Satish Babu Korada

Polar Codes are Optimal for Lossy Source Coding

We consider lossy source compression of a binary symmetric source using polar codes and a low-complexity successive encoding algorithm. It was recently shown by Arikan 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 multiterminal problems including the binary Wyner-Ziv and the binary Gelfand-Pinsker problems. Our results extend to general versions of these problems.

Polar Codes: Characterization of Exponent, Bounds, and Constructions

Polar codes were recently introduced by Arıkan. They achieve the symmetric capacity of arbitrary binary-input discrete memoryless channels under a low complexity successive cancellation decoding strategy. The original polar code construction is closely related to the recursive construction of Reed-Muller codes and is based on the 2 × 2 matrix of the given equation. It was shown by Arıkan and Telatar that this construction achieves an error exponent of 1/2, i.e., that for sufficiently large blocklengths the error probability decays exponentially in the square root of the length. It was already mentioned by Arıkan that in principle larger matrices can be used to construct polar codes. A fundamental question then is to see whether there exist matrices with exponent exceeding 1/2. We characterize the exponent of a given square matrix and derive upper and lower bounds on achievable exponents. Using these bounds we show that there are no matrices of size less than 15 with exponents exceeding 1/2. Further, we give a general construction based on BCH codes which for large matrix sizes achieves exponents arbitrarily close to 1 and which exceeds 1/2 for size 16.

Performance of polar codes for channel and source coding

Polar codes, introduced recently by Arıkan, are the first family of codes known to achieve capacity of symmetric channels using a low complexity successive cancellation decoder. Although these codes, combined with successive cancellation, are optimal in this respect, their finite-length performance is not record breaking. We discuss several techniques through which their finite-length performance can be improved. We also study the performance of these codes in the context of source coding, both lossless and lossy, in the single-user context as well as for distributed applications.

Polar codes: Characterization of exponent, bounds, and constructions

Polar codes were recently introduced by Arikan. They achieve the symmetric capacity of arbitrary binary-input discrete memoryless channels under a low complexity successive cancellation decoding scheme. The original polar code construction is closely related to the recursive construction of Reed-Muller codes and is based on the 2 × 2 matrix [1 0 : 1 1]. It was shown by Arikan Telatar that this construction achieves an error exponent of 1/2, i.e., that for sufficiently large blocklengths the error probability decays exponentially in the square root of the blocklength. It was already mentioned by Arikan that in principle larger matrices can be used to construct polar codes. In this paper, it is first shown that any ℓ × ℓ matrix none of whose column permutations is upper triangular polarizes binary-input memoryless channels. The exponent of a given square matrix is characterized, upper and lower bounds on achievable exponents are given. Using these bounds it is shown that there are no matrices of size smaller than 15×15 with exponents exceeding 1/2. Further, a general construction based on BCH codes which for large I achieves exponents arbitrarily close to 1 is given. At size 16 × 16, this construction yields an exponent greater than 1/2.

Polar codes are optimal for lossy source coding

We consider lossy source compression of a binary symmetric source with Hamming distortion function. We show that polar codes combined with a low-complexity successive cancellation encoding algorithm achieve the rate-distortion bound. The complexity of both the encoding and the decoding algorithm is O(N log(N)), where N is the blocklength of the code. Our result mirrors Arikans capacity achieving polar code construction for channel coding.

Robust synchronization of absolute and difference clocks over networks

We present a detailed re-examination of the problem of inexpensive yet accurate clock synchronization for networked devices. Based on an empirically validated, parsimonious abstraction of the CPU oscillator as a timing source, accessible via the TSC register in popular PC architectures, we build on the key observation that the measurement of time differences, and absolute time, requires separate clocks, both at a conceptual level and practically, with distinct algorithmic, robustness, and accuracy characteristics. Combined with round-trip time based filtering of network delays between the host and the remote time server, we define robust algorithms for the synchronization of the absolute and difference TSCclocks over a network. We demonstrate the effectiveness of the principles, and algorithms using months of real data collected using multiple servers. We give detailed performance results for a full implementation running live and unsupervised under numerous scenarios, which show very high reliability, and accuracy approaching fundamental limits due to host system noise. Our synchronization algorithms are inherently robust to many factors including packet loss, server outages, route changes, and network congestion.

The compound capacity of polar codes

We consider the compound capacity of polar codes under successive cancellation decoding for a collection of binary-input memoryless output-symmetric channels. By deriving a sequence of upper and lower bounds, we show that in general the compound capacity under successive decoding is strictly smaller than the unrestricted compound capacity.

Tight Bounds on the Capacity of Binary Input Random CDMA Systems

In this paper, we consider code-division multiple-access (CDMA) communication over a binary input additive white Gaussian noise (AWGN) channel using random spreading. For a general class of symmetric distributions for spreading sequences, in the limit of a large number of users, we prove an upper bound to the capacity. The bound matches the formula obtained by Tanaka using the replica method. We also show concentration of various relevant quantities including mutual information and free energy. The mathematical methods are quite general and allow us to discuss extensions to other multiuser scenarios.

Polar codes for Slepian-Wolf, Wyner-Ziv, and Gelfand-Pinsker

Polar codes, combined with successive cancellation algorithms, are known to be asymptotically optimal for both the channel as well as the lossy source coding problem. The complexity of the encoding and the decoding algorithm in both cases is O(N log(N)), where N is the blocklength of the code. We show that polar codes also achieve optimum performance for the Slepian-Wolf, the Wyner-Ziv, and the Gelfand-Pinsker problem. The optimality of polar codes for these scenarios rests on the fact that polar codes are optimal for both the channel and the lossy source coding problems. Our results extend to general versions of these problems.

Lossless source coding with polar codes

In this paper lossless compression with polar codes is considered. A polar encoding algorithm is developed and a method to design the code and compute the average compression rate for finite lengths is given. It is shown that the scheme achieves the optimal compression rate asymptotically. Furthermore, the proposed scheme has a very good performance at finite lengths. Both the encoding and decoding operations can be accomplished with complexity O(N log N) where N denotes the length of the code.