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 constraints which are essential to model in any practical transceiver. In particular, this model allows studying the impact of quantization on out-of-band emission constraints. In the limit of a large random unitary transform, it is shown that the effect of quantization can be precisely described via an additive Gaussian noise model. This model in turn leads to simple and intuitive expressions for the power spectrum of the transmitted signal and a lower bound to the capacity with quantization. Comparison with non-quantized capacity and a capacity upper bound that does not make linearity assumptions suggests that while low resolution quantization has minimal impact on the achievable rate at typical parameters in 5G systems today, satisfying out-of-band emissions are potentially much more of a challenge.
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 $…
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…
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…
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…
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…
To relax power consumption requirements in multigigabit/s communications systems low resolution quantization can be used. Information-theoretic results have shown that systems employing 1-bit quantization and oversampling are a viable option for this. This work investigates such a structure under the influence of additive Gaussian noise and two matched pulse shaping filters. It is described how a BCJR algorithm, based on a finite-state channel assumption, can be used to reconstruct symbols of higher order modulation schemes that allow the transmission of more than one bit per symbol. Furthermo…
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 address secure vehicle communication using secrecy capacity. In particular, we research the relationship between secrecy capacity and various types of parameters that determine secrecy capacity in the vehicular wireless network. For example, we examine the relationship between vehicle speed and secrecy capacity, the relationship between the response time and secrecy capacity of an autonomous vehicle, and the relationship between transmission power and secrecy capacity. In particular, the autonomous vehicle has set the system modeling on the assumption that the speed of the vehicle is relate…
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 …
Similar to the energy flowing process in traditional heat engines, information could be considered to flow in the communication systems with the form of energy and entropy. Combining the thermodynamic Carnot machine and the classical Shannon information theory, a generalized thermodynamic MIMO (multiple input multiple outputs) communication system is established to analyze the channel capacity using forward error correction codes. Based on the concepts of freedom and entropy in the communication system, the generalized channel capacity is proposed under the thermodynamic theory. Furthermore, t…
Due to hardware limitations, the phase shifts of the reflecting elements of reconfigurable intelligent surfaces (RISs) need to be quantized into discrete values. This letter aims to unveil the minimum required number of phase quantization levels $L$ in order to achieve the full diversity order in RIS-assisted wireless communication systems. With the aid of an upper bound of the outage probability, we first prove that the full diversity order is achievable provided that $L$ is not less than three. If $L=2$, on the other hand, we prove that the achievable diversity order cannot exceed $(N+1)/2$,…
The paper is devoted to the analysis of problems appearing in optimisation and improvement of the power-bandwidth efficiency of digital communication feedback systems (FCS). There is shown that unlike digital systems, adaptive FCS with the analogue forward transmission allow full optimisation and derivation of optimal transmission-reception algorithm approaching their efficiency to the Shannon boundary. Differences between the forward channel capacity and capacity of adaptive FCS as communication unit, as well as their influence of the power-bandwidth efficiency of transmission are considered.
