RNS-To-Binary Converter for a New Three-Moduli Set {2n+1−1,2n,2n−1}

In this brief, the design of residue number system (RNS) to binary converters for a new powers-of-two related three-moduli set {2ⁿ⁺¹ - 1, 2ⁿ, 2ⁿ - 1} is considered. This moduli set uses moduli of uniform word length (n to n + 1 bits). It is derived from a previously investigated four-moduli set {2ⁿ - 1, 2ⁿ, 2ⁿ + 1, 2^{n +1} - 1}. Three RNS-to-binary converters are proposed for this moduli set: one using mixed radix conversion and the other two using Chinese remainder theorem. Detailed architectures of the three converters as well as comparison with some earlier proposed converters for three-moduli sets with uniform word length and the four-moduli set {2ⁿ - 1, 2ⁿ, 2ⁿ + 1, 2ⁿ⁺¹ - 1} are presented.     

RNS-to-Binary Converters for Two Four-Moduli Sets {2n−1,2n,2n+1,2n+1−1} and {2n−1,2n,2n+1,2n+1+1}

In this paper, reverse converters for two recently proposed four-moduli sets {2ⁿ - 1,2ⁿ,2ⁿ + 1,2ⁿ+1 - 1} and {2ⁿ - 1, 2ⁿ, 2ⁿ + 1, 2ⁿ+1 + 1} are described. The reverse conversion in the three-moduli set {2ⁿ - 1,2ⁿ,2ⁿ + 1} has been optimized in literature. Hence, the proposed converters are based on two new moduli sets {(2ⁿ(2²ⁿ-1)),2ⁿ⁺¹-1} and {(2ⁿ(2²ⁿ-1)), 2ⁿ⁺¹+1} and use mixed radix conversion. The resulting designs do not require any ROM. Both are similar in their architecture except that the converter for the moduli set {2ⁿ - 1, 2ⁿ, 2ⁿ + 1, 2ⁿ+1 + 1} is slightly complicated due to the difficulty in performing reduction modulo (2ⁿ⁺¹+1) as compared with modulo (2ⁿ⁺¹-1). The proposed conversion techniques are compared with earlier realizations described in literature with regard to conversion time as well as area requirements.     

A Residue-to-Binary Converter for a New Five-Moduli Set

The efficiency of the residue number system (RNS) depends not only on the residue-to-binary converters but also the operand sizes and the modulus in each residue channel. Due to their special number theoretic properties, RNS with a moduli set consisting of moduli in the form of 2 ⁿplusmn1 is more attractive than those with other forms of moduli. In this paper, a new five-moduli set RNS {2ⁿ-1,2ⁿ,2ⁿ+1,2ⁿ+1-1,2 ^n-1-1} for even n is proposed. The new moduli set has a dynamic range of (5n-1) bits. It incorporates two additional moduli to the celebrated three-moduli set, {2ⁿ-1,2ⁿ,2ⁿ +1} with VLSI efficient implementations for both the binary-to-residue conversion and the residue arithmetic units. This extension increases the parallelism and reduces the size of each residue channel for a given dynamic range. The proposed residue-to-binary converter relies on the properties of an efficient residue-to-binary conversion algorithm for {2ⁿ-1,2ⁿ,2ⁿ+1,2ⁿ+1-1} and the mixed-radix conversion (MRC) technique for the two-moduli set RNS. The hardware implementation of the proposed residue-to-binary converter employs adders as the primitive operators. Besides, it can be easily pipelined to attain a high throughput rate     

On the binary quadratic residue system with noncoprime moduli

The residue number system (RNS) appropriate for implementing fast digital signal processors since it can support parallel, carry-free, high-speed arithmetic. A development in residue arithmetic is the quadratic residue number system (QRNS), which can perform complex multiplications with only two integer multiplications instead of four. An RNS/QRNS is defined by a set of relatively prime integers, called the moduli set, where the choice of this set is one of the most important design considerations for RNS/QRNS systems. In order to maintain simple QRNS arithmetic, moduli sets with numbers of forms 2ⁿ+1 (n is even) have been considered. An efficient such set is the three-moduli set (2^2k-2+1.2^2k+1.2^2k+2+1) for odd k. However, if large dynamic ranges are desirable, QRNS systems with more than three relatively prime moduli must be considered. It is shown that if a QRNS set consists of more than four relatively prime moduli of forms 2ⁿ+1, the moduli selection process becomes inflexible and the arithmetic gets very unbalanced. The above problem can be solved if nonrelatively prime moduli are used. New multimoduli QRNS systems are presented that are based on nonrelatively prime moduli of forms 2ⁿ +1 (n even). The new systems allow flexible moduli selection process, very balanced arithmetic, and are appropriate for large dynamic ranges. For a given dynamic range, these new systems exhibit better speed performance than that of the three-moduli QRNS system     

