$2^m$-ary modulation creates $m$ bit channels which are neither independent nor identical, and this causes problems when applying polar coding because polar codes are designed for independent identical channels. Different from the existing multi-level coding (MLC) and bit-interleaved coded modulation (BICM) schemes, this paper provides a convolutional polar coded modulation (CPCM) method that preserves the low-complexity nature of BICM while offering improved spectral efficiency. Numerical results are given to show the good performance of the proposed method.
Coded modulation is a key technique to increase the spectral efficiency of coherent optical communication systems. Two popular strategies for coded modulation are turbo trellis-coded modulation (TTCM) and bit-interleaved coded modulation (BICM) based on low-density parity-check (LDPC) codes. Although BICM LDPC is suboptimal, its simplicity makes it very popular in practice. In this work, we compare the performance of TTCM and BICM LDPC using information-theoretic measures. Our information-theoretic results show that for the same overhead and modulation format only a very small penalty (less th…
A practical $2^m$-ary polar coded modulation (PCM) scheme with optimal constellation labeling is proposed. To efficiently find the optimal labeling rule, the search space is reduced by exploiting the symmetry properties of the channels. Simulation results show that the proposed PCM scheme can outperform the bit-interleaved turbo coded modulation scheme used in the WCDMA (Wideband Code Division Multiple Access) mobile communication systems by up to 1.5dB.
We consider the joint design of polar coding and higher-order modulation schemes for ever increased spectral efficiency. The close connection between the polar code construction and the multi-level coding approach is described in detail. Relations between different modulation schemes such as bit-interleaved coded modulation (BICM) and multi-level coding (MLC) in case of polar-coded modulation as well as the influence of the applied labeling rule and the selection of frozen channels are demonstrated.
We consider probabilistic shaping to maximize the achievable information rate of coded modulation (CM) with hard decision decoding. The proposed scheme using binary staircase codes outperforms its uniform CM counterpart by more than 1.3 dB for 64-QAM and 5 bits/symbol.
We discuss the application of coded modulation for power-line communications. We combine M-ary FSK with diversity and coding to make the transmission robust against permanent frequency disturbances and impulse noise. We give a particular example of the coding/modulation scheme that is in agreement with the existing CENELEC norms. The scheme can be considered as a form of coded Frequency Hopping and is thus extendable to any frequency range.
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.
An LDPC coded modulation scheme with probabilistic shaping, optimized interleavers and noniterative demapping is proposed. Full-field simulations show an increase in transmission distance by 8% compared to uniformly distributed input.
A framework is proposed that allows for a joint description and optimization of both binary polar coding and the multilevel coding (MLC) approach for $2^m$-ary digital pulse-amplitude modulation (PAM). The conceptual equivalence of polar coding and multilevel coding is pointed out in detail. Based on a novel characterization of the channel polarization phenomenon, rules for the optimal choice of the bit labeling in this coded modulation scheme employing polar codes are developed. Simulation results for the AWGN channel are included.
A polar-coded modulation scheme for deep-space optical communication is proposed. The photon counting Poisson channel with pulse position modulation (PPM) is considered. We use the fact that PPM is particularly well suited to be used with multilevel codes to design a polar-coded modulation scheme for the system in consideration. The construction of polar codes for the Poisson channel based on Gaussian approximation is demonstrated to be accurate. The proposed scheme uses a cyclic redundancy check outer code and a successive cancellation decoder with list decoding and it is shown that it outper…
The use of multiple frequency shift keying modulation with permutation codes addresses the problem of permanent narrowband noise disturbance in a power line communications system. In this paper, we extend this coded modulation scheme based on permutation codes to general codes and introduce an additional new parameter that more precisely captures a code's performance against permanent narrowband noise. As a result, we define a new class of codes, namely, equitable symbol weight codes, which are optimal with respect to this measure.
A commonly assumed drawback of multi-level coding, compared to a bit-interleaved coded modulation, is its high latency: Indeed, the levels must be decoded sequentially. In this paper, we consider polar codes to code each level. We show that the decoding time complexity of the multi-level scheme, using successive-cancellation list decoding for each polar code, is only 1.5 times the one of a single polar code, regardless of the signal-to-noise ratio and the number of levels.
