# Optimal Waveform of the Partial-Respond Signal Based on Minimum Out-of-Band Radiation Criterion
## 2 Optimized Mathematical Modeling of the Optimized Waveform
### Additional Constraints of the Pan-Function
#### 2.2.4 Restriction on the Minimum Cross-Correlation Function  
The intersymbol interference (ISI) lowers the correlation performance in a partial-response system, increasing the bit error rate (BER). In order to improve the receiving reliability, additional restrictions are introduced, including single-symbol energy, boundary condition, constant-envelope, and minimum cross-correlation function. Three random baseband symbols and eight types of waveforms exist in the interval \([-2T,2T]\), in orthogonal or in-phase channel. Eight different waveforms yield 36 combinations in Euclidean distances. We may express the minimum Euclidean distance between each two waveforms as  
\[d_{\min}^{2}=\min{\left\{{d_{\left\{{a_{i},{j}}\right\}}^{2}}\right\}}=\min_{ i,j}\int\limits_{-2T}^{2T}{\left({a_{i}\left(t\right)-a_{j}\left(t\right)} \right)^{2}}dt=\min{\left\{{d_{i,j}^{2}}\right\}} \tag{15}\]  
where \(a_{i}\left(t\right)\) is the waveform in the interval \([-2T,2T]\), \(i=1,2,\ldots,8\). Figure 2 shows all possible composed modes of three random baseband symbols in the interval \([-2T,2T]\). In contrast, Figure 3 plots the synthetic waveform of three random baseband symbols in the interval \([-2T,2T]\). Table 1 lists the Euclidean distance between two waveforms.  
\[d^{2}{}_{i,j} = \int\limits_{-2T}^{2T}{(a_{i}\left(t\right)-a_{j}\left(t\right))^{2}}dt\] \[K = \int\limits_{-2T}^{2T}{a(t)\cdot a(t-2T)}dt\] \[E = \int\limits_{-2T}^{2T}{a^{2}(t)}dt \tag{16}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(d_{ij}\) & \(j=1\) & \(j=2\) & \(j=3\) & \(j=4\) & \(j=5\) & \(j=6\) & \(j=7\) & \(j=8\) \\ \hline \(i=1\) & 0 & \(2E\) & \(4E\) & \(6E+8K\) & \(2E\) & \(4E\) & \(6E+8K\) & \(8E+8K\) \\ \(i=2\) & - & 0 & \(6E-8K\) & \(4E\) & \(4E\) & \(2E\) & \(8E\) & \(6E+8K\) \\ \(i=3\) & - & - & 0 & \(2E\) & \(6E-8K\) & \(8E-16K\) & \(2E\) & \(4E\) \\ \(i=4\) & - & - & - & 0 & \(8E\) & \(6E-8K\) & \(4E\) & \(2E\) \\ \(i=5\) & - & - & - & - & 0 & \(2E\) & \(4E\) & \(6E+8K\) \\ \(i=6\) & - & - & - & - & - & 0 & \(6E-8K\) & \(4E\) \\ \(i=7\) & - & - & - & - & - & - & 0 & \(2E\) \\ \(i=8\) & - & - & - & - & - & - & - & 0 \\ \hline \end{tabular}
\end{table}
Table 1: Euclidean distance between any two waveforms.  
Figure 3: Synthetic waveforms of three bits random baseband symbols in the interval \([-2T,2T]\).  
Figure 2: Composed modes of three random baseband symbols in the interval \([-2T,2T]\).  
We obviously see that decreasing \(K\) leads to increasing the minimum Euclidean distance, for a constant symbol energy \(E\). Therefore, the minimum cross-correlation function \(K\) may restrict the improvement of the BER performance. The autocorrelation function in \([0,2T]\) can be defined as  
\[K=\int\limits_{0}^{2T}a\left(t\right)\cdot a\left(t-2T\right)dt \tag{17}\]  
we have  
\[\int\limits_{0}^{2T}\left(a\left(t\right)\cdot a\left(t-2T\right)-K/2T\right) dt=0 \tag{18}\]  
Introducing the Fourier series and additional restrictions, single-symbol energy, constant envelope condition after modulation and minimum cross-correlation function, the Pan-function can be expressed as  
\[H = J+\lambda\left(\int\limits_{-2T}^{2T}a^{2}\left(t\right)dt-E \right)+\mu\int\limits_{0}^{2T}[a(t)\cdot a(t-2T)-K/2T]dt \tag{19}\] \[+ \gamma\int\limits_{0}^{T}\{[I_{11}^{2}(t)+Q_{21}^{2}(t)-a^{2}(0) ]^{2}+[I_{12}^{2}(t)+Q_{22}^{2}(t)-a^{2}(0)]^{2}\] \[+ [I_{13}^{2}(t)+Q_{23}^{2}(t)-a^{2}(0)]^{2}+[I_{14}^{2}(t)+Q_{24}^{ 2}(t)-a^{2}(0)]^{2}\}dt\]  
where \(\lambda\),\(\mu\), and \(\gamma\) are Lagrange constants. \(H\) is the Lagrange function.