# Joint Radar and Communications Waveform Design Based on Complementary Sequence Sets
## 3 JRC-CWG and DR-JRC-CWG Design
### Dr-JRC-CWG Design
#### 3.2.1 Doppler Sensitivity Analysis  
According to (3), the sum of matched filter outputs of the \(g\)-th JRC-CWG with Doppler shift \(f_{d}\) is  
\[y_{g}^{\tau}(n,f_{d})=\delta_{g}\rho\sum\limits_{m=1}^{M}e^{j2\pi(m-1)f_{d}T}{ \bf g}_{g,m}^{H}{\bf U}_{n}{\bf s}_{g,m}+\sum\limits_{m=1}^{M}{\bf n\cdot}_{g,m} \tag{17}\]  
As shown in (17), the Doppler frequency \(f_{d}\) will affect the range sidelobe level of the JRC-CWG. It is necessary to design the DR-JRC-CWG when \(f_{d}\) cannot be accurately acquired.  
According to (17), the AF of the \(g\)-th JRC-CWG is defined as  
\[\begin{array}{c}y_{g}(n,\varphi_{d})=\sum\limits_{m=1}^{M}e^{j2\pi(m-1) \varphi_{d}}{\bf s}_{g,m}^{H}{\bf U}_{n}{\bf s}_{g,m},\\ \mbox{for }n=0,1,\ldots,N-1\end{array} \tag{18}\]  
and \(y_{g}(n,\varphi_{d})=y_{g}^{*}(-n,\varphi_{d}),\mbox{ for }n=1-N,2-N,\ldots,-1\), where \(\varphi_{d}=f_{d}T\) is the normalized Doppler frequency.  
Next, the AF of the \(g\)-th JRC-CWG is analyzed using the Taylor formula, and we will show how the nonzero-order Taylor expansion terms affect the Doppler resilience of the JRC-CWG. The Taylor expansion of the AF of the \(g\)-th JRC-CWG around \(\varphi_{d}=0\) can be represented as  
\[y_{g}(n,\varphi_{d})=\sum\limits_{q=0}^{\infty}\frac{\sum\limits_{m=1}^{M}(m- 1)^{q}{\bf s}_{g,m}^{H}{\bf U}_{n}{\bf s}_{g,m}}{q!}(j2\pi\varphi_{d})^{q} \tag{19}\]  
Introduce \(\beta_{g,q}=\left[\beta_{g,q}(1-N),\ldots,\beta_{g,q}(0),\ldots\beta_{g,q}(N- 1)\right]^{\top}\), where  
\[\begin{array}{c}\beta_{g,q}(n)=\frac{1}{q!}\sum\limits_{m=1}^{M}(m-1)^{q}{ \bf s}_{g,m}^{H}{\bf U}_{n}{\bf s}_{g,m}\\ \mbox{for }n=0,1,\ldots,N-1\end{array} \tag{20}\]  
and \(\beta_{g,q}(n)=\beta_{g,q}^{*}(-n),\mbox{ for }n=1-N,2-N,\ldots,-1\), and then, (19) can be rewritten as  
\[y_{g}(n,\varphi_{d})=\sum\limits_{q=0}^{\infty}\beta_{g,q}(n)(j2\pi\varphi_{ d})^{q} \tag{21}\]  
Let  
\[P_{g,q}(n,\varphi_{d})=\beta_{g,q}(n)(j2\pi\varphi_{d})^{q} \tag{22}\]  
Then, (21) can be rewritten as  
\[y_{g}(n,\varphi_{d})=\sum\limits_{q=0}^{\infty}P_{g,q}(n,\varphi_{d}). \tag{23}\]  
In (23), \(y_{g}(n,\varphi_{d})\) can be regarded as the summation of \(P_{g,q}(n,\varphi_{d})\) for different \(q\). When \(q=0\), we will obtain  
\[P_{g,0}(n,\varphi_{d})=\sum\limits_{m=1}^{M}{\bf s}_{g,m}^{H}{\bf U}_{n}{\bf s }_{g,m} \tag{24}\]  
In (24), it can be seen that \(P_{g,0}(n,\varphi_{d})\) is the SACF of the \(g\)-th JRC-CWG. This indicates that only the zero-order Taylor expansion term is considered in the optimization problem (10). However, according to (23), if only the zero-order Taylor expansion term \(P_{g,0}(n,\varphi_{d})\) is considered in the JRC-CWG design, the nonzero-order Taylor expansion terms of \(y_{g}\) will deteriorate the Doppler resilience of the JRC-CWGs. Hence, to suppress the sidelobe level of the JRC-CWG around \(\varphi_{d}=0\), more Taylor expansion terms of \(y_{g}\), i.e., \(P_{g,q}\), should beoptimized. Moreover, according to (20) and (22), as \(q\) increases, the value of \(P_{g,q}(n,\varphi_{d})\) decreases because of the term \(1/q!\). This means that some lower order of \(P_{g,q}\) has the major effect on the Doppler resilience of the JRC-CWG. Furthermore, according to (22), when optimizing \(P_{g,q}\), the term \((j2\pi\varphi_{d})^{q}\) can be regarded as a constant. Hence, it is reasonable to optimize \(P_{g,q}\) by optimizing \(\beta_{g,q}\). In summary, to design the DR-JRC-CWG, \(\beta_{g,q}\) for \(q=0,1,\ldots,Q\), should be jointly optimized.