Non-orthogonal multiple access (NOMA) schemes have been proved to increase the multiple-access achievable rate with respect to orthogonal multiple access (OMA). In this paper we propose a novel communication system that combines multi-level coded modulation and polar codes in a NOMA scenario. Computational complexity decreases with the proposed scheme with respect to state-of-the-art solutions. We also highlight the trade-off between error rate performance and computational complexity.
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.
Coded caching is an information theoretic scheme to reduce high peak hours traffic by partially prefetching files in the users local storage during low peak hours. This paper considers heterogeneous decentralized caching systems where cache of users and content library files may have distinct sizes. The server communicates with the users through a Gaussian broadcast channel. The main contribution of this paper is a novel modulation strategy to map the multicast messages generated in the coded caching delivery phase to the symbols of a signal constellation, such that users can leverage their ca…
Two decoder structures for coded modulation over the Gaussian and flat fading channels are studied: the maximum likelihood symbol-wise decoder, and the (suboptimal) bit-wise decoder based on the bit-interleaved coded modulation paradigm. We consider a 16-ary quadrature amplitude constellation labeled by a Gray labeling. It is shown that the asymptotic loss in terms of pairwise error probability, for any two codewords caused by the bit-wise decoder, is bounded by 1.25 dB. The analysis also shows that for the Gaussian channel the asymptotic loss is zero for a wide range of linear codes, includin…
The wideband regime of bit-interleaved coded modulation (BICM) in Gaussian channels is studied. The Taylor expansion of the coded modulation capacity for generic signal constellations at low signal-to-noise ratio (SNR) is derived and used to determine the corresponding expansion for the BICM capacity. Simple formulas for the minimum energy per bit and the wideband slope are given. BICM is found to be suboptimal in the sense that its minimum energy per bit can be larger than the corresponding value for coded modulation schemes. The minimum energy per bit using standard Gray mapping on M-PAM or …
We consider the problem of using polar codes with higher order modulation over AWGN channels. Unlike prior work, we focus on using modulation independent polar codes. That is, the polar codes are not re-designed based on the modulation used. Instead, we propose bit-permuted coded modulation (BPCM): a technique for using the multilevel coding (MLC) approach for an arbitrary polar code. The BPCM technique exploits a natural connection between MLC and polar codes. It involves applying bit permutations prior to mapping the polar code to a higher order modulation. The bit permutations are designed,…
Bit-interleaved coded modulation (BICM) and multilevel coded modulation (MLC) are commonly used to combine polar codes with high order modulation. While BICM benefits from simple design and the separation of coding and modulation, MLC shows better performance under successive-cancellation decoding. In this paper we propose a hybrid polar coded modulation scheme that lies between BICM and MLC, wherein a fraction of bits are assigned to set-partition (SP) labeling and the remaining bits are assigned for Gray labeling. The SP labeled bits undergo sequential demodulation, using iterative demodulat…
The intuitive sphere-packing argument is used to obtain analytically-tractable closed-form approximations for achievable information rates of coded modulation transmission systems, for which only analytically-intractable expressions are available in the literature. These approximations provide a number of insights, possess useful properties and facilitate design/optimization of such systems. They apply to constellations of various cardinalities (including large ones), are simple yet reasonably accurate over the whole signal-to-noise ratio range, and compare favorably to the achieved rates of r…
A pragmatic coded modulation system is presented that incorporates signal shaping and exploits the excellent performance and efficient high-speed decoding architecture of staircase codes. Reliable communication within 0.62 bits/s/Hz of the estimated capacity (per polarization) of a system with L=2000 km is provided by the proposed system, with an error floor below 1E-20. Also, it is shown that digital backpropagation increases the achievable spectral efficiencies---relative to linear equalization---by 0.55 to 0.75 bits/s/Hz per polarization.
