The polar receiver architecture is a receiver design that captures the envelope and phase information of the signal rather than its in-phase and quadrature components. Several studies have demonstrated the robustness of polar receivers to phase noise and other nonlinearities. Yet, the information-theoretic limits of polar receivers with finite-precision quantizers have not been investigated in the literature. The main contribution of this work is to identify the optimal signaling strategy for the additive white Gaussian noise (AWGN) channel with polar quantization at the output. More precisely, we show that the capacity-achieving modulation scheme has an amplitude phase shift keying (APSK) structure. Using this result, the capacity of the AWGN channel with polar quantization at the output is established by numerically optimizing the probability mass function of the amplitude. The capacity of the polar-quantized AWGN channel with $b_1$-bit phase quantizer and optimized single-bit magnitude quantizer is also presented. Our numerical findings suggest the existence of signal-to-noise ratio (SNR) thresholds, above which the number of amplitude levels of the optimal APSK scheme and their respective probabilities change abruptly. Moreover, the manner in which the capacity-achieving input evolves with increasing SNR depends on the number of phase quantization bits.
This paper provides new insight into the classical problem of determining both the capacity of the discrete-time channel with uniform output quantization and the capacity achieving input distribution. It builds on earlier work by Gallager and Witsenhausen to provide a detailed analysis of two particular quantization schemes. The first is saturation quantization where overflows are mapped to the nearest quantization bin, and the second is wrapping quantization where overflows are mapped to the nearest quantization bin after reduction by some modulus. Both the capacity of wrapping quantization a…
Several information-theoretic studies on channels with output quantization have identified the capacity-achieving input distributions for different fading channels with 1-bit in-phase and quadrature (I/Q) output quantization. However, an exact characterization of the capacity-achieving input distribution for channels with multi-bit phase quantization has not been provided. In this paper, we consider four different channel models with multi-bit phase quantization at the output and identify the optimal input distribution for each channel model. We first consider a complex Gaussian channel with $…
Low-resolution digital-to-analog and analog-to-digital converters (DACs and ADCs) have attracted considerable attention in efforts to reduce power consumption in millimeter wave (mmWave) and massive MIMO systems. This paper presents an information-theoretic analysis with capacity bounds for classes of linear transceivers with quantization. The transmitter modulates symbols via a unitary transform followed by a DAC and the receiver employs an ADC followed by the inverse unitary transform. If the unitary transform is set to an FFT matrix, the model naturally captures filtering and spectral const…
We investigate the limits of communication over the discrete-time Additive White Gaussian Noise (AWGN) channel, when the channel output is quantized using a small number of bits. We first provide a proof of our recent conjecture on the optimality of a discrete input distribution in this scenario. Specifically, we show that for any given output quantizer choice with K quantization bins (i.e., a precision of log2 K bits), the input distribution, under an average power constraint, need not have any more than K + 1 mass points to achieve the channel capacity. The cutting-plane algorithm is employe…
We consider a channel with discrete binary input X that is corrupted by a given continuous noise to produce a continuous-valued output Y. A quantizer is then used to quantize the continuous-valued output Y to the final binary output Z. The goal is to design an optimal quantizer Q* and also find the optimal input distribution p*(X) that maximizes the mutual information I(X; Z) between the binary input and the binary quantized output. A linear time complexity searching procedure is proposed. Based on the properties of the optimal quantizer and the optimal input distribution, we reduced the searc…
The capacity of a Multiple-Input Multiple-Output (MIMO) channel in which the antenna outputs are processed by an analog linear combining network and quantized by a set of threshold quantizers is studied. The linear combining weights and quantization thresholds are selected from a set of possible configurations as a function of the channel matrix. The possible configurations of the combining network model specific analog receiver architectures, such as single antenna selection, sign quantization of the antenna outputs or linear processing of the outputs. An interesting connection between the ca…
Motivated by recent high bandwidth communication systems, Inter-Symbol Interference (ISI) channels with 1-bit quantized output are considered under an average-power-constrained continuous input. While the exact capacity is difficult to characterize, an approximation that matches with the exact channel output up to a probability of error is provided. The approximation does not have additive noise, but constrains the channel output (without noise) to be above a threshold in absolute value. The capacity under the approximation is computed using methods involving standard Gibbs distributions. Mark…
