Hilbert Transform and Its Engineering Applications

The Hilbert transform is useful in calculating instantaneous attributes of a time series, especially the envelope amplitude and instantaneous frequency. The instantaneous envelope is the amplitude of the complex Hilbert transform; the instantaneous frequency is the time rate of change of the instantaneous phase angle. These properties can be applied to identify dynamic characteristics of a linear as well as a nonlinear system. However, the conventional discrete Hilbert transform, which is based on fast Fourier transform and inverse Fourier transform, has shown the lack of accuracy for time-derivative calculations. In this paper, we first introduce the Hilbert transform and its applications to the nonlinear system parameter identification. Then we address the practical issues in applying the Hilbert transform to engineering applications. To increase the accuracy of the envelope detection, we propose a Hilbert transform technique based on local-maxima interpolation. Analyses and simulations are carried out to demonstrate the advantages of the proposed technique. Finally, we employ the proposed local-maxima-interpolation technique in identifying the nonlinear dynamic characteristics of industrial examples.