Coding Gain Research Articles

Advanced fine-tuning procedures to enhance DNN robustness in visual coding for machines

Video Coding for Machines (VCM) is gaining momentum in applications like autonomous driving, industry manufacturing, and surveillance, where the robustness of machine learning algorithms against coding artifacts is one of the key success factors. This work complements the MPEG/JVET standardization efforts in improving the resilience of deep neural network (DNN)-based machine models against such coding artifacts by proposing the following three advanced fine-tuning procedures for their training: (1) the progressive increase of the distortion strength as the training proceeds; (2) the incorporation of a regularization term in the original loss function to minimize the distance between predictions on compressed and original content; and (3) a joint training procedure that combines the proposed two approaches. These proposals were evaluated against a conventional fine-tuning anchor on two different machine tasks and datasets: image classification on ImageNet and semantic segmentation on Cityscapes. Our joint training procedure is shown to reduce the training time in both cases and still obtain a 2.4% coding gain in image classification and 7.4% in semantic segmentation, whereas a slight increase in training time can bring up to 9.4% better coding efficiency for the segmentation. All these coding gains are obtained without any additional inference or encoding time. As these advanced fine-tuning procedures are standard-compliant, they offer the potential to have a significant impact on visual coding for machine applications.

Rate 5/6 TCM Code Having 64 States with 64 QAM for Fading Channel

Trellis Coded Modulation (TCM) is powerful technique employed in digital communication systems to improve the reliability and efficiency of data transmission. TCM combines error control coding and modulation schemes to achieve superior performance in challenging channel conditions. In TCM, information bits are encoded using a trellis encoder, which generates a sequence of encoded symbols. These symbols are then mapped onto a modulation scheme, such as Quadrature Amplitude Modulation (QAM) or Phase Shift Keying (PSK), to create the modulated signal. At the receiver, the received signal is demodulated and decoded using a trellis decoder, which employs maximum likelihood decoding to recover the original information bits. The trellis structure allows for efficient error correction and makes TCM particularly suitable for channels with fading, noise, and other impairments. TCM has been widely used in various communication standards and applications to enhance data transmission reliability and spectral efficiency. TCM is a coding technique that merges coding and modulation to attain substantial coding gain without compromising bandwidth efficiency. The emergence of TCM occurred in the late 1970s, and ever since, it has found extensive utilization in various contemporary information transmission systems. In this paper, a new methodology is introduced for the design of a 64-state rate 5/6 TCM code optimized for fading channels. The outcomes obtained from the study are remarkably encouraging, demonstrating coding gain of around 10 dB in comparison to uncoded 32 PSK. This substantial gain plays a pivotal role in boosting data rates, especially in situations where bandwidth-limited channels are involved.

Feature of the forward error correction constructions for optical transport networks

Due to the rapid increase of network traffic in the last few years, many telecommunication operators have started transitions to 100G optical networks and beyond. However, high speed optical networks need more efficient Forward Error Correction (FEC) codes to deal with the optical impairments. Forward error correction is widely used in optical communications to improve error correction and extend optical transmission distance. It is also used to reduce optical transmitter power and system costs. In response to the rapid growth of optical communications, the ITU-T has started researching FEC coding and has recommended ITU-T G.709 and G.975.1. As optical transmission systems evolve towards longer transmission distance, greater capacity, and speeds of 100G and beyond a variety of unwanted effects are seriously affecting smooth evolution. In optical transport network, FEC is getting more and more important as spectral efficiency is increasing. As 100G system is commercialized, there are some trends in the evolution of FEC, for example overhead is increased from 7% to 20% or even more. High-performance FEC codes are therefore being developed so that higher net coding gain (NCG) and better error correction can be achieved. The Optical Internetworking Forum (OIF) suggests that soft-decision forward-error correction (SD-FEC) with a redundancy of 20% and beyond be used in a 100G DWDM coherent system. The use of advanced modulation formats is a promising solution for attaining such high data rates in optical networks. However, as a signal constellation grows in size, so does the optical signal to signal-to-noise ratio it requires to achieve a certain bit error rate (BER). This will degrade the error correction capability. Decoding is moving from hard to soft. Vendors are developing their own advanced codes instead of using codes recommended by ITU-T. Because different application systems have different decoding performance, latency, power and cost requirements, FEC systems with various transmission overhead, implementation complexity, burst-error correction ability and error floor are available in today’s market. Soft-decision FEC often uses turbo product codes and low density parity check (LDPC) codes. In research area, new codes such as LDPC codes and their different varieties are introduced into optical communication. Optimized codeword structure and decoding algorithm that provides low BER. LDPC codes can be very powerful, but their practical implementation for optical communications links at ultra-high data rates remains a challenge since the decoding of the code requires soft-decision bits about the received bits. A family of structured codes known as quasi-cyclic LDPC codes, whose common structural properties can be exploited by unified encoding and decoding architectures to reduce the complexity in implementation.

