This paper considers the cognitive interference channel (CIC) with two transmitters and two receivers, in which the cognitive transmitter non-causally knows the message and codeword of the primary transmitter. We first introduce a discrete memoryless more capable CIC, which is an extension to the more capable broadcast channel (BC). Using superposition coding, we propose an inner bound and an outer bound on its capacity region. The outer bound is also valid when the primary user is under strong interference. For the Gaussian CIC, this outer bound applies for $|a| \geq 1 $, where $a$ is the gain of interference link from secondary user to primary receiver. These capacity inner and outer bounds are then applied to the Gaussian cognitive Z-interference channel (GCZIC) where only the primary receiver suffers interference. Upon showing that jointly Gaussian input maximizes these bounds for the GCZIC, we evaluate the bounds for this channel. The new outer bound is strictly tighter than other outer bounds on the capacity of the GCZIC at strong interference ($a^2 \geq 1 $). Especially, the outer bound coincides with the inner bound for $|a| \geq \sqrt{1 + P_1}$ and thus, establishes the capacity of the GCZIC at this range. For such a large $a$, superposition encoding at the cognitive transmitter and successive decoding at the primary receiver are capacity-achieving.
There are two variants of superposition coding schemes. Cover's original superposition coding scheme has code clouds of the identical shape, while Bergmans's superposition coding scheme has code clouds of independently generated shapes. These two schemes yield identical achievable rate regions in several scenarios, such as the capacity region for degraded broadcast channels. This paper shows that under the optimal maximum likelihood decoding, these two superposition coding schemes can result in different rate regions. In particular, it is shown that for the two-receiver broadcast channel, Cove…
Proving capacity for networks without feedback or cooperation usually involves two fundamental random coding techniques: superposition coding and binning. Although conceptually very different, these two techniques often achieve the same performance, suggesting an underlying similarity. In this correspondence we propose a new random coding technique that generalizes superposition coding and binning and provides new insight on relationship among the two With this new theoretical tool, we derive new achievable regions for three classical information theoretical models: multi-access channel, broad…
We present a superposition coding scheme for communication over a network, which combines partial decode and forward and noisy network coding. This hybrid scheme is termed as superposition noisy network coding. The scheme is designed and analyzed for single relay channel, single source multicast network and multiple source multicast network. The achievable rate region is determined for each case. The special cases of Gaussian single relay channel and two way relay channel are analyzed for superposition noisy network coding. The achievable rate of the proposed scheme is higher than the existing…
This paper shows that the capacity region of the continuous-time Poisson broadcast channel is achieved via superposition coding for most channel parameter values. Interestingly, the channel in some subset of these parameter values does not belong to any of the existing classes of broadcast channels for which superposition coding is optimal (e.g., degraded, less noisy, more capable). In particular, we introduce the notion of effectively less noisy broadcast channel and show that it implies less noisy but is not in general implied by more capable. For the rest of the channel parameter values, we…
We develop practical coding schemes for the cognitive overlay radios as modeled by the cognitive interference channel, a variation of the classical two user interference channel where one of the transmitters has knowledge of both messages. Inspired by information theoretical results, we develop a coding strategy for each of the three parameter regimes where capacity is known. A key feature of the capacity achieving schemes in these regimes is the joint decoding of both users' codewords, which we accomplish by performing a posteriori probability calculation over a combined trellis. The schemes …
In this paper, we propose a low-complexity, secrecy capacity achieving polar coding scheme for the cognitive interference channel with confidential messages (CICC) under the strong secrecy criterion. Existing polar coding schemes for interference channels rely on the use of polar codes for the multiple access channel, the code construction problem of which can be complicated. We show that the whole secrecy capacity region of the CICC can be achieved by simple point-to-point polar codes due to the cognitivity, and our proposed scheme requires the minimum rate of randomness at the encoder.
