# Optimal Waveform of the Partial-Respond Signal Based on Minimum Out-of-Band Radiation Criterion
## 2 Optimized Mathematical Modeling of the Optimized Waveform
### Signal Solving Process Using Fourier Series  
A periodic function in the interval \([-2T,2T]\) may be expressed by Fourier series expansion [13]. Therefore, the efficient spectrum signal \(a(t)\)--as an even function with the width of \(4T\)--can be written as  
\[a(t)=\frac{a_{0}}{2}+\sum\limits_{k=1}^{m}\left(a_{k}\cos(\frac{\pi}{2T}kt)\right) \tag{20}\]  
where  
\[a_{0}=(1/2T)\int\limits_{2T}^{2T}a(t)dt\]  
\[a_{k}=(1/2T)\int\limits_{-2T}^{2T}a(t)\cos(\pi kt/2T)dt\]  
Then, substituting Equation (20) into Equation (19), yielding  
\[H = 2T\sum\limits_{k=1}^{m}a_{k}^{2}(\frac{\pi}{2T}k)^{2n}+\lambda \{\frac{E}{2T}-[2\sum\limits_{k=1}^{m}\left(-1\right)^{k+1}a_{k}]^{2}-\sum \limits_{k=1}^{m}a_{k}^{2}\} \tag{21}\] \[+ \mu\int\limits_{0}^{2T}[a(t)\cdot a(t-2T)-K/2T]dt+\gamma\int \limits_{0}^{T}((I_{11}^{2}(t)+Q_{21}^{2}(t)-a^{2}(0))^{2}\] \[+ (I_{12}^{2}(t)+Q_{22}^{2}(t)-a^{2}(0))^{2}+(I_{13}^{2}(t)+Q_{23}^{ 2}(t)-a^{2}(0))^{2}\] \[+ (I_{14}^{2}(t)+Q_{24}^{2}(t)-a^{2}(0))^{2})dt\]Based on the extreme value theory of the variational calculus [14], Equation (21) should satisfy the following partial differential equations  
\[\frac{\partial H}{\partial a_{k}} = 0;k=1,2,\ldots,m\] \[\frac{\partial H}{\partial\lambda} = 0,\frac{\partial H}{\partial\mu}=0,\frac{\partial H}{\partial\gamma }=0 \tag{22}\]  
The normalizations are defined as \(E=1\) and \(T=1\). Thus, the coefficients, \(a_{0}\), \(a_{1}\), \(\ldots\), \(a_{m}\) of the Fourier series, and the Lagrange constants, \(\lambda\) and \(\mu\), are the unknown parameters. The solution of simultaneous equations is used to determine the values of the unknown coefficients, \(a_{0}\), \(a_{1}\), \(\ldots\), \(a_{m}\) of the Fourier series. We introduce envelope fluctuation coefficient to determine the constant envelope properties. The envelope fluctuation coefficient is defined as  
\[\varepsilon=\frac{e_{\max}-e_{\rm ave}}{e_{\rm ave}} \tag{23}\]  
where \(e_{\max}\) represents the maximum value of the envelope and \(e_{\rm ave}\) denotes the average value of the envelope. Table 2 shows the coefficient of the Fourier series for different \(m\) and \(n\). Figure 4 shows the corresponding time domain curves. Figure 5 shows the normalized power spectra curves of optimized signal, where curve 1 corresponds to the MSK and curves 2-5 correspond to \([n=1,m=4]\), \([n=3,m=5]\), \([n=5,m=6]\), and \([n=7,m=7]\), respectively. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(n\) & \(m\) & \(a_{0}\) & \(a_{1}\) & \(a_{2}\) & \(a_{3}\) & \(a_{4}\) & \(a_{5}\) & \(a_{6}\) & \(a_{7}\) \\ \hline
1 & 4 & 0.660 & 0.5000 & 0.1793 & \(-0.001865\) & \(-0.01111\) & - & - & - \\
3 & 5 & 0.6364 & 0.4998 & 0.2152 & 0.01092 & \(-0.03402\) & \(-0.01126\) & - & - \\
5 & 6 & 0.6280 & 0.4992 & 0.2266 & 0.02026 & \(-0.03790\) & \(-0.02006\) & \(-0.003333\) & - \\
7 & 7 & 0.6310 & 0.4995 & 0.2224 & 0.01635 & \(-0.03726\) & \(-0.01671\) & \(-0.001131\) & 0.0004888 \\ \hline \end{tabular}
\end{table} 
Table 2: The unknown coefficient of the Fourier series in Equation (20). Figure 4: Waveform of the optimized baseband signals. Table 3 considers different pairs of \(n\) and \(m\), and presents the bandwidth, where the normalized power spectra of the optimized signals decreased as \(-40\) dB (\(\Delta F_{-40\text{dB}}\)) and \(-60\) dB (\(\Delta F_{-60\text{dB}}\)). Table 3 also lists the value of the bandwidth, where \(90\%\) and \(99\%\) of the signal energy is achieved (i.e., \(\Delta F_{90\%}\) and \(\Delta F_{99\%}\), respectively). Besides, the corresponding envelope fluctuation coefficient \(\varepsilon\) and the cross-correlation function \(K\) of the optimized signal are presented in the last two columns of Table 3.  
Figure 5 and Table 3 reveal that for the minimum out-of-band energy radiation criterion with the restrictions, the optimized efficient spectrum signal presents the bandwidth of \(1.09/T\) (for \(n=~{}1\) and \(m=4\), calculated according to \(-40\) dB) and \(1.21/T\) (for \(n=1\) and \(m=4\), calculated according to \(-60\) dB).