Millimeter wave (mmWave) spectrum has drawn attention due to its tremendous available bandwidth. The high propagation losses in the mmWave bands necessitate beamforming with a large number of antennas. Traditionally each antenna is paired with a high-speed analog-to-digital converter (ADC), which results in high power consumption. A hybrid beamforming architecture and one-bit resolution ADCs have been proposed to reduce power consumption. However, analog beamforming and one-bit quantization make channel estimation more challenging. In this paper, we propose a narrowband channel estimation algo…
Variable length communication over a compound channel with feedback is considered. Traditionally, capacity of a compound channel without feedback is defined as the maximum rate that is determined before the start of communication such that communication is reliable. This traditional definition is pessimistic. In the presence of feedback, an opportunistic definition is given. Capacity is defined as the maximum rate that is determined at the end of communication such that communication is reliable. Thus, the transmission rate can adapt to the channel chosen by nature. Under this definition, feed…
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…
This paper considers secure communication over free-space optical (FSO) links suffering from turbulence-induced fading. In particular, we study the average secrecy capacity of applying physical layer security to achieve security in an FSO communication system with binary-level on-off keying intensity modulation and threshold detection, a widely used system in FSO communication. We first define the instantaneous secrecy capacity for our system and then provide a lower bound of it, which is achieved based on a simple design principle for practical applications and yields a simpler equation. We n…
The effective capacity of communication systems over generalized $κ$-$μ$ shadowed fading channels is investigated in this letter. A novel and analytical expression for the exact effective capacity is derived in terms of extended generalized bivariate Meijer's-$G$ function. To intuitively reveal the impact of the system and channel parameters on the effective capacity, we also derive closed-form expressions for the effective capacity in the asymptotically high signal-to-noise ratio regime. Our results demonstrate that the effective capacity is a monotonically increasing function of channel fadi…
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…
This paper focuses on the capacity of point-to-point relay communications wherein the transmitter is assisted by an intermediate relay. We detail the mathematical model of cutset and amplify and forward (AF) relaying strategy. We present the upper bound capacity of each relaying strategy from information theory viewpoint and also in networks with Gaussian channels. We exemplify various outer region capacities of the addressed strategies with two different case studies. The results exhibit that in low signal-to-noise ratio (SNR) environments the cutset performance is better than amplify and for…
Channel capacity bounds are derived for a point-to-point indoor visible light communications (VLC) system with signal-dependent Gaussian noise. Considering both illumination and communication, the non-negative input of VLC is constrained by peak and average optical intensity constraints. Two scenarios are taken into account: one scenario has both average and peak optical intensity constraints, and the other scenario has only average optical intensity constraint. For both two scenarios, we derive closed-from expressions of capacity lower and upper bounds. Specifically, the capacity lower bound …
We characterize the capacity region of a two-user multiple access channel (MAC) in a reconfigurable intelligent surface (RIS)-aided communication system with side information (SI) at the transmitters. We consider two uplink communication scenarios: (i) a double dirty MAC where the interferences are known non-causally to both users; and (ii) a single dirty MAC model where only one of the users knows the interference and the other one is not aware of the interference. Considering that each of the users is assisted by a different RIS with Mi elements, we derive closed-form expressions for the cap…
This paper considers the quantization problem on the Grassmann manifold \mathcal{G}_{n,p}, the set of all p-dimensional planes (through the origin) in the n-dimensional Euclidean space. The chief result is a closed-form formula for the volume of a metric ball in the Grassmann manifold when the radius is sufficiently small. This volume formula holds for Grassmann manifolds with arbitrary dimension n and p, while previous results pertained only to p=1, or a fixed p with asymptotically large n. Based on this result, several quantization bounds are derived for sphere packing and rate distortion tr…
We consider the communication scenario where K transmitters are each connected to a common receiver with an orthogonal noiseless link. One of the transmitters has a message for the receiver, who is prohibited from learning anything in the information theoretic sense about which transmitter sends the message (transmitter anonymity is guaranteed). The capacity of anonymous communications is the maximum number of bits of desired information that can be anonymously communicated per bit of total communication. For this anonymous communication problem over a parallel channel with K transmitters and …
We analyze the impact of arbitrary dependence between the forward and backward links in backscatter communication systems. Specifically, we quantify the effect of positive and negative dependence between these fading links on channel capacity, using Copula theory. The benefits of this approach are highlighted over the classical framework of linear dependence based on Pearson's correlation coefficient, which is also analyzed. Results show that for a fixed transmit power budget, capacity grows with positive dependence as well as with fading severity in the low signal-to-noise ratio (SNR) regime.…
