An amplitude-modulated radio signal with carrier frequency C, which includes frequencies from 0 to F, will use output frequencies in the range C-F to C+F, or a total bandwidth of 2F. A modulation approach called single-sideband modulation omits either the frequencies below C or those above C, and simply transmits the others, on the basis that the frequencies on the other side of C are "redundant".
It would seem, though, that there is information content in the seemingly "redundant" frequencies. For example, if the signal to be modulated on a 1MHz carrier was a sine waves at 100Hz, an AM signal would contain two frequencies: 999,900Hz and 1,000,100Hz. Receiving both frequencies and demodulating them would a 100Hz signal whose phase matched that of the original.
If the signal were single-sideband modulated (let's assume upper), then the modulated signal would simply be a continuous 1,000,100Hz signal. Although a receiver which was tuned to precisely 1,000,000Hz would be able to detect that the signal was a 100Hz signal, I see no means by which it could determine anything about the phase of it.
On the other hand, it would seem like it would be possible to have two signals amplitude-modulated in the same bandwidth if the carrier waves were 90 degrees out of phase, provided that the receiver could discern which carrier wave was which. If the signals to be modulated were devoid of DC content, one could obtain such a result by having the base level of one carrier substantially exceed that of the other. The receiver would be phase-locked onto the first signal when the primary (0 degrees) carrier strength was at maximum.
If one can make use of two simultaneous analog communications channels, would amplitude modulation of two signals with carrier frequencies 90 degrees out of phase provide the same level of bandwidth efficiency as single-sideband modulation? What other tricks exist?
(BTW, I'm pondering the notion of performing spread-spectrum transmission by amplitude-modulating a medium-frequency signal (e.g. 100,000-250,000Hz) on a ~900Mhz carrier. Most "spread-spectrum" receivers I've seen are limited to receiving a single channel at once, but I would think that using analog modulation and demodulation would allow for a DSP to process many channels simultaneously). To get optimal results, however, one would probably have to be able to accurately determine the relative phases of the signals one was receiving.
Your perfectly single-sideband suppressed-carrier modulated sinusoid certainly has a phase which can be measured. However, what you cannot tell is what the contributions of that measured phase from the audio input and the RF oscillator were.
There is another form of single-sideband modulation, in which not only one sideband but also the carrier component is transmitted. This provides a reference which can be used to synchronize the receive LO to the transmit one - normally done to insure exact tuning, but it would also give you the ability to recover the original audio phase.
It is also quite possible, especially with modern DSP gear, to transmit two separate audio channels, one on each side band. This is commonly called independent sideband modulation (ISB).
Many spread spectrum implementations are DSP based and capable of receiving multiple channels at once - GPS being a good example.