# Low-Pass Filtering Method for Poisson Data Time Series
## 2 Method
### Quasi-Gaussian Digital Low-Pass Filter
#### 2.1.2 Synthesis Requirements  
The problem of synthesis of the supposed FIR filter is solved based on the approximation of the FR function \(H_{0g}(w_{l},w_{c})\) (6) in discrete points \(w_{l},\ l=1,\ldots,L\) with a FR function \(H_{g}(w_{l},0)\) according to (5). A functional \(S(H_{0g},a,w_{c})\) is formed:  
\[S(H_{0g},a,w_{c})=\sum_{l=1}^{L}[(\sum_{s=0}^{s_{0}}a_{s}C_{s}(w_{l}))^{2}+( \sum_{s=0}^{s_{0}}a_{s}S_{s}(w_{l}))^{2}-H_{0g}^{2}(w_{l},w_{c})]^{2}. \tag{7}\]  
Obviously, the FR (5) represents a function which is polyharmonic in frequency \(w_{l}\). In case the prototype filter FR frequency derivative has discontinuities or is subject to strong alternations, e.g., if FR is a trapezoidal function, then the FR of the synthesized FIR filter, obtained based on approximation, will contain fluctuations due to the so-called Gibbs effect. Elimination and reduction of these fluctuations are usually achieved by choosing a suitable smooth prototype filter FR function. The smoothness requirement is largely satisfied by the Gaussian function (6).It should be noted that the Gaussian function is naturally suitable for the FR of a low-pass filter, since its values (6) practically differ from zero only in the region of low frequencies.  
The requirements listed in the Introduction lead to formalized requirements:* Ensuring that the gain at zero frequencies is equal to one: \[1=H(0,a)=\sum_{s=0}^{s_{0}}a_{s},a\in A_{1},\ \ A_{1}=\{a:(1=\sum_{s=0}^{s_{0}}a_{s})\};\] (8)
* Ensuring non-negativity of coefficients: \[a\in A_{0},\ \ A_{0}=\{a:(0\leq a_{s},s=0,1,\ldots,s_{0})\};\] (9)  
For the synthesis procedure, it is assumed to set a small value \(r_{0}\), based on the a priori known duration of fluctuations, and some small cutoff frequency value \(w_{c}\) for a prototype filter. The quasi-Gaussian filter synthesis procedure, consisting of finding the optimal coefficients \(a_{s}^{\circ}\), \(s=0,1,\ldots,s_{0}\), taking into account the requirements \(\mathbf{a},\mathbf{b}\), Equations (8) and (9) the predefined \(r_{0}\)and \(w_{c}\), is performed on the basis of the approximation problem, which reduces to the implementation of conditional minimization:  
\[a^{\circ}(w_{c})=\arg\{\min_{a\in A_{0},a\in A_{1}}S(H_{0g},a,w_{c})\}. \tag{10}\]  
For a given small dimension \(r_{0}\) of the synthesized quasi-Gaussian filter and a given small cutoff frequency \(w_{c}\) for a prototype filter, the value for cutoff frequency to be found for a quasi-Gaussian filter is \(w_{cg}\), and the filter FR for the frequencies \(w_{l}\) is denoted as \(H_{g}(w_{l},w_{cg},a^{\circ})\), \(l=1,\ldots,L\).  
The minimization of (10) could be performed based on modified direct zero-order optimization methods, taking into account the restrictions (8) and (9). However, because the (7) functional is multi-extremal, traditional modified direct methods, for example, using the coordinate descent method, the Hook-Jeeves method, the random descent method, etc. [13] do not provide successful minimization. The listed methods, as a rule, lead to \"getting stuck\" with search procedures in local minima.