We consider communication over the block noncoherent AWGN channel with low-precision Analog-to-Digital Converters (ADCs) at the receiver. For standard uniform Phase Shift Keying (PSK) modulation, we investigate the performance of a receiver architecture that quantizes only the phase of the received signal; this has the advantage of being implementable without automatic gain control, using multiple 1-bit ADCs preceded by analog multipliers. We study the structure of the transition density of the resulting channel model. Several results, based on the symmetry inherent in the channel, are provide…
We consider finite-level, symmetric quantization procedures for construction and decoding of polar codes. Whether polarization occurs in the presence of quantization is not known in general. Hassani and Urbanke have shown that a simple three-level quantization procedure polarizes and they have proposed a calculation method to obtain a lower bound for achievable rates. We find an improved calculation method for achievable rates and also the exact asymptotic behavior of the block error probability under the aforementioned simple case. We then prove that certain D-level quantization schemes polar…
Base stations in 5G and beyond use advanced antenna systems (AASs), where multiple passive antenna elements and radio units are integrated into a single box. A critical bottleneck of such a system is the digital fronthaul between the AAS and baseband unit (BBU), which has limited capacity. In this paper, we study an AAS used for precoded downlink transmission over a multi-user multiple-input multiple-output (MU-MIMO) channel. First, we present the baseline quantization-unaware precoding scheme created when a precoder is computed at the BBU and then quantized to be sent over the fronthaul. We p…
We consider the strong secret key (SK) agreement problem for the satellite communication setting, where a satellite chooses a common binary phase shift keying modulated input for three statistically independent additive white Gaussian noise measurement channels whose outputs are observed by two legitimate transceivers (Alice and Bob) and an eavesdropper (Eve), respectively. Legitimate transceivers have access to an authenticated, noiseless, two-way, and public communication link, so they can exchange multiple rounds of public messages to agree on a SK hidden from Eve. Without loss of essential…
Polar codes provably achieve the capacity of a wide array of channels under successive decoding. This assumes infinite precision arithmetic. Given the successive nature of the decoding algorithm, one might worry about the sensitivity of the performance to the precision of the computation. We show that even very coarsely quantized decoding algorithms lead to excellent performance. More concretely, we show that under successive decoding with an alphabet of cardinality only three, the decoder still has a threshold and this threshold is a sizable fraction of capacity. More generally, we show that …
In this paper, we leverage polar codes and the well-established channel polarization to design capacity-achieving codes with a certain constraint on the weights of all the columns in the generator matrix (GM) while having a low-complexity decoding algorithm. We first show that given a binary-input memoryless symmetric (BMS) channel $W$ and a constant $s \in (0, 1]$, there exists a polarization kernel such that the corresponding polar code is capacity-achieving with the \textit{rate of polarization} $s/2$, and the GM column weights being bounded from above by $N^s$. To improve the sparsity vers…
Millimeter wave systems suffer from high power consumption and are constrained to use low resolution quantizers --digital to analog and analog to digital converters (DACs and ADCs). However, low resolution quantization leads to reduced data rate and increased out-of-band emission noise. In this paper, a multiple-input multiple-output (MIMO) system with linear transceivers using low resolution DACs and ADCs is considered. An information-theoretic analysis of the system to model the effect of quantization on spectrospatial power distribution and capacity of the system is provided. More precisely…
Arıkan's polar coding, is by now a well studied technique that allows achieving the symmetric capacity of binary input memoryless channels with low complexity encoding and decoding, provided that the polar decoding architecture is used and the decoding metric is matched to the true channel. In this paper, we analyze communication rates that are achievable when the polar coding/decoding architecture is used with the decoder using an incorrect model of the channel. We define the `polar mismatched capacity' as an analogue of the classical mismatched capacity, give an expression for it, and derive…
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.
It is shown that nested polar codes achieve the Shannon rate-distortion function for arbitrary (binary or non-binary) discrete memoryless sources and the Shannon capacity of arbitrary discrete memoryless channels.
