Electronic – Time-limited signal Fourier transform

The Fourier transform is linear, which means that if you sum two signals, then also their spectrum will be the sum.

$$ \mathscr{F} [x(t) + y(t)] = \mathscr{F}[x(t)] + \mathscr{F}[y(t)] $$

If you consider the power, you can have two cases:

1. The signals have different frequency
   Then their power just sums up, because: $$ P_{tot} = P_1 cos (\omega _1 t + \phi _1) + P_2 cos (\omega _2 t + \phi _2) $$

2. The signals have the same frequency
   Then if they are in phase you have: $$ P_{tot} = P_1 cos (\omega _0 t + \phi _0) + P_2 cos (\omega _0 t + \phi _0) = (P_1 + P_2) \cdot cos(\omega _0 t + \phi_0) $$ If they're out of phase, the resulting power will be lower and precisely: $$ P_{tot} = P_1 cos (\omega _0 t + \phi _1) + P_2 cos (\omega _0 t + \phi _2) $$ and setting arbitrarily \$\phi_1 = 0\$ we obtain: $$ P_{tot} = P_1 cos (\omega _0 t) + P_2 cos (\omega _0 t + \Delta \phi) $$ which for \$\Delta \phi = \pi\$ gives the subtraction of the signals.

It seems like your question is about data transmission, so I'll try to answer it in this context.

You can't transmit information on a single frequency, because technically "single frequency" implies that the sender is turned on forever. As soon as it is switched on/off (corner case of AM) additional frequencies are added. For example a 1 MHz carrier switched on/off (this is called on/off keying or OOK) at 1kHz causes two sharp spikes at 0.999 MHz and 1.001 MHz in the spectrum. The difference between the highest and the lowest frequency is usually called the bandwidth of your signal and in the case of OOK it is twice the modulation frequency. The max. data rate would be 2 kBit/s, because the highest frequency sequence that is possible is 1010101....

If you want to composite such transmission systems onto a shared medium (cable, air) you need to make sure that they don't overlap. in the above example you could place additional transmitters at .998 MHz, 1.002 MHz etc. A practical example is FM radio, the stations are separated by a couple of 100 kHz for this reason.

Multiplexing is about combining N serial bit streams into one with a N-times higher bitrate. Lets say that you want to multiplex 10 2kBit/s (example above) bit streams into one with 20kBit/s. The bandwidth of the signal is than 20kHz and the transmitter will occupy the same amount of spectrum on your shared medium as ten individual transmitters.

At this point, I recommend you to learn more about Fourier Transformation and Shannon Coding Theorem, it will help you to get the connection between time and frequency domain in a data transmission system.