Is there an optimal way to quantize log-likelihood ratios of an AWGN channel? My understanding is that given an AWGN channel and BPSK modulation, an LDPC decoder that uses message passing takes as input log-likelihood ratios $L$ of the following form by Bayes' rule:
$$
L=\frac{P(a=1|r)}{P(a=-1|r)}=\frac{2}{\sigma^2}r
$$
where $a\in\{\pm1\}$, $n\sim \mathcal{N}(0,\sigma^2)$, and $r=a+n$.
I have been referencing page 22 of this report, which uses a uniform quantizer and mentions that a larger dynamic range results in larger step sizes assuming a fixed number of bits.
Are there other quantization methods that might be better suited to the Gaussian nature of this channel? Said differently, are there quantization methods that better account for $\sigma^2$?
It depends on your actual decoder architecture, but if memory serves me right, uniform quantization is optimal for Min-Sum or Sum-Product decoders. (Intuitively, kind of makes sense – any other quantization could be modelled as &quot;let's first apply a function to the LLRs and then quantize the result of that uniformly&quot;, and LLRs are what asymptotically achieves maximum likelihood performance.)
The question is where you put your quantization boundaries: If you have a fixed number of bits for your LLRs, do you quantize the region from, say, $-2\sigma$ to $+2\sigma$, because everything larger is sufficiently &quot;safely&quot; the sign-indicated value, anyway? Or do you have a large code word length, and you want to make sure that in sea of large LLRs, where some represent bit errors, the &quot;righter&quot; ones are
still understood as such?
That's a question of looking your design SNR and your target FER; no general answer or formula is known, far as I'm aware.