Signal component extraction method based on polynomial chirp Fourier transform

Abstract

As frequency modulated (FM) signals widely exist in the natural world as well as in different artificial applications, it is of great practical significance to explore the ways to extract such signal components in the complex and noisy environment. To extract one component from the noisy multicomponent signal effectively, a component extraction method based on polynomial chirp Fourier transform (PCFT) is presented in this paper. First, the physical meanings of Fourier transform (FT) and fractional Fourier transform (FRFT) are analyzed and their internal relations are expounded from the perspective of signal energy accumulation. Essentially, the FT accumulates signal energy along the time-frequency beelines parallel to the time axis and obtains an energy-concentrated spectrum from the narrow-band stationary signals whose frequency does not change, whereas it fails to process non-stationary signals with changeable frequencies. By rotating the time-frequency axis, the FRFT changes the energy accumulation mode of the signal in the old time-frequency plane and achieves a more concentrated spectrum for the linear frequency modulated (LFM) signal, but with larger error or even invalidation when dealing with nonlinear frequency modulated (NLFM) signal. Using FT and FRFT, in this paper we attempt to improve the energy accumulation mode of the conventional transform method and propose the PCFT. In this transform, the beeline families in the traditional transform, independent of time (or v) axes, are replaced by a family of polynomial chirping curves in the time-frequency plane. These polynomial chirping curves are capable of approaching more closely to the instantaneous frequency curve of FM signal so as to obtain a more concentrated transform spectrum and thereby extend the application of PCFT from LFM signal to NLFM signal. When selecting the polynomial chirping curve, we build up a nonlinear optimization model guided by the principle of energy spectrum concentration and in this way convert the problem of determining the polynomial curve families into the one of optimizing the polynomial parameters. Then particle swarm optimization algorithm is employed to search for the optimal polynomial parameters so as to concentrate the energy of one component in the new transform domain, i.e., the polynomial chirp Fourier domain. After doing that, each component is separated into its concentrated spectrum with a narrow-band filter and reconstructed with the inverse PCFT. Moreover, to extract components from a noisy multicomponent signal successfully, an iteration involving parameter estimation, PCFT, filter and recovery is introduced. To verify the effectiveness of the PCFT-based method, a series of examples, including simulated and real-world signals, is chosen for simulations and experiments. The experimental results indicate that compared with FT and FRFT, the proposed method overcomes the shortcoming of distributed energy spectrum for NLFM components in the traditional transforms and obtains a concentrated energy spectrum in the polynomial chirp Fourier domain, therefore realizing component separation and time-frequency characteristic extraction. The PCFT-based method not only has the capability of dealing with the extraction of LFM components, but also performs well in the separation of crossed NLFM components, and with little extraction error.