Channel coding over arbitrarily-permuted parallel channels was first studied by Willems et al. (2008). This paper introduces capacity-achieving polar coding schemes for arbitrarily-permuted parallel channels where the component channels are memoryless, binary-input and output-symmetric.
We show that the mismatched capacity of binary discrete memoryless channels can be improved by channel combining and splitting via Arıkan's polar transformations. We also show that the improvement is possible even if the transformed channels are decoded with a mismatched polar decoder.
High-capacity NAND flash memories use multi-level cells (MLCs) to store multiple bits per cell and achieve high storage densities. Higher densities cause increased raw bit error rates (BERs), which demand powerful error correcting codes. Low-density parity-check (LDPC) codes are a well-known class of capacity-approaching codes in AWGN channels. However, LDPC codes traditionally use soft information while the flash read channel provides only hard information. Low resolution soft information may be obtained by performing multiple reads per cell with distinct word-line voltages. We select the val…
The use of 1-bit analog-to-digital converters (ADCs) is seen as a promising approach to significantly reduce the power consumption and hardware cost of multiple-input multiple-output (MIMO) receivers. However, the nonlinear distortion due to 1-bit quantization fundamentally changes the optimal communication strategy and also imposes a capacity penalty to the system. In this paper, the capacity of a Gaussian MIMO channel in which the antenna outputs are processed by an analog linear combiner and then quantized by a set of zero threshold ADCs is studied. A new capacity upper bound for the zero t…
We consider universal quantization with side information for Gaussian observations, where the side information is a noisy version of the sender's observation with noise variance unknown to the sender. In this paper, we propose a universally rate optimal and practical quantization scheme for all values of unknown noise variance. Our scheme uses Polar lattices from prior work, and proceeds based on a structural decomposition of the underlying auxiliaries so that even when recovery fails in a round, the parties agree on a common "reference point" that is closer than the previous one. We also pres…
We construct a channel coding scheme to achieve the capacity of any discrete memoryless channel based solely on the techniques of polar coding. In particular, we show how source polarization and randomness extraction via polarization can be employed to "shape" uniformly-distributed i.i.d. random variables into approximate i.i.d. random variables distributed ac- cording to the capacity-achieving distribution. We then combine this shaper with a variant of polar channel coding, constructed by the duality with source coding, to achieve the channel capacity. Our scheme inherits the low complexity e…
Polar codes under successive cancellation decoding proposed by Arıkan provably achieve the symmetric capacity of any given binary-input discrete memoryless channel. The successive cancellation list decoder for polar codes was described by Tal and Vardy as a generalization of the successive cancellation decoder of Arıkan. The performance of the successive cancellation list decoder is encouraging in practice. In this paper, we formalize the successive cancellation list decoder in our notation and prove that polar codes under successive cancellation list decoding achieve the symmetric capacity of…
We present a method of constructing rate-compatible polar codes that are capacity-achieving with low-complexity sequential decoders. The proposed code construction allows for incremental retransmissions at different rates in order to adapt to channel conditions. The main idea of the construction exploits the common characteristics of polar codes optimized for a sequence of successively degraded channels. The proposed approach allows for an optimized polar code to be used at every transmission thereby achieving capacity.
A multilevel coded modulation scheme is studied that uses solely binary polar codes and Honda-Yamamoto probabilistic shaping. The scheme is shown to achieve the capacity of discrete memoryless channels with input alphabets of cardinality a power of two. The performance of finite-length implementations is compared to polar-coded probabilistic amplitude shaping and constant composition distribution matching.