The capacity of the AWGN broadcast channel is achieved by superposition coding, but superposition of individual coded modulations expands the modulation alphabet and distorts its configuration. Coded modulation over a broadcast channel subject to a specific channel-input modulation constraint remains an important open problem. Some progress has been made in the related area of unequal-error protection modulations which can be considered single-user broadcast transmission, but it does not approach all points on the boundary of the capacity region. This paper studies broadcast coded modulation u…
Coded modulation with probabilistic amplitude shaping (PAS) is considered for intensity modulation/direct detection channels with a transmitter peak-power constraint. PAS is used to map bits to a uniform PAM-6 distribution and outperforms PAM-8 for rates up to around 2.3 bits per channel use. PAM-6 with PAS also outperforms a cross-shaped QAM-32 constellation by up to 1 dB and 0.65 dB after bit-metric soft- and hard decoding, respectively. An alternative PAM-6 scheme based on a framed cross-shaped QAM-32 constellation is proposed that shows similar gains.
Caire, Taricco and Biglieri presented a detailed analysis of bit interleaved coded modulation, a simple and popular technique used to improve system performance, especially in the context of fading channels. They derived an upper bound to the probability of error, called the expurgated bound. In this correspondence, the proof of the expurgated bound is shown to be flawed. A new upper bound is also derived. It is not known whether the original expurgated bound is valid for the important special case of square QAM with Gray labeling, but the new bound is very close to, and slightly tighter than,…
A new variant of bit interleaved coded modulation (BICM) is proposed. In the new scheme, called Parallel BICM, L identical binary codes are used in parallel using a mapper, a newly proposed finite-length interleaver and a binary dither signal. As opposed to previous approaches, the scheme does not rely on any assumptions of an ideal, infinite-length interleaver. Over a memoryless channel, the new scheme is proven to be equivalent to a binary memoryless channel. Therefore the scheme enables one to easily design coded modulation schemes using a simple binary code that was designed for that binar…
In this paper, we investigate network nest coded modulation schemes for multiple access relay channels. The performance of the distributed systems which are based on distributed convolutional codes with network coded modulation is presented. An analytical upper bound on bit error probability performance for the studied distributed systems with Maximum Likelihood Sequence Detection (MLSD) is derived. The constructured bounds for the investigated systems are shown to be asymptotically tight for increasing channel Signal-to-Noise Ratio (SNR) values
In this paper, we propose a novel joint coding-modulation technique based on serial concatenation of orthogonal linear transform, such as discrete Fourier transform (DFT) or Walsh-Hadamard transform (WHT), with memoryless nonlinearity. We demonstrate that such a simple signal construction may exhibit properties of a random code ensemble, as a result approaching channel capacity. Our computer simulations confirm that if the decoder relies on a modified approximate message passing algorithm, the proposed modulation technique exhibits performance on par with state-of-the-art coded modulation sche…
This paper investigates the design and performance of delayed bit-interleaved coded modulation (DBICM) with low-density parity-check (LDPC) codes. For Gray labeled square $M$-ary quadrature amplitude modulation (QAM) constellations, we investigate the optimal delay scheme with the largest spectrum efficiency of DBICM for a fixed maximum number of delayed time slots and a given signal-to-noise ratio. When analyzing the capacity of DBICM, we find two important properties: the capacity improvement due to delayed coded bits being mapped to the real and imaginary parts of the transmitted symbols ar…
In this paper, we present a low-complexity joint detection-decoding algorithm for nonbinary LDPC codedmodulation systems. The algorithm combines hard-decision decoding using the message-passing strategy with the signal detector in an iterative manner. It requires low computational complexity, offers good system performance and has a fast rate of decoding convergence. Compared to the q-ary sum-product algorithm (QSPA), it provides an attractive candidate for practical applications of q-ary LDPC codes.
