# 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
#### 2.2.3 Restriction on the Constant-Envelope Condition of the Quadrature-Modulated Signal  
The constant-envelope modulation represents that the envelope of the modulated wave should be constant, the generated modulation signal is band-limited by the transmitter, and only a small spectrum spreading is produced after the signal passes through the nonlinear processing components [12]. Thus, the following conditions should be satisfied with \(t\) in the interval \([0,4T]\)  
\[I^{2}(t)+Q^{2}(t)=const \tag{5}\]  
Since the signal is symmetric, we only need to satisfy Equation (5) for \(t\) in the interval \([0,T]\). Then, the condition will be satisfied in the rest of the initial interval; i.e., \([T,4T]\). In this case, the four symbols in the interval are defined as \(d_{-1},\ d_{0},\ d_{1},\ d_{2}\). The baseband signal of \(I(t)\) and \(Q(t)\) in the interval \([0,T]\) can be written as  
\[I(t)=d_{-1}a(t+T)+d_{1}a(t-T) \tag{6}\]  
\[Q(t)=d_{0}a(t)+d_{2}a(t-2T) \tag{7}\]  
where \(d_{-1}\), \(d_{0}\), \(d_{1}\), \(d_{2}\) are the uncorrelated binary data symbols, each taking one of the values \(\pm 1\). The different conditions are as follows: For \(d_{-1}d_{1}=1\), the adjacent baseband symbol signals take the same sign in the \(I\) channel.  
If \(d_{0}d_{2}=1\), the adjacent baseband symbol signals take the same sign in the \(Q\) channel, and to meet the constant-envelope condition, we should introduce the attenuation factor \(r\) in \([0,1]\), yielding  
\[I_{11}^{2}(t)=r^{2}a^{2}(t-T)+2 ra(t+T)a(t-T)+a^{2}(t+T)\] \[Q_{21}^{2}(t)=r^{2}a^{2}(t)+a^{2}(t-2T)+2 ra(t)a(t-2T) \tag{8}\]  
If \(d_{0}d_{2}=-1\), the adjacent baseband symbol signals take opposite signs in the \(Q\) channel, we have  
\[I_{12}^{2}(t)=a^{2}(t-T)+2a(t+T)a(t-T)+a^{2}(t+T)\] \[Q_{22}^{2}(t)=r^{2}a^{2}(t)+a^{2}(t-2T)-2 ra(t)a(t-2T) \tag{9}\]The constant value of the envelope is set to be \(a(0)\), which is the value of \(a(t)\) at time \(t=0\), and is also the maximum value of the signal \(a(t)\). To ensure that the modulated signal always has a constant envelope in the interval \([0,T]\), the constant-envelope condition is rewritten as  
\[\int\limits_{0}^{T}{\left({I_{11}^{2}(t)+Q_{21}^{2}(t)-a^{2}(0)}\right)^{2}}dt=0 \tag{10}\]  
\[\int\limits_{0}^{T}{\left({I_{12}^{2}(t)+Q_{22}^{2}(t)-a^{2}(0)}\right)^{2}}dt=0 \tag{11}\]  
Similarly, if \(d_{-1}d_{1}=1\), the adjacent baseband symbol signals take the opposite signs in the \(I\) channel. The constant envelope conditions may be obtained as  
\[\int\limits_{0}^{T}{\left({I_{13}^{2}(t)+Q_{23}^{2}(t)-a^{2}(0)}\right)^{2}}dt=0 \tag{12}\]  
\[\int\limits_{0}^{T}{\left({I_{14}^{2}(t)+Q_{24}^{2}(t)-a^{2}(0)}\right)^{2}}dt=0 \tag{13}\]  
Thus, the constant-envelope conditions in the interval \([0,T]\) can be expressed as follows:  
\[\int\limits_{0}^{T}{\left\{{\begin{array}{*{20}{c}}{\left[{I_{11}^{2}(t)+Q_{2 1}^{2}(t)-a^{2}(0)}\right]^{2}+\left[{I_{12}^{2}(t)+Q_{22}^{2}(t)-a^{2}(0)} \right]^{2}}\\ {+\left[{I_{13}^{2}(t)+Q_{23}^{2}(t)-a^{2}(0)}\right]^{2}+\left[{I_{14}^{2}(t)+ Q_{24}^{2}(t)-a^{2}(0)}\right]^{2}}\end{array}}\right\}dt=0 \tag{14}\]