Improving the power efficiency and spectral efficiency of communication systems has been one of the major research goals over the recent years. However, there is a tradeoff in achieving both goals at the same time. In this work, we consider the joint optimization of the power amplifier and a pulse shaping filter over a single-input single-output (SISO) additive white Gaussian noise (AWGN) channel using 1-bit analog-todigital (ADC) and digital-to-analog (DAC) converters. The goal of the optimization is the selection of the optimal system parameters in order to maximize the desired figure-of-mer…
An upper bound to the information capacity of a wavelength-division multi- plexed optical fiber communication system is derived in a model incorporating the nonlinear propagation effects of cross-phase modulation (XPM). This work is based on the paper by Mitra et al., finding lower bounds to the channel capacity, in which physical models for propagation are used to calculate statistical properties of the conditional probability distribution relating input and output in a single WDM channel. In this paper we present a tractable channel model incorporating the effects of cross phase modulation. …
Integrated sensing and communication (ISAC) has emerged as a key technology for future communication systems. In this paper, we provide a general framework to reveal the fundamental tradeoff between sensing and communication in OFDM systems, where a unified ISAC waveform is exploited to perform both tasks. In particular, we define the Capacity-Bayesian Cramer Rao Bound (BCRB) region in the asymptotically case when the number of subcarriers is large. Specifically, we show that the asymptotically optimal input distribution that achieves the Pareto boundary point of the Capacity-BCRB region is Ga…
In this paper, the capacity of a diffusion based molecular communication network under the model of a Linear Time Invarient-Poisson (LTI-Poisson) channel is studied. Introduced in the context of molecular communication, the LTI-Poisson model is a natural extension of the conventional memoryless Poisson channel to include memory. Exploiting prior art on linear ISI channels, a computable finite-letter characterization of the capacity of single-hop LTI-Poisson networks is provided. Then, the problem of finding more explicit bounds on the capacity is examined, where lower and upper bounds for the …
Communications over power lines in the frequency range above 2 MHz, commonly referred to as broadband (BB) power line communications (PLC), has been the focus of increasing research attention and standardization efforts in recent years. BB-PLC channels are characterized by a dominant colored non-Gaussian additive noise, as well as by periodic variations of the channel impulse response and the noise statistics. In this work we study the fundamental rate limits for BB-PLC channels by bounding their capacity while accounting for the unique properties of these channels. We obtain explicit expressi…
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 derive upper and lower bounds as well as a simple closed-form approximation for the capacity of the continuous-time, bandlimited, additive white Gaussian noise channel in a three-dimensional free-space electromagnetic propagation environment subject to constraints on the total effective antenna aperture area of the link and a total transmitter power constraint. We assume that the communication range is much larger than the radius of the sphere containing the antennas at both ends of the link, and we show that, in general, the capacity can only be achieved by transmitting mult…
This paper focuses on the capacity of peer-to-peer relay communications wherein the transmitter are assisted by an arbitrary number of parallel relays, i.e. there is no link and cooperation between the relays themselves. We detail the mathematical model of different relaying strategies including cutset and amplify and forward strategies. The cutset upper bound capacity is presented as a reference to compare another realistic strategy. We present its outer region capacity which is lower than that in the existing literature. We show that a multiple parallel relayed network achieves its maximum c…
In this work, we study a full-duplex (FD) cloud radio access network (C-RAN) from the aspects of infrastructure sharing and information secrecy, where the central unit utilizes FD remote radio units (RU)s belonging to the same operator, i.e., the trusted RUs, as well as the RUs belonging to other operators or private owners, i.e., the untrusted RUs. Furthermore, the communication takes place in the presence of untrusted external receivers, i.e., eavesdropper nodes. The communicated uplink (UL) and downlink (DL) waveforms are quantized in order to comply with the limited capacity of the frontha…
We investigate design and performance of communications over the bandlimited Gaussian channel with one-bit output quantization. A transceiver structure is proposed, which generates the channel input using a finite set of time-limited and approximately bandlimited waveforms, and performs oversampling on the channel output by an integrate-and-dump filter preceding the one-bit quantizer. The waveform set is constructed based on a specific bandlimited random process with certain zero-crossing properties which can be utilized to convey information. In the presence of the additive white Gaussian noi…
In this paper, we investigated the capacity and bit error rate (BER) performance of Multiple Input Multiple Output (MIMO) satellite systems with single and multiple dual polarized satellites in geostationary orbit and a mobile ground receiving station with multiple antennas. We evaluated the effects of both system parameters such as number of satellites, number of receive antennas, and SNR and environmental factors including atmospheric signal attenuations and signal phase disturbances on the overall system performance using both analytical and spatial models for MIMO satellite systems.
