CONCLUSION

We proposed to compute the time-frequency map of an input signal based on NPM coupled with Hilbert spectral analysis. The proposed method is an empirical mode decomposition-like method, but using NPM to compute its intrinsic mode functions. Compared with the Fourier transform, the proposed method is data-driven and needs much less base functions to approximate the original signal. Since the NPM results an under-determined linear system, we use shaping regularization to regularize it. The regularization makes the intrinsic mode functions more smooth with respect to the amplitudes and frequencies compared with the intrinsic mode functions of the empirical mode decomposition. There are many time-frequency methods, which one is the best? This is a difficult question to answer. Methods are good for some type signals, maybe not good for other type signals.

Yung-Huang et al. (2014) pointed out that the complexity of empirical mode decomposition/ensemble empirical mode decomposition is $41*\mathbf{N}_E*\mathbf{N}_S*n(\log_2n)=O(n\log n)$ , where $n$ is the data length and the parameters $\mathbf{N}_E$ and $\mathbf{N}_S$ are the ensemble and sifting numbers respectively. For the non-stationary Prony method, the computation complexity is mainly attributed to the polynomial zero-finding. We used the pseudo-zeros method to compute the pseudo-spectra of the associated balanced companion matrix (Toh and Trefethen, 1994), which requires approximate $\mathbf{N}^3$ works, where $\mathbf{N}$ is the polynomial degree number. Therefore, the total computation complexity is $\mathbf{N}^3*\frac{n}{\mathbf{N}}=n*\mathbf{N}^2$, where $n$ is the data length. In this paper, we choose $\mathbf{N}=5$, and therefore the total computation complexity is approximate $n*5^2 = O(n)$.