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Register a new userOn the Rate of Channel Polarization
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Added by
Erdal Arikan
Publication date
2009
fields
Informatics Engineering
and research's language is
English
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It is shown that for any binary-input discrete memoryless channel $W$ with symmetric capacity $I(W)$ and any rate $R <I(W)$, the probability of block decoding error for polar coding under successive cancellation decoding satisfies $P_e le 2^{-N^beta}$ for any $beta<frac12$ when the block-length $N$ is large enough.
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Reconfigurable intelligent surface (RIS) assisted radio is considered as an enabling technology with great potential for the sixth-generation (6G) wireless communications standard. The achievable secrecy rate (ASR) is one of the most fundamental metrics to evaluate the capability of facilitating secure communication for RIS-assisted systems. However, the definition of ASR is based on Shannons information theory, which generally requires long codewords and thus fails to quantify the secrecy of emerging delay-critical services. Motivated by this, in this paper we investigate the problem of maximizing the secrecy rate under a delay-limited quality-of-service (QoS) constraint, termed as the effective secrecy rate (ESR), for an RIS-assisted multiple-input single-output (MISO) wiretap channel subject to a transmit power constraint. We propose an iterative method to find a stationary solution to the formulated non-convex optimization problem using a block coordinate ascent method (BCAM), where both the beamforming vector at the transmitter as well as the phase shifts at the RIS are obtained in closed forms in each iteration. We also present a convergence proof, an efficient implementation, and the associated complexity analysis for the proposed method. Our numerical results demonstrate that the proposed optimization algorithm converges significantly faster that an existing solution. The simulation results also confirm that the secrecy rate performance of the system with stringent delay requirements reduce significantly compared to the system without any delay constraints, and that this reduction can be significantly mitigated by an appropriately placed large-size RIS.
For general memoryless systems, the typical information theoretic solution - when exists - has a single-letter form. This reflects the fact that optimum performance can be approached by a random code (or a random binning scheme), generated using independent and identically distributed copies of some single-letter distribution. Is that the form of the solution of any (information theoretic) problem? In fact, some counter examples are known. The most famous is the two help one problem: Korner and Marton showed that if we want to decode the modulo-two sum of two binary sources from their independent encodings, then linear coding is better than random coding. In this paper we provide another counter example, the doubly-dirty multiple access channel (MAC). Like the Korner-Marton problem, this is a multi-terminal scenario where side information is distributed among several terminals; each transmitter knows part of the channel interference but the receiver is not aware of any part of it. We give an explicit solution for the capacity region of a binary version of the doubly-dirty MAC, demonstrate how the capacity region can be approached using a linear coding scheme, and prove that the best known single-letter region is strictly contained in it. We also state a conjecture regarding a similar rate loss of single letter characterization in the Gaussian case.
We give a unified treatment of some inequalities that are used in the proofs of channel polarization theorems involving a binary-input discrete memoryless channel.
The cutoff rate $R_0(W)$ of a discrete memoryless channel (DMC) $W$ is often used as a figure of merit, alongside the channel capacity $C(W)$. Given a channel $W$ consisting of two possibly correlated subchannels $W_1$, $W_2$, the capacity function always satisfies $C(W_1)+C(W_2) le C(W)$, while there are examples for which $R_0(W_1)+R_0(W_2) > R_0(W)$. This fact that cutoff rate can be ``created by channel splitting was noticed by Massey in his study of an optical modulation system modeled as a $M$ary erasure channel. This paper demonstrates that similar gains in cutoff rate can be achieved for general DMCs by methods of channel combining and splitting. Relation of the proposed method to Pinskers early work on cutoff rate improvement and to Imai-Hirakawa multi-level coding are also discussed.
In this paper we consider the secure transmission over the fast fading multiple antenna Gaussian broadcast channels with confidential messages (FMGBC-CM), where a multiple-antenna transmitter sends independent confidential messages to two users with information theoretic secrecy and only the statistics of the receivers channel state information are known at the transmitter. We first use the same marginal property of the FMGBC-CM to classify the non-trivial cases, i.e., those not degraded to the common wiretap channels. We then derive the achievable rate region for the FMGBC-CM by solving the channel input covariance matrices and the inflation factor. Due to the complicated rate region formulae, we resort to low SNR analysis to investigate the characteristics of the channel. Finally, the numerical examples show that under the information-theoretic secrecy requirement both users can achieve positive rates simultaneously.
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