Focusing on the downlink, we consider a base station (BS) and the cell it covers. The Superposition Coding (SC), also referred to as Non Orthogonal Multiple Access (NOMA), is implemented. We propose an analytical model of the wireless cell covered by the BS. Based on this model, we establish a closed form expression of the minimum transmit power of the base station, needed to achieve a given SINR (signal to interference plus noise ratio), whatever the users locations, on the area covered by the base station. The closed form expression of the BS transmit power allows to establish quality of ser…
A low-complexity coding scheme is developed to achieve the rate region of maximum likelihood decoding for interference channels. As in the classical rate-splitting multiple access scheme by Grant, Rimoldi, Urbanke, and Whiting, the proposed coding scheme uses superposition of multiple codewords with successive cancellation decoding, which can be implemented using standard point-to-point encoders and decoders. Unlike rate-splitting multiple access, which is not rate-optimal for multiple receivers, the proposed coding scheme transmits codewords over multiple blocks in a staggered manner and reco…
Multilevel diversity coding is a classical coding model where multiple mutually independent information messages are encoded, such that different reliability requirements can be afforded to different messages. It is well known that {\em superposition coding}, namely separately encoding the independent messages, is optimal for symmetric multilevel diversity coding (SMDC) (Yeung-Zhang 1999). In the current paper, we consider weakly secure SMDC where security constraints are injected on each individual message, and provide a complete characterization of the conditions under which superposition co…
A general inner bound is given for the discrete memoryless broadcast channel with an arbitrary number of users and general message sets, a setting that accounts for the most general form of concurrent groupcasting, with messages intended for any set of subsets of receivers. Achievability is based on superposition coding and rate-splitting without and with binning, where each receiver jointly decodes both its desired messages as well as the partial interference assigned to it via rate-splitting. The proof of achievability builds on the techniques for the description and analysis of superpositio…
We derive universal classical-quantum superposition coding and universal classical-quantum multiple access channel code by using generalized packing lemmas for the type method. Using our classical-quantum universal superposition code, we establish the capacity region of a classical-quantum compound broadcast channel with degraded message sets. Our universal classical-quantum multiple access channel codes have two types of codes. One is a code with joint decoding and the other is a code with separate decoding. The former universally achieves corner points of the capacity region and the latter u…
In this paper, we consider a primary and a cognitive user transmitting over a wireless fading interference channel. The primary user transmits with a constant power and utilizes an adaptive modulation and coding (AMC) scheme satisfying a bit error rate requirement. We propose a link adaptation scheme to maximize the average spectral efficiency of the cognitive radio, while a minimum required spectral efficiency for the primary user is provisioned. The resulting problem is constrained to also satisfy a bit error rate requirement and a power constraint for the cognitive link. The AMC mode select…
Existing polar coding schemes for the two-user interference channel follow the original idea of Han and Kobayashi, in which component messages are encoded independently and then mapped by some deterministic functions (i.e., homogeneous superposition coding). In this paper, we propose a new polar coding scheme for the interference channel based on the heterogeneous superposition coding approach of Chong, Motani and Garg. We prove that fully-joint decoding (the receivers simultaneously decode both senders' common messages and the intended sender's private message) in the Han-Kobayashi strategy c…
We study the $K=3$ user multiple input single output (MISO) broadcast channel (BC) with $M=3$ antennas at the transmitter and $1$ antenna at each receiver, from the generalized degrees of freedom (GDoF) perspective, under the assumption that the channel state information at the transmitter (CSIT) is limited to finite precision. In particular, our goal is to identify a parameter regime where a simple layered superposition (SLS) coding scheme achieves the entire GDoF region. With $α_{ij}$ representing the channel strength parameter for the link from the $j^{th}$ antenna of the transmitter to the…
We recently showed in [1] the superiority of certain structured coding matrices ensembles (such as partial row-orthogonal) for sparse superposition codes when compared with purely random matrices with i.i.d. entries, both information-theoretically and under practical vector approximate message-passing decoding. Here we generalize this result to binary input channels under generalized vector approximate message-passing decoding [2].We focus on specific binary output channels for concreteness but our analysis based on the replica symmetric method from statistical physics applies to any memoryles…
