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.


  • $\begingroup$ Consider what happens to the SNR of an individual symbol of you send it twice. $\endgroup$ – Dan Boschen Mar 28 '20 at 2:31
  • $\begingroup$ Multiplexing is about sharing a channel. It's unrelated to the SNR. $\endgroup$ – MBaz Mar 28 '20 at 2:33
  • $\begingroup$ @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) $\endgroup$ – Dan Boschen Mar 28 '20 at 2:41
  • $\begingroup$ @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? $\endgroup$ – user53203 Mar 28 '20 at 3:14
  • 1
    $\begingroup$ @DanBoschen Agreed! :-) I just wanted the OP to be aware that repetition requires additional bandwidth and energy to be of benefit. $\endgroup$ – MBaz Mar 28 '20 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.

See @MBaz's good comments under the OP's question clarifying that there really is no actual SNR gain if you consider the total signal power of all messages sent, since the total signal power would need to increased to realize the gain listed above.

  • $\begingroup$ This is very revealing thank you. Ultimately, however, is the gain in SNR limited by the number of quantization levels? I ran some simulations for 16 QAM and under near-perfect detection and no noise, using PCM encoding I was able to achieve a maximum SNR of ~18dB. If I was to increase this further is my only option to increase the number of quantization levels, or can this be achieved through more advanced coding techniques also (I assume more advanced coding only helps in aiding detection) $\endgroup$ – user53203 Mar 28 '20 at 17:41
  • $\begingroup$ No the gain is not limited but the number of levels can introduce an additional noise source. Look into the SNR for an A/D converter based on number of bits used and note how you can equally increase SNR by oversampling (similarly trading bandwidth for SNR) $\endgroup$ – Dan Boschen Mar 28 '20 at 17:47
  • $\begingroup$ @user53203 this post may help you further in understanding that: dsp.stackexchange.com/questions/40259/… $\endgroup$ – Dan Boschen Mar 28 '20 at 18:42
  • $\begingroup$ ok thank you very much. i'll have a look into that link. $\endgroup$ – user53203 Mar 28 '20 at 18:46
  • $\begingroup$ and this one dsp.stackexchange.com/questions/60035/… specifically the plot about "Maximum ADC Signal AGC" where for your case with an 18dB SNR signal you could use 6 bits and you would set the rms of your signal level about -11 dB below full scale to minimize SNR degradation to 0.4 dB. (Putting the quantization noise 10 dB below your system noise floor and balancing that with possible clipping noise assuming your signal is Gaussian distributed) Or use more than 6 bits to reduce the 0.4 dB hit. $\endgroup$ – Dan Boschen Mar 28 '20 at 18:50

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