Amplitude Modulation

Explore how multiplying a carrier signal $c(t)$ with a modulating signal $m(t)$ creates a modulated signal $s(t)$ with new frequencies. Use the controls below to set the carrier and modulating wave frequencies. Watch the envelope shape update in the time domain, inspect the sidebands in the frequency domain, and listen to the resulting modulated sound.

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Time-Domain Waveforms
Frequency-Domain Spectrum (Sidebands)
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Amplitude Modulation is a process based on the multiplication of two signals: a low-frequency modulator $m(t)$ and a high-frequency carrier $c(t)$.

The modulated signal is mathematically defined as:

$$s(t) = m(t) c(t) = \cos(2\pi f_m t) \sin(2\pi f_c t)$$

Frequency Sidebands

Using the trigonometric identity: $$\sin(A)\cos(B) = \frac{1}{2}\left[\sin(A-B) + \sin(A+B)\right]$$

We can rewrite the modulated signal as: $$s(t) = \frac{1}{2}\sin(2\pi (f_c - f_m)t) + \frac{1}{2}\sin(2\pi (f_c + f_m)t)$$

Thus, the output signal contains exactly two frequencies: the lower sideband $f_c - f_m$ and the upper sideband $f_c + f_m$ with half the amplitude. The carrier frequency $f_c$ itself is completely suppressed.

Auditory Effect: When $f_m$ is below 20 Hz, you hear a rhythmic pulsation in volume (tremolo). As $f_m$ enters the audio range, you hear a complex, metallic, and often dissonant double-tone consisting of the sidebands.