Amplitude Modulation with large carrier and suppressed carrier From what I have read so far Amplitude modulation with large carrier is favaroble because envelope detector can be used at the receiver to derive the message signal and therefore another mixer is not needed at the receiver.

According to my text book one way to find the envelope of a signal is to lowpass filter the absolute value of the modulated signal.

Using low pass filter on the frequency spectrum $\mathcal F\left\{\lvert (m(t)+1)\cos(\omega t)\rvert\right\}$ will give $m(t)+1$ which is the envelope offset by constant unity.


My question is why can't we use the same technique for AM with suppressed carrier modulation? 
Wouldn't low pass filtering of the spectrum of $\mathcal F\left\{\lvert m(t)\cos(\omega t)\rvert\right\}$ give us $m(t)$?
In short: you can use envelope detection on a suppressed-carrier signal, but you'll recover $|m(t)|$, not $m(t)$.

One easy way to see it is by drawing the signals. Draw an arbitrary signal $m(t)$, which takes positive and negative values. Now draw the signal $s(t)=\cos(2\pi f_ct)\cdot m(t)$. To do this, draw $-m(t)$ on top of $m(t)$, and draw a cosine wave that is "enveloped" by $m(t)$ and $-m(t)$.

To find the envelope of $s(t)$, take its positive part only and draw lines connecting the peaks of the carrier. You'll see that what you obtain is $|m(t)|$.

By the way, the same is true for large carrier modulation; it just turns out that $1+m(t)$ is always positive, so $|1+m(t)|=1+m(t)$.