Labeling-Based Recipient Identification with Low-Order Modulation

Labeling-Based Recipient Identification (LABRID) brings the possibility of representing the destination station address in a Bit-Interleaved Coded Modulation with Iterative Decoding (BICM-ID) system by a signal labeling rule. Low-order modulations, such as BPSK or QPSK, pose a general problem for BICM-ID due to a limited convergence of iterative decoding. In the context of LABRID, they have one more drawback—a small number of different labeling rules in general; the number of the optimal ones, which exhibit the maximal asymptotic coding gain, is reduced even further. Meanwhile, LABRID needs a sizable collection of different optimal labeling rules to serve many users in large wireless networks. In this paper, the author suggests the use of hypercube BPSK or QPSK labeling to overcome all these challenges. By means of the Reactive Tabu Search (RTS) algorithm, more than 1500 equivalent optimal hypercube labeling rules are found. Analytical error bounds of the system are developed and supported by simulation experiments. Then, the focus is moved to the criterion to determine the frame destination at the LABRID receiver; a simple threshold-based method is proposed to keep the incorrect decision probability below 10−5. Finally, it is shown that LABRID outperforms a reference BICM-ID system in terms of computational complexity.

Analysis of the asymptotical average packet error rate over Rician fading channels

In the paper, we develop a simple and rather accurate method to analyze and research the average packet error rate (PER) in the high SNR region for the packet transmission system over Rician channels. By introducing the concept of inverse coding gain, the calculation expression of the average PER under the Rician fading channel can be obtained. Numerical calculations and simulations validate our theoretical analysis and approximation.

Strategies for Optimal Transmission and Delay Reduction in Dynamic Index Coding Problem

Dynamic index coding is a practical generalization of conventional index coding that deals with real dynamic traffic streams. We identify the code-constrained capacity region of a dynamic index coding problem with a complete bi-directional side information graph and introduce the performance metric of dynamic index coding gain to measure how dynamic index coding reduces the required data transmissions. A greedy dynamic index coding scheme is proposed that achieves the maximum coding gain almost everywhere in the identified capacity region. Although the greedy scheme attains the maximum coding gain, its selfish nature may unacceptably increase transmission delay. To address this issue, a time-shared friendly dynamic index coding scheme is introduced that achieves the maximum coding gain over the entire capacity region and offers a lower delay than its greedy counterpart. To obtain the minimum delay, a constrained optimization problem is formulated to tune time-sharing weights in the friendly scheme. The closed-form solution of the optimization is derived for the special two-flow case. Furthermore, the results and analysis are extended to dynamic index coding problems with arbitrary side information graphs. We also use analytical and simulation results to provide graphical intuition for the obtained results.

