Walsh-function generator

In recent years several papers dealing with the mathematical theory as well as the technical applications of Walsh functions have been published. One method of defining Walsh functions is by using the Rademacher functions and the multiplication law of Walsh functions which is in fact a binary addition modulo 2 (no carry). This letter, however, describes in detail a method of defining Walsh functions by using orthogonal code blocks or the so-called Hadamard matrices. Some advantages of this method are its simplicity and its straightforward hardware implementation.     

Simplification of the Computation of Real Hadamard Transfors

The computation of a real Hadamard (or Walsh) transform can be simplified by first computing a pseudotransform, employing an easily derived auxiliary matrix each of whose elements is either l or 0 (instead of 1 or -1), and then applyimg se very simple linear corrections to obtain the; desired transform. The auxiliary matrix is not orthogonal, so its square, unlike that of the Hadamard (or Walsh) matrix, is, not a scalar matrix. However, sibject to one easilyg satisfied restrietion, it has an inverse, Which, within a scalar. multiplier, differs from ihgt of the Hadamard (or Wtalh) matrix of the same order in only a single element.     

DS-CDMA系统正交码调制的抗多址干扰性能及其改进

本文推导非同步 Ｃ Ｄ Ｍ Ａ 系统多址干扰表达式,计算正交码调制码组相关性对多址干扰的影响。提出一种改进型 Ｗ ａｌｓｈ＋ Ｇ Ｍ Ｗ 正交码组,分析比较 Ｗ ａｌｓｈ 正交码组和 Ｗ ａｌｓｈ＋ Ｇ Ｍ Ｗ 正交码组应用于非同步 Ｃ Ｄ Ｍ Ａ 系统的性能。计算机模拟结果表明应用改进型正交码组时系统性能获得改善。     

New covering codes from an ADS-like construction

A covering code construction is presented. Using this construction it is shown that t[52,39]=3, t[36,21]=4, t[58,32]=7, K(32,2)/spl les/62/spl middot/2/sup 18/, and K(62,5)/spl les/31/spl middot/2/sup 37/, where t[n,k] is the minimum covering radius among all binary [n,k] codes and K(n,R) is the minimum cardinality of a binary code of length n and covering radius R. Four new linear codes found by computer search are also given. These include a [23,9]5 code, a [32,8]10 code, a [51,41]2 code, and a [45,20]8 code.     

Using Walsh code selection to reduce the power variance ofband-limited forward-link CDMA waveforms

This paper investigates the effects of Walsh code selection and baseband filtering on the statistical moments of the band-limited forward-link CDMA signal. A normalized moment, the “power variance,” is proposed as a measure of the signal's sensitivity to nonlinear amplification. The estimation of the power variance for the case of a band-limited QPSK-modulated random waveform is presented as a baseline fur illustrating the effect of filtering on a sampled signal. The band-limited CDMA signal case illustrates the interaction between the Walsh codes and the wave-shaping filter, and quantifies the influence of “missing Hadamard basis functions. ”Results verify the accuracy of the statistical models and illustrate the importance of Walsh code selection in the reduction of the power variance of a band-limited CDMA waveform     

Unified algorithm to generate Walsh functions in four different orderings and its programmable hardware implementations

The paper presents an algorithm to generate Walsh functions in four different orderings: Hadamard, Harmuth, Paley and strict sequencing. By the analysis of the properties and mutual relations among these four orderings, the authors found a unified approach to generate any of the orderings from the primary set of Rademacher functions. By using these properties, the authors developed a programmable Walsh function generator for 64 outputs by both field programmable gate arrays and lookup table cascades to estimate the amount of hardware and performance. Such a programmable Walsh function generator can be used in VLSI testing, CDMA, pattern recognition, as well as image and signal processing.     

