A kind of biorthogonal functions based on Fourier-Legendre series and its application in wireless transmission system

In wireless communications, security transmission is hoped to be guaranteed. In this paper, we suggest a new wireless transmission system based on biorthogonal signal. Biorthogonal functions are often obtained from an orthogonal system with some algorithm. With the theorems of Mobius inverse and Dirichlet inverse, we analyze the algorithm to obtain biorthognal functions based on Fourier series analysis. Then, we propose an algorithm to obtain a kind of new biorthogonal functions based on Legendre polynomial analysis, called Generalized Fourier series analysis. The obtained biorthogonal functions are used as scramble modulation signal in wireless transmission system and the transmitted signal has outstanding security performance. Compared with other systems including biorthogonal system based on Fourier series analysis, orthogonal MCM and OFDM, the suggested biorthogonal modulation system based on Fourier-Legendre series has better BER and not bad spectrum efficiency performance than OFDM.