Derivation of Wavelet Soft Threshold_Soft Threshold Calculation

The basic method of wavelet decomposition is to use the Mallat tower algorithm to reduce the decomposition of the signal f(x). The decomposition process is shown in Figure 1 (generally C0=f(x)).

Figure 1 Schematic diagram of wavelet decomposition algorithm

The wavelet decomposition algorithm decomposes the signal into the profile component Ci and the detail component Di at each scale i. In the higher-order wavelet decomposition, the profile component Ci of the previous stage is decomposed into the profile component Ci with lower frequency component. +1 and detail component Di+1. The profile component Ci mainly contains low frequency components in the signal; the detail component Di contains only the high frequency portion of the signal, which also includes high frequency noise. As mentioned above, the wavelet transform coefficient of noise decreases with the increase of the scale, and the wavelet coefficient of the continuous signal increases with the increase of the scale. In this way, a threshold value can be set, and the threshold value is used to perform threshold adjustment on the wavelet coefficients according to a certain rule. The wavelet coefficients of the wavelet coefficients after threshold adjustment are reconstructed by the wavelet transform inversion algorithm to obtain the denoised signal.

Due to the randomness of the noise signal strength and the propagation characteristics of the signal and noise during the wavelet decomposition process, the threshold used by each layer of the wavelet decomposition coefficient should be changed with the change of the wavelet coefficient. The method that can achieve this variation threshold is the soft threshold denoising method.

Let the detected signal be:

There are many ways to calculate soft thresholds. By comparison, the soft threshold calculation in this paper uses the Hein unbiased risk threshold calculation method of heuristic method, which is calculated based on the general threshold and the unbiased risk threshold. The specific calculation principle and calculation steps are as follows.

The noise intensity can be calculated using the following formula:

Where: Dki is the k-th wavelet coefficient (1 "k" M); N is the number of wavelet coefficients of the layer; M is the highest number of wavelet decomposition (see Figure 1).

The theoretical basis for the general threshold calculation is that the probability that the maximum of the N standard Gaussian variables with independent and identical distributions is less than T1 tends to 1 as N increases. Among them, T1 is calculated by the following formula:

Where: R is the noise intensity; N is the number of variables processed.

The square of a certain layer of wavelet coefficients is arranged from small to large to obtain a vector: W = [w1, w2, ..., wN], where w1 ≤ w2 ≤ ... ≤ wN, where N is the number of wavelet coefficients. From this, the risk vector R = [r1, r2, ..., rN] is calculated, where:

Taking the minimum value rb of the R element as the risk value and finding the corresponding wb from the subscript variable b of rb, the threshold value T2 is:

The final soft threshold is T3.

The significance of the soft threshold algorithm is not only to achieve signal denoising, detection, but also contribute to data compression. For example, when the sampling frequency is 6400 Hz, C0 has 128 data in each fundamental period, and after wavelet decomposition, C1 and D1 each have 64 data; and after soft threshold denoising, there are only 6 in D1. The value of ~15 points is non-zero. In this way, the data is greatly compressed, and the amount of data storage and transmission efficiency is improved.

Power Storage LiFePO4 Battery Solar Energy Systems For Home

Energy Storage Lithium Battery Cbinet For Home

Jiangsu Zhitai New Energy Technology Co.,Ltd , https://www.zhitainewenergy.com