Channel Polarization under Channel Noise with Memory

The channel polarization behavior of polar codes under noise with memory is investigated. By introducing a genie-aided channel model, we first show that the polarized subchannels still converge to extremal channels under the standard polar coding framework. More importantly, we explicitly quantify the gap between the mutual information achieved by ignoring memory effects and the actual capacity attained after sufficient polarization. It is proven that the channel capacity remains achievable even without prior knowledge of the channel noise. Furthermore, we demonstrate that the polarization rate is slower than that in the binary-input memoryless channel (BMC) case, provided that the channel transition function satisfies certain conditions. In particular, the Bhattacharyya parameter is asymptotically upper-bounded and lower-bounded by a polynomial function and an exponential function with respect to the block length, respectively.