Generalized Kasami Sequences: The Large Set

In this correspondence, new binary sequence families F^k of period 2ⁿ-1 are constructed for even n and any k with gcd(k,n)=2 if n/2 is odd or gcd(k,n)=1 if n/2 is even. The distribution of their correlation values is completely determined. These families have maximum correlation 2^n/2+1 and family size 2^3n/2 + 2^n/2 for odd n/2 or 2^3n/2+2^n/2-1 for even n/2. The proposed families include the large set of Kasami sequences, where the k is taken as k=n/2+1.     

Path diversity for FFH/PSK spread-spectrum communication systems

A fast frequency hopping (FFH) method which uses path-diversity combining is proposed. Diversity techniques are realized when a signal is received from diverse independent paths, each of which carries identical information but suffers from independent fading variations. The improvement of communication performance of FFH systems is possible as the delayed paths are used and path-diversity combination is realized. The advantages of this method, operating in Rayleigh fading channels are confirmed by theoretical analyses. The improvement is in order of 2~3 dB at bit error rate (BER) of 10^-3. This method can be also effective against interferences from other users in a code division multiple access (CDMA) environment. The performance of this system in a CDMA environment is evaluated by theoretical analyses and is shown to be superior to non-combining method. At BER of 10^-3 the required E_b/N₀ of the proposed system is 5 dB lower. If E_b/N_o is fixed, a greater number of users can be accommodated using the proposed system     

Spread-space holographic CDMA technique: basic analysis andapplications

We analyze the basic principles of the spread-space holographic code division multiple access (CDMA) technique. We describe the structure of the encoder and decoder and present a basic mathematical analysis based on spatial harmonic decomposition for the receiver's processing module, which is comprised of a Fourier transform lens and a holographic matched filter. Subsequently, we study two applications of optical holographic CDMA, namely, free-space holographic CDMA photonic switch and free-space (wireless) multiaccess optical (infrared) indoor communications. For both techniques, we describe the two-dimensional imaging techniques, namely user's code transmission, and highlight the basic parameters and components that need to be optimized. To obtain the bit-error rate (BER) of the proposed application, we first evaluate the probability density function of multiuser interference and then evaluate the BER as a function of the processing gain, number of users, the received power, and the optimum threshold. For BER ≈ 10^-9, the result shows an extraordinary number of users that can be supported via holographic CDMA, for both applications     

Optical CDMA network codecs with merged-M-coded wavelength-hopping and prime-coded time-spreading

This paper proposes a new hybrid code for optical CDMA systems based on modified maximal-length sequence (M-sequence) codes for wavelength-hopping and prime codes for time spreading. Using the proposed hybrid code, two optical encoder/decoders are constructed using either fiber-Bragg gratings (FBGs) and optical delay lines, or arrayed-waveguide gratings (AWGs) and optical delay lines, respectively. The system performance is compared with that provided by the well-known prime-hop sequence code. The results show that the merged-M-sequence codes not only provide a higher capacity, but can also be realized via a simpler configuration. Furthermore, the proposed code is less expensive than prime-hop code under virtually identical BER conditions.     

Bias-correlated branched-waveguide cascaded light intensitymodulator in the scattering matrix formalism

The authors describe the concept of a composite interferometric light intensity modulator constructed in the form of a cascade of 2ⁿ branched-waveguide electrooptic modulators in which the DC bias of the individual modulators are provided in a certain phasing required to yield a high overall modulation efficiency. The analysis of this cascaded modulator, presented in the scattering matrix formalism, predicts that the overall modulation efficiency of the cascade is increased by the factor 2^n/2 compared with that of a single constituent modulator when both are supplied with the same amount of modulating signal power     

