the message. The read-outs give the modulation index \(\mu = A_m/A_c\), the bandwidth \(B = 2f_m\) and the share of power carried by the sidebands. In FM mode, the amplitude stays at \(A_c\). Instead, the cycles bunch together where the dashed message is high. The read-outs give \(\Delta f\), \(\beta\), the range of \(f_i\), and Carson's bandwidth \(B \approx 2(\Delta f + f_m)\). The lower plot is the amplitude spectrum. AM has a line at \(f_c\) and one at each of \(f_c \pm f_m\). FM has lines at \(f_c \pm n f_m\) with heights \(A_c|J_n(\beta)|\). The gold bar marks the bandwidth.

The frequencies are scaled down so that single carrier cycles are visible. Real broadcast stations use far higher carriers, near 1 MHz for AM and 88–108 MHz for FM. Analog modulation is not a separate learning-graph concept. It builds on Sinusoidal Signals, Signal Amplitude and Signal Frequency in Chapter 2: Introduction to Signals and Systems. It also applies the Frequency Domain view of Chapter 6: Fourier Analysis Fundamentals. There, multiplying by a carrier shifts a spectrum and creates sidebands.

Lesson Plan

Learning Objectives

By the end of this lesson, students will be able to:

  1. Calculate the AM modulation index \(\mu = A_m/A_c\) and predict the envelope limits \(A_c \pm A_m\)
  2. Predict the frequencies and heights of the AM spectral lines (\(f_c\) with height \(A_c\), and \(f_c \pm f_m\) with height \(A_m/2\)) and the AM bandwidth \(2f_m\)
  3. Calculate the FM peak deviation \(\Delta f = k_f A_m\), the modulation index \(\beta = \Delta f/f_m\) and Carson's bandwidth \(2(\Delta f + f_m)\)
  4. Compare AM and FM in the time domain and the frequency domain: which property of the carrier carries the message, and how the bandwidth depends on \(A_m\) and \(f_m\)
  5. Explain over-modulation (\(\mu > 1\)) in AM and the vanishing FM carrier line at \(\beta \approx 2.405\)

Target Audience

College undergraduates in a first signals and systems or communications course, after sinusoids and the Fourier transform have been introduced. Plan 25–35 minutes.

Prerequisites

  • Amplitude, frequency and phase of a sinusoid
  • The product-to-sum identity \(\sin a \sin b = \tfrac{1}{2}[\cos(a-b) - \cos(a+b)]\)
  • Reading a one-sided amplitude spectrum (line height = amplitude of that sinusoid)

Activities

Activity 1: AM lines (6 min). Select AM and set \(A_c\) = 100, \(A_m\) = 50, \(f_c\) = 20 Hz, \(f_m\) = 2 Hz. The envelope swings between 50 and 150, \(\mu\) = 0.50, and the spectrum shows 25.0 at 18 Hz, 100.0 at 20 Hz and 25.0 at 22 Hz, so \(B\) = 4.0 Hz. Change \(f_m\) to 4 Hz: the sidebands move to 16 and 24 Hz and \(B\) = 8.0 Hz. Then set \(A_m\) = 80: the sidebands grow to 40.0 and the sideband power rises to 24.2%, but the lines do not move and the carrier stays at 100.0.

Activity 2: Over-modulation (5 min). Keep AM, set \(f_m\) = 2 Hz and raise \(A_m\) to 100. Now \(\mu\) = 1.00, the envelope just touches zero, and the sidebands carry 33.3% of the power. Raise \(A_m\) to 150 (\(\mu\) = 1.50). \(A_c + m(t)\) now dips to \(-50\), the dashed envelope lines cross, and the waveform no longer has the shape of the message. The spectrum still has only three lines (75.0, 100.0, 75.0), so the spectrum alone does not reveal the problem.

Activity 3: FM lines and the carrier null (8 min). Select FM with \(A_c\) = 100, \(A_m\) = 50, \(f_c\) = 20 Hz, \(f_m\) = 2 Hz. Then \(\Delta f\) = 2.50 Hz, \(\beta\) = 1.25 and \(f_i\) runs from 17.50 to 22.50 Hz. The carrier line is 64.6. The pairs at 18/22, 16/24 and 14/26 Hz are 51.1, 17.1 and 3.7, and Carson's \(B\) = 9.0 Hz. Raise \(A_m\) slowly to 96 (\(\Delta f\) = 4.80 Hz, \(\beta\) = 2.40). The carrier line falls to about 0.3, while the time plot still peaks at 100. Then set \(A_m\) = 50 and \(f_m\) = 5 Hz. \(\Delta f\) stays 2.50 Hz, but \(\beta\) drops to 0.50, the carrier line rises to 93.9, and only the 15 and 25 Hz pair (24.2) stays large.

Activity 4: AM and FM side by side (6 min). Select AM + FM and set \(f_c\) = 10 Hz, \(f_m\) = 2 Hz, \(A_m\) = 100. The AM envelope swings from 0 to 200. The FM signal keeps a peak of 100, while its frequency swings from 5 to 15 Hz (\(\Delta f\) = 5.00 Hz). Find the moments where the message is highest. AM is tallest there, and FM has its shortest cycles.

