Frequency analysis based on general periodic functions

Abstract

Following the sine–cosine function, the sawtooth wave, square wave, triangular wave, trapezoidal wave, and so on become new easily generated periodic functions in modern electronics. Similar to Fourier’s idea, a natural question is whether a signal can be considered as a superposition of easily generated functions with different frequencies. Therefore it is necessary to generalize Fourier analysis based on sine–cosine functions into frequency analysis based on general periodic functions. In this paper, we introduce the frequency series and frequency transformation based on general periodic functions. We discuss when a frequency system is a complete system or an unconditional basis in L2[−π,π], and when a frequency transformation can be carried out in L2(−∞,+∞). For practical convenience almost all easily generated functions in electronics are considered carefully as examples. As a new and practical generalization of classical Fourier analysis, these results will become a theoretical foundation for the technique of easily generated function analysis in signal processing.