Random noise attenuation by a selective hybrid approach using f-x empirical mode decomposition

EMD

The aim of empirical mode decomposition (EMD) is to empirically decompose a non-stationary signal into a finite set of sub-signals, which are termed as intrinsic mode functions (IMF) and are considered to be stationary. The IMFs satisfy two conditions: (1) in the whole data set, the number of extrema and the number of zero crossings must either be equal or differ at most by one; and (2) at any point, the mean value of the envelopes defined by the local maxima and the local minima is reasonably close to zero (Huang et al., 1998).

Provided that $s(t)$ , $c_n(t)$ , $r(t)$ , and $N$ denote the original non-stationary signal, the separated IMFs, the residual and the number of IMFs, respectively, EMD can be expressed as:

$$s(t)=\sum_{n=1}^{N}c_n(t)+r(t).$$ (1)

For a non-stationary signal $s(t)$ , using equation 1, we get a finite set of sub-signals $c_n(t)$ ,( $n=1,2,\cdots,N$ ).

A special property of EMD is that the IMFs represent different oscillations embedded in the data, where the mean frequency for each sub-signal $c_n(t)$ decreases with IMF order increasing.

Figures 1 and 2 give a example of EMD for a non-stationary signal. Figure 1 shows a non-stationary signal and its corresponding instantaneous frequency, which is similar to that used by Herrera et al. (2013). From the instantaneous frequency we can see that the signal can be divided into four temporal regions. From 0 s - 4 s, the signal is composed of two frequency components corresponding to 5 Hz and 15 Hz. From 4 s - 6 s, the signal is composed of three frequency components corresponding to 5 Hz, 15 Hz, and 34 Hz, respectively. From 6 s- 7.8 s, the signal is composed of two frequency components corresponding to 10 Hz and 34 Hz. The last part from 7.8 s to 10 s is a monotonic signal, whose frequency is 10 Hz. Figure 2 shows the separated signal component using EMD. The separated components are consistent with the instantaneous frequency map as shown in Figure 1. Each individual frequency components have been separated out, with negligible edge effects and residual.

s-sf
Figure 1. Non-stationary signal and its instantaneous frequency. Up: Non-stationary signal. Down: Corresponding instantaneous frequency.

Imf
Figure 2. EMD separated signal. From up to down: first IMF, second IMF, third IMF, and the residual.

Random noise attenuation by a selective hybrid approach using f-x empirical mode decomposition