## 2 Theoretical Considerations of the Digital Filter
### Creating the aIMF Algorithm  
[PERSON] and [PERSON] [17] explained that a meaningful instantaneous frequency (IF) decomposed by the EMD algorithm must be nearly monochromatic, of which is a condition that is not guaranteed by the algorithm and fails to be met clearly when the signals is corrupted by noise. Several reports demonstrated that the EMD performance is likely to be sensitive and produces a large quantity of noise [12, 18]. Moreover, the accuracy of HHT analysis suffers from several mathematical and numerical effects that require further studies. This is mainly because HHT signals emerged from EMD algorithm are not shift-invariant in times (stationary) and is likely be full narrow band [18]. To reduce noise, one of the solutions was to normalise the analytic signal prior to Hilbert transform [8, 19]. There were many studies to improve the HHT algorithms, and those at least are; adoption of wavelet packet transform as the pre-processes to decompose the signals into a set of narrow-band signals prior to the application of the EMD [20, 21], and an ensemble empirical mode decomposition (EEMD) which consists of shifting an ensemble of white-added noise signals and treats the mean as a final true result, using a windowed average. One of the publications of the EEMD was used to add noise to provide stoppage criteria for the EMD process [9]. [PERSON] [22] mentioned that Fourier spectral analysis and its derivatives such as WT encountered with a limited time window width by sliding a predetermined window(s) along the time axis. Moreover, there is a trade-off between the time consumed in the window width and the frequency resolution, and this phenomenon has been considered by the uncertainty principle Heisenberg [21]. In the WT, the window width must be preselected and it is known as the primary or mother wavelet which is a prototype that provides less flexibility when handling datasets where the mean and variances are highly volatile.  
It is imperative to restate that two conditions of EMD process defined in Section 2.1. However, with more iteration, the more residual \(r_{n}(t)\) becomes either over-distorted or a monochromatic function from which no further IMF can be decomposed [11]. The other study confirmed that the real complex quantity of all IMFs decomposed _i.e._, IMF1 to IMF7, which are in the amplitude-frequency domain, shifted to the lower region [19].  
To reduce noise produced in the EMD process, we propose to extract noise spreading to all of the spectrums by averaging all IMFs and subtract them from the original signals in which is represented by  
\[c_{a}(t)=\frac{\sum_{i=1}^{n}c_{i}(t)}{n} \tag{16}\]  
Next, we subtract the averaged IMF in Equation (16), \(c_{a}(t)\), from the original nonlinear, nonstationary times series datasets, \(x_{n}(t)\). Thus, a new digital filter is created, which is given by  
\[\text{aIMF}(t)=y_{n}(t)=x_{i}(t)-c_{a}(t) \tag{17}\]  
where \(y_{n}(t)\) is a function of the aIMF algorithm.  
In Figure 2, the aIMF process starts by inputting the nonlinear, nonstationary times series data, Euro and US dollars (EUR-USD), exchange rates into the EMD process. The next block uses the results from the EMD process whereas the aIMF algorithm begins in the third block where all of the IMFs are averaged and the averaged IMFs are subtracted from the original datasets.  
Referring to Figure 2, the proposed algorithm aIMF has an advantage which provides stoppage criteria once the two conditions of the EMD process defined in Section 2.1 are accomplished. This is because with more iteration, the more residual \(r_{n}(t)\) becomes either over-distorted or a monochromatic function from which no further IMF can be decomposed.