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