Amplitude Modulation

An interactive demo which enables you to both see and hear the result of multiplying a sine wave with a cosine amplitude envelope.

Physics Music Engineering Waves Signals

The demo above shows three waveforms. From top to bottom, they are coloured blue, red and purple, and are mathematically described by the following equations

\[ y_1 = \sin{(2\pi f_1 t)} \] \[ y_2 = 1 + A_2\cos{(2\pi f_2 t)} \] \[ y_{1\times2} = y_1 \times y_2 = \sin{(2\pi f_1 t)} \left(1 + A_2\cos{(2\pi f_2 t)}\right) \]

where \( f1 \) and \( f2 \) are the frequencies of each wave, \( A_2 \) is the amplitude of the second wave, and \( t \) is the time. The amplitude of the first wave is 1.

You can check the "Sound on/off" checkbox to hear what \( y_{1\times2} \) sounds like (this feature may not work in older browsers or Internet Explorer). You should be able to hear a note that has a frequency of \( f_1 \), that is going up and down in loudness at a rate of \( f_2 \). For example, if the value of \( f_2 \) is 1Hz, the volume should go to zero once every second.

The \( y_2\) waveform is acting as a modulator and creates an amplitude envelope. This means its magnitude is determining the magnitude of \( y_{1\times2} \). When the value of \( A_2 \) is zero, the modulator has a constant value of 1, and the line \( y_{1\times2} \) is exactly equal to \( y_1 \) meaning you will not hear any variation in loudness.

If you increase \( f_2 \) higher and higher, you may begin to hear two discernible frequencies, and the tone sounds a little like a dial tone. This can be explained by understanding that the trigonometric identity

\[ \sin{\left(\frac{x+y}{2}\right)}\cos{\left(\frac{x-y}{2}\right)} = \frac{1}{2} \left[\sin{(x)} + \sin{(y)}\right] \]

allows us to write the equation for \( y_{1\times2} \) as a sum of sine waves, which is equivalent to playing two tones of different frequencies. The fact that rapidly modulating the amplitude of one wave results in a waveform identical to playing two notes of different frequencies is quite remarkable and for more information on such wave interference, please see our wave interference and beat frequency demo.