Frame And Applied In Signal

Fourier analysis is the most widely-applied embranchment in modern mathematics. Since mid 1960s', a period of inventing the fast algorithm of Fourier transform, the influencing range of Fourier analysis has been being bigger and bigger. To now, nearly all fields of modern science and technology use the method of Fourier analysis. However, its limitation becomes more extrusive with its development. Since 1940s', many new methods have been put forward and been improved, such as the STFT, Gabor and the FRFT. Although they all make new improvements theoretically and even in applications, the direct answers to nonorthogonality has still not been given out.Frames that play an important role in wavelet analysis were firstly defined by R.J.Duffin and A.G.Schaeffer in 1952 when they studied the nonharmonious Fourier series.In this paper, the trigonometric series with fractional frequency intervals (TSFFI) is proposed just as a universal development of the Fourier series (FS). TSFFI consists of sine and cosine functions as FS, but frequency intervals of the trigonometric functions don't have to be integers. Therefore, the processing of signal by TSFFI is more flexible and precise. in this paper is to provide a study direction for further research in theory.This thesis consists four chapters:The first chapter is prepared knowledge, discussed the Fourier series, space, operator and the basic knowledge of Bessel sequence, after each chapter is essential foundation.Chapter II on the basis of previous work from the following aspects: Frame for systematically summing up the theory:1. The concept of the Frame;2. Frame and operator, Bessel sequence;3. The dual Frame;4,. The Frame of decomposition and reconstruction; 5, Special Frame;6, The Frame of other issues.Chapter III presents the concept of a one-dimensional, two-dimensional fractional Fourier series, and noted that fractional Fourier series in signal processing applications;Chapter IV presents the specified fractional Fourier series research direction.