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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.

Jonah F
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  • Consider what happens to the SNR of an individual symbol of you send it twice. – Dan Boschen Mar 28 '20 at 02:31
  • Multiplexing is about sharing a channel. It's unrelated to the SNR. – MBaz Mar 28 '20 at 02:33
  • @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) – Dan Boschen Mar 28 '20 at 02:41
  • @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? – Jonah F Mar 28 '20 at 03:14
  • @DanBoschen What you describe is not multiplexing, but coding. – MBaz Mar 28 '20 at 15:40
  • @DanBoschen Repetition coding provides zero SNR gain. – MBaz Mar 28 '20 at 15:40
  • @MBaz Agreed but to his question of using FDM or any other multiplexing scheme for this purpose of increasing SNR that is valid, no? – Dan Boschen Mar 28 '20 at 15:41
  • In my opinion, that is stretching the definition of multiplexing too far, and there's already a term for what you describe (repetition coding). But it's a matter of taste I guess. – MBaz Mar 28 '20 at 15:41
  • Repetition certainly does provide SNR gain = that is what processing gain is and predetection combining is essentially. Consider combining two antenna "channels" and getting a 3 dB gain. The distinction is doing the operations pre-detection. – Dan Boschen Mar 28 '20 at 15:43
  • @DanBoschen You only get a gain if you transmit $N$ symbols each with the same energy as the non-coded symbol. If you keep the total energy the same (that is, you transmit symbols with energy $E_s/\sqrt{N}$), then you get zero gain. – MBaz Mar 28 '20 at 15:46
  • To obtain an SNR gain you need to use a better code than repetition. For example, when using a (7,4) Hamming code you transmit the 7-bit code word using the same energy as the original 4-bit data word, and you still get an SNR increase. – MBaz Mar 28 '20 at 15:49
  • @MBaz Agreed-- that would be the intention. (So is trading bandwidth with SNR and as you point out transmitting more energy but this would be the same energy as transmitting M messages over M channels---- so if we choose to use the M channels for the same user with his 1 message it would increase the SNR by 10Log(M) if it was a AWGN channel-- I assumed this was the core question the OP had) – Dan Boschen Mar 28 '20 at 15:52
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    @DanBoschen Agreed! :-) I just wanted the OP to be aware that repetition requires additional bandwidth and energy to be of benefit. – MBaz Mar 28 '20 at 16:00
  • @MBaz but I see your good distinction in comparing coding gain such as your example of a repetition code versus Hamming code. I am considering a layer lower where we get SNR increase from spread spectrum and multiple antenna channel combining for example and assume that was the line of thinking of the OP’s question. – Dan Boschen Mar 28 '20 at 16:02
  • @DanBoschen Yes, of course, those techniques would work too. – MBaz Mar 28 '20 at 16:09
  • @MBaz I updated my answer with your good clarification – Dan Boschen Mar 28 '20 at 16:14

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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.

MBaz
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Dan Boschen
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  • 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) – Jonah F Mar 28 '20 at 17:41
  • 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) – Dan Boschen Mar 28 '20 at 17:47
  • @user53203 this post may help you further in understanding that: https://dsp.stackexchange.com/questions/40259/what-are-advantages-of-having-higher-sampling-rate-of-a-signal/40261#40261 – Dan Boschen Mar 28 '20 at 18:42
  • ok thank you very much. i'll have a look into that link. – Jonah F Mar 28 '20 at 18:46
  • and this one https://dsp.stackexchange.com/questions/60035/how-to-adjust-receiver-gains-to-avoid-saturation-and-quantization-noise-to-optim/63086#63086 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. – Dan Boschen Mar 28 '20 at 18:50
  • Hm, not sure I quite understand (I'm a first-year EE for context). perhaps more context might help. currently, I have a 100s message signal (max freq. ~1.5hz), which I sample at 3Hz with 16 quantization levels. I then use PCM/grey coding to encode the quantized signal and then modulate via 16-QAM . The message signal is subject to AWGN upon transmission. With this setup and ideal conditions (zero noise) I see the following IQ diagram, SNR = 18.8, whereas with the additional noise I see the following IQ diagram, SNR = 15.3 ... – Jonah F Mar 28 '20 at 20:55
  • ... so could you ELI(A first-year EE student) how one might more effectively optimize the quantisation process to generate a higher SNR. I'm ultimately aiming for around 30dB SNR, which Im guessing would require 64-QAM in addition to quantization optimization gains. sorry to keep asking questions, i'm really interested in wireless comms. ! – Jonah F Mar 28 '20 at 20:59
  • Ok got you- we are probably biting off too much. In simpler terms when you quantize you can picture that as simply adding an additional uncorrelated noise source (in most cases). This noise adds in power to the AWGN. If you choose enough quantization levels so that the quantization noise is 10 dB below AWGN (for example) then you only regard the noise by 0.4 dB (sum those two in power to see this: 10Log10(1+ 10^(-10/10)). But read through my links related to this that I listed above and ask any other q’s there – Dan Boschen Mar 28 '20 at 21:02
  • We also want to avoid a lengthy back and forth on here as that is discouraged on SE. So do your research and ask any additional questions as new specific ones if possible. – Dan Boschen Mar 28 '20 at 21:05
  • Ok Thanks, ill look into it more and ask more questions under those threads if needs be – Jonah F Mar 28 '20 at 21:15