Hadamard Codes Research Articles

Equivalences among [formula omitted]-linear Hadamard codes

The Z2s-additive codes are subgroups of Z2sn, and can be seen as a generalization of linear codes over Z2 and Z4. A Z2s-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-additive code. A partial classification of these codes by using the dimension of the kernel is known. In this paper, we establish that some Z2s-linear Hadamard codes of length 2t are equivalent, once t is fixed. This allows us to improve the known upper bounds for the number of such nonequivalent codes. Moreover, up to t=11, this new upper bound coincides with a known lower bound (based on the rank and dimension of the kernel). Finally, when we focus on s∈{2,3}, the full classification of the Z2s-linear Hadamard codes of length 2t is established by giving the exact number of such codes.

Analysis of Outage Probability for MC-CDMA Systems Using Different Spread Codes

Modern communication systems demand proper utilisation of bandwidth, high throughput, integration of services and flexibility. To meet these requirements, spread spectrum code-division multiple access (CDMA) techniques have been proposed for various wireless communication systems. Here the individual user is assigned a unique binary code called spreading code used to increase capacity and provide higher robustness to interference. This paper investigates outage probability performance of MIMO multicarrier spread spectrum code-division multiple access (CDMA) with different robust spreading codes namely Walsh-Hadamard (WH), Gold and Kasami codes. Outage probability Poutage is significant performance measure to evaluate the effect of co-channel interference. System’s performance is initially evaluated in terms of outage probability by varying spreading factor (code length L) L= 4, 16, 64,256 for the number of subcarriers (NC) NC =4. In the second case spread factor SF is kept constant at L=64 and Poutage is analysed for varying number of subcarriers NC = 4, 16, 64. Outage probability analysis for system shows that under similar load conditions Kasami spread sequences outperforms Gold codes and WH codes in terms of outage probability for different spreading factor (SF) of codes and for varying number of subcarriers. This is because of its improved peak isolation and low cross-correlation than other.

Novel OFDM System Using Orthogonal Pilot Symbols with Subcarrier Index Modulation

In this work, two new transmission schemes are proposed to increase the spectral efficiency of orthogonal frequency division multiplexing (OFDM). In practical OFDM systems, channel estimation is usually performed by employing pilot symbols which is based on inserting known symbols in the time-frequency domain. However, pilot symbol designing is one of the bottlenecks of OFDM systems limiting the increase of spectral efficiency. We apply new pilot design structures, which are use orthogonal Walsh-Hadamard codes. To increase the spectral efficiency of OFDM systems, we assign the extra bits to the index of the each orthogonal Walsh-Hadamard codes. Simulation and theoretical results show that proposed methods have better performance than conventional OFDM with higher spectral efficiency. Moreover, no more energy is spent for additional information carried in the indices compared to classical OFDM systems. As a result, proposed methods provide both spectral efficiency and energy efficiency.

Multi-Carrier $M$-ary DCSK System With Code Index Modulation: An Efficient Solution for Chaotic Communications

A new multi-carrier $M$ -ary differential chaos shift keying system with code index modulation, referred to as CIM-MC- $M$ -DCSK , is proposed in this paper. In the proposed CIM-MC- $M$ -DCSK system, the reference and information-bearing signals for each subcarrier can be transmitted simultaneously by using the orthogonal sinusoidal carriers, where the information-bearing signal adopts the $M$ -DCSK modulation to further increase the data rate. With an aim to making full use of the system energy resources, the reference signals in all subcarriers are coded by a Walsh code to carry additional information bits. The analytical bit-error-rate (BER) expressions of the proposed CIM-MC- $M$ -DCSK system are derived over additive white Gaussian noise as well as multipath Rayleigh fading channels. Furthermore, a noise-reduction scheme and a hierarchical-modulation scheme are designed for the proposed system. In particular, the former scheme can significantly improve the BER performance while the latter scheme can provide different quality of service for the transmitted bits according to their different levels of importance. Simulation results verify the accuracy of the analytical expressions and the superiority of the proposed systems.