Secrecy Performance for RIS-Based Integrated Satellite Vehicle Networks With a UAV Relay and MRC Eavesdropping

This work investigates the secrecy issues for the secrecy reconfigurable intelligent surface-based integrated satellite unmanned aerial vehicle (UAV) relay networks with multiple vehicle eavesdroppers. Particularly, the maximal ratio combining (MRC) eavesdropping is utilized to overhear the legitimate signal information. The UAV is applied to forward the legitimate signal to the destination user. Reconfigurable intelligent surface is applied to ensure the transmission. Relied on the proposed secrecy system model and MRC eavesdropping scheme, this paper gets the detailed investigations for the secrecy outage probability (SOP) through the whole signal-to-noise ratios (SNRs) to valuate the effects of important parameters on the SOP. Moreover, to achieve the further impacts of main parameters on the SOP at high SNRs, the approximate analysis for the SOP is also gotten together with the secrecy diversity order (SDO) and secrecy coding gain (SCG). At last, several typical Monte Carlo (MC) results are obtained to present the rightness of obtained theoretical analysis.

Secrecy Outage Probability for Two-Way Integrated Satellite Unmanned Aerial Vehicle Relay Networks with Hardware Impairments

In this paper, we investigate the secrecy outage performance for the two-way integrated satellite unmanned aerial vehicle relay networks with hardware impairments. Particularly, the closed-form expression for the secrecy outage probability is obtained. Moreover, to get more information on the secrecy outage probability in a high signal-to-noise regime, the asymptotic analysis along with the secrecy diversity order and secrecy coding gain for the secrecy outage probability are also further obtained, which presents a fast method to evaluate the impact of system parameters and hardware impairments on the considered network. Finally, Monte Carlo simulation results are provided to show the efficiency of the theoretical analysis.

Phase-only hologram video compression using a deep neural network for up-scaling and restoration.

This paper proposes a coding method for compressing a phase-only hologram video (PoHV), which can be directly displayed in a commercial phase-only spatial light modulator. Recently, there has been active research to use a standard codec as an anchor to develop a new video coding for 3D data such as MPEG point cloud compression. The main merit of this approach is that if a new video codec is developed, the performance of relative coding methods can be increased simultaneously. Furthermore, compatibility is increased by the capability to use various anchor codecs, and the developing time is decreased. This paper uses a currently used video codec as an anchor codec and develops a coding method including progressive scaling and a deep neural network to overcome low temporal correlation between frames of a PoHV. Since it is difficult to temporally predict a correlation between frames of a PoHV, this paper adopts a scaling function and a neural network in the encoding and decoding process, not adding complexity to an anchor itself to predict temporal correlation. The proposed coding method shows an enhanced coding gain of an average of 22%, compared with an anchor in all coding conditions. When observing numerical and optical reconstructions, the result images by the proposed show clearer objects and less juddering than the result by the anchor.

Unselfish Coded Caching Can Yield Unbounded Gains Over Selfish Caching

The original coded caching scenario assumes a content library that is of interest to all receiving users. In a realistic scenario though, the users may have diverging interests which may intersect to various degrees. What happens for example if each file is of potential interest to, say, 40% of the users and each user has potential interest in 40% of the library? In this work, we investigate the so- called symmetrically selfish coded caching scenario, where each user only makes requests from a subset of the library that defines its own file demand set (FDS), each user caches selfishly only contents from its own FDS, and where the different FDSs symmetrically overlap to some extent. In the context of various traditional prefetching scenarios (prior to the emergence of coded caching), selfish approaches were known to be potentially very effective. On the other hand — with the exception of some notable works — little is known about selfish coded caching. We here present a new information-theoretic converse that proves, in a general setting of symmetric FDS structures, that selfish coded caching, despite enjoying a much larger local caching gain and a much smaller set of possible demands, introduces an unbounded load increase compared to the unselfish case. In particular, in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> -user broadcast channel where each user stores a fraction <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\gamma $ </tex-math></inline-formula> of the library, where each file (class) is of interest to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> users, and where any one specific file is of interest to a fraction <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> of users, the optimal coding gain of symmetrically selfish caching is at least <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(K - \alpha)\gamma + 1$ </tex-math></inline-formula> times smaller than in the unselfish scenario. This allows us to draw the powerful conclusion that the optimal selfish coding gain is upper bounded by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1/(1 - \delta)$ </tex-math></inline-formula> , and thus does not scale with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> . These derived limits are shown to be exact for different types of demands. In the end, this work provides, in a unified manner, the strong conclusion that selfish caching can cause unbounded performance deterioration in coded caching systems.

