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We consider finite-level, symmetric quantization procedures for constructionand decoding of polar codes. Whether polarization occurs in the presence ofquantization is not known in general. Hassani and Urbanke have shown that asimple three-level quantization procedure polarizes and they have proposed acalculation method to obtain a lower bound for achievable rates. We find animproved calculation method for achievable rates and also the exact asymptoticbehavior of the block error probability under the aforementioned simple case.We then prove that certain D-level quantization schemes polarize and give alower bound on achievable rates. Furthermore, we show that a broad class ofquantization procedures result in a weaker form of the polarization phenomenon.

Finite-Level Quantization Procedures for Construction and Decoding of Polar Codes