In this paper, we propose a new class of lattices constructed from polar codes, namely polar lattices, to achieve the capacity $\frac{1}{2}\log(1+\SNR)$ of the additive white Gaussian-noise (AWGN) channel. Our construction follows the multilevel approach of Forney \textit{et al.}, where we construct a capacity-achieving polar code on each level. The component polar codes are shown to be naturally nested, thereby fulfilling the requirement of the multilevel lattice construction. We prove that polar lattices are \emph{AWGN-good}. Furthermore, using the technique of source polarization, we propos…
While the impact of finite-rate feedback on the capacity of fading channels has been extensively studied in the literature, not much attention has been paid to this problem under secrecy constraint. In this work, we study the ergodic secret capacity of a multiple-input multiple-output multiple-antenna-eavesdropper (MIMOME) wiretap channel with quantized channel state information (CSI) at the transmitter and perfect CSI at the legitimate receiver, under the assumption that only the statistics of eavesdropper CSI is known at the transmitter. We refine the analysis of the random vector quantizati…
The performance of cloud radio access network (C-RAN) is constrained by the limited fronthaul link capacity under future heavy data traffic. To tackle this problem, extensive efforts have been devoted to design efficient signal quantization/compression techniques in the fronthaul to maximize the network throughput. However, most of the previous results are based on information-theoretical quantization methods, which are hard to implement due to the extremely high complexity. In this paper, we consider using practical uniform scalar quantization in the uplink communication of an orthogonal freq…
This paper establishes the capacity of additive white Gaussian noise (AWGN) channels with phase-quantized output. We show that a rotated $2^b$-phase shift keying scheme is the capacity-achieving input distribution for a complex AWGN channel with $b$-bit phase quantization. The result is then used to establish the expression for the channel capacity as a function of average power constraint $P$ and quantization bits $b$. The outage performance of phase-quantized system is also investigated for the case of Rayleigh fading when the channel state information (CSI) is only known at the receiver. Ou…
We study quantization of log-likelihood ratios (LLR) in bit-interleaved coded modulation (BICM) systems in terms of an equivalent discrete channel. We propose to design the quantizer such that the quantizer outputs become equiprobable. We investigate semi-analytically and numerically the ergodic and outage capacity over single- and multiple-antenna channels for different quantizers. Finally, we show bit error rate simulations for BICM systems with LLR quantization using a rate 1/2 low-density parity-check code.
Due to heavy reliance of millimeter-wave (mmWave) wireless systems on directional links, Beamforming (BF) with high-dimensional arrays is essential for cellular systems in these frequencies. How to perform the array processing in a power efficient manner is a fundamental challenge. Analog and hybrid BF require fewer analog-to-digital converters (ADCs), but can only communicate in a small number of directions at a time,limiting directional search, spatial multiplexing and control signaling. Digital BF enables flexible spatial processing, but must be operated at a low quantization resolution to …
A rateless coding scheme transmits incrementally more and more coded bits over an unknown channel until all the information bits are decoded reliably by the receiver. We propose a new rateless coding scheme based on polar codes, and we show that this scheme is capacity-achieving, i.e. its information rate is as good as the best code specifically designed for the unknown channel. Previous rateless coding schemes are designed for specific classes of channels such as AWGN channels, binary erasure channels, etc. but the proposed rateless coding scheme is capacity-achieving for broad classes of cha…
We present a rate-compatible polar coding scheme that achieves the capacity of any family of channels. Our solution generalizes the previous results [1], [2] that provide capacity-achieving rate-compatible polar codes for a degraded family of channels. The motivation for our extension comes from the fact that in many practical scenarios, e.g., MIMO systems and non-Gaussian interference, the channels cannot be ordered by degradation. The main technical contribution of this paper consists in removing the degradation condition. To do so, we exploit the ideas coming from the construction of univer…
We define a new phenomenon for communication over noisy quantum channels. The investigated solution is called polaractivation and based on quantum polar encoding. Polaractivation is a natural consequence of the channel polarization effect in quantum systems and makes possible to open the hidden capacity regions of a noisy quantum channel by using the idea of rate increment. While in case of a classical channel only the rate of classical communication can be increased, in case of a quantum channel the channel polarization and the rate improvement can be exploited to open unreachable capacity re…
We construct concatenated capacity-achieving quantum codes for noisy optical quantum channels. We demonstrate that the error-probability of capacity-achieving quantum polar encoding can be reduced by the proposed low-complexity concatenation scheme.
Recently, it has been shown that a quantize-map-and-forward scheme approximately achieves (within a constant number of bits) the Gaussian relay network capacity for arbitrary topologies. This was established using Gaussian codebooks for transmission and random mappings at the relays. In this paper, we show that the same approximation result can be established by using lattices for transmission and quantization along with structured mappings at the relays.