Groups of algebraic integers used for coding QAM signals

Linear block codes over Gaussian integers and Eisenstein integers were used for coding over two-dimensional signal space. A group of Gaussian integers with 2²ⁿ elements was constructed to code quadrature amplitude modulation (QAM) signals such that a differentially coherent method can be applied to demodulate the QAM signals. This paper shows that one subgroup of the multiplicative group of units in the algebraic integer ring of any quadratic number field with unique factorization, modulo the ideal (Pⁿ), can be used to obtain a QAM signal space of 2p^2n-2 points, where p is any given odd prime number. Furthermore, one subgroup of the multiplicative group of units in the quotient ring Z[ω]/(pⁿ) can be used to obtain a QAM signal space of 6p^2n-2 points; one subgroup of the multiplicative group of units in the quotient ring Z[i](pⁿ) can be used to obtain a QAM signal space of 4p^2n-2 points which is symmetrical over the quadrants of the complex plane and useful for differentially coherent detection of QAM signals; the multiplicative group of units in the quotient ring Z[ω]/(2ⁿ) can be used to obtain a QAM signal space of 3·2^2n-2 points, where i=√-1, ω=(-1+√-3)/2=(-1+i√3)/2, p is any given odd prime number, Z[i] and Z[ω] are, respectively, the Gaussian integer ring and the Eisenstein integer ring. These multiplicative groups can also be used to construct block codes over Gaussian integers or Eisenstein integers which are able to correct some error patterns     

Channel reuse strategies for indoor infrared wirelesscommunications

We examine systems of fixed-channel reuse for base stations in an indoor infrared wireless communication system. The following techniques are compared: time-division multiple access (TDMA) using on-off keying (OOK) or pulse-position modulation (PPM); frequency-division multiple access (FDMA) using binary phase-shift keying (BPSK) or quadrature phase-shift keying (QPSK); code-division multiple access (CDMA) using OOK with direct-sequence spreading by m-sequences or optical orthogonal codes (OOCs). We define a parameter γ, which equals the signal-to-noise ratio (SNR) for unit optical path gain and is proportional to the square of the transmitted average optical power. Using measured pathloss data, it is found that in a system using hexagonal cells and a reuse factor of three, for cell radii above 3 m, TDMA with OOK or 2-PPM, and CDMA using OOCs all require approximately the same γ to achieve a worst-case bit-error rate (BER) of 10^-9 within a cell. Using TDMA with 4-PPM results in a 6-dB decrease in the required value of γ. CDMA using m-sequences requires an increase in γ of 5 dB over TDMA using OOK, and FDMA with BPSK requires an increase of 12 dB. For a given reuse factor N in the noise-limited regime, the required value of γ decreases in inverse proportion to N² for TDMA schemes and inversely with N for FDMA and CDMA schemes. For cell radii below 3 m, cochannel interference dominates the systems using TDMA, FDMA, and CDMA with an OOC, resulting in an irreducible BER above 10^-9 at cell radii below 1.5 m. Only CDMA with m-sequences does not develop an irreducible BER, making it the only choice for cell radii below 1.5 m     

Fast conversion between binary and residue numbers

New, efficient hardware implementations are considered which perform code conversions between the three moduli (2ⁿ-1, 2 ⁿ, 2ⁿ+1) residue number systems and their binary representations. Significant hardware saving together with high-speed throughput is achieved by using only n and (n+1)-bit binary adders     

Quadtree and symmetry in FFT computation of digital images

The discrete Fourier transform (DFT) of a real sequence f[x, y] of size N×N, where N=2ⁿ, can be computed by a two-dimensional (2-D) FFT of size N/4, or smaller if f[x, y] is known to have certain symmetries. This paper presents theorems that identify the symmetry in f[x, y] based on the depth of the quadtree to expedite 2-D FFT computation of coherent digital images. In principle, it establishes that if the quadtree of f[x, y] has maximum depth kn , then the DFT can be computed by a 2-D FFT of size K/2. An algorithm is given, and its performance analyzed. Finally, applications are considered in transform coding systems and lossy compression of images     

2n prime-sequence codes and coding architecture foroptical code-division multiple-access

Recent study shows that optical code-division multiple-access (CDMA) networks cannot be evaluated or designed by only considering the performance (i.e., correlation properties) of the optical pseudo-orthogonal codes selected. The structures of optical encoders and decoders are another important factors to consider and are needed to coordinate with the selected optical codes as much as possible. A special family of 2ⁿ codes, so-called 2ⁿ prime-sequence codes, is constructed. A general theorem on the cardinality of the new codes is provided. The properties and performance of the codes are also studied. Since these codes pose the algebraic properties of both prime-sequence and 2ⁿ codes, new optical encoding and decoding structures are designed to optimize the system parameters (e.g., power budget and cost) of these optical CDMA networks. This new configuration is particularly attractive for ultrafast optical processing and waveguide implementation for tile future high-capacity, low-loss, all-optical CDMA networks     

Enhancing spectral efficiency and capacity in synchronous OCDMA by transposed-MPC

