Research on perfect dyadic binary sequence pair

In this paper, the perfect dyadic binary sequence pair with one-value dyadic correlation functions is presented. That is, the perfect dyadic binary sequence pair is a perfect discrete signal, for its dyadic relative function is δ-function. The transformation features and some existing admissibility conditions of perfect dyadic binary sequence pair are discussed, and the properties for this kind of code in Walsh transformation spectrum and weight spectrum are also analyzed. From above, It is found that the perfect dyadic binary sequence pair can easily differentiate from its dyadic shifting. So these good signals can used in engineering as synchronization code, multi-user code and so on.     

Walsh Spectrum Measurement in Natural, Dyadic, and Sequency Ordering

Walsh spectra may have natural, dyadic, or sequency ordering. This paper describes an improved processor design to provide spectra in any selected ordering. The incoming data are stored either directly or in a permuted sequence, and then are followed by a fast Walsh-Hadamard transform. For sequency ordering, the permutation operation is that of a Gray-to- binary code conversion, followed by bit reversal. For dyadic ordering, the operation is only a bit reversal. First a Hadamard processor that uses long shift registers is described. It is adapted to yield coefficients in either of the other two orderings by way of the permuting module (PM).     

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.     

Group Codes and Walsh Functions

The set of Walsh functions, wal(j,?), is the character group of the dyadic group. For O?j?2k it is shown that they may also be derived from the character table of the abstract Abelian group Ck generated by k elements of order two. The method uses Slepians modular representation table[3] to compute the 2k irreducible representations (each of degree one) of Ck. The character table, K, is a 2kx2k square array of +1's and -l's and, considered as a matrix, the orthogonality relationships for characters show that K has the Hadamard property, [K][K]T = 2K [I]. In fact, for the proper ordering of the group elements in the construction of the modular representation table it is the Hadamard matrix, the entries of whose ith row take on the values of the Walsh function wal (i,?) in each of ?/2k subintervals. In a similar way other permutations of the modular representation table define different functions taking on the values +l, -l, also orthogonal and in a one to one relationship to the Walsh functions. Since an n place binary group code with k information places is isomorphic to Ck,[3] each code can thus be used to generate real functions orthogonal over a given interval or period ?. In the special case of cyclic codes where the elements of the code interpreted as polynomials form an ideal in a polynomial ring of characteristic two, the group operation used in deriving the character table is of course, addition.     

On Moments and Walsh Characteristic Functions

This correspondence treats the derivation of natural moments from their corresponding Walsh characteristic function via the dyadic derivative operator. The derivation of a result concerning Walsh transforms of dyadic derivatives of functions is also considered. However, some established ideas such as Walsh transform, dyadic stationarity, and dyadic correlation are introduced first.     

并元序列研究

对并元码偶作出进一步研究,首先提出了一类新的区组设计——并元加集偶的概念,研究了并元加集偶的一些特性。然后给出了并元加集偶与并元码偶的等价关系,应用这一等价关系给出了并元码偶存在的必要条件,也为并元加集偶的构造提供了理论依据。最后还研究了并元码偶的Walsh谱特性和重量分布特性。通过这些问题的研究,展示了Walsh变换新的应用。这些存在条件在这类码的理论研究和计算机搜索中有重要的作用。     

A Nonrecursive Equation for the Fourier Transform of a Walsh Function

Convolution of a discrete Walsh function with a rectangular pulse simplifies the derivation of an expression for the Fourier transform of a Walsh function. The nonrecursive transform equation that is developed is a function of the bits of the Gray code number for the order of the Walsh function.     

Short-time spectral and autocorrelation analysis in the Walsh domain

A short-time dyadic autocorrelation function (dacf) and a short-time Walsh energy spectrum of the first kind are defined in the Walsh-Fourier domain. The "natural" choice of the short-time functions does not lead to a Walsh-Fourier transform pair (dyadic Wiener-Khintchine theorem), and thus a second kind of short-time dacf and short-time Walsh energy spectrum are defined as the Walsh-Fourier transforms of the first kind. This leads to a meaningful and convenient Walsh transform pair between the first short-time Walsh energy spectrum and the second short-time dacf. The measurement procedures for both kinds of functions are discussed, and the mean values of these short-time functions are shown to be related to the corresponding long-time functions.     

Time-Shift Theorems for Walsh Transforms and Solution of Difference Equations

Several time-shift theorems for Walsh transforms of functions subject to nondyadic as well as dyadic time displacements are presented. Spectrum-conversion matrices are defined and a relation between a function with an ordinary shift and that with a dyadic shift is established. Procedures for solving difference equations by Walsh transformation are given.     

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.     

复制生成序列的自相关函数研究

介绍了以二进制码为复制信息,用平移复制生成序列的方法。讨论了复制生成序列的性质和复制生成序列的自相关函数的性质。依据序列的复制特性,给出了一个复制生成序列自相关函数的计算公式,并给出了自相关函数值为零的一个充要条件,及相应的数学证明。最后讨论了Paley顺序的离散Walsh函数的自相关函数。     

Comparison of Walsh and Fourier spectroscopy of geomagneticreversals and nonsinusoidal palaeoclimatic time series

Higher resolving capabilities and theoretical appropriateness of Walsh spectral techniques as compared to Fourier spectral analyses are presented for synthetic and nonsinusoidal geotime series. Theoretical developments of Walsh transform techniques and a comparative study of Walsh and Fourier spectral estimates are presented. The Walsh spectral technique is applied specifically to two actual time series data of geomagnetic reversals in binary telegraphic wave form and nonsinusoidal palaeomagnetic and palaeoclimate time series. Walsh spectra reveal periodicities in Milankovitch frequency bands and provide exceptionally well-resolved spectral lines. The possible physical significance of these orbital periodicities is discussed. A comparative example of autocorrelation analysis in the real time domain and dyadic time domain is also presented using a telegraphic signal model of actual geomagnetic reversal time series. and the result is briefly discussed. The computational efficiency of the Walsh function could be exploited further for many other binary and nonsinusoidal geophysical/geological time series     