Applications of massive machine-type communications, such as sensor networks, smart metering, 'internet-of-things', or process and factory automation, are forecast to have great economic impact in the next five to ten years. Low-complexity energy- and cost-efficient communication schemes are essential to enable large-scale deployments. To target these requirements, we study decoding of polar codes with coarse quantization in the short block length regime. In particular, we devise schemes to mitigate the impact of coarse quantization, which has not been adequately explored in prior works. We in…
This work considers the downlink of a cloud radio access network (C-RAN), in which a control unit (CU) encodes confidential messages, each of which is intended for a user equipment (UE) and is to be kept secret from all the other UEs. As per the C-RAN architecture, the encoded baseband signals are quantized and compressed prior to the transfer to distributed radio units (RUs) that are connected to the CU via finite-capacity fronthaul links. This work argues that the quantization noise introduced by fronthaul quantization can be leveraged to act as "artificial" noise in order to enhance the rat…
In this paper, we design polar codes and polar lattices for i.i.d. fading channels when the channel state information is only available to the receiver. For the binary input case, we propose a new design of polar codes through single-stage polarization to achieve the ergodic capacity. For the non-binary input case, polar codes are further extended to polar lattices to achieve the egodic Poltyrev capacity, i.e., the capacity without power limit. When the power constraint is taken into consideration, we show that polar lattices with lattice Gaussian shaping achieve the egodic capacity of fading …
In this paper we study codes with sparse generator matrices. More specifically, codes with a certain constraint on the weight of all the columns in the generator matrix are considered. The end result is the following. For any binary-input memoryless symmetric (BMS) channel and any epsilon > 2 epsilon*, where epsilon^* = \frac{1}{6}-\frac{5}{3}\log{\frac{4}{3}} \approx 0.085, we show an explicit sequence of capacity-achieving codes with all the column wights of the generator matrix upper bounded by (\log N)^{1+epsilon}, where N is the code block length. The constructions are based on polar code…
Regular perturbation is applied to the Manakov equation and motivates a generalized correlated phase-and-additive noise model for wavelength-division multiplexing over dual-polarization optical fiber channels. The model includes three hidden Gauss-Markov processes: phase noise, polarization rotation, and additive noise. Particle filtering is used to compute lower bounds on the capacity of multi-carrier communication with frequency-dependent powers and delays. A gain of 0.17 bits/s/Hz/pol in spectral efficiency or 0.8 dB in power efficiency is achieved with respect to existing models at their p…
One problem with MAC polar codes that are based on MAC polarization is that they may not achieve the entire capacity region. The reason behind this problem is that MAC polarization sometimes induces a loss in the capacity region. This paper provides a single letter necessary and sufficient condition which characterizes all the MACs that do not lose any part of their capacity region by polarization.
We demonstrate, for a widely used model of channels with polarization dependent loss (PDL), that channel capacity is achieved by a simple interference cancellation scheme in conjunction with a universal precoder. Crucially, the proposed scheme is not only information-theoretically optimal, but it is also exceptionally simple and concrete. It transforms the PDL channel into separate scalar additive white Gaussian noise channels, allowing off-the-shelf coding and modulation schemes designed for such channels to approach capacity. The signal-to-noise ratio (SNR) penalty incurred under 6 dB of PDL…
We present simple coding strategies, which are variants of the Schalkwijk-Kailath scheme, for communicating reliably over additive white noise channels in the presence of corrupted feedback. More specifically, we consider a framework comprising an additive white forward channel and a backward link which is used for feedback. We consider two types of corruption mechanisms in the backward link. The first is quantization noise, i.e., the encoder receives the quantized values of the past outputs of the forward channel. The quantization is uniform, memoryless and time invariant (that is, symbol-by-…
The common approach of designing a communication device is to maximize a well-defined objective function, e.g., the channel capacity and the cut-off rate. We propose easy-to-implement solutions for Gaussian channels that approximate the optimal results for these maximization problems. Three topics are addressed. First, we consider the case where the channel output is quantized, and we find the quantization thresholds that maximize the mutual information. The approximation derived from the asymptotic solution has a negligible loss on the entire range of SNR when 2-PAM modulation is used, and it…
Analog-to-digital conversion (ADC) is a key bottleneck in scaling DSP-centric receiver architectures to multiGigabit/s speeds. Recent information-theoretic results, obtained under ideal channel conditions (perfect synchronization, no dispersion), indicate that low-precision ADC (1-4 bits) could be a suitable choice for designing such high speed systems. In this work, we study the impact of employing low-precision ADC in a {\it carrier asynchronous} system. Specifically, we consider transmission over the block noncoherent Additive White Gaussian Noise (AWGN) channel, and investigate the achieva…
We propose a nonuniform quantized decoder for polar codes. The design metric of the quantizers is to minimize the distortion incurred by quantization. The quantizers are obtained via dynamic programming and the optimality of the quantizer is proved as well. Simulation results show that the error correction performance degradation of the proposed nonuniform quantized decoder is less than 0.1 dB compared to conventional float-point decoders under 5-bit nonuniform quantization, outperforming previous uniform quantized decoders for polar codes significantly.
