Polar Codes are Optimal for Lossy Source Coding

Abstract

Abstract-We consider lossy source compression of a binary symmetric source using polar codes and the low-complexity successive encoding algorithm. It was recently shown by Arıkan that polar codes achieve the capacity of arbitrary symmetric binary-input discrete memoryless channels under a successive decoding strategy. We show the equivalent result for lossy source compression, i.e., we show that this combination achieves the rate-distortion bound for a binary symmetric source. We further show the optimality of polar codes for various problems including the binary Wyner-Ziv and the binary Gelfand-Pinsker problem.

On Polar Coding for Binary Dirty Paper

“…The performance is compared to the best achievable rate of any coding scheme for binary DP using nested codes. We observed that typically the set F c ∩ F c s , that needs to be retransmitted in the second phase of the scheme [1] is zero for D − p larger than some small threshold. Our main theoretical contribution is an improved analysis compared to that presented in [1] on |F c ∩ F c s | and on its scaling with respect to the blocklength when D is sufficiently large or p sufficiently small.…”

“…The encoder then observes s and obtains a polar codeword s ∈ C s (F s , u Fs (m)) that satisfies the power constraint (e.g., under the individual codeword power constraint, w H (s⊕s ) ≤ D) using a successive cancellation (SC) encoding algorithm which is a randomized version of the standard SC decoding algorithm [2] (as explained in [1], in practice the standard SC decoding algorithm can be used by the encoder without modification, but the proof requires the randomized version of the algorithm). Now the encoder transmits x = s ⊕ s .…”

“…When the assumption F c ⊆ F s is violated, it was suggested [1] to use a two phase transmission scheme. The first phase is identical to the one described above.…”

“…, respectively) is the Bhattacharyya parameter of the i-th sub-channel after 1 n = log N polarization steps of a BSC(D) (BSC(p)) channel. In [1], δ N (D) = 1 − δ 2 III. ANALYSIS OF |F c ∩ F c s | As was explained above, for low delay communications with polar codes using our SCL coding variant of the method in [1], |F c ∩ F c s | needs to be small (ideally zero).…”

“…where h 2 (·) is the binary entropy function (base 2). Following [1], we will assume in this paper that D > p since for the other case, D < p, the capacity as a function of D can be achieved by time sharing with the point with D = 0 and R = 0. In [1, Section VIII] Arikan's polar codes [2] were extended to the problem of binary DP using a polar nested codes structure.…”