Quantizers with uniform encoders and channel optimized decoders

Scalar quantizers with uniform encoders and channel optimized decoders are studied for uniform sources and binary symmetric channels. It is shown that the natural binary code (NBC) and folded binary code (FBC) induce point density functions that are uniform on proper subintervals of the source support, whereas the Gray code (GC) does not induce a point density function. The mean-squared errors (MSE) for the NBC, FBC, GC, and for randomly chosen index assignments are calculated and the NBC is shown to be mean-squared optimal among all possible index assignments, for all bit-error rates and all quantizer transmission rates. In contrast, it is shown that almost all index assignments perform poorly and have degenerate codebooks.     

Bandwidth requirements for Walsh functions (Corresp.)

The power-density spectrum for the Walsh function of binary orderkis calculated. The model analyzed considers the Walsh function as a random sequence modulating akth-order Rademacher function. The total number of Walsh functions of binary orderkand less is found to be aboutfrac{1}{6}of the overbound 2.4TW.     

High-speed gray-binary and binary-Gray code convertors using electro-optic light modulators

A high-speed Gray-binary code convertor (g.b.c.) using electro-optic light modulators, which translates all bits of Gray code into binary code simultaneously, and a binary-Gray code convertor, which translates binary into Gray code, are described. A digital-analogue convertor using g.b.c. is also presented.     

Sinusoids versus Walsh functions

In most of the applications contemplated for Walsh functions these binary waveforms would replace the more usual sinusoids, as the fast-Walsh-transform algorithm appears to make them very attractive for many kinds of signal processing. This paper begins with a brief review of the characteristics of Walsh functions and of their applications. Some old and some new interrelations are presented between sinusoids and Walsh functions, but the principal aim of the paper is to investigate the truncation and roundoff errors associated with the use of Fourier and of Walsh series. By employing simplifying approximations it is found that, for long samples of smooth signals, far more terms are required in the Walsh-series representation and greater accuracy is required of their coefficients for a given rms total error. Even for discontinuous signals the Walsh series may require substantially more terms, thus counterbalancing the computational advantage of the fast Walsh transform. This relative inefficiency of the Walsh-series representation of long waveforms may explain why it has not proven particularly effective in applications.     

Walsh-Transform Analysis of Discrete Dyadic-Invariant Systems

This short paper shows how the sampled output of a dyadic-invariant linear system with a given sequency-domain transfer function, in response to a sampled input, can be determined by 1) a term-wise multiplication of the sampled transfer function and the discrete Walsh transform of the sampled input function, followed by an inverse Walsh transform, or 2) a discrete dyadic convolution of the sampled impulse response and the sampled input directly in the time domain. Functions in both time and sequency domains are represented by column matrices, and discrete Walsh transformation is effected simply by the multiplication with a Walsh matrix. An example is included to illustrate both procedures. The validity of the solutions is further verified by showing that the governing dyadic differential equation of the system is satisfied.     

Cyclic codes over Z4, locator polynomials, and Newton'sidentities

Certain nonlinear binary codes contain more codewords than any comparable linear code presently known. These include the Kerdock (1972) and Preparata (1968) codes that can be very simply constructed as binary images, under the Gray map, of linear codes over Z₄ that are defined by means of parity checks involving Galois rings. This paper describes how Fourier transforms on Galois rings and elementary symmetric functions can be used to derive lower bounds on the minimum distance of such codes. These methods and techniques from algebraic geometry are applied to find the exact minimum distance of a family of Z ₄. Linear codes with length 2^m (m, odd) and size 2(2^m+1-5m-2). The Gray image of the code of length 32 is the best (64, 2³⁷) code that is presently known. This paper also determines the exact minimum Lee distance of the linear codes over Z₄ that are obtained from the extended binary two- and three-error-correcting BCH codes by Hensel lifting. The Gray image of the Hensel lift of the three-error-correcting BCH code of length 32 is the best (64, 2³²) code that is presently known. This code also determines an extremal 32-dimensional even unimodular lattice     

Noise Analysis of Soft Errors in Combinational Digital Circuits Via Walsh Transforms

The noise produced at the output of combinational logic circuits by individual gate failures is analyzed through the use of Walsh functions. Soft errors are modeled by allowing the output of each gate in a particular realization to fail temporarily, possibly introducing an error in the single binary output. The input variables also are allowed to be stochastically driven. The output probability of error contains the Walsh transform of an extended logic function and the Walsh characteristic functions of the input variables as well as the individual gate failure variables. These results are specialized to the case where the inputs are statistically independent of the soft errors. A discussion of the transform of the extended logic function is included.     

Quasi-orthogonal sequences for code-division multiple-accesssystems

The notion of quasi-orthogonal sequence (QOS) as a means of increasing the number of channels in synchronous code-division multiple-access (CDMA) systems that employ Walsh sequences for spreading information signals and separating channels is introduced. It is shown that a QOS sequence may be regarded as a class of Bent (almost Bent) functions possessing, in addition, a certain window property. Such sequences while increasing the system capacity, minimize interference to the existing set of Walsh sequences. The window property gives the system the ability to handle variable data rates. A general procedure of constructing QOSs from well-known families of binary sequences with good correlation, including the Kasami and Gold (1967) sequence families, as well as from the binary Kerdock code is provided. Examples of QOSs are presented for small lengths. Some examples of quaternary QOSs drawn from Family A are also included