Achieving security against adversaries with unlimited computational power is of great interest in a communication scenario. Since polar codes are capacity achieving codes with low encoding-decoding complexity and they can approach perfect secrecy rates for binary-input degraded wiretap channels in symmetric settings, they are investigated extensively in the literature recently. In this paper, a polar coding scheme to achieve secrecy capacity in non-symmetric binary input channels is proposed. The proposed scheme satisfies security and reliability conditions. The wiretap channel is assumed to b…
Polar codes, invented by Arikan in 2009, are known to achieve the capacity of any binary-input memoryless output-symmetric channel. One of the few drawbacks of the original polar code construction is that it is not universal. This means that the code has to be tailored to the channel if we want to transmit close to capacity. We present two "polar-like" schemes which are capable of achieving the compound capacity of the whole class of binary-input memoryless output-symmetric channels with low complexity. Roughly speaking, for the first scheme we stack up $N$ polar blocks of length $N$ on top of…
In this paper, we first propose an universal polar coding scheme for parallel Gaussian channels with non-binary inputs. It is assumed that the encoder knows only the sum capacity of M parallel channels instead of the capacity of any single channel. By decomposing each parallel channel into T = [log2r] sub channels, we therefore obtain MT binary sub-channels. A super polar coding scheme that across all sub-channels is then proposed. This scheme can achieve the sum capacity when the block length is sufficiently large. We have also discussed the applications of parallel polar coding design for bo…
We propose two coding schemes for the two-receiver discrete memoryless broadcast channel (BC) with rate-limited feedback from one or both receivers. They improve over the nofeedback capacity region for a large class of channels, including the class of \emph{strictly essentially less-noisy BCs} that we introduce in this article. Examples of strictly essentially less-noisy BCs are the binary symmetric BC (BSBC) or the binary erasure BC (BEBC) with unequal cross-over or erasure probabilities at the two receivers. When the feedback rates are sufficiently large, our schemes recover all previously k…
This paper proposes a dimension reduction-based signal compression scheme for uplink distributed MIMO cloud radio access networks (C-RAN) with an overall excess of receive antennas, in which users are jointly served by distributed multi-antenna receivers connected to a central processor via individual finite-capacity fronthaul links. We first show that, under quantization noise-limited operation, applying linear dimension reduction at each receiver before compressing locally with a uniform quantization noise level results in a sum capacity that scales approximately linearly with fronthaul capa…
In this work, an explicit wiretap coding scheme based on polar lattices is proposed to achieve the secrecy capacity of the additive white Gaussian noise (AWGN) wiretap channel. Firstly, polar lattices are used to construct secrecy-good lattices for the mod-$Λ_s$ Gaussian wiretap channel. Then we propose an explicit shaping scheme to remove this mod-$Λ_s$ front end and extend polar lattices to the genuine Gaussian wiretap channel. The shaping technique is based on the lattice Gaussian distribution, which leads to a binary asymmetric channel at each level for the multilevel lattice codes. By emp…
