## 2 Optimization Problem with Constraint on PAPR  
Consider the representation of a random sequence of signals with offset quadrature phase shift keying (OQPSK). The transmitted binary message consisting of \(N\) characters with the symbol rate \(R=1/T\) can be expressed as:  
\[u(t)=\sum_{k=0}^{N-1}\psi_{T}(t-kT)d_{r}^{(k)}, \tag{1}\]  
where \(d_{r}^{(k)}\) denotes a channel alphabet symbol and \(\psi_{T}(t)\) denotes a rectangular pulse shape:  
\[d_{r}^{(k)}= \begin{cases}1,&r=1\\ -1,&r=2\end{cases} ,\ \psi_{T}(t)= \begin{cases}1,&t\in[0,T]\\ 0,&t\
otin[0,T]\end{cases} . \tag{2}\]  
A signal sequence with OQPSK having amplitude \(A_{0}\), carrier frequency \(f_{0}\), and an arbitrary pulse shape \(a(t)\) can be expressed as:  
\[y(t)=y_{1}(t)-y_{2}(t), \tag{3}\]  
\[y_{1}(t)=(A_{0}/\sqrt{2})u_{1}(t)\cos(2\pi f_{0}t+\phi_{0}),\ y_{2}(t)=(A_{0}/ \sqrt{2})u_{2}(t)\sin(2\pi f_{0}t+\phi_{0}), \tag{4}\]  
\[u_{1}(t)=\sum_{k=0}^{(N-1)/2}a(t-2 kT)d_{r}^{(2k)},\ u_{2}(t)=\sum_{k=0}^{(N-1)/ 2}a(t-2 kT-T)d_{r}^{(2k+1)}. \tag{5}\]  
To simplify calculations, the initial phase is set to \(\phi_{0}=\pi/4\). Note that the sequence (1) can be divided into two quadrature streams consisting of even and odd symbols transmitted at the rate of \(R=1/2T\). The shape of function \(a(t)\) in time interval \(T_{s}=LT\) (\(L=1,\,2,\,\dots\)) is determined as a solution to the optimization problem [8; 19]. The main requirement of the optimal signal generation is the requirement for the maximum spectral filling of the allocated frequency band. This requirement is met by using the optimization criterion of ensuring the specified reduction rate of out-of-band emissions level. Solutions to this problem are considered in [20] for signals with binary phase shift keying (BPSK). The natural requirement for the cross-correlation coefficient (the condition of the guaranteed level of the BER performance), the pulse duration, and signal energy represent the optimization problem constraints. Consider the application of such signals with OQPSK and constraint on PAPR. The optimization criterion of ensuring the specified reduction rate of the level of out-of-band emissions is adopted. The optimization function can be expressed as [19; 20]:  
\[\arg\{\min_{a(t)}J\},\ J=\int_{-\infty}^{\infty}g(f)G(f)df,G(f)=\left|\int_{- \infty}^{\infty}a(t)\exp(-j2\pi ft)dt\right|^{2}. \tag{6}\]In (6), the symmetrical weighting function \(g(f)=f^{2n}\) (\(n=1,2,3,\dots\)), together with the boundary conditions that are given below, determines the reduction rate of the energy spectrum level of the synthesized signals. The shapes of function \(g(f)\) are presented in Figure 1, where it can be seen that, by adjusting the value of \(n\), it is possible to achieve the required reduction rate of the energy spectrum level. With a proper choice of the energy spectrum parameters of synthesized signals, the occupied frequency band \(\Delta F\) can be determined based on the criterion of the energy spectrum level \(G(f)\) relative to the maximum value of the energy spectrum. For instance, \(\Delta F_{-30\text{ dB}}\) corresponds to the occupied bandwidth for the power spectrum level of \(-30\text{ dB}\). The analytical solution to the optimization problem (6) is possible only for the parameters \(n=1\) and the pulse duration equal to \(T\)[21]. In [22], the analytical solution to more complex optimization cases with quadrature signals and \(T_{s}=4T\) is brought to the Lagrange equations, which can be solved numerically. When the number of optimization constraints or the values of optimization parameters (i.e., \(n\), \(T_{s}\)) increase, the analytical closed-form solution cannot be obtained. To solve (6), numerical methods for solving the Lagrange equations can be applied [23]. The desired pulse shape \(a(t)\) is represented as the expansion in a limited Fourier series, where \(m\) denotes the number of expansion coefficients.