An orthogonal technique for empirical mode decomposition in Hilbert-Huang transform

First, it is indicated that the intrinsic mode functions (IMF) obtained by the empirical mode decomposition (EMD) are not orthogonal. Then an orthogonal technique based on Gram-Schmidt method is proposed to obtain the complete orthogonal intrinsic mode functions. Three ways of the orthogonal processes are suggested and compared in the paper. Finally, the suggested method is validated through the decomposition of a typical time history. The numerical results show that the orthogonal IMF can be obtained easily by the technique proposed in the paper. 1 Intrinsic mode functions Hilbert-Huang transform is applied widely in the signal analysis process as well as in Earthquake Engineering. After the empirical mode decomposition (EMD), the signal X t can be expressed as 1 n X t c t r t j n j (1) where j c t is the j intrinsic mode function (IMF), r t n is the residual function. The integral orthogonal index IOT and the orthogonal index IOjk between the j IMF and the k IMF are designated respectively for checking the orthogonality of the IMFs. 1 1 2 0 0 1 1 ( ) ( ) ( ) n n T T j k j k k j IOT c t c t dt X t dt (2) 0 2 2 0 0 ( ) ( ) ( ) ( ) T j k jk T T j k c t c t dt IO c t dt c t dt (3) If all of IMFs are orthogonal mutually, the values of IOT and IOjk are zero. However, the orthogonality of the IMFs can not be verified by mathematical technique. The following simple example shows the IMFs are not orthogonal. The signal X t is assumed as the sum of three sinusoidal waves 3 3 1 1 ( ) ( ) sin(2 ) j j j j X t x t f t (4) where 1 1 f Hz , 2 5 f Hz , 3 10 f Hz . Its time history curve with 5 seconds is shown in Figure 1. Obviously, three sinusoidal waves are orthogonal, the theoretical values of IOT and IOjk are zero. The numerical values of IOT and IOjk are very small if they are calculated by computer, such as 2.5545e-016 IOT and 0.5 8.1685e-017 5.8467e-017 8.1685e-017 0.5 2.4303e-016 5.8467e-017 2.4303e-016 0.5 IO 0 1 2 3 4 5 -3 -2 -1 0 1 2 3 Figure 1. Time history curve of a simple signal


Intrinsic mode functions
Hilbert-Huang transform is applied widely in the signal analysis process as well as in Earthquake Engineering [1][2][3] . After the empirical mode decomposition (EMD), the signal X t can be expressed as 1 n X t c t r t j n j ¦ (1) where j c t is the j th intrinsic mode function (IMF), r t n is the residual function.
The integral orthogonal index IOT and the orthogonal index IO jk between the j th IMF and the k th IMF are designated respectively for checking the orthogonality of the IMFs.
If all of IMFs are orthogonal mutually, the values of IOT and IO jk are zero. However, the orthogonality of the IMFs can not be verified by mathematical technique. The following simple example shows the IMFs are not orthogonal. The signal X t is assumed as the sum of three sinusoidal waves 3 3 where 1   The signal X t can be decomposed as five IFMs and one residual function when applying the EMD, shown in Figure 2. These five IMFs are defined as original IMFs.
The orthogonal index 0.06633 IOT and IO jk are shown in Table 1. Comparing with the orthogonal level of three sinusoidal components, the orthogonal level of the original IMFs drops a lot.

Orthogonal technique for original IMFs
Gram-Schmidt method is applied usually for obtaining the orthogonal eigenvectors or Ritz vectors in structural dynamics [4] . This method is used here to achieve the orthogonal IMFs from the original IMFs. The process to obtain the orthogonal IMFs is defined as improved EMD (IEMD). There are three different ways to get the orthogonal IMFs

Forward sequence orthogonalization
The original IMFs obtained from Eq.(1) are sequenced from high frequency to low frequency. In Gram-Schmidt orthogonalization, the sequence order is not changed. The computing steps of the orthogonalization are introduced as follows [5] .
(1) The first basic orthogonal function is taken as the first original IMF, that is (2) The second basic orthogonal function is formed from the second original IMF (3) Recurrent formula for solving high order basic orthogonal functions, Utilizing the orthogonality of the first j basic orthogonal functions and equation when k=i, we can obtain is the j th orthogonal IMF that are sequenced from high frequency to low frequency.

Backward sequence orthogonalization
In Gram-Schmidt orthogonalization, the sequence order is changed reversely. It means that the computing steps are modified as follows.
(1) The first basic orthogonal function is taken as the last original IMF, that is (2) The second basic orthogonal function is formed from the last second original IMF Obviously, if the work of rearranging reversely the sequence order of the original IMFs is finished at first, the orthogonal computations can be completed by Eqs. (6)-(12).

Arbitrary sequence orthogonalization
In Gram-Schmidt orthogonalization, the sequence order is changed arbitrarily. It means that any one of these n orthogonal IMFs can be selected as the first basic orthogonal function and the sequence order of these n original IMFs can be rearranged if necessary. After the sequence order rearrangement, the computing steps of the Gram-Schmidt orthogonalization are the same introduced in section 2.1. The difference is that the orthogonal IMFs * j c t are not sequenced from low frequency to high frequency or from high frequency to low frequency.

Numerical example
The signal X t shown in Eq.(4) is continuously used as the example. The comprisons of the numerical results obtained from different orthogonal methods are shown as follows. In arbitrary sequence method, the 3 th original IMF is selected as the first basic orthogonal IMF. The following is the 2 nd , the 1 st , the 4 th and 5 th IMF.
It is obvious that the orthogonal level of the IMFs has been greatly improved when the orthogonal technique is applied to the original IMFs.

Importance of orthogonality
The orthogonality of the IMFs has an important effect on its applications. It is illustrated by a simple example. If the signal X t is a seismic wave, its vibrating energy can be computed by The energy of each component in the seismic wave can be expressed