Recently, we proposed a spectrum-sharing paradigm for single-carrier cognitive radio (CR) networks, where a secondary user (SU) is able to maintain or even improve the performance of a primary user (PU) transmission, while also obtaining a low-data rate channel for its own communication. According to such a scheme, a simple multiplication is used to superimpose one SU symbol on a block of multiple PU symbols.The scope of this paper is to extend such a paradigm to a multicarrier CR network, where the PU employs an orthogonal frequency-division multiplexing (OFDM) modulation scheme. To improve i…
The introduction of 5G has changed the wireless communication industry. Whereas previous generations of cellular technology are mainly based on communication for people, the wireless industry is discovering that 5G may be an era of communications that is mainly focused on machine-to-machine communication. The application of Ultra Reliable Low Latency Communication in factory automation is an area of great interest as it unlocks potential applications that traditional wired communications did not allow. In particular, the decrease in the inter-device distance has led to the discussion of coding…
Sparse superposition codes were originally proposed as a capacity-achieving communication scheme over the gaussian channel, whose coding matrices were made of i.i.d. gaussian entries.We extend this coding scheme to more generic ensembles of rotational invariant coding matrices with arbitrary spectrum, which include the gaussian ensemble as a special case. We further introduce and analyse a decoder based on vector approximate message-passing (VAMP).Our main findings, based on both a standard replica symmetric potential theory and state evolution analysis, are the superiority of certain structur…
We analyze the problem of transmitting information to multiple users over a shared wireless channel. The problem of resource allocation (RA) for the users with the knowledge of their channel state information has been treated extensively in the literature where various approaches trading off the users' throughput and fairness were proposed. The emphasis was mostly on the time-sharing (TS) approach, where the resource allocated to the user is equivalent to its time share of the channel access. In this work, we propose to take advantage of the broadcast nature of the channel and we adopt superpo…
Symmetrical multilevel diversity coding (SMDC) is a classical model for coding over distributed storage. In this setting, a simple separate encoding strategy known as superposition coding was shown to be optimal in terms of achieving the minimum sum rate (Roche, Yeung, and Hau, 1997) and the entire admissible rate region (Yeung and Zhang, 1999) of the problem. The proofs utilized carefully constructed induction arguments, for which the classical subset entropy inequality of Han (1978) played a key role. This paper includes two parts. In the first part the existing optimality proofs for classic…
This letter investigates the role of index coding in the capacity of AWGN broadcast channels with receiver message side information. We first show that index coding is unnecessary where there are two receivers; multiplexing coding and superposition coding are sufficient to achieve the capacity region. We next show that, for more than two receivers, multiplexing coding and superposition coding alone can be suboptimal. We give an example where these two coding schemes alone cannot achieve the capacity region, but adding index coding can. This demonstrates that, in contrast to the two-receiver ca…
In this paper, a joint cross-layer design of adaptive modulation and coding (AMC) and cooperative automatic repeat request (C-ARQ) scheme is proposed for a secondary user in a shared-spectrum environment. First, based on the statistical descriptions of the channel, closed-form expressions of the average spectral efficiency (SE) and the average packet loss rate (PLR) are presented. Then, the cross-layer scheme is designed, with the aim of maximizing the average SE while maintaining the average PLR under a prescribed level. An optimization problem is formed, and a sub-optimal solution is found: …
We propose in this paper a new coding scheme called twisted-pair superposition transmission (TPST). The encoding is to "mix together" a pair of basic codes by superposition, while the decoding can be implemented as a successive cancellation list decoding algorithm. The most significant features of the TPST code are its predictable performance that can be estimated numerically from the basic codes and its flexible construction in the sense that it can be easily adapted to different coding rates. To construct good TPST codes in the finite length regime, we propose two design approaches-rate allo…
An emerging requirement for 5G systems is the ability to provide wireless ultra-reliable communication (URC) services with close-to-full availability for cloud-based applications. Among such applications, a prominent role is expected to be played by mobile cloud computing (MCC), that is, by the offloading of computationally intensive tasks from mobile devices to the cloud. MCC allows battery-limited devices to run sophisticated applications, such as for gaming or for the "tactile" internet. This paper proposes to apply the framework of reliable service composition to the problem of optimal tas…
In this paper, it is shown that for any given single-hop communication network with two receivers, splitting messages into more than two sub-messages in a random coding scheme is redundant. To this end, the Broadcast Channel with Cognitive Relays (BCCR) is considered. A novel achievability scheme is designed for this network. Our achievability design is derived by a systematic combination of the best known achievability schemes for the basic building blocks included in the network: the Han-Kobayashi scheme for the two-user interference channel and the Marton coding scheme for the broadcast cha…
