I am trying to understand this paper says.

I have found this paragraph to be vague:

(a) Preprocessing Stage. The first step in this stage was to apply the Discrete Wavelet Transform (DWT). This tool is based on the decomposition of a signal in subbands by means of the use of a pair of digital filters (low pass and high pass filters). The outputs of the low pass filter are named approximation coefficients (), while the outputs of the high pass filter are named detail coefficients (), where represents the level of the subband. This process of decomposition through filtering is repeated times. In each iteration the signal is subsampled by a factor of 2. In practice, the DWT is implemented with the Mallat pyramid algorithm [28]. Some studies have shown that the use of a four order Daubechies wavelet is one of the most effective when processing ECG signals [29]. Afterwards, the energy percentage of each level was calculated (see (1)-(3) and Figure 3), then the four ones with more energy were selected to reconstruct the signal (Figure 4). In this way it was assured that the levels with more information of the ECG signal were selected because noise or some interferences such as those of the electrical network or the artifacts are usually found at low energy levels (high frequencies, generally between and ) (see Figure 3). Therefore, using energy levels to discriminate the noise of QRS complex was a good option. Finally we proceeded to remove the offset of the signal by leaving out the approximation coefficients (). In the present study a db4 mother wavelet with 7 levels of decomposition was selected. The number of levels was selected because the data were processed in buffers of 1024 and thus the number of iterations allowed was 7.

Specially this part:

Afterwards, the energy percentage of each level was calculated (see (1)-(3) and Figure 3), then the four ones with more energy were selected to reconstruct the signal (Figure 4).

the energy percentage of each level was calculated - the energy of what, of the low frequency coefficients or the high frequency coefficients or both? I guess the high coefficients, or detail, right?

then the four ones with more energy were selected to reconstruct the signal

how?

On the text the researcher says the signal was decomposed in seven levels, so, he have obtained seven low frequency coefficients and seven high frequency coefficients.

He says he have used detail level 4, 5, 6, and 7 but he mentions these numbers in the following order: 5, 4, 6, and 7. Why?

What is he talking about? Reconstructing the signal just using 4 levels of IDWT? If he does that the reconstructed signal will have 3 levels missing, or in other words, it will be 1/8 of the original frequency. I the signal is being sampled at 30 Hz, the final reconstruction will have 3.75Hz.

Is this what the paper is saying? and why have the researcher mention the levels out of order?

Basically, the method decomposes a signal into one approximation $$A_7$$ and seven details $$D_7$$,..., $$D_1$$. Since the wavelet is orthogonal, the energy of the signal is equal to the sum of the energy of each subband. To me, equation (1) is not correct, one should not sum approximations over levels. Approximation coefficients $$A_7$$ are discarded, they kind of remove the low frequency or baseline parts. They keep the four most energetic detail subbands, which they order by decreasing energy. Then they set the others to zero, and reconstruct the signal by inverse wavelet transform. So no level is missing, they are just erased. This turns out to be a very crude form of denoising, as they finally only keep the central frequencies, in the interval $$[1/2^7\; 1/2^3]$$. This could be a misuse of wavelets, as a simple bandpass filter could have been better.