Multi‐carrier chaotic communication scheme for underwater acoustic communications

Multi-carrier (MC) and spread-spectrum (SS) are two good technologies for underwater acoustic (UWA) communications. In this study, the authors propose a novel MC chaos modulation scheme, which combines code-shifted differential chaos shift keying (CS-DCSK) with orthogonal frequency division multiplexing (OFDM), namely MC-CS-DCSK. The proposed scheme exploits the orthogonal characteristic of the Walsh code to superimpose the chaotic reference chips and the information bearing chips in the same time/frequency unit. Besides, a cyclic-shift interleaver is used on chips to harvest frequency diversity. To the in-depth understanding of the proposed system, the bit error rate (BER) of the proposed system under Gaussian channels and multipath Rayleigh fading channels are derived and verified by simulation. Furthermore, the spectral efficiency of the proposed system is analysed and compared with some existing chaos-based communication systems. Finally, they study BER performance of the system under UWA channels and compare it with traditional MC direct sequence SS and OFDM-based MC-DCSK systems. Simulation results show that the system has good robustness under time-varying UWA channels.

Doubly Orthogonal Wavelet Packets for Multi-Users Indoor Visible Light Communication Systems

Visible Light Communication (VLC) is a data communication technology that modulates the intensity of the light to transmit the information mostly by means of Light Emitting Diodes (LEDs). The data rate is mainly throttled by the limited bandwidth of the LEDs. To combat, Multi-carrier Code Division Multiple Access (MC-CDMA) is a favorable technique for achieving higher data rates along with reduced Inter-Symbol Interference (ISI) and easy access to multi-users at the cost of slightly reduced compromised spectral efficiency and Multiple Access Interference (MAI). In this article, a multi-user VLC system is designed using a Discrete Wavelet Transform (DWT) that eradicates the use of cyclic prefix due to the good orthogonality and time-frequency localization properties of wavelets. Moreover, the design also comprises suitable signature codes, which are generated by employing double orthogonality depending upon Walsh codes and Wavelet Packets. The proposed multi-user system is simulated in MATLAB software and its overall performance is assessed using line-of-sight (LoS) and non-line-of-sight (NLoS) configurations. Furthermore, two sub-optimum multi-users detection schemes such as zero forcing (ZF) and minimum-mean-square-error (MMSE) are also used at the receiver. The simulated results illustrate that the doubly orthogonal signature waveform-based DWT-MC-CDMA with MMSE detection scheme outperforms the Walsh code-based multi-user system.

An $M$ -Ary Orthogonal Multilevel Differential Chaos Shift Keying System With Code Index Modulation

In this paper, an $M$ -ary orthogonal multilevel differential chaos shift keying system with code index modulation (CIM-OM-MDCSK) is proposed. In the proposed system, the multiple information bearing signals are modulated by the selected Walsh codes with specific indices and the $M$ -ary information signals which are composed of modulated bits, chaotic signals, and their Hilbert transform. Since the reference signal along with the information bearing signals are overlapping in time domain and orthogonal in code domain by using Walsh code sequences, the spectral efficiency of the CIM-OM-MDCSK system is improved largely. By combining the $M$ -ary modulation, orthogonal multilevel modulation, and code index modulation, the proposed CIM-OM-MDCSK system can achieve high data rate and superior bit error rate (BER) performance compared to its rivals. Moreover, we derive the BER expressions of the proposed system over additive white Gaussian noise and multipath Rayleigh fading channels. Finally, we make a performance comparison between the proposed system and other state-of-the-art non-coherent chaotic communication systems. The results confirm the competitive benefits of the proposed system.

Key Distribution System Based on Hadamard Designs

The problem of key distribution in a community for providing secure communication between its participants is studied. To solve this problem, key predistribution systems can be used, in which each user receives some key information that can later be used to independently calculate required shared secret keys for conferences they participate in. Such key distribution systems can be based on different structures, such as error-correcting codes and combinatorial designs. The drawback of such systems is the possibility of collusive attacks, when traitors within the system can form a coalition and use their key information to try to calculate shared secret keys of other users. But the secrecy of keys is guaranteed by the system when the number of traitors in the coalition does not exceed a threshold defined by the system structure. In this paper, a key distribution system is based on combinatorial designs and, in particular, on Hadamard 3-design that guarantees the secrecy of communications in the presence of coalitions with less than three users. New notions of combinatorial span and combinatorial rank of a subset of Hadamard code that are required for the study of the resilience of the system to collusive attacks are introduced. The probability of successful collusive attack on an arbitrary conference against the cardinality of coalition is calculated for this system.