Subspace codes were introduced by Kötter and Kschischang for error control in random linear network coding. In this paper, a layered type of subspace codes is considered, which can be viewed as a superposition of multiple component subspace codes. Exploiting the layered structure, we develop two decoding algorithms for these codes. The first algorithm operates by separately decoding each component code. The second algorithm is similar to the successive interference cancellation (SIC) algorithm for conventional superposition coding, and further permits an iterative version. We show that both al…
This paper shows that the Maddah-Ali--Tse (MAT) scheme which establishes the symmetric capacity of two example broadcast channels with strictly causal state information at the transmitter is a simple special case of the Shayevitz--Wigger scheme for the broadcast channel with generalized feedback, which involves block Markov coding, compression, superposition coding, Marton coding, and coded time sharing. Focusing on the class of symmetric broadcast channels with state, we derive an expression for the maximum achievable symmetric rate using the Shayevitz--Wigger scheme. We show that the MAT res…
Previous works established the capacity region for some special cases of discrete memoryless degraded cognitive interference channel (CIC) with unidirectional destination cooperation (UDC). In this letter, we characterize the capacity region of the general discrete memoryless degraded CIC-UDC. The obtained results imply that the capacity region is achieved by the Gel'fand-Pinsker coding at the cognitive transmitter, superposition coding at the primary transmitter and decode-and-forward at the relay. Furthermore, using this general result and a novel converse analysis, we establish the capacity…
This paper studies multiuser random coding techniques for channel coding with a given (possibly suboptimal) decoding rule. For the mismatched discrete memoryless multiple-access channel, an error exponent is obtained that is tight with respect to the ensemble average, and positive within the interior of Lapidoth's achievable rate region. This exponent proves the ensemble tightness of the exponent of Liu and Hughes in the case of maximum-likelihood decoding. An equivalent dual form of Lapidoth's achievable rate region is given, and the latter is shown to extend immediately to channels with infi…
We consider a cognitive heterogeneous network (HetNet), in which multiple pairs of secondary users adopt sensing-based approaches to coexist with a pair of primary users on a certain spectrum band. Due to imperfect spectrum sensing, secondary transmitters (STs) may cause interference to the primary receiver (PR) and make it difficult for the PR to select a proper modulation and/or coding scheme (MCS). To deal with this issue, we exploit deep reinforcement learning (DRL) and propose an intelligent MCS selection algorithm for the primary transmission. To reduce the system overhead caused by MCS …
We establish the capacity region for a class of discrete memoryless cognitive interference channel (DM-CIC) called cognitive-more-capable channel, and we show that superposition coding is the optimal encoding technique. This is the largest capacity region for the DM-CIC to date, as the existing capacity results are explicitly shown to be its subsets.
We consider the full-duplex two-way relay channel with direct link between two users and propose two coding schemes: a partial decode-forward scheme, and a combined decode-forward and compute-forward scheme. Both schemes use rate-splitting and superposition coding at each user and generate codewords for each node independently. When applied to the Gaussian channel, partial decode-forward can strictly increase the rate region over decode-forward, which is opposite to the one-way relay channel. The combined scheme uses superposition coding of both Gaussian and lattice codes to allow the relay to…
We consider the problem of efficient ultra-massive multiple-input multiple-output (UM-MIMO) data detection in terahertz (THz)-band non-orthogonal multiple access (NOMA) systems. We argue that the most common THz NOMA configuration is power-domain superposition coding over quasi-optical doubly-massive MIMO channels. We propose spatial tuning techniques that modify antenna subarray arrangements to enhance channel conditions. Towards recovering the superposed data at the receiver side, we propose a family of data detectors based on low-complexity channel matrix puncturing, in which higher-order d…
A Dirty Paper Coding (DPC) based transmission scheme for the Gaussian multiple-input multiple-output (MIMO) cognitive radio channel (CRC) is studied when there is imperfect and perfect channel knowledge at the transmitters (CSIT) and the receivers, respectively. In particular, the problem of optimizing the sum-rate of the MIMO CRC over the transmit covariance matrices is dealt with. Such an optimization, under the DPC-based transmission strategy, needs to be performed jointly with an optimization over the inflation factor. To this end, first the problem of determination of inflation factor ove…
