## 2 Optimization Problem with Constraint on PAPR
### Optimization Problem-Solving Method  
The procedure of numerical optimization problem solving was carried out in Matlab environment with the help of Optimization Toolbox. The functional has a ravine shape due to the presence of constraints (10) and (13). Functional (6) needs to be found for five optimization parameters: pulse duration \(T_{s}\), reduction rate of out-of-band emissions \(1/\omega^{2(n+1)}\), cross-correlation coefficient \(K_{0}\), maximum peak-to-average power ratio max{PAPR}, and packet length \(N_{p}\). The packet length is taken into account in the constraint on PAPR. The optimization problem-solving method is as follows. The optimization procedure is divided into two parts. In the first part, only three parameters (\(T_{s}\), \(n\), and \(K_{0}\)) are considered. In the second part, their values are fixed, and the five-dimensional space is transformed into the two-dimensional space with parameters max{PAPR} and \(N_{p}\).  
The first part of the optimization procedure is presented in Figure 2. At the start, a rectangular pulse with duration \(T_{s}=T\) is used. For this signal, it holds that \(n=0\), and the cross-correlation coefficient is \(K_{0}=0\) at the symbol rate \(R=1/T\). At the first stage, the optimal pulse shape with a duration \(T_{s}=2T\) has to be found. An increase in the duration leads to the appearance of ISI and to an increase in the correlation coefficient to a certain value that does not exceed 0.5. The solution obtained at each stage is used as the initial value at the next stage. At the second stage, the required reduction rate of the out-of-band emissions is achieved, for example \(1/\omega^{6}\) at \(n=2\). At the third stage, a constraint on the cross-correlation coefficient \(K_{0}<0.1\) is introduced. Thus, after the third stage, the optimal pulse shape with a duration \(T_{s}=2T\), a reduction rate of \(1/\omega^{6}\) and correlation coefficient less than or equal to 0.1 is obtained. To obtain a pulse of a longer duration for a given correlation coefficient, the first and third stages are repeated. After completing the first part of the optimization, a pulse of duration \(T_{s}\) = 4\(T\) with \(n\) = 2 and \(K_{0}\) = 0.1 is obtained.  
In the second part of the optimization procedure, the limiting parameters max{PAPR} and \(N_{p}\) are considered. By setting \(T_{s}\) = 4\(T\), \(n\) = 2, and \(K_{0}\) = 0.1, a plane is obtained. In Figure 2, different arrows correspond to different packet lengths \(N_{p}\) used in the constraint on PAPR (13).  
The iterative process of minimizing function (8) results in the expansion of coefficients \(a_{k}\) into a limited Fourier series, where \(k\) = 0...M. The surfaces of function (8) for several coefficients \(a_{k}\) are presented in Figure 3.  
In Figure 3, the signal parameters are set as follows: pulse duration \(T_{s}\) = 4\(T\), packet length \(N_{p}\) = 5, and the maximum value of PAPR reduction \(\Delta\)maxPAPR = 0.5 dB. Figure 3 shows two-dimensional plots of the local minima of the function depending on the pair of expansion coefficients: (_a_) \(a_{2}\), \(a_{3}\); (_b_) \(a_{5}\), \(a_{6}\).  
Figure 2: The optimization problem-solving procedure.  
Using the proposed technique for the optimization problem solving, we consider the results of optimizing the shape of the function \(a(t)\) when using OQPSK in (3) with the constraint (13).