A new efficient sensing matrix for cluster structured sparse signals recovery

Compressive sensing is a new sensing technique aiming to recover the original sparse signal from a much fewer number of samples compared to the conventional sensing technique of Shannon-Nyquist. Recently, several works have considered the problem of recovering sparse signals when their nonzero coefficients tend to cluster into blocks. We refer to this new type of signals as cluster structured sparse signals. In this paper, we deal with the problem of cluster structured sparse signals recovery by proposing a novel sensing matrix based on Bernoulli sensing matrix and full-orthogonal Hadamard codes. We demonstrate theoretically and experimentally that the proposed sensing matrix performs well in the case of cluster structured sparse signals recovery. Indeed, the new sensing matrix provides a considerable gain in terms of the rate of exact recovery and error support rate. Also, an application on natural images is conducted exploiting their cluster structured sparse property in the discrete cosine transform domain.

Optimization of power to achieve intended capacity in inter-cell interference of visible light communication network

In this article, a novel algorithm is proposed to eliminate the Inter-Cell Interference (ICI) for Visible Light Communication (VLC). The proposed algorithm is based on Hadamard Code (HC) and Pulse Amplitude Modulation (PAM). Four schemes of HCs distribution between nine cells are separately designed and tested within a room to mitigate the ICI, whereas PAM is used to improve data rate. The primary objective of this research is to evaluate the amount of power utilized by four proposed schemes to obtain the Channel Capacity Without Interference (CCWI). However, the higher order of PAM (HOP) uses high average power. This increases the ICI which is diminished by using additional power to achieve the CCWI. Therefore, predetermined power is derived for the receivers based on ICI. Moreover, theoretical SER is derived to achieve the trade-off between SER performance and flickering in the room. This is done by choosing positive and negative amplitude from DC bias (PND). The designed four schemes are validated by using computer-based simulation on MATLAB for VLC. The simulated results show that the overall performance of the 4th scheme is power efficient as compared to other three schemes. Keywords: Hadamard Code, Inter-cell Interference, Visible Light Communication

Novel multispectral face descriptor using orthogonal walsh codes

This paper proposes a novel high-performance multispectral local descriptor that uses orthogonal Walsh codes during the generation of the discriminative feature set. Rotational variance and noise are compelling factors that significantly affect the distinctive performance of a facial identifier. The descriptor proposed in this article handles these challenges favourably sacrificing any distinctive performance. Orthogonal Walsh codes are used during the generation of the local descriptor. A Walsh code is assigned to each neighbour of a reference pixel. Before the assignment of an orthogonal code to each neighbouring pixel, these pixels are sorted in ascending order that consolidates the robustness of the method against rotational variances. Almost all methods proposed so far focus on grayscale images. However, colour bands contain important information about the relationship between pixels. Therefore, authors’ method considers RGB colour bands of pixels to improve distinctive performance. The results of extensive simulations show the remarkable and competitive performance of the proposed method regarding recognition accuracy and robustness against rotational variances as well as noise effects.

Analysis of Variable Bit Rate WH/CI‐Spread OFDM based Integrated Satellite‐Terrestrial Broadcast System

In this study, the exact closed-form signal-to-interference-noise ratio (SINR) expression, for the impact of Doppler frequency shift on minimum mean square error receiver is derived for Walsh Hadamard (WH) and polyphase carrier interferometry (CI) code based multicarrier spread orthogonal frequency division multiplexing (SOFDM) transmission scheme over integrated satellite–terrestrial broadcast system. Additionally, bit error rate and spectral efficiency are analysed based on the obtained SINR expression. To examine that, three-state Fontan land mobile satellite, and six-path typical urban and rural area terrestrial channel models are employed. This analysis is performed to substantiate variable bit rate (VBR) quality of service at the physical layer of digital video broadcast satellite handheld (DVB-SH)-A architecture, the provision of which is absent in DVB-SH standard. VBR is assumed for SOFDM. The results illustrate that WH-SOFDM outperforms CI-SOFDM under both performance measures in all the three channel environments. Further, pilot-based channel estimation technique is performed and the estimated error variance is evaluated for VBR WH/CI-SOFDM. Later, block-wise system computational complexity of WH/CI-SOFDM schemes is evaluated. It is shown that, CI-SOFDM is a relatively more complex system than WH-SOFDM. Thus, a trade-off is present between the achieved performance gain and system complexity.

