communication
I understand what SSB and DSB mean in frequency domain.
For SSB, the signals appear at one side of the carrier frequency.
For DSB, the signals appear at both sides of the carrier frequency.
So now suppose the signal is a sinusoidal Acos(w1t) at baseband and the carrier frequency is a sinusoidal Bcos(w2*t) where w2 > w1. If these two signals are applied to an SSB mixer and a DSB mixer, then how the resulting SSB and DSB signals look like in time domain?
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.
Regular double sideband with unsuppressed carrier is used by AM radio stations around the world mainly because the detector in the thousands of listener's radios is as simple as a diode, a capacitor and a resistor. The modulation difficulty is unimportant given that there is only one transmitter but, as it happens, this sort of modulation is fairly easy to implement - it's just 2-quadrant multiplication: -
A refinement of this is DSBSC (double sideband suppressed carrier) and is modulated by using a 4-quadrant multiplier. To receive this type of modulation requires more sophisticated electronics than a simple diode detector and hence it makes receivers of this type more expensive. As can be seen below, at certain points in the waveform the carrier reverses phase and this makes the diode detector demodulate incorrectly.
Sideband modulation is more complicated again but, like DSBSC offers better signal-to-noise ratios than regular AM because the transmit power can be focused into just one sideband. Or look at it another way - the overall transmit power can be reduced significantly. Also because the signal occupies less spectrum, with the appropriate filters in the receiver, the received unwanted noise is less.
Best Answer
@ChrisStratton has described time-domain properly. Since OP has asked what it looks like, here's an example of two sinusoidal waves after DSB modulator.
Calling them carrier vs. modulation is arbitrary - usually we consider the lower frequency to be modulated onto the higher frequency.
For DSB, note that there is a phase inversion of the carrier wave when the modulating signal passes through a zero-crossing. If a DC offset is added to the 1kHz signal (in this case \$\pm1V\$) then you end up with AM rather than DSB.
It may look like there is significant energy at 20kHz: there is not. For this sinusoidal case, there is energy only at 19kHz, and an equal amount at 21kHz. Perhaps this is why time-domain plots of DSB or SSB are not particularly useful to your eye & brain.
If you pass this DSB waveform through a bandpass filter, whose passband includes 21000Hz, and whose stopband rejects 19000Hz, you get SSB: a sinewave @ 21000Hz.
If you pass this DSB waveform through a bandpass filter whose passband includes 19000Hz, and whose stopband rejects 21000Hz, you get similar SSB: a sinewave @ 19000Hz.
There are a few other ways to generate SSB than the filter method described above: phasing method, and FM (Weaver method).