A framework is proposed that allows for a joint description and optimization of both binary polar coding and $2^m$-ary digital pulse-amplitude modulation (PAM) schemes such as multilevel coding (MLC) and bit-interleaved coded modulation (BICM). The conceptual equivalence of polar coding and multilevel coding is pointed out in detail. Based on a novel characterization of the channel polarization phenomenon, rules for the optimal choice of the labeling in coded modulation schemes employing polar codes are developed. Simulation results regarding the error performance of the proposed schemes on th…
We study the problem of constructing coded modulation schemes over multidimensional signal sets in Nakagami-$m$ block-fading channels. In particular, we consider the optimal diversity reliability exponent of the error probability when the multidimensional constellation is obtained as the rotation of classical complex-plane signal constellations. We show that multidimensional rotations of full dimension achieve the optimal diversity reliability exponent, also achieved by Gaussian constellations. Multidimensional rotations of full dimension induce a large decoding complexity, and in some cases i…
In this letter, we propose a progressive rate-filling method as a framework to study agile construction of multilevel polar-coded modulation. We show that the bit indices within each component polar code can follow a fixed, precomputed ranking sequence, e.g., the Polar sequence in the 5G standard, while their allocated rates (i.e., the number of information bits of each component polar code) can be fast computed by exploiting the target sum-rate approximation and proper rate-filling methods. In particular, we develop two rate-filling strategies based on the capacity and the rate considering th…
In this paper we present a block coded modulation scheme for a 2 x 2 MIMO system over slow fading channels, where the inner code is the Golden Code. The scheme is based on a set partitioning of the Golden Code using two-sided ideals whose norm is a power of two. In this case, a lower bound for the minimum determinant is given by the minimum Hamming distance. The description of the ring structure of the quotients suggests further optimization in order to improve the overall distribution of determinants. Performance simulations show that the GC-RS schemes achieve a significant gain over the unco…
The Golden space-time trellis coded modulation (GST-TCM) scheme was proposed in \cite{Hong06} for a high rate $2\times 2$ multiple-input multiple-output (MIMO) system over slow fading channels. In this letter, we present the performance analysis of GST-TCM over block fading channels, where the channel matrix is constant over a fraction of the codeword length and varies from one fraction to another, independently. In practice, it is not useful to design such codes for specific block fading channel parameters and a robust solution is preferable. We then show both analytically and by simulation t…
The polar codes are proven to be capacity-achieving and are shown to have equivalent or even better finite-length performance than the turbo/LDPC codes under some improved decoding algorithms over the additive white Gaussian noise (AWGN) channels. Polar coding is based on the so-called channel polarization phenomenon induced by a transform over the underlying binary-input channel. The channel polarization is found to be universal in many signal processing problems and has been applied to the coded modulation schemes. In this paper, the channel polarization is further extended to the multiple a…
We develop a low-complexity coding scheme to achieve covert communications over binary-input discrete memoryless channels (BI-DMCs). We circumvent the impossibility of covert communication with linear codes by introducing non-linearity through the use of pulse position modulation (PPM) and multilevel coding (MLC). We show that the MLC-PPM scheme exhibits many appealing properties; in particular, the channel at a given index level remains stationary as the number of level increases, which allows one to use families of channel capacity- and channel resolvability-achieving codes to concretely ins…
We consider the special case of index coding over the Gaussian broadcast channel where each receiver has prior knowledge of a subset of messages at the transmitter and demands all the messages from the source. We propose a concatenated coding scheme for this problem, using an index code for the Gaussian channel as an inner code/modulation to exploit side information at the receivers, and an outer code to attain coding gain against the channel noise. We derive the capacity region of this scheme by viewing the resulting channel as a multiple-access channel with many receivers, and relate it to t…
In this paper, code pairs based on trellis coded modulation are proposed over PSK signal sets for a two-user Gaussian multiple access channel. In order to provide unique decodability property to the receiver and to maximally enlarge the constellation constrained (CC) capacity region, a relative angle of rotation is introduced between the signal sets. Subsequently, the structure of the \textit{sum alphabet} of two PSK signal sets is exploited to prove that Ungerboeck labelling on the trellis of each user maximizes the guaranteed minimum squared Euclidean distance, $d^{2}_{g, min}$ in the \texti…