Physical Layer Security Analysis of Multi-Hop Hybrid RF/FSO System in Presence of Multiple Eavesdroppers

In this paper, the physical layer security (PLS) analysis for a decode-and-forward (DF) protocol based multi-hop hybrid radio frequency (RF)/ free space optics (FSO) is presented. Herein, two different scenarios are considered for detecting the secured information at the destination. In the first scenario, at each hop, the received signals with higher secrecy capacity are selected. However, in the second scenario, the RF and FSO signals are simply decoded and forwarded to the next hop and selection is done at the last hop only. Each node in the system is connected to its subsequent node through parallel RF and FSO links. The FSO links and RF links are characterized by Málaga ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {M}$</tex-math></inline-formula> ) and Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> composite distributions respectively. For both scenarios, the secrecy outage probability (SOP), strictly positive secrecy capacity (SPSC), intercept probability (IP), and effective secrecy throughput (EST) are obtained as performance metrics. These performance metrics are obtained by considering pointing error, different optical signal detection methods, turbulence effect, and the RF link fading parameter ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$m$</tex-math></inline-formula> ). Using the results, the security and reliability trade-off analysis is discussed for two scenarios. In addition, the asymptotic results are also obtained to study the system in depth. Further, the secrecy diversity order (SDO) and secrecy coding gain (SCG) are obtained to analyse the system in depth. Finally, Monte-Carlo Simulation is performed to verify the obtained results. The results manifest that the scenario with selection at each hop provides better secrecy performance than the scenario with selection at the last hop.

OpenVVC Decoder Parameterized and Interfaced Synchronous Dataflow (PiSDF) Model: Tile Based Parallelism.

The emergence of the new video coding standard, Versatile Video Coding (VVC), has resulted in a 40-50% coding gain over its predecessor HEVC for the same visual quality. However, this is accompanied by a sharp increase in computational complexity. The emergence of the VVC standard and the increase in video resolution have exceeded the capacity of single-core architectures. This fact has led researchers to use multicore architectures for the implementation of video standards and to use the parallelism of these architectures for real-time applications. With the strong growth in both areas, video coding and multicore architecture, there is a great need for a design methodology that facilitates the exploration of heterogeneous multicore architectures, which automatically generates optimized code for these architectures in order to reduce time to market. In this context, this paper aims to use the methodology based on data flow modeling associated with the PREESM software. This paper shows how the software has been used to model a complete standard VVC video decoder using Parameterized and Interfaced Synchronous Dataflow (PiSDF) model. The proposed model takes advantage of the parallelism strategies of the OpenVVC decoder and in particular the tile-based parallelism. Experimental results show that the speed of the VVC decoder in PiSDF is slightly higher than the OpenVVC decoder handwritten in C/C++ languages, by up to 11% speedup on a 24-core processor. Thus, the proposed decoder outperforms the state-of-the-art dataflow decoders based on the RVC-CAL model.

PTR-CNN for in-loop filtering in video coding

A deep learning method called PTR-CNN (Predicted frame with Transform unit partition and prediction Residual aided CNN) is proposed for in-loop filtering in video compression. To reduce the computational complexity of an end-to-end CNN in-loop filter, a non-learning method of reference frame selection is designed to select the highest quality frame based on the frame’s blurriness and smoothiness scores. The transform unit (TU) partition and the prediction residual (PR) of the current frame are used as extra inputs to the neural network as the filtering guidance. The selected similar and high quality reference frame (RF) and the current unfiltered frame (CUF) are input to a CNN based motion compensation module to generate a predicted frame (PF). Finally input the PF, the CUF, the CUF’s TU partition and the CUF’s PR into the main CNN to reconstruct the filtered frame. The model is implemented in Tensorflow and tested in HEVC and AV1. Experimental results show that the complexity of proposed PTR-CNN is less than SOTA CNN-based reference aided in-loop filtering methods and slightly outperforms their RD performance. The scheme introduces a complexity overhead of 7% on the encoder. In particular, for random access, the proposed model achieves 11.78% coding gain over HEVC with DBF/SAO off, while has a gain of 4.76% over HEVC with DBF/SAO on. Ablation study demonstrates that the RF contributes about 10% of the total gain, and the TU and PR contribute over 4% of the total one, proving the effectiveness of each module. Moreover, it is observed that the proposed method can restore detailed structures and textures and hence improve the subjective quality.

