Amplitude Modulation Phenomona

[dmorr]

Hey Everyone,

Can someone explain to me the natural of amplitude modulation in regards to sound. As in one modulator frequencies' amplitude modulating the amplitude of a carrier frequency. Because at low frequencies the modulator only affects the carrier in a way that makes its volume go up and down like tremolo (0.5hz). But at higher speeds like when the carrier is being modulated at 65hz the carrier starts to produce new frequencies known as sum and difference.

Why? What is the nature of this process that makes it happen? Why do you start to hear new frequencies just by turning the volume up and down of a signal at super high speeds?

 

[Bob S]

The product of a small amplitude modulation on a carrier can be represented by $\left(1+A\cos\omega_1t \right)\cos\omega_2t$. This can be rewritten as
$$\left(1+A\cos\omega_2t \right)\cos\omega_1t=\cos\omega_1t +A\cos\omega_2t\cos\omega_1t =\cos\omega_1t + \frac{A}{2} \cos\left(\omega_1-\omega_2 \right)t+\frac{A}{2} \cos\left(\omega_1+\omega_2 \right)t$$
So the tremulo modulation produces sum and difference sidebands at any amplitude modulation frequency. So not being able to hear the sidebands as new frequencies might be just due to sound perception (see http://en.wikipedia.org/wiki/Psychoacoustics.)

 

[dmorr]

Ok, I understand that the sidebands wouldn't be audible to the human ear as they would be only a few hz or less away from the carrier at low volumes.

However, I still can't explain why it happens this phenomena happens, because I don't know how to do trigonometry, so I can't see it from the numbers' perspective.

Could you explain to me where the sidebands get their energy from? Do they take away energy from the carrier the farther they start to move away? Do the sidebands get created because the modulation is so fast that the beats start to shape the signal fundamentally, as opposed to them being so wide that the tone of the carrier is much more clear?

thanks everyone.

 

[Bob S]

The energy in the sidebands does not depend on the modulation frequency. In the above equation for the sidebands, the power in the three frequencies (carrier, lower sideband, upper sideband) is proportional to 1 + A²/4 + A²/4. So a higher amplitude modulation increases power, but changing the frequency of modulation does not.

The cochlea in the inner ear is a mechanical spectrum analyzer, in the sense that different frequencies stimulate different hair cells. This is due to the shape of the cochlea, and its stiffness gradient along its length. Different frequencies have different resonant excitation sites, and excite different hair cells. I can only surmise that when the sidebands are too close to the carrier, the stimulated hair cells are too close to the hair cells stimulated by the carrier, so discerning the sidebands becomes difficult.