BER Performance of Walsh–Hadamard Like Kronecker Product Codes in a DS-CDMA and Cognitive Underlay System

Walsh–Hadamard transform, a discrete unitary transform is widely used in many applications such as signature codes in the current wireless standards IS-95 CDMA, WCDMA, CDMA2000 and image transform applications. It is simple to implement this transform since they can be generated by a single Kronecker product recursion formula. In this paper, a new set of binary code families similar to Walsh codes are obtained based on the concept of code concatenation and permutation. It is shown that these codes can be generated by reconfiguring the Walsh–Hadamard code generator. Hence it can be utilized in reconfigurable radios such as underlay cognitive radio (UCR). Theoretical results showing the BER performance due to MAI between primary users and secondary users in an UCR is also obtained. Simulation results showing the BER performance of these codes in a direct sequence spread spectrum system and UCR system with quadrature multiplexing operating in the individual decoding mode under AWGN plus flat fading Rayleigh channel conditions is also obtained.     

On the covering radius of codes

The covering radiusRof a code is the maximal distance of any vector from the code. This work gives a number of new results concerningt[n, k], the minimal covering radius of any binary code of lengthnand dimensionk. For examplet[n, 4]andt[n, 5]are determined exactly, and reasonably tight bounds ont[n, k]are obtained for anykwhennis large. These results are found by using several new constructions for codes with small covering radius. One construction, the amalgamated direct sum, involves a quantity called the norm of a code. Codes with normleq 2 R + 1are called normal, and may be combined efficiently. The paper concludes with a table giving bounds ont[n, k]forn leq 64.     

Prove the Minimal Delay of Complex Orthogonal Space-Time Block Code by Hadamard Matrix

1IntroductionWith the development of mobile communications inthe world Space-Ti me Block Code(STBC)fromcom-plex orthogonal designs have received extensive interestin Code Division Multiple Access(CDMA)system de-sign recently,see for example[1~3].Since the…     

正交特征群

本文在证明了有限Ａｂｅ１群Ｇ与它的特征群Ｇ√存在同构的一个必要充分条件之后，构造了一类正交特征群，从而概括了相当一大类正交变换（它包括了离散付里叶变换、阿达玛变换、数论变换、Ｃｈｒｅｓｔｅｎｓｏｎ变换）这对构成有限域与整环上的正交变换具有指导作用。     

On Representations of Walsh Functions

A new recursive formula for defining Walsh functions on the real line is presented. It leads, in a natural way, to an explicit representation of Walsh futctions. The explicit representation can be identified with. the representation of Walsh functions by the products of Rademacher functions. It can also lead to other representations of Walsh functions in a straight-forward manner. Using the new-formulation, one can prove many known properties of Walsh functions easily and systematically and gain new insight.     

Design of a Walsh Function Generator

To develop a sequency-division multiplex system for telemetry at the Beijing Institute of Aeronautics and Astronautics, it was necessary to design a Walsh function generator that produces ?almost pure? Walsh functions, just as a generator for almost pure sinusoidal functions is needed for frequency-division multiplex system. In this paper, a symbol function s(n, t) is discussed, and the dyadic increment d(k, t) of the Gray code is defined. The relation between s(n, t) and d(h, t) is deduced. A method for the design of a Walsh function generator based on the symbol function s(n, t) is presented. This method has been used to design a Walsh function generator with order number up to n = 64.     

Multicarrier orthogonal CDMA signals for quasi-synchronouscommunication systems

Proposes a multicarrier orthogonal CDMA signaling scheme for a multiple-access communication system, such as the reverse channel of a cellular network, as an alternative to the multi-user interference cancellation approach. The average variance of cross-correlations between sequences is used as a measure for sequence design. The authors search for sets of sequences that minimize the probability of symbol detection error, given that there is imperfect synchronization among the signals, that is, the signals are quasi-synchronous. Orthogonal sequences based on the Sylvester-type Hadamard matrices (Walsh functions) are shown to provide a significant improvement over the case where a Hadamard (orthogonal) matrix is chosen at random. Computer searches suggest that this set of codes is optimal with respect to the above measure. The issue of chip pulse shaping is investigated. Optimal pulses designed to minimize multiple-access interference in quasi-synchronous systems are obtained for various bandwidths and are shown to provide a large improvement over the raised cosine pulses. A multicarrier signaling scheme is introduced in order to reduce chip level synchronization offsets between the users     

Walsh Functions and Gray Code

Gray code is a natural way of ordering binary vectors in dyadic space, hence it appears frequently in connetion with Walsh functions. In Paley's definition of Walsh functions their sequencies are arranged in Gray code. Gray code also appears in a new Walsh function generation algorithm which obtains a function by locating all its sign changes. There are certain computational advantages in using Gray code rather than sequency ordering. Examples in fast Walsh transform, dyadic convolution and digital filtering are given. Methods of Gray code to binary conversion are discussed.     

