A Wavelet Transform Approach to the Design of Complementary Sequences for Communications

A wavelet transform approach to the design of complementary sequences for communications

Todor Cooklev, Keh-Gang Lu

School of Engineering, San Francisco State University, San Francisco CA 94132, USA

Abstract: 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.

Keywords: Correlation, Discrete Fourier transforms, Orthogonal functions, Sequences, Transforms, Wavelet transforms.

1. Introduction

There 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].

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