This paper investigates a newly proposed spreading code-set based on the prime code (PC) families referred to as ‘transposed modified prime codes (T-MPC)’ for the enhancement of spectral efficiency of synchronous incoherent optical code-division multiple-access (OCDMA) networks. This code family increased the code-set cardinality up to twice of existing PC families’ size. This also implies that a greater number of users can be accommodated by the network. Since there is no longer a time-shift feature in T-MPC like in conventional modified prime codes (MPC), the code is not predictable and thus even more secure. Since the code structure is similar to MPC, its deployment in a system/network already running MPC will not require hardware modification. Due to the higher code utilization factor of T-MPC, a greater number of users are accommodated under certain bit-error rate (BER) resulting in remarkable improvement in the spectral efficiency (SE) and capacity. The T-MPC compatibility with low-weight energy-efficient MPC construction is also investigated. The BER and SE performances are analyzed and compared with existing code families. The results indicated that the T-MPC employment can improve up to 50% higher spectral efficiency.     

New single asymmetric error-correcting codes

New single asymmetric error-correcting codes are proposed. These codes are better than existing codes when the code length n is greater than 10, except for n=12 and n=15. In many cases one can construct a code C containing at least [2ⁿ/n] codewords. It is known that a code with |C|⩾[2ⁿ/(n+1)] can be easily obtained. It should be noted that the proposed codes for n=12 and n=15 are also the best known codes that can be explicitly constructed, since the best of the existing codes for these values of n are based on combinatorial arguments. Useful partitions of binary vectors are also presented     

Turbo-SPC codes

This letter is concerned with a family of modified turbo-type codes, referred to as turbo-SPC (single parity check) codes. A technique based on the SPC code is introduced to replace puncturing for rate adjustment. A noticeable feature of the proposed scheme is the significantly reduced decoding complexity compared with the standard punctured turbo code. The cost reduction factor becomes more noticeable as rate increases (over ten times for rate →1). The error-rate floor problem known to turbo codes can also be improved using the proposed scheme. For moderate to high rates, performances of about 0.5 dB from the theoretical limits at BER=10^-5 have been observed for turbo-SPC codes     

基于改进LSOOC的SAC-OCDMA通信系统性能分析

提出了一种基于改进的latin方阵光正交码(MLSOOC)、适合光谱幅度编码(SAC)光码分多址(OCDMA)系统的地址码构造方法,码字间有理想的互相关,容量达到了SAC的上限.分析了SAC编解码的原理、光纤Bragg光栅(FBGs)谱幅编解码的工作过程及在考虑相位引起的强度噪声(PIIN)、热噪声和散弹噪声情况下的系统信噪比(SNR)和误码率(BER).结果表明:在光源和信道理想条件下,多址干扰(MAI)可以消除;在考虑噪声的情况下,可以有效抑制PIIN减小BER;在满足一定BER下,系统承载的同步上路用户多容量比用Hadamard码大大增加.     

对联接杂凑函数的“特洛伊”消息攻击

“特洛伊”消息攻击是Andreeva等针对MD结构杂凑函数提出的一种攻击方法,首次将其应用于不同于MD结构的一类杂凑函数,即联接杂凑。结合联接杂凑的特点,综合利用Joux的多碰撞和深度为n?l的“钻石树”结构多碰撞,构造出了2n-bit联接杂凑函数的长度为 块的“特洛伊”消息,并据此首次提出了对其的固定前缀“特洛伊”消息攻击,其存储复杂性为 块消息,时间复杂性为 次压缩函数运算,远低于理想的时间复杂性 。     

Hamming Coding of DCT-Compressed Images Over Noisy Channels

Theoretical and simulation results of using Hamming codes with the two-dimensional discrete cosine transform (2D-DCT) at a transmitted data rate of 1 bit/pixel over a binary symmetric channel (BSC) are presented. The design bit error rate (BER) of interest is 10^-2. The (7, 4), (15, 11), and (31, 26) Hamming codes are used to protect the most important bits in each 16 by 16 transformed block, where the most important bits are determined by calculating the mean squared reconstruction error (MSE) contributed by a channel error in each individual bit. A theoretical expression is given which allows the number of protected bits to achieve minimum MSE for each code rate to be computed. By comparing these minima, the best code and bit allocation can be found. Objective and subjective performance results indicate that using the (7, 4) Hamming code to protect the most important 2D-DCT coefficients can substantially improve reconstructed image quality at a BER of 10^-2. Furthermore, the allocation of 33 out of the 256 bits per block to channel coding does not noticeably degrade reconstructed image quality in the absence of channel errors.