A New Walsh-Like Near Orthogonal (WNO) Sequence for Asynchronous CDMA System

Present generation mobile communication system employs one of the most popular wireless access technologies called code division multiple access (CDMA). Design of CDMA spreading codes has drawn significant attention amongst the researchers over the last few decades. CDMA code family is generally categorized into purely orthogonal and non-orthogonal (near-orthogonal) members which have established their application in synchronous (downlink) and asynchronous (uplink) CDMA system respectively. Walsh code has been regarded as the most useful spreading code to be used in synchronous link because of its orthogonality property. However, the performance of Walsh code is significantly inferior in asynchronous surroundings. A number of codes have consequently been proposed with an aim to mitigate the shortcomings of Walsh code. This paper makes an innovative attempt to enhance the correlation properties of existing Walsh code through one simple yet powerful algorithm. Proposed code of length ‘N’ has been generated from code sets of length ‘N/4’ and thus makes the code generation algorithm recursive in nature. Performance of the proposed code has subsequently been compared with some existing orthogonal and semi orthogonal codes in terms of various performance metrics and finally the supremacy of our proposition has been established.     

Orthogonal variable spreading codes for wideband CDMA

Orthogonal codes of variable lengths are essential for multirate services in the Third-Generation Partnership Project (3GPP). We propose an alternative to Walsh functions for variable spreading codes. There are three important points. First, we design an orthogonal code of constant length by mapping the points of a set of finite projective planes to chips of high/low amplitudes. Second, we develop variable spreading codes using multiple layers of recursion. Last, we compare and contrast the proposed variable spreading technique with Walsh functions in terms of ease of synchronization, blind-rate detection and wideband characteristics     

Cubic Boolean functions with highest resiliency

We classify those cubic m-variable Boolean functions which are (m-4)-resilient. We prove that there are four types of such functions, depending on the structure of the support of their Walsh spectra. We are able to determine, for each type, the Walsh spectrum and, then, the nonlinearity of the corresponding functions. We also give the dimension of their linear space. This dimension equals m-k where k=3 for the first type, k=4 for the second type, k=5 for the third type, and 5/spl les/k/spl les/9 for the fourth type.     

Construction of multilevel Hadamard matrices with small alphabet

Multilevel Hadamard matrices, which are related to orthogonal design and binary Hadamard matrices, are investigated. It is assumed that all elements of the matrix are integers. Construction of multilevel Hadamard matrices with a small alphabet, such as three, four or eight different elements, is presented.     

On the existence of[M, n]group codes for the Gaussian channel with[M, n]odd

The question of the existence of nonplanar[M,n]group codes for the Gaussian channel has been settled except in the case ofnodd andModd and composite. For this unsettled case, it is shown that the existence of a nonplanar[M,n]group code is implied by the existence of a groupGsatisfying i)|G|is even, ii)Ghas a faithful complex irreducible representationTof the first kind, and iii)Trestricted to the two-Sylow subgroup ofGcontains the identity representation. A partial converse of this existence result is also given. Finally, it is shown that for each oddnnot of the form2^{m} - 1, there exists a nonplanar[M,n ]group code withModd and composite.     

A Simple Recursive Definition for Walsh Functions

This paper presents a simple recursive definition for Walsh functions, which overcomes the shortcomings of other recursive definitions. A rule of thumb for writing down the explicit representation for Walsh functions of any order is also devised. In addition, this recursive formula is used to generate discrete Walsh functions in matrix form.