Signal Matched Multirate Filter Bank Design for Optimum Coding Gain and its Application in Real-Time Sleep Apnea Detection

Signal Matched Multirate Filter Bank Design for Optimum Coding Gain and its Application in Real-Time Sleep Apnea Detection

PUNCTURED NCC CODES FOR INFORMATION PROTECTION IN INFORMATION AND MEASUREMENT SYSTEMS

During development and design of information-measuring systems, enabling to carry out collection, processing and transmission of information, one of the main problems is the choice of effective methods of information protection against defects in noisy communication channels. Effective use of frequency-time resources of information communication channels, as the most valuable part of the information transmission system, is the key to provide reliable delivery of transmitted messages. One of effective directions of reliability increase and information transfer reliability in information-measuring communication networks is implementation of methods and algorithms of noise-resistant coding, providing for detection and coping with errors, arising due to interferences in the communication channel. In this case, the choice in favor of one or another coding method depends on the information characteristics of the data channel. Parameters of the noise coder must be coordinated with the source of the message, the communication channel, as well as the requirements for the reliability of bringing information to the recipient. The problem of obtaining a wide range of codec parameters with simultaneous preservation of the unified macrostructure of the codec in communication systems causes the need for research on the development of adaptive algorithms for error information protection. In the article the research results of the characteristics of variable rate slot convolutional codes for adaptive coding/decoding in information-measuring systems of information transmission are proposed. Consequently, when creating communication networks, there is no need to use a large number of different codecs, even with completely different requirements to the code rate, channel rate and gain due to coding. In addition, there is a real opportunity to create terminal equipment, working on unified algorithms of protection against errors and access.

Improving Interference Cancellation in Spatial Division Multiplexing Using a Combination of the Channel Coding and QR-Successive Interference Cancellation

This paper introduces QR-SIC detector to minimize the probability of error comparing to SDM system. Spatial-division-multiplexing (SDM) system is a multiple-inputs-multiple-outputs (MIMO) system used to increase data transmission rate in modern communication systems. The major disadvantage of SDM system is the interference between the received symbols. In this paper, a new trans-receiver architecture for SDM system is introduced. This receiver uses a Successive-Interference-Cancellation (SIC) detector based on QR-decomposition of the channel matrix and a Forward-Error- Correction (FEC) channel code. The proposed architecture merges the interference cancellation process in QR-SIC detector and the forward error correction process in the channel decoder. The feedback symbols in the QR-SIC detector are the output symbols from the channel decoder after they are re-encoded and re-modulated and more robust. The new architecture minimizes the probability of error in the demodulated symbols more than the conventional SDM receiver. If the conventional SDM receiver needs 100 MWATT transmitted power the proposed system needs only 64 MWATT transmitted power. So The new architecture of the SDM receiver can save up to 36% of the transmitted signal power used with the conventional SDM receiver. At BER of 10-4, the convolution codes of code rates 3/4, 1/2, and 1/4 achieve coding gains of 15 dB, 24 dB, and 30 dB with respect to the uncoded 4-antennas SDM receiver. the proposed SDM receiver at BER of 10-4 is 2dB, 1.5 dB, and 0.9 dB better than the BER performance of the conventional coded SDM receiver at the same code rates and the same number of antenna. The enhancement in the BER performance of the proposed SDM receiver increases as the code gain of the used code. This increase in the power efficiency and bandwidth comes at the expense of an increase in the receiver complexity and latency

