Frequency of the carrier wave changed during amplitude modulation

[Pranav Jha]

While reading about frequency modulation, I found that it was clearly written that the amplitude of the carrier wave remains unchanged. However, i didn't find a statement stating that the frequency of the carrier wave remains unchanged for amplitude modulation. So, is the frequency of the carrier wave changed during amplitude modulation? Also, please explain the relation between the frequency of the modulated wave and the frequency of the information signal (the sidebands confused me)

 

[rcgldr]

During transitions in amplitude, the frequency "appears" to change. An increase in amplitude while the amplitude itself is also increasing appears to be an increase in frequency and vice versa for the other combinations, with the end result that during a change in amplitude, the result is a fuzzy zone of frequency (bandwidth) that gets wider depending the rate of increase or decrease. This is the reason why morse code transmitters ramp up the amplitude over a few milliseconds instead of instantly turning the signal on and off.

 

[Studiot]

While reading about frequency modulation, I found that it was clearly written that the amplitude of the carrier wave remains unchanged. However, i didn't find a statement stating that the frequency of the carrier wave remains unchanged for amplitude modulation. So, is the frequency of the carrier wave changed during amplitude modulation? Also, please explain the relation between the frequency of the modulated wave and the frequency of the information signal (the sidebands confused me)

1) Not only is the amplitude of an FM signal theoretically constant, additional stabilising circuitry is employed to keep it so in better equipment.

2) When amplitude modulation is employed, 2 new frequencies appear, that were not in either the original carrier or the modulating signal. These are called sidebands. I don't know if you have studied yet beats in physics but this is the same phenomenon. When two waves of nearly equal frequency combine, beats occur at the difference between their frequencies. You can here this in the thrumming of engines in an enclosed space, and the beat occurs in the audio spectrum.
With radio transmission the modulating audio signal and the carrier frequencies are quite different so the effect is given a different name.

The amplitude (the quantity we wish to vary) of the carrier is

$$v = {V_c}\sin \left( {{\omega _c}t} \right)$$

If we add a modulating signal to V_c this becomes

$$v = \left( {{V_c} + {V_m}\sin \left( {{\omega _m}t} \right)} \right)\sin \left( {{\omega _c}t} \right)$$

A bit of trigonometry turns this into

$$v = {V_c}\sin \left( {{\omega _c}t} \right) + \frac{{{V_m}}}{2}\cos \left( {{\omega _c} - {\omega _m}} \right)t - \frac{{{V_m}}}{2}\cos \left( {{\omega _c} + {\omega _m}} \right)t$$

This shows that a sinusoidal wave, sinusoidally modulated contains three frequencies.

The original carrier

$${f_c} = {\omega _c}/2\pi$$

The lower side frequency or sideband

$${f_c} - {f_m} = \left( {{\omega _c} - {\omega _m}} \right)/2\pi$$

the upper side frequency or sideband

$${f_c} + {f_m} = \left( {{\omega _c} + {\omega _m}} \right)/2\pi$$

The modulating frequency is not present.

It is worth noting that amplitude modulation represents addition of two waves, frequency modulation represents multiplication.

go well

 

[Carrock]

The amplitude (the quantity we wish to vary) of the carrier is

$$v = {V_c}\sin \left( {{\omega _c}t} \right)$$

If we add a modulating signal to V_c this becomes

$$v = \left( {{V_c} + {V_m}\sin \left( {{\omega _m}t} \right)} \right)\sin \left( {{\omega _c}t} \right)$$

It is worth noting that amplitude modulation represents addition of two waves, frequency modulation represents multiplication.

go well

A minor clarification.

Amplitude modulation involves the the multiplication of the modulation frequency plus a constant offset (to produce the carrier), and the carrier frequency.

 

[Pranav Jha]

1) Not only is the amplitude of an FM signal theoretically constant, additional stabilising circuitry is employed to keep it so in better equipment.

2) When amplitude modulation is employed, 2 new frequencies appear, that were not in either the original carrier or the modulating signal. These are called sidebands. I don't know if you have studied yet beats in physics but this is the same phenomenon. When two waves of nearly equal frequency combine, beats occur at the difference between their frequencies. You can here this in the thrumming of engines in an enclosed space, and the beat occurs in the audio spectrum.
With radio transmission the modulating audio signal and the carrier frequencies are quite different so the effect is given a different name.

The amplitude (the quantity we wish to vary) of the carrier is

$$v = {V_c}\sin \left( {{\omega _c}t} \right)$$

If we add a modulating signal to V_c this becomes

$$v = \left( {{V_c} + {V_m}\sin \left( {{\omega _m}t} \right)} \right)\sin \left( {{\omega _c}t} \right)$$

A bit of trigonometry turns this into

$$v = {V_c}\sin \left( {{\omega _c}t} \right) + \frac{{{V_m}}}{2}\cos \left( {{\omega _c} - {\omega _m}} \right)t - \frac{{{V_m}}}{2}\cos \left( {{\omega _c} + {\omega _m}} \right)t$$

This shows that a sinusoidal wave, sinusoidally modulated contains three frequencies.

The original carrier

$${f_c} = {\omega _c}/2\pi$$

The lower side frequency or sideband

$${f_c} - {f_m} = \left( {{\omega _c} - {\omega _m}} \right)/2\pi$$

the upper side frequency or sideband

$${f_c} + {f_m} = \left( {{\omega _c} + {\omega _m}} \right)/2\pi$$

The modulating frequency is not present.

It is worth noting that amplitude modulation represents addition of two waves, frequency modulation represents multiplication.

go well

Thanks but could you clarify the math further. I think i am starting to get the concept.

 

[Pranav Jha]

A minor clarification.

Amplitude modulation involves the the multiplication of the modulation frequency plus a constant offset (to produce the carrier), and the carrier frequency.

Please state that mathematically