Advanced Modulation Scheme for Visible Light Communication

In this paper, an improved version of existing modulation schemes is investigated by inserting Hadamard codes (HC) at the peak slot of each symbol for the Visible Light Communication (VLC) system. The performance of the proposed scheme is analyzed and compared with existing schemes On-Off Keying (OOK), Pulse Position Modulation (PPM), Differential PPM (DPPM), Pulse Amplitude Modulation (PAM), Differential Amplitude PPM (DAPPM) and Differential Pulse Interval Modulation (DPIM). Moreover, the symbol error rate (SER) for kth PAM is derived for ds distances between symbols. The performance in Symbol Error Rate (SER) is improved by introducing the Positive and Negative Amplitudes (PNA) levels with respect to DC-bias. The theoretical and simulation results show that the proposed scheme outperforms the existing schemes in terms of bandwidth, and transmission capacity. Keywords: Hadamard Code Matrix, Modulation, Visible Light Communication

Hybrid Chaos Communication With Code Index Modulation

Hybrid chaos communication (HCC) is promising for covert transmission by delivering confidential information via chaotic signals. To enhance the transmission rate and the energy efficiency while maintaining a low probability of interception, in this paper, we propose a code index modulation-aided HCC (CIM-HCC) scheme. Specifically, the input bit stream is split into the chaotic signaling stream, the modulating stream, and the code indexing stream. The chaotic signaling stream is modulated by the hybrid chaos signal and the code indexing stream is mapped onto the Walsh codes. The modulating stream is multiplied to the Walsh codes. Thus, the proposed CIM-HCC achieves higher data rate and lower power consumption than that of the conventional HCC. Another advantage is that, since part of the information bits are conveyed by the indices of the Walsh codes, the CIM-HCC outperforms the conventional HCC, and binary phase-shift keying (BPSK) in terms of reliability. In particular, when the number of indexing bits equals to that of the signaling bits, the proposed CIM-HCC doubles the data rate and saves about 50% power consumption. In addition, the numerical results show that the proposed scheme outperforms the code index modulation differential chaos shift keying in terms of bit error rate over both the additive white Gaussian noise channel and the multipath Rayleigh fading channels.

Efficient and robust M ‐ary differential chaos shift keying scheme with code index modulation

In this study, a new chaos modulation scheme which combines M -ary differential chaos shift keying (DCSK) with code-index modulation (CIM-MDCSK) is proposed. In the CIM-MDCSK scheme, the two Walsh codes that are used to generate two orthogonal chaos components are selected from the candidate codes, whose code indices are determined by information symbols. Thus, the information bits are conveyed not only by M -ary DCSK signal but also by the indices of the Walsh codes. Theoretical symbol error rate (SER) and bit error rate (BER) of the proposed scheme are derived and confirmed by simulations. Furthermore, by applying the noise-reduction to reference signals of CIM-MDCSK, the SER/BER expressions are also derived. Simulation results show that the proposed CIM-MDCSK outperforms DCSK and M -ary DCSK based on Hilbert transform. Especially, the CIM-MDCSK with noise-reduction shows significant advantages. Moreover, the performance of CIM-MDCSK scheme is compared with direct sequences spread-spectrum system with code-index modulation over flat and frequency selective fading channel. The proposed CIM-MDCSK can offer good performance over frequency selective fading channel with a low complexity non-coherent receiver which does not require any channel state information and channel equalisation.

A Series <inline-formula> <tex-math notation="LaTeX">${N}$ </tex-math> </inline-formula>-Path Code Selective Filter for Transmitter Rejection in Full-Duplex Communication

A series code selective filter (CSF) for full-duplex communication is presented. The CSF passes a desired signal in a frequency band only when the signal is tagged with a pseudonoise (PN) code that matches the code at the filter. This approach is shown to reject a transmit signal in the same band as a desired receive signal. The coded transmitter signal is orthogonal through appropriate code selection from a codebook and the PN sequence spreads the signal energy outside the filter bandwidth (BW). The CSF achieves a self-interference rejection of 23.2 dB over a 10-MHz BW using on-chip generated Walsh codes. The CSF operates from 300 to 675 MHz and consumes 28.2 mA from a 1-V supply at 500 MHz. The CSF achieves an IIP3 of 15.7 dBm. The CSF was implemented in a 45-nm CMOS silicon on insulator process and occupies an area of 0.9 mm2.