The maximum capacity of fractal D2D (device-to-device) social networks with both direct and hierarchical communications is studied in this paper. Specifically, the fractal networks are characterized by the direct social connection and the self-similarity. Firstly, for a fractal D2D social network with direct social communications, it is proved that the maximum capacity is $ Θ\left(\frac{1}{\sqrt{n\log n}}\right) $ if a user communicates with one of his/her direct contacts randomly, where $ n $ denotes the total number of users in the network, and it can reach up to $ Θ\left(\frac{1}{\log n}\ri…
The maximum capacity of fractal D2D (device-to-device) social networks with both direct and hierarchical communications is studied in this paper. Specifically, the fractal networks are characterized by the direct social connection and the self-similarity. Firstly, for a fractal D2D social network with direct social communications, it is proved that the maximum capacity is $ Θ\left(\frac{1}{\sqrt{n\log n}}\right) $ if a user communicates with one of his/her direct contacts randomly, where $ n $ denotes the total number of users in the network, and it can reach up to $ Θ\left(\frac{1}{\log n}\ri…
In physical-layer security, one of the most fundamental issues is the secrecy capacity. The objective of this paper is to determine the secrecy capacity for an indoor visible light communication system consisting of a transmitter, a legitimate receiver and an eavesdropping receiver. In such a system, both signal-independent and signal-dependent Gaussian noises are considered. Under non-negativity and average optical intensity constraints, lower and upper secrecy capacity bounds are first derived by the variational method, the dual expression of the secrecy capacity, and the concept of "the opt…
Effective capacity (EC) determines the maximum communication rate subject to a particular delay constraint. In this work, we analyze the EC of ultra reliable Machine Type Communication (MTC) networks operating in the finite blocklength (FB) regime. First, we present a closed form approximation for EC in quasi-static Rayleigh fading channels. Our analysis determines the upper bounds for EC and delay constraint when varying transmission power. Finally, we characterize the power-delay trade-off for fixed EC and propose an optimum power allocation scheme which exploits the asymptotic behavior of E…
We review how Shannon's classical notion of capacity is not enough to characterize a noisy communication channel if the channel is intended to be used as part of a feedback loop to stabilize an unstable scalar linear system. While classical capacity is not enough, another sense of capacity (parametrized by reliability) called ``anytime capacity'' is shown to be necessary for the stabilization of an unstable process. The required rate is given by the log of the unstable system gain and the required reliability comes from the sense of stability desired. A consequence of this necessity result is …
Capacity Analysis and Joint Gaussian Beam Pattern Optimization for Positioning-Assisted Communications
National Mobile Communications Research Laboratory, Frontiers Science Center for Mobile Information Communication and Security, Southeast University（东南大学）；Purple Mountain Laboratories（紫金山实验室）
The communication complexity of achieving secret key (SK) capacity in the multiterminal source model of Csisz$á$r and Narayan is the minimum rate of public communication required to generate a maximal-rate SK. It is well known that the minimum rate of communication for omniscience, denoted by $R_{\text{CO}}$, is an upper bound on the communication complexity, denoted by $R_{\text{SK}}$. A source model for which this upper bound is tight is called $R_{\text{SK}}$-maximal. In this paper, we establish a sufficient condition for $R_{\text{SK}}$-maximality within the class of pairwise independent n…
This paper considers the quantization problem on the Grassmann manifold with dimension n and p. The unique contribution is the derivation of a closed-form formula for the volume of a metric ball in the Grassmann manifold when the radius is sufficiently small. This volume formula holds for Grassmann manifolds with arbitrary dimension n and p, while previous results are only valid for either p=1 or a fixed p with asymptotically large n. Based on the volume formula, the Gilbert-Varshamov and Hamming bounds for sphere packings are obtained. Assuming a uniformly distributed source and a distortion …
In this work we study the capacity advantage achieved by adding a single bit of communication -- not a link of capacity 1 but a single bit over all time -- to a memoryless network. Specifically, we present a memoryless network in which adding a single bit of communication strictly increases the capacity region.
