# The use of FDM for a single message signal

Suppose I have a single message signal, m(t), that is subjected to AWGN upon transmission.

Is there any way that FDM (or any multiplexing strategy) can be used to ultimately improve the SNR of the demodulated signal at the other end of the transmission line. I'm open to using any sort of modulation technique.

Thanks.

• Consider what happens to the SNR of an individual symbol of you send it twice. Mar 28, 2020 at 2:31
• Multiplexing is about sharing a channel. It's unrelated to the SNR.
– MBaz
Mar 28, 2020 at 2:33
• @MBaz Consider that you could use a multiplexing scheme (such as FMD) to send the same message to the same user N times and in that process achieve the related processing gain (so ultimately a trade of bandwidth with SNR) Mar 28, 2020 at 2:41
• @DanBoschen ahhh yes, so lets say I send the message 3 times over a given bandwidth Since they are all subject to different WGN I could effectively average the three demodulated message signals at the other end. Is there any particular way to process the three messages so that the signals least affected by the channel noise contribute most to the sampled signal at the recieving end? Mar 28, 2020 at 3:14
• @DanBoschen Agreed! :-) I just wanted the OP to be aware that repetition requires additional bandwidth and energy to be of benefit.
– MBaz
Mar 28, 2020 at 16:00

You can simply send the message multiple ($$N$$) times and if all N messages were received at the same SNR you would coherently average the messages for a processing gain in SNR equal to $$10\log_{10}(N)$$ in dB. This is effectively trading bandwidth for SNR as you are using more resources to send the same message. To coherently add you remove the complex carrier phase for each message prior to adding in the average. If the messages were not received at the same SNR (such as if it was a fading channel) you would optimally weight each message by the SNR of each message prior to averaging. This latter point is similar to what occurs in a matched filter in that each sample within a symbol duration is optimally weighted by the SNR for that sample prior to averaging over the symbol duration.