Hilbert Space Filling Curve Based Scan-Order for Point Cloud Attribute Compression

Point cloud is a set of three-dimensional points in arbitrary order, which is a popular representation of 3D scene in autonomous navigation and immersive applications in recent years. Compression becomes an inevitable issue due to the huge data volume of point cloud. In order to effectively compress attributes of those points, proper reordering is important. The existing voxel-based point cloud attributes compression scheme uses a naive scan for points reordering. In this paper, we theoretically analyzed 3C properties of point cloud, i.e., Compactness, Clustering and Correlation, of different scan-orders defined by different space filling curves and disclosed that the Hilbert curve can provide the best spatial correlation preservation compared with Z-order and Gray-coded curves. It is also statistically verified that the Hilbert curve always has the best ability of attributes correlation preservation for point clouds with different sparsity. We also proposed a fast and iterative Hilbert address code generation method to implement points reordering. The Hilbert scan-order could be combined with various point cloud attribute coding methods. Experiments show that the correlation preservation feature of the proposed scan-order can bring us 6.1% and 6.5% coding gain for prediction and transform coding, respectively.

Outage Performance Analysis of Widely Linear Receivers in Uplink Multi-User MIMO Systems

This paper considers the application of widely linear (WL) receivers in an uplink multi-user system using real-valued modulation schemes, where the cellular base station (BS) with multiple antennas provides connectivity for randomly deployed single-antenna users. The targeted use case is massive machine type communication (mMTC) with grant-free access in the uplink, where the network is required to host a large number of low data rate devices transmitting in an uncoordinated fashion. Four types of WL receivers are investigated, namely the WL zero-forcing (ZF) and the WL minimum mean-squared error (MMSE) receivers, along with their enhanced versions employing successive interference cancellation (SIC) with channel-dependent ordering, i.e., the WL-ZF-SIC and WL-MMSE-SIC receivers. The outage performances of these receivers are analytically characterized in the high signal-to-noise ratio (SNR) regime and compared to those of conventional linear (CL) receivers using complex-valued modulation schemes. For the non-SIC receivers, we show that, when compared to the CL counterparts, the WL receivers yield a higher diversity gain when decoding the same number of users and have the same diversity gain but a decreased coding gain when the number of users is nearly doubled. The outage performance analysis of WL-SIC receivers is facilitated by the marginal distribution of ordered eigenvalues of a real-valued Wishart matrix. It is shown that the SIC operation with channel-dependent ordering brings no additional diversity gain to the WL receivers but instead increases the coding gain. Moreover, the coding gain of WL-SIC receivers grows as the number of users increases and even exceeds that of CL-SIC receivers under suitable conditions. For the mMTC scenario with grant-free transmission, it is demonstrated that the WL receivers outperform their CL counterparts in terms of offering a lower outage (and packet drop) probability and a higher system throughput for a given packet drop probability.

Transform Coding in the VVC Standard

In the past decade, the development of transform coding techniques has achieved significant progress and several advanced transform tools have been adopted in the new generation Versatile Video Coding (VVC) standard. In this paper, a brief history of transform coding development during VVC standardization is presented, and the transform coding tools in the VVC standard are described in detail together with their initial design, incremental improvements and implementation aspects. To improve coding efficiency, four new transform coding techniques are introduced in VVC, which are namely Multiple Transform Selection (MTS), Low-Frequency Non-separable Secondary Transform (LFNST) and Sub-Block Transform (SBT), as well as a large (64-point) type-2 DCT. The experimental results on VVC reference software (VTM-9.0) show that average 4.5&#x0025; and 3.6&#x0025; overall coding gain can be achieved by the VVC transform coding tools for All Intra and Random Access configurations, respectively.

Improvement of channel coding gain of chaos modulation using logistic maps

Improvement of channel coding gain of chaos modulation using logistic maps