Partial Permutation Decoding for Several Families of Linear and <inline-formula> <tex-math notation="LaTeX">${\mathbb{Z}}_4$ </tex-math> </inline-formula>-Linear Codes

A general criterion to obtain $s$ -PD-sets of minimum size $s+1$ for partial permutation decoding, which enable correction up to $s$ errors, for systematic codes over a finite field ${ {F}}_{q}$ and ${ {Z}}_{4}$ -linear codes is provided. We show how this technique can be easily applied to linear cyclic codes over ${ {F}}_{q}$ , ${ {Z}}_{4}$ -linear codes which are the Gray map image of a quaternary linear cyclic code, and some related codes such as quasi-cyclic codes. Furthermore, specific results for some linear and nonlinear binary codes, including simplex, Kerdock, Delsarte-Goethals, and extended dualized Kerdock codes are given. Finally, applying this technique, new $s$ -PD-sets of size $s+1$ for ${ {Z}}_{4}$ -linear Hadamard codes of type $2^\gamma 4^\delta $ , for all $\delta \geq 4$ and $1 ; and for ${ {Z}}_{4}$ -linear simplex codes of type $4^{m}$ , for all $m\geq 2$ and $1 , are also provided.

ANALISIS KINERJA KOMBINASI SISTEM CODE DIVISION MULTIPLE ACCESS MULTIPLE INPUT MULTIPLE OUTPUT DENGAN KODE PENEBAR WALSH PADA KANAL RAYLEIGH FADING

The increasing human population are same to the need for telecommunications. Code Division Multiple Access (CDMA) is one of the most widely used telecommunications technologies. But with the increasing number of users, this technology is no longer able to meet the needs of users. With the required capacity requirements, the CDMA system and Multiple Input Multiple Output (MIMO) technology are merged.To find out the performance of a combined CDMA MIMO system, the system was tested by increasing the number of users and Signal to Noise Ratio (SNR) where the spreading code used was the Walsh code and the channel used was Rayleigh Fading modeling. The reference value used as material for consideration are the value of Bit Error Rate (BER).The method used in testing the CDMA MIMO system is the increase number of users, starting with a single user up to 16 users and SNR values from -15 dB to a maximum of 15 dB.Where all users will be combined with the CDMA technique then it will be sent using MIMO technology. The system will be tested using a simulation of the MatLab R2012a software.The results of the simulation show the performance of the CDMA MIMO system when using SNR that is increasing high, the BER will be lower or better used for communication. Whereas for simulations using a user that increases it results in a BER value that is almost the same because of the nature of the orthogonal Walsh code so that each user does not experience interference with other users.

Spectral Amplitude Coding Optical CDMA: Performance Analysis on Free Space Optical Channel

In this paper, we investigate the performance of Spectral Amplitude Coding Optical Code Division Multiple Access technique based on Free Space Optical channel. SAC OCDMA is one of the most popular multiplexing techniques since many years and FSO is gaining popularity and is very useful especially in point to point communication. The system presented in this paper utilizes Walsh Hadamard code as signature code. The coder and decoder structures are based on optical filters of fiber Bragg gratings (FBGs). This system focuses on the performance analysis of FSO based systems for subtropical regions. This paper demonstrates the error rate performance in the form of eye diagrams/BER under varying channel gain and link distances. Five cases have been taken which shows that SAC OCDMA FSO system is reliable even for distances sufficient to overcome the last mile problem.

On the Rank of [formula omitted]-linear Hadamard Codes

The Z2s-additive codes are subgroups of Z2sn, and can be seen as a generalization of linear codes over Z2 and Z4. A Z2s-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-additive code. It is known that either the rank or the dimension of the kernel can be used to give a complete classification for the Z4-linear Hadamard codes. However, when s>2, the dimension of the kernel of Z2s-linear Hadamard codes of length 2t only provides a complete classification for some values of t and s. In this paper, the rank of these codes is given for s=3. Moreover, it is shown that this invariant, along with the dimension of the kernel, provides a complete classification, once t≥3 is fixed. In this case, the number of nonequivalent such codes is also established.