Emre Telatar

On the rate of channel polarization

A bound is given on the rate of channel polarization. As a corollary, an earlier bound on the probability of error for polar coding is improved. Specifically, it is shown that, for any binary-input discrete memoryless channel W with symmetric capacity I(W) and any rate R ≪ I(W), the polar-coding block-error probability under successive cancellation decoding satisfies P<inf>e</inf>(N, R) ≤ 2^−Nβ for any β ≪ 1/2 when the block-length N is large enough.

Polarization for arbitrary discrete memoryless channels

Channel polarization, originally proposed for binary-input channels, is generalized to arbitrary discrete memoryless channels. Specifically, it is shown that when the input alphabet size is a prime number, a similar construction to that for the binary case leads to polarization. This method can be extended to channels of composite input alphabet sizes by decomposing such channels into a set of channels with prime input alphabet sizes. It is also shown that all discrete memoryless channels can be polarized by randomized constructions. The introduction of randomness does not change the order of complexity of polar code construction, encoding, and decoding. A previous result on the error probability behavior of polar codes is also extended to the case of arbitrary discrete memoryless channels. The generalization of polarization to channels with arbitrary finite input alphabet sizes leads to polar-coding methods for approaching the true (as opposed to symmetric) channel capacity of arbitrary channels with discrete or continuous input alphabets.

Polar Codes for the Two-User Multiple-Access Channel

Arikans polar coding method is extended to two-user multiple-access channels. It is shown that if the two users of the channel use Arikans construction, the resulting channels will polarize to one of five possible extremals, on each of which uncoded transmission is optimal. The sum rate achieved by this coding technique is the one that corresponds to uniform input distributions. The encoding and decoding complexities and the error performance of these codes are as in the single-user case: O(nlogn) for encoding and decoding, and o(2-n1/2-ε) for the block error probability, where n is the blocklength.

Polar Codes for the m-User MAC

In this paper, polar codes for the m-user multiple access channel (MAC) with binary inputs are constructed. It is shown that Arikans polarization technique applied individually to each user transforms independent uses of an m-user binary input MAC into successive uses of extremal MACs. This transformation has a number of desirable properties: 1) the “uniform sum-rate” of the original MAC is preserved, 2) the extremal MACs have uniform rate regions that are not only polymatroids but matroids, and thus, 3) their uniform sum-rate can be reached by each user transmitting either uncoded or fixed bits; in this sense, they are easy to communicate over. A polar code can then be constructed with an encoding and decoding complexity of O(n log n) (where n is the block length), a block error probability of o(exp (- n1/2 - ε)), and capable of achieving the uniform sum-rate of any binary input MAC with arbitrary many users. Applications of this polar code construction to channels with a finite field input alphabet and to the additive white Gaussian noise channel are also discussed.

On the construction of polar codes

We consider the problem of efficiently constructing polar codes over binary memoryless symmetric (BMS) channels. The complexity of designing polar codes via an exact evaluation of the polarized channels to find which ones are “good” appears to be exponential in the block length. In [3], Tal and Vardy show that if instead the evaluation if performed approximately, the construction has only linear complexity. In this paper, we follow this approach and present a framework where the algorithms of [3] and new related algorithms can be analyzed for complexity and accuracy. We provide numerical and analytical results on the efficiency of such algorithms, in particular we show that one can find all the “good” channels (except a vanishing fraction) with almost linear complexity in block-length (except a polylogarithmic factor).

Information theoretic upper bounds on the capacity of large extended ad-hoc wireless networks

We derive an information theoretic upper bound on the maximum achievable rate per communication pair in a large extended ad-hoc wireless network. We show that under a reasonably weak assumption on the attenuation due to environment, this rate tends to zero as the number of users gets large

Polar codes for q-ary source coding

Polar coding is a recent channel coding technique invented by Arikan to achieve the ‘symmetric capacity’ of binary-input memoryless channels. Subsequently it was observed by Korada and Urbanke that such codes are also good for lossy channel coding, achieving the ‘symmetric rate distortion’ bound, when the representation alphabet is binary. In this note we extend this result to the case when the representation alphabet is q-ary, for q a prime number.

Dense multiple antenna systems

We consider multiple antenna systems in which a large number of antennas occupy a given physical volume. In this regime the assumptions of the standard models of multiple antennas systems become questionable. We show that for such spatially dense multiple antenna systems one should expect the behavior of the capacity to be qualitatively different than what the standard multiple antenna models predict.

A Simple Converse of Burnashev's Reliability Function

In a remarkable paper published in 1976, Burnashev determined the reliability function of variable-length block codes over discrete memoryless channels (DMCs) with feedback. Subsequently, an alternative achievability proof was obtained by Yamamoto and Itoh via a particularly simple and instructive scheme. Their idea is to alternate between a communication and a confirmation phase until the receiver detects the codeword used by the sender to acknowledge that the message is correct. We provide a converse that parallels the Yamamoto-Itoh achievability construction. Besides being simpler than the original, the proposed converse suggests that a communication and a confirmation phase are implicit in any scheme for which the probability of error decreases with the largest possible exponent. The proposed converse also makes it intuitively clear why the terms that appear in Burnashevs exponent are necessary.

An empirical scaling law for polar codes

Using scaling laws, we obtain estimates of the block error probability of polar codes under successive cancellation decoding. For the binary erasure channel we present an upper and a lower bound for the scaling parameter. Numerically these two bounds match. We also present a scaling law for general binary discrete memoryless channels.