# 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. (My memory was not fully correct, though. For Message-passing decoders using LLRs with bit widths larger than 4 bits, uniform quantization is indeed approximately optimal. For smaller lengths, you can find better quantizers¹.)

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 "safely" 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 "righter" 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.

¹ Geiselhart, Marvin, et al. “Learning Quantization in LDPC Decoders.” 2022 IEEE Globecom Workshops (GC Wkshps), Dec. 2022. Crossref, https://doi.org/10.1109/gcwkshps56602.2022.10008635.

• This makes sense, thank you for your answer! Commented Jan 3, 2021 at 3:26
• I think OP might have been asking whether it would be better to use a non-linear ADC, such as \$\mu\$-law (en.wikipedia.org/wiki/%CE%9C-law_algorithm) Commented Apr 25, 2023 at 14:56
• @BenVoigt I address that, don't I, in the first sentence! Commented Apr 25, 2023 at 14:58
• @BenVoigt that you find on hardware take shortcuts that do introduce observable bias at BERs that are commonly below 10⁻⁸. At the same time where being true to a correct normal distribution close to zero, we needed to design the optimal "clipping" point (if your received noise value is more than six standard deviations away, it doesn't really matter that much whether it's six or seven or eight standard deviations), so to make best use of our quantized randomness. Sooo long story short, we asked Laurent Schmalen and IIRC he derived that uniformly quantized LLRs are the best. Now, because 3/N Commented Apr 25, 2023 at 15:23
• our LLR happens to be a scalar multiple of the received value, that implies that the same is true for the received value quantization. Now, I must admit I wish I could reproduce that derivation for you here, but I'll be honest and say, I can't. arxiv.org/abs/2208.05186 claims "implementation of a message passing LDPC decoder (e.g., min-sum decoder) the messages are usually quantized using a uniform quantization", and that's the best I can do right now. (By the way, the student went on and implemented a 38 GS/s Box-Muller-based AWGN generator+BER-Tester with phenomenally white noise)4/4 Commented Apr 25, 2023 at 15:32