This paper focuses on the signal design in a Gaussian amplify-and-forward (AF) relay system with superposition coding (SC) being applied at the relay, which allows the source and relay to transmit their own information within two time slots. Practical quadrature amplitude modulation (QAM) constellations are adopted at the source and relay. To improve the system error performance, we optimize the weight coefficients adopted at the source and relay to maximize the minimum Euclidean distance of the received composite constellation, subject to their individual average power constraints. The formul…
The secrecy problem in the state-dependent cognitive interference channel is considered in this paper. In our model, there are a primary and a secondary (cognitive) transmitter-receiver pairs, in which the cognitive transmitter has the message of the primary one as side information. In addition, the channel is affected by a channel state sequence which is estimated partially at the cognitive transmitter and the corresponding receiver. The cognitive transmitter wishes to cooperate with the primary one, and it sends its individual message which should be confidential at the primary receiver. The…
In this paper, we present the cognitive radio system design with partial channel state information known at the transmitter (CSIT).We replace the dirty paper coding (DPC) used in the cognitive radio with full CSIT by the linear assignment Gel'fand-Pinsker coding (LA-GPC), which can utilize the limited knowledge of the channel more efficiently. Based on the achievable rate derived from the LA-GPC, two optimization problems under the fast and slow fading channels are formulated. We derive semianalytical solutions to find the relaying ratios and precoding coefficients. The critical observation is…
It is well known that {\em superposition coding}, namely separately encoding the independent sources, is optimal for symmetric multilevel diversity coding (SMDC) (Yeung-Zhang 1999). However, the characterization of the coding rate region therein involves uncountably many linear inequalities and the constant term (i.e., the lower bound) in each inequality is given in terms of the solution of a linear optimization problem. Thus this implicit characterization of the coding rate region does not enable the determination of the achievability of a given rate tuple. In this paper, we first obtain clos…
We study a new class of codes for Gaussian multi-terminal source and channel coding. These codes are designed using the statistical framework of high-dimensional linear regression and are called Sparse Superposition or Sparse Regression codes. Codewords are linear combinations of subsets of columns of a design matrix. These codes were recently introduced by Barron and Joseph and shown to achieve the channel capacity of AWGN channels with computationally feasible decoding. They have also recently been shown to achieve the optimal rate-distortion function for Gaussian sources. In this paper, we …
In this paper, we introduce the Causal Cognitive Interference Channel With Delay (CC-IFC-WD) in which the cognitive user transmission can depend on $L$ future received symbols as well as the past ones. Taking the effect of the link delays into account, CC-IFC-WD fills the gap between the genie-aided and causal 1cognitive radio channels. We study three special cases: 1) Classical CC-IFC (L=0), 2) CC-IFC without delay (L=1) and 3) CC-IFC with a block length delay (L=n). In each case, we obtain an inner bound on the capacity region. Our coding schemes make use of cooperative strategy by generaliz…
A novel class of achievable rate regions is obtained for the K-receiver broadcast channel with two groupcast messages. The associated achievability schemes are parameterized by an expansion of the message set which then determines how random coding techniques are employed, which include generalized versions of {\em up-set} message-splitting, the generation of possibly multiple auxiliary codebooks for certain compositions of split messages using superposition coding, partial interference decoding at all receivers, and joint unique and non-unique decoding. New capacity results are established fo…
A new achievable rate region is given for the Gaussian cognitive many-to-one interference channel. The proposed novel coding scheme is based on the compute-and-forward approach with lattice codes. Using the idea of decoding sums of codewords, our scheme improves considerably upon the conventional coding schemes which treat interference as noise or decode messages simultaneously. Our strategy also extends directly to the usual many-to-one interference channels without cognitive messages. Comparing to the usual compute-and-forward scheme where a fixed lattice is used for the code construction, t…
The spectrum sharing has recently passed into a mainstream Cognitive Radio (CR) strategy. We investigate the core issue in this strategy: interference mitigation at Primary Receiver (PR).We propose a linear precoder design which aims at alleviating the interference caused by Secondary User (SU) from the source for Orthogonal Space-Time Block Coding (OSTBC) based CR. We resort to Minimum Variance (MV) approach to contrive the precoding matrix at Secondary Transmitter (ST) in order to maximize the Signal to Noise Ratio (SNR) at Secondary Receiver (SR) on the premise that the orthogonality of OST…
Coding for the causal cognitive radio channel, with the cognitive source subjected to a half-duplex constraint, is studied. A discrete memoryless channel model incorporating the half-duplex constraint is presented, and a new achievable rate region is derived for this channel. It is proved that this rate region contains the previously known causal achievable rate region of \cite{Devroye06} for Gaussian channels.
