Polar Coded Distribution Matching for Probabilistic Shaping and Stealth Communication

We present a polar coded one-to-many distribution matching scheme. The scheme uses a randomized encoding approach to approximate, in an invertible manner, binary discrete memoryless sources (B-DMSs) using a binary symmetric source. This is accomplished by using polar codes as lossy source codes, and by leveraging their linear structure. Due to the special recursive structure of polar codes, the encoding and decoding complexity of the scheme is of order $\mathcal{O}(N \log N)$, where N denotes the output blocklength. Leveraging the rate-distortion optimality of polar codes, the scheme is shown to be asymptotically optimal for probabilistic shaping and stealth communication over binary output alphabets. Namely, in the limit of large blocklength N, the polar coded scheme is shown to approximate any B-DMS with vanishing Kullback–Leibler divergence and with rate approaching the entropy of the B-DMS.