We consider the problem of using polar codes with higher order modulation over AWGN channels. Unlike prior work, we focus on using modulation independent polar codes. That is, the polar codes are not re-designed based on the modulation used. Instead, we propose bit-permuted coded modulation (BPCM): a technique for using the multilevel coding (MLC) approach for an arbitrary polar code. The BPCM technique exploits a natural connection between MLC and polar codes. It involves applying bit

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... ons prior to mapping the polar code to a higher order modulation. The bit permutations are designed, via density evolution, to match the rates provided by various bit levels of the higher order modulation to that of the polar code. We demonstrate performance of the BPCM technique using link simulations and density evolution for the AWGN channel. We compare the BPCM technique with the bit-interleaved coded modulation (BICM) technique. When using polar codes designed for BPSK modulation, we show gains for BPCM over BICM with random interleaver of up to 0.2 dB, 0.7 dB and 1.4 dB for 4-ASK, 8-ASK, and 16-ASK respectively.