# AM/FM Modulation Explorer

Interactive AM and FM modulation lab — waveforms, spectra, IQ traces and demodulators for analog communications.

> Canonical page: https://elysiatools.com/en/visualizations/am-fm-modulation

- **Category:** Signal Processing

## Overview

Interactive AM and FM modulation lab covering the two foundations of analog communications, complementing the passive spectrogram-analyzer, the abstract z-transform, and the digital-keying morse-code-oscillator (which uses OOK, the one-bit AM case). Amplitude modulation s_AM(t) = [1 + m·cos(2π f_m t)]·cos(2π f_c t) produces a carrier plus two sidebands (bandwidth 2·f_m), with the index m≤1 constraint to avoid overmodulation; frequency modulation s_FM(t) = cos(2π f_c t + β·sin(2π f_m t)) produces an infinite Bessel-function sideband family at f_c ± n·f_m with peak deviation Δf = β·f_m and Carson-rule bandwidth B ≈ 2(β+1)·f_m; double-sideband suppressed-carrier s_DSB(t) = m·cos(2π f_m t)·cos(2π f_c t) demonstrates coherent detection. The model implements all signal generators, a radix-2 Cooley–Tukey FFT for the magnitude spectrum, and three demodulators: an envelope detector (Hilbert analytic-signal magnitude, carrier DC removed) for AM, a frequency discriminator (instantaneous frequency via the phase derivative of the analytic signal) for FM, and a product/low-pass coherent detector for DSB-SC. Four visualization panels: (1) Time-domain plot of the message (cyan), carrier (amber, faint), and the live modulated signal (white) with an animated cursor. (2) Magnitude spectrum around the carrier showing AM's symmetric sideband pair or FM's Bessel sideband ladder, with the carrier line marked. (3) IQ constellation trace (I=s(t), Q=Hilbert(s)(t)) — AM draws a breathing ring whose radius tracks the message, FM draws a rotating phasor of constant radius, with a live phasor dot. (4) Demodulator output comparing the recovered message against the original (dashed). Adjustable parameters: message frequency f_m (10–2000 Hz), carrier f_c (4–20 kHz), modulation index m or β (0–5), and channel noise (0–1). Six presets: Standard AM (m=0.8), Overmodulated AM (m=1.3, envelope inversion), Narrowband FM (β=0.5), Wideband FM (β=3), DSB-SC with coherent detection, and Noisy FM (noise=0.3). Real-time diagnostics: Carson bandwidth, FM frequency deviation, output SNR, and the FM improvement factor (3/2)·β² illustrating FM's constant-envelope noise immunity. Educational content covers AM sideband structure and the overmodulation limit, FM Bessel spectra and Carson's rule, the noise-immunity trade-off and threshold effect, and the digital descendants (ASK/FSK/PSK/QAM, Wi-Fi OFDM, Bluetooth GFSK, GPS BPSK). Multi-language support (zh, en, es, fr, de, ru, pt).

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