When FrFT meets quadratic frequency modulation functions—A novel tool for nonstationary signals and time-varying systems

The fractional Fourier transform (FrFT) is a general form of the Fourier transform, which has provided a theoretical framework for linear frequency modulation signal processing. The quadratic frequency modulation (QFM) function is a generalization of the linear frequency modulation function, which balances the precision and complexity of nonstationary signals in practical applications. However, there is no framework for analyzing and processing QFM signals, which limits the applications of QFM signal model. The QFMFrFT, a generalization of the FrFT, is developed in this study to tackle this issue. The impact of the QFMFrFT on the time-frequency distribution explains its usefulness in sampling, denoising, and filter design of QFM signals. Two one-place statistical functions are defined for QFM-type stochastic signals, of which the input-output relationships of a linear time-varying system are discussed. Using these relations, a linear time-varying matched filter is designed. Two types of generalized convolution operators are designed associated with the QFMFrFT, which allow a multiplicative realization of a time-varying system. The QFM signal is demonstrated as a singular function of this type of linear time-varying system, which inspires a multi-carrier technique in spread-spectrum communications. Simulations verify the applications of the QFMFrFT in parameter estimation, matched filter design, and communications.