Message signal and the Envelope of AM wave

Question

Amplitude modulated signal is represented by

$$ x(t) = (E_c + E_m \cdot \cos(2\pi f_m t)) \cdot \cos(2\pi f_c t). $$

At t=0, we have max. amplitude i.e, Ec + Em. Since the envelope is periodic, I expected Ec+Em amplitude to repeat.

But clearly from the equation the cosine is 1 (carrier) only when $t = \frac{(4n+1)pi}{4f_c}$ ,n is integer. And this is not equal to $\frac{(4n+1)pi}{4f_m}$ . So if both cosines (that of message and carrier signals) aren't equal to 1, we will not get same max. amplitude again.

How does the envelope represent the message signal?

Answer 1

The carrier itself need not be aligned with the peaks of the message; the message for AM modulation is the envelope itself and the carrier will have an amplitude that independently changes with time according to the message. The following graphic will help clear this up:

Notice the first two peaks of the envelope indicated with the arrows, how in the first peak the carrier is not aligned with the envelope peak, while the second peak (in this example) it comes closer to the peak. In general the envelope waveform is independent of the carrier, and the envelope itself is the smoothed waveform going from peak to peak of the carrier.