Assessment

  1. An AM signal has \(A_c\) = 100, \(A_m\) = 80, \(f_c\) = 20 Hz and \(f_m\) = 4 Hz. Give \(\mu\), the line frequencies and heights, and the bandwidth. (Answer: \(\mu\) = 0.8; lines at 16, 20 and 24 Hz with heights 40, 100 and 40; \(B = 2f_m\) = 8 Hz. Check it in the sim.)
  2. Show that the sidebands of a tone-modulated AM signal carry the fraction \(\mu^2/(2+\mu^2)\) of the total power, and evaluate it at \(\mu = 1\). (Answer: the carrier power is \(A_c^2/2\), and each sideband has amplitude \(\mu A_c/2\) and power \(\mu^2 A_c^2/8\). The ratio is \((\mu^2 A_c^2/4)/(A_c^2/2 + \mu^2 A_c^2/4) = \mu^2/(2+\mu^2)\), which is 1/3 (33.3%) at \(\mu\) = 1.)
  3. With \(k_f\) = 0.05 Hz per unit, \(A_m\) = 120 and \(f_m\) = 3 Hz, find \(\Delta f\), \(\beta\) and Carson's bandwidth. (Answer: \(\Delta f\) = 6 Hz, \(\beta\) = 2.0, \(B \approx 2(6 + 3)\) = 18 Hz. At \(f_c\) = 20 Hz the carrier line is \(100\,|J_0(2)|\) = 22.4.)
  4. Why does the FM signal's peak stay at \(A_c\) while its carrier line shrinks as \(A_m\) grows? (Answer: FM changes only the phase, so the envelope and the total power \(A_c^2/2\) stay constant. A larger \(\beta\) moves that fixed power from the carrier into more sideband pairs, \(A_c J_n(\beta)\).)
  5. With \(f_m\) = 1 Hz, which message amplitude makes the FM carrier line vanish? (Answer: the first zero of \(J_0\) is at \(\beta \approx 2.405\), so \(\Delta f = 2.405\) Hz and \(A_m = 2.405/0.05 \approx 48\). At \(A_m\) = 48 the sim gives \(\beta\) = 2.40 and a carrier line of about 0.3.)

Common Misconceptions

"A louder message widens an AM signal." In AM the sidebands always sit at \(f_c \pm f_m\). Raising \(A_m\) from 50 to 150 (Activities 1 and 2) makes them taller but leaves \(B = 2f_m\) unchanged. In FM, the same change raises \(\Delta f\) from 2.50 to 7.50 Hz and Carson's bandwidth (at \(f_m\) = 2 Hz) from 9.0 to 19.0 Hz.

"The carrier is always the strongest line." In AM the carrier line is always \(A_c\). In FM its height is \(A_c|J_0(\beta)|\), which can be zero. Activity 3 makes the carrier vanish at \(\beta\) = 2.40 while the waveform keeps its full amplitude.

References

  1. Amplitude modulation - Wikipedia - the AM equation, modulation index, sidebands and over-modulation
  2. Frequency modulation - Wikipedia - instantaneous frequency, peak deviation and the Bessel-function spectrum of tone FM
  3. Carson bandwidth rule - Wikipedia - the estimate \(B \approx 2(\Delta f + f_m)\) shown by the gold bar
  4. Bessel function - Wikipedia - \(J_n(\beta)\), which sets the heights of the FM lines, and the zeros of \(J_0\)
  5. Sideband - Wikipedia - why multiplying by a carrier creates lines at \(f_c \pm f_m\)
  6. The Scientist and Engineer's Guide to Digital Signal Processing - Steven W. Smith - free textbook on the Fourier transform and spectra used to read the lower plot
  7. Analogue Modulation - CDT design guide - the introduction cited by the original version of this page

Development Notes

Prompt history

The original sketch (2025) was generated from this prompt:

Develop a p5.js simulation that demonstrates analog modulation examples, focusing on amplitude or frequency modulation schemes. Incorporate sliders for adjusting variables such as Ac (amplitude of the carrier signal), Am (amplitude of the modulating signal), fc (frequency of the carrier signal), and fm (frequency of the modulating signal). Ensure the ranges for these variables are selected carefully to facilitate seamless interactivity.

2026-10-14 upgrade to p5.js 2.3.2. The canvas is now responsive, and the entry file analog-mod.html now redirects to main.html. The unused p5.sound library was removed, because the sketch never made or analyzed sound. The slider ranges and defaults are unchanged. The amplitudes were pixel heights in the original. They are now amplitude units on labeled axes, and the "cycles per screen" frequencies are hertz over a 1 s window. An AM / FM / AM + FM selector, the read-outs and the amplitude spectrum are new. The FM equation was corrected. The original used the phase term \((A_m/f_c)\sin(2\pi f_m t)\), which made the modulation index depend on the carrier frequency (10 rad at the default settings). The new equation is the textbook form, with \(\beta = k_f A_m/f_m\) and \(k_f\) = 0.05 Hz per unit.