AM Modulation: Understanding Envelopes and Detection

[Khaled Kord]

i just had my first Signals processing Lecture, during AM modulation part my prof said that:
V_AM(t) = (V_main(t) + 1) * V_Carrier
while V_Carrier = sin (omega * t)

1-is there a proof that we get the envelop of a function by adding one then multiplying by some trig value?
2- what's the difference between main message function m(t) = A Cos(omega *t) and the term V(t) = A Sin(omega *t) ?

 

[Charles Link]

A quick google of the topic was somewhat helpful with a diagram that it showed. To answer the first question, it appears if a carrier signal with steady amplitude is received, this will be seen (after demodulation) as a steady DC signal with zero ac (audio frequency) signal. The "1" in your equation is some arbitrary $V_{DC}$, and $V_{AM}(t)$ is the r-f signal. The envelope is the sum of $V_{main}(t)$ ,which is the ac (audio) signal, and $V_{DC}$. When the signal is demodulated, the envelope including the $V_{DC}$ is recovered (e.g. with a half-wave rectifier), but that demodulated waveform can always be put into an ac coupled circuit where the $V_{DC}$ gets blocked by the capacitor of the C-R (ac coupled) circuit and the ac (audio) portion of the envelope remains without the $V_{DC}$. I am not a communications or r-f expert, but I think I correctly answered your first question, at least in explaining where the equation comes from. Hopefully this was helpful. editing... To answer your second question, the main message function $V_{main}(t)$ $\$ is at audio frequencies. Your "omega" in that equation will be limited to 60 kHz or thereabouts (i.e. $20 \ Hz<f_{audio}<10 \ kHz$ and $\omega=2 \pi f$). The carrier "omega" for $V_{carrier}(t)$ meanwhile is in the Megahertz(radio frequency) range.

 

[Baluncore]

1-is there a proof that we get the envelop of a function by adding one then multiplying by some trig value?

If the modulation, V_main(t), is restricted in amplitude to be between +/–1 then, when the one is added, the modulation becomes unipolar. After the multiplication by the sinewave carrier the phase of the modulated carrier is never reversed, so the envelope of the signal peaks follows the modulation amplitude.

 

[Khaled Kord]

If the modulation, V_main(t), is restricted in amplitude to be between +/–1 then, when the one is added, the modulation becomes unipolar. After the multiplication by the sinewave carrier the phase of the modulated carrier is never reversed, so the envelope of the signal peaks follows the modulation amplitude.

i think i didn't explain my first question right, i was asking that: who said the peaks of the V_AM(t) are tangent to the main signal V_main(t)? who said V_main(t) is an envelop? why doesn't it intersect in some arbitrary points like i draw in red (attached picture) ?

 

[Baluncore]

Three features of AM detection regenerate the original AC modulation waveform.
1. Automatic Gain Control stabilises the amplitude of the received AM signal.
2. A peak follower tracks and so detects the envelope of the AM signal.
3. The detected envelope is AC coupled.

Your diagram shows peaks with amplitude Ac+Am and Ac-Am, that is addition and subtraction, but AM is multiplication.