The causal cognitive relay channel (CRC) has two sender-receiver pairs, in which the second sender obtains information from the first sender causally and assists the transmission of both senders. In this paper, we study both the full- and half-duplex modes. In each mode, we propose two new coding schemes built successively upon one another to illustrate the impact of different coding techniques. The first scheme called partial decode-forward binning (PDF-binning) combines the ideas of partial decode-forward relaying and Gelfand-Pinsker binning. The second scheme called Han-Kobayashi partial de…
Massive machine type communication (mMTC) has attracted new coding schemes optimized for reliable short message transmission. In this paper, a novel deep learning based near-orthogonal superposition (NOS) coding scheme is proposed for reliable transmission of short messages in the additive white Gaussian noise (AWGN) channel for mMTC applications. Similar to recent hyper-dimensional modulation (HDM), the NOS encoder spreads the information bits to multiple near-orthogonal high dimensional vectors to be combined (superimposed) into a single vector for transmission. The NOS decoder first estimat…
In conventional communication systems, any interference between two communicating points is regarded as unwanted noise since it distorts the received signals. On the other hand, allowing simultaneous transmission and intentionally accepting the interference of signals and even benefiting from it have been considered for a range of wireless applications. As prominent examples, non-orthogonal multiple access (NOMA), joint source-channel coding, and the computation codes are designed to exploit this scenario. They also inspired many other fundamental works from network coding to consensus algorit…
We study a two-user cognitive interference channel (CIC) where one of the transmitters (primary) has knowledge of a linear combination (over an appropriate finite field) of the two information messages. We refer to this channel model as Network-Coded CIC, since the linear combination may be the result of some linear network coding scheme implemented in the backbone wired network.In this paper, we characterize the generalized degrees of freedom (GDoF) for the Gaussian Network-Coded CIC. For achievability, we use the novel Precoded Compute-and-Forward (PCoF) and Dirty Paper Coding (DPC), based o…
In many information-theoretic channel coding problems, adding an input cost constraint to the operational setup amounts to restricting the optimization domain in the capacity formula. This paper shows that, in contrast to common belief, such a simple modification does not hold for the cost-constrained (CC) wiretap channel (WTC). The secrecy-capacity of the discrete memoryless (DM) WTC without cost constraints is described by a single auxiliary random variable. For the CC DM-WTC, however, we show that two auxiliaries are necessary to achieve capacity. Specifically, we first derive the secrecy-c…
In this paper, we consider three transmit strategies for the fading three-node, two-way relay network (TWRN) -- physical-layer network coding (PNC), digital network coding (DNC) and codeword superposition (CW-Sup). The aim is to minimize the total average energy needed to deliver a given pair of required average rates. Full channel state information is assumed to be available at all transmitters and receivers. The optimization problems corresponding to the various strategies in fading channels are formulated, solved and compared. For the DNC-based strategies, a simple time sharing of transmiss…
The capacity region of the Multicast Cognitive Interference Channel (CIFC) is investigated. This channel consists of two independent transmitters that wish to multicast two different messages, each of them to a different set of users. In addition, one of the transmitters --commonly referred to as the cognitive transmitter-- has prior non-causal knowledge of both messages to be transmitted. This scenario combines difficulties and challenges arising in the Interference Channel, the Broadcast Channel and multicasting communications. Our aim concerns the derivation of optimal interference mitigati…
We consider quantum channels with one sender and two receivers, used in several different ways for the simultaneous transmission of independent messages. We begin by extending the technique of superposition coding to quantum channels with a classical input to give a general achievable region. We also give outer bounds to the capacity regions for various special cases from the classical literature and prove that superposition coding is optimal for a class of channels. We then consider extensions of superposition coding for channels with a quantum input, where some of the messages transmitted ar…
This paper studies the transmission of Gaussian sources through additive white Gaussian noise (AWGN) channels in bandwidth expansion regime, i.e., the channel bandwidth is greater than the source bandwidth. To mitigate the error propagation phenomenon of conventional digital transmission schemes, we propose in this paper a new capacity-approaching joint source channel coding (JSCC) scheme based on partially block Markov superposition transmission (BMST) of nested lattice codes. In the proposed scheme, first, the Gaussian source sequence is discretized by a lattice-based quantizer, resulting in…