It is shown that while the mutual information curves for coded modulation (CM) and bit interleaved coded modulation (BICM) overlap in the case of a single input single output channel, the same is not true in multiple input multiple output (MIMO) channels. A method for mitigating fading in the presence of multiple transmit antennas, named coordinate interleaving (CI), is presented as a generalization of component interleaving for a single transmit antenna. The extent of any advantages of CI over BICM, relative to CM, is analyzed from a mutual information perspective; the analysis is based on an…
In this paper, we propose a novel non-orthogonal multiple access (NOMA) scheme based on trellis-coded modulation (TCM). Different from those in the traditional code-domain NOMA, the incoming data streams of multiple users are jointly coded and mapped to the codewords so as to improve the coding gain of the system without any bandwidth extension. Both codeword design and multi-user detection are considered to optimize the TCM for NOMA. The multi-dimensional (MD) constellation is carefully selected and a new set partitioning algorithm is developed for MD signal set labeling. To achieve the trade…
This paper investigates the bit-interleaved coded generalized spatial modulation (BICGSM) with iterative decoding (BICGSM-ID) for multiple-input multiple-output (MIMO) visible light communications (VLC). In the BICGSM-ID scheme, the information bits conveyed by the signal-domain (SiD) symbols and the spatial-domain (SpD) light emitting diode (LED)-index patterns are coded by a protograph low-density parity-check (P-LDPC) code. Specifically, we propose a signal-domain symbol expanding and re-allocating (SSER) method for constructing a type of novel generalized spatial modulation (GSM) constella…
We propose a hybrid coded modulation scheme which composes of inner and outer codes. The outer-code can be any standard binary linear code with efficient soft decoding capability (e.g. low-density parity-check (LDPC) codes). The inner code is designed using a deep neural network (DNN) which takes the channel coded bits and outputs modulated symbols. For training the DNN, we propose to use a loss function that is inspired by the generalized mutual information. The resulting constellations are shown to outperform the conventional quadrature amplitude modulation (QAM) based coding scheme for modu…
A flexible incremental redundancy hybrid auto- mated repeat request (IR-HARQ) scheme for polar codes is proposed based on dynamically frozen bits and the quasi-uniform puncturing (QUP) algorithm. The length of each transmission is not restricted to a power of two. It is applicable for the binary input additive white Gaussian noise (biAWGN) channel as well as higher-order modulation. Simulation results show that this scheme has similar performance as directly designed polar codes with QUP and outperforms LTE-turbo and 5G-LDPC codes with IR-HARQ.
Delayed bit-interleaved coded modulation (DBICM) generalizes bit-interleaved coded modulation (BICM) by modulating differently delayed sub-blocks of codewords onto the same signals. DBICM improves transmission reliability over BICM due to its capability of detecting undelayed sub-blocks with the extrinsic information of the decoded delayed sub-blocks. In this work, we propose a novel windowed decoding algorithm for DBICM, which uses the extrinsic information of both the decoded delayed and undelayed sub-blocks, to improve the detection on all sub-blocks. Numerical results show that the propose…
Trellis-coded modulation (TCM) is a power and bandwidth efficient digital transmission scheme which offers very low structural delay of the data stream. Classical TCM uses a signal constellation of twice the cardinality compared to an uncoded transmission with one bit of redundancy per PAM symbol, i.e., application of codes with rates $\frac{n-1}{n}$ when $2^{n}$ denotes the cardinality of the signal constellation. Recently published work allows rate adjustment for TCM by means of puncturing the convolutional code (CC) on which a TCM scheme is based on. In this paper it is shown how punctured …
In classic trellis-coded modulation (TCM) signal constellations of twice the cardinality are applied when compared to an uncoded transmission enabling transmission of one bit of redundancy per PAM-symbol, i.e., rates of $\frac{K}{K+1}$ when $2^{K+1}$ denotes the cardinality of the signal constellation. In order to support different rates, multi-dimensional (i.e., $\mathcal{D}$-dimensional) constellations had been proposed by means of combining subsequent one- or two-dimensional modulation steps, resulting in TCM-schemes with $\frac{1}{\mathcal{D}}$ bit redundancy per real dimension. In contras…
We consider a relay channel for which the following assumptions are made. (1) The source-destination and relay-destination channels are orthogonal (frequency division relay channel). (2) The relay implements the decode-and-forward protocol. (3) The source and relay implement the same channel encoder, namely, a onvolutional encoder. (4) They can use arbitrary and possibly different modulations. In this framework, we derive the best combiner in the sense of the maximum likelihood (ML) at the destination and the branch metrics of the trellis associated with its channel decoder for the ML combiner…
