A Wavelet Transform Approach to the Design of Complementary Sequences for CommunicationsEssay Preview: A Wavelet Transform Approach to the Design of Complementary Sequences for CommunicationsReport this essayA wavelet transform approach to the design of complementary sequences for communicationsTodor Cooklev, Keh-Gang LuSchool of Engineering, San Francisco State University, San Francisco CA 94132, USAAbstract: In this paper we study the relationship between filter banks and complementary sequences. Non-periodic and periodic complementary sequences are identified to be special cases of non-periodic and periodic (or cyclic) wavelet transforms. These wavelet transforms are non-regular. A systematic approach for the generation of periodic symmetric and anti-symmetric sequences is advanced. The novel approach is based on analytic formulae. A systematic approach for the generation of all Golay sequences of a given length is also described.
The Structure of a Scalable Composed Set of Coded Set of Coded Sets of Fibonacci Sequences (Aschedular) in the Nature of Computers (Part I, Chapter 14)A wavelet transformation approach to the design of sequence types is the best alternative path for data mining. The fundamental concept of a wavelet transform approach is to make use of the wavelet transform algorithm along with an algorithm that is similar to that of the algorithm used to generate Poisson arrays, Poisson binomial numbers, and many other algorithms. We propose that an algorithm to efficiently build and generate sequences is a non-interactive search algorithm which will be possible because of its inherent simplicity. The purpose of this paper is to investigate how a non-interactive search method can be implemented. We first review a series of computational issues associated with the search for non-interactive search algorithms in the context of the search process. The work of this paper is described through theoretical, conceptual, and experimental research. Finally, a discussion of the limitations and potential consequences of the non-interactive search algorithm is introduced. While this paper was originally focused on the algorithm performance of non-interactive search algorithms, it is more relevant to an introduction on the search for non-interactive random searching (SNR). With the advent of computer algorithms, researchers have developed and applied new techniques and algorithms to a wide variety of data. In recent years we have seen the application of several new algorithms within the non-computer science literature, and the availability of more specific and specialized algorithms for analysis of this number of applications. A wavelet transform approach to the design of an N-dimensional collection of computationally related sequences may thus help to build a collection of sequences of large lengths with high power and speed. This research focuses on the sequence generation in N-dimensional combinatorics in the domain of random sequences to produce non-interactive random sequences. We propose that a non-interactive search algorithm is an efficient (albeit non-optimized) and stable way to derive information about some of the features of natural numbers. Our main objective is to evaluate its computational feasibility and validity across the various natural numbers. Our paper will discuss the problems we identify for a non-interactive search and describe some of the specific problems that we face for a non-interactive search algorithm. We hope that this paper will provide empirical support for our findings and present the results for use in an article about the non-interactive search. This is where our research is presented. We hope to continue the discussion about data mining to investigate the problem of N-dimensional combinatorics. Future information about the computational utility of non-interactive search algorithms will be presented throughout this paper. References: P. V. Kulkarni, Z. M. Li, K. Wu, P. V. Kulkarni, P. R. V. Kulkarni. 2016. Non-Interactive Search for Recursive N-Class Search for Compressed N-Lines in the Large Sequences of Computationally Unsparring Discrete Random Numbers (NSL-3). Proceedings of the National Academy Meeting at NIST, September 23-25, 2016, http://nist.nist.usda.gov/events/nis/NSL-3/papers/NSL-03.pdf.
Keywords: Correlation, Discrete Fourier transforms, Orthogonal functions, Sequences, Transforms, Wavelet transforms.1. IntroductionThere is a wealth of literature on the theory and design of pseudo-random (or pseudo-noise) sequences for communications with different properties of their autocorrelation and cross-correlation functions (ACF and CCF) [1-4], [11-26].
The theory of filter banks was developed completely independently and it is widely believed that it dates back to 1976, when Croisier, Esteban and Galand designed the first aliasing-free filter bank. Perfect-reconstruction was initially thought to be impossible, and was achieved by three research groups independently around 1984 (for a collection of references see [6]). The discovery of I. Daubechies that orthogonal filter banks provide orthogonal bases for the Hilbert space of square-summable sequences stimulated a tremendous research activity in the area. Furthermore I. Daubechies showed that provided the filters satisfy constraints additional to PR, regular (or smooth) continuous-time functions (scaling functions and wavelets) can be obtained, which are orthogonal bases for the space of square-integrable functions [5].
The main purpose of this paper is firstly to demonstrate the relationship between wavelet transform theory and the theory of complementary sequences, and secondly to develop novel formulae for the analytic construction of complementary sequences using wavelet (or filter bank) theory. We shall consider one important class of sequences, namely complementary sequences. These sequences were recently found to be efficient in a new modulation for wireless communications, called spread-signature CDMA [10]. The connection between two-channel orthogonal FIR filter banks and aperiodic complementary sequences was observed by several researchers [7, 9] and is not novel. Periodic complementary sequences were advanced in [26]. It is shown here that they can be approached using the cyclic wavelet transform. This allows us to develop systematic algorithms for their generation. These two new sets of orthogonal sequences are generalizations of the Golay sequences in the sense that the Golay sequences are members of both of these sets. The novel approach allows to derive formulae for the systematic generation of Golay sequences, which are also given. Previously these sequences could only be generated using computer searches.
The paper is organized as follows. In Section 2 we review filter bank theory. Section 3 is a review of Golay (non-periodic complementary sequences) and Section 4 is a review of various generalizations of these sequences. Section 5 is devoted to orthogonal periodic symmetric codes, and Section 6 – to anti-symmetric codes. Section 7 is devoted to a systematic synthesis of Golay complementary pairs.
2. Two-channel orthogonal FIR filter banksTwo-channel orthogonal FIR filter banks are the most fundamental and widely used class of filter banks [5, 6]. They consist of two parts (Fig. 1): an analysis part of two filters and , each followed by downsampling, and a synthesis part, consisting of upsampling in each channel followed by two filters and . It is easily shown that the output signal, is given by
(1)In perfect-reconstruction (PR) filter banks we have = X(z) and therefore(2)(3)The transform which represents the computation of the two subband signals and from x[n] is called a forward wavelet transform. The transform which computes the signal (which is equal to x[n]) is called an inverse wavelet transform. In orthogonal filter banks the impulse response together with its integer translates forms an orthogonal basis for the Hilbert space of square summable sequences. The aperiodic auto-correlation function (ACF) of the impulse responses and are half-band functions:
(4)(5)while the cross-correlation is identically zero(6)Any two sequences and with the auto-correlation and cross-correlation properties in (5), (6) and (7) define an orthogonal wavelet transform and the two sequences are an orthogonal basis for the Hilbert space of square-summable sequences. The synthesis filters are completely determined from the analysis filters:
(7)(8)where the operation means transposition, conjugation of the coefficients and replacing z by z−1. In the time-domain is related to according to(9)where N is the order of the filters and is necessarily odd. A necessary and sufficient condition for perfect-reconstruction is that the product P(z) = = = be half-band:
(10)Splitting the even-indexed and odd-indexed coefficients is called a polyphase decomposition:(11)(12)