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…
This paper, mostly tutorial in nature, deals with the problem of characterizing the capacity of fading channels in the high signal-to-noise ratio (SNR) regime. We focus on the practically relevant noncoherent setting, where neither transmitter nor receiver know the channel realizations, but both are aware of the channel law. We present, in an intuitive and accessible form, two tools, first proposed by Lapidoth & Moser (2003), of fundamental importance to high-SNR capacity analysis: the duality approach and the escape-to-infinity property of capacity-achieving distributions. Furthermore, we app…
Training a machine learning model with federated edge learning (FEEL) is typically time-consuming due to the constrained computation power of edge devices and limited wireless resources in edge networks. In this paper, the training time minimization problem is investigated in a quantized FEEL system, where the heterogeneous edge devices send quantized gradients to the edge server via orthogonal channels. In particular, a stochastic quantization scheme is adopted for compression of uploaded gradients, which can reduce the burden of per-round communication but may come at the cost of increasing …
Today's communication systems typically use high resolution analog-to-digital converters (ADCs). However, considering future communication systems with data rates in the order of 100Gbit/s the ADC power consumption becomes a major factor due to the high sampling rates. A promising alternative are receivers based on 1-bit quantization and oversampling w.r.t. the signal bandwidth. Such an approach requires a redesign of modulation, receiver synchronization, and demapping. A zero crossing modulation is a natural choice as the information needs to be carried in the zero crossing time instants. The…
We show that for product sources, rate splitting is optimal for secret key agreement using limited one-way communication at two terminals. This yields an alternative proof of the tensorization property of a strong data processing inequality originally studied by Erkip and Cover and amended recently by Anantharam et al. We derive a `water-filling' solution of the communication-rate--key-rate tradeoff for two arbitrarily correlated vector Gaussian sources, for the case with an eavesdropper, and for stationary Gaussian processes.
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…
The wireless communication systems has gone from different generations from SISO systems to MIMO systems. Bandwidth is one important constraint in wireless communication. In wireless communication, high data transmission rates are essential for the services like tripple play i.e. data, voice and video. At user end the capacity determines the quality of the communication systems. This paper aims to compare the different RF wireless communication systems like SISO, MISO, SIMO and MIMO systems on the capacity basis and explaining the concept as today, the wireless communication has evolved from 2…
This paper starts by assuming a 1-2-1 network, the abstracted noiseless model of mmWave networks that was shown to closely approximate the Gaussian capacity in [1], and studies secure communication. First, the secure capacity is derived for 1-2-1 networks where a source is connected to a destination through a network of unit capacity links. Then, lower and upper bounds on the secure capacity are derived for the case when source and destination have more than one beam, which allow them to transmit and receive in multiple directions at a time. Finally, secure capacity results are presented for d…
This paper studies achievable information rates of backscatter communication systems where the tag performs load modulation with a freely adaptable passive termination. We find that the complex phasor of the tag current is constrained to a disk and that the capacity problem can therefore be described with existing results on peak-power-limited quadrature channels. This allows us to state the channel capacity and the capacity-achieving distribution of the load impedance, which is described by non-concentric circles in the right half-plane. For the low-SNR case (SNR < 4.8 dB) we find that channe…