The space-time bit-interleaved coded modulation (ST-BICM) is an efficient technique to obtain high diversity and coding gain on a block-fading MIMO channel. Its maximum-likelihood (ML) performance is computed under ideal interleaving conditions, which enables a global optimization taking into account channel coding. Thanks to a diversity upperbound derived from the Singleton bound, an appropriate choice of the time dimension of the space-time coding is possible, which maximizes diversity while minimizing complexity. Based on the analysis, an optimized interleaver and a set of linear precoders,…
A new coded modulation scheme is proposed. At the transmitter, the concatenation of a distribution matcher and a systematic binary encoder performs probabilistic signal shaping and channel coding. At the receiver, the output of a bitwise demapper is fed to a binary decoder. No iterative demapping is performed. Rate adaption is achieved by adjusting the input distribution and the transmission power. The scheme is applied to bipolar amplitude shift keying (ASK) constellations with equidistant signal points and it is directly applicable to two-dimensional quadrature amplitude modulation (QAM). Th…
Bit-interleaved coded modulation (BICM) has attracted considerable attention from the research community in the past three decades, because it can achieve desirable error performance with relatively low implementation complexity for a large number of communication and storage systems. By exploiting the iterative demapping and decoding (ID), the BICM is able to approach capacity limits of coded modulation over various channels. In recent years, protograph low-density parity-check (PLDPC) codes and their spatially-coupled (SC) variants have emerged to be a pragmatic forward-error-correction (FEC…
We present lower bounds on the minimum pseudocodeword effective Euclidean distance (or minimum "pseudodistance") for coded modulation systems using linear codes with $q$-ary phase-shift keying (PSK) modulation over the additive white Gaussian noise (AWGN) channel. These bounds apply to both binary and nonbinary coded modulation systems which use direct modulation mapping of coded symbols. The minimum pseudodistance may serve as a first-order measure of error-correcting performance for both linear-programming and message-passing based receivers. In the case of a linear-programming based receive…
This paper develops a low-complexity near-optimal non-coherent receiver for a multi-level energy-based coded modulation system. Inspired by the turbo processing principle, we incorporate the fundamentals of bit-interleaved coded modulation with iterative decoding (BICM-ID) into the proposed receiver design. The resulting system is called bit-interleaved coded energy-based modulation with iterative decoding (BICEM-ID) and its error performance is analytically studied. Specifically, we derive upper bounds on the average pairwise error probability (PEP) of the non-coherent BICEM-ID system in the …
In this paper we consider noisy index coding problem over AWGN channel. We give an algorithm to map the index coded bits to appropriate sized PSK symbols such that for the given index code, in general, the receiver with large amount of side information will gain in probability of error performance compared to the ones with lesser amount, depending upon the index code used. We call this the \textbf{PSK side information coding gain}. Also, we show that receivers with large amount of side information obtain this coding gain in addition to the bandwidth gain whereas receivers with lesser amount of…
Visible light communication (VLC) systems using the indoor lighting system to also provide downlink communications require high average optical powers in order to satisfy the illumination needs. This can cause high amplitude signals common in higher-order modulation schemes to be clipped by the peak power constraint of the light emitting diode (LED) and lead to high signal distortion. In this paper we introduce Hadamard coded modulation (HCM) to achieve low error probabilities in LED-based VLC systems needing high average optical powers. This technique uses a fast Walsh-Hadamard transform (FWH…
This paper investigates the design of spatially coupled low-density parity-check (SC-LDPC) codes constructed from connected-chain ensembles for bit-interleaved coded modulation (BICM) schemes. For short coupling lengths, connecting multiple SC-LDPC chains can improve decoding performance over single-chains and impose structured unequal error protection (UEP). A joint design of connected-chain ensembles and bit mapping to further exploit the UEP from codes and high-order modulations is proposed. Numerical results demonstrate the superiority of the proposed design over existing connected-chain e…
In this paper, we present a concatenated coding scheme for a high rate $2\times 2$ multiple-input multiple-output (MIMO) system over slow fading channels. The inner code is the Golden code \cite{Golden05} and the outer code is a trellis code. Set partitioning of the Golden code is designed specifically to increase the minimum determinant. The branches of the outer trellis code are labeled with these partitions. Viterbi algorithm is applied for trellis decoding. In order to compute the branch metrics a lattice sphere decoder is used. The general framework for code optimization is given. The per…
