# Joint Transmit Waveform and Receive Mismatched Filter Design to Suppress Range Sidelobe
## 2 Design of NLFM-CPM Integrated Waveform and Signal Processing Methods
### LPG Analysis  
LPG is characterized as the ratio between the SNR enhancement obtained via EWMF and that of the MF. The expression for this relationship is outlined below. \[\mathrm{LPG}=10\log_{10}\biggl{(}\frac{\mathrm{SNR}_{\mathrm{MF}}}{\mathrm{ SNR}_{\mathrm{EWMF}}}\biggr{)} \tag{37}\] \begin{table}
\begin{tabular}{l l} \hline Input & \(\mathbf{w}_{\mathrm{ini}}\), \(\mathbf{\Lambda}(\mathbf{s})\), \(\mathbf{C}\), \(\mathbf{g}\), \(b_{\max}\), \(k=0\), \(K\), and auxiliary variables. \\ \hline Output & EWMF coefficients: \(\mathbf{h}\), \(\mathbf{x}\). \\ \hline Processing Steps & \\ Step 1 & Set \(k=k+1\). Compute \(\mathbf{B}\) using Equation (24) and \(\mathbf{y}_{\mathrm{ISL}}\) using Equation (25). \\ Step 2 & Compute \(\mathbf{u}\), \(\mathbf{x}\), and \(\mathbf{h}\) by minimizing the metric using Equations (33), (34) and (35), respectively. Update other variables using Equation (33). \\ Step 3 & Identify the lowest peak level \(\mathrm{Pr}^{k}\) as the reference level. Form a new reference template based on \(\mathrm{Pr}^{k}\) and compute \(w(i)\) using Equation (36). \\ Step 4 & Repeat Step 1 until \(k=K\) or \(\left|w^{k+1}(i)-w^{k}(i)\right|<\epsilon\), where \(\epsilon\to 0\). \\ \hline \end{tabular}
\end{table} 
Table 2: The algorithm steps for the joint design of NLFM-CPM waveform and EWMF based on ADMM. The total noise power \(P_{n}\) and the maximal signal power output \(P_{s}\) of the EWMF output are respectively  
\[P_{n}=\frac{N_{0}}{4\pi}\sum_{n=-N+1}^{N-1}w^{2}(n)*|x(n)|^{2} \tag{38}\]  
\[P_{s}=\frac{1}{4\pi^{2}}\bigg{|}\sum_{n=-N+1}^{N-1}w(n)*|x(n)|^{2}\bigg{|}^{2}. \tag{39}\]  
where \(*\) represents the convolution operation; \(\frac{N_{0}}{2}\) is the power spectral density of the white noise. We can get the output SNR of the EWMF as  
\[\text{SNR}_{\text{EWMF}}=\frac{P_{s}}{P_{n}}=\frac{1}{\pi N_{0}}\cdot\frac{ \left|\sum_{n=-N+1}^{N-1}w(n)*|x(n)|^{2}\right|^{2}}{\sum_{n=-N+1}^{N-1}w^{2}(n )*|x(n)|^{2}} \tag{40}\]  
The output SNR of the MF can be easily derived from Equation (40) by setting \(W(k)\equiv 1\)  
\[\text{SNR}_{\text{MF}}=\text{SNR}_{\text{EWMF}}\Big{|}_{w(n)\equiv 1}=\frac{1}{ \pi N_{0}}\cdot\frac{\left|\sum_{k=-N+1}^{N-1}|x(n)|^{2}\right|^{2}}{\sum_{k=- N+1}^{N-1}|x(n)|^{2}}=\frac{1}{\pi N_{0}}\sum_{n=-N+1}^{N-1}|x(n)|^{2} \tag{41}\]  
For a given \(N_{0}\), the \(\text{SNR}_{\text{MF}}\) is determined exclusively by the power of the waveform itself, in accordance with the principles of the matched filter. Subsequently, the SNR loss is defined as the ratio of \(\text{SNR}_{\text{MF}}\) to \(\text{SNR}_{\text{EWMF}}\). \[\text{LPG}=10\log_{10}\biggl{(}\frac{\text{SNR}_{\text{MF}}}{\text{SNR}_{ \text{EWMF}}}\biggr{)}=10\log_{10}\left(\frac{\sum_{n=-N+1}^{N-1}|x(n)|^{2} \cdot\sum_{n=-N+1}^{N-1}w^{2}(n)*|x(n)|^{2}}{\left|\sum_{n=-N+1}^{N-1}w(n)*|x(n )|^{2}\right|^{2}}\right) \tag{42}\]  
[PERSON]'s inequality states that for two signals \(f(n)\) and \(g(n)\)[33], we have  
\[\|f*g\|_{r}\leq\|f\|_{p}\cdot\|g\|_{q} \tag{43}\]  
where \(\frac{1}{r}=\frac{1}{p}+\frac{1}{q}-1\). Let \(r=q=p=1\), \(f(n)=w(n)*|x(n)|\), and \(g(n)=|x(n)|\). Substituting these into (43), we obtain  
\[ \begin{split}\left|\sum_{n=-N+1}^{N-1}w(n)*|x(n)|^{2}\right|^{2 }&=\left|\sum_{n=-N+1}^{N-1}w(n)*|x(n)|*|x(n)|\right|^{2}\\ &\leq\sum_{n=-N+1}^{N-1}\Bigl{|}w^{2}(n)*|x(n)|^{2}\Bigr{|}\cdot \sum_{n=-N+1}^{N-1}|x(n)|^{2}.\end{split}  \tag{44}\]  
Therefore, according to [PERSON]'s inequality, we can conclude that  
\[ \begin{split}\text{SNR}_{\text{MF}}&\geq\text{ SNR}_{\text{EWMF}}\\ \text{LPG}&\geq 0.\end{split}  \tag{45}\]  
The only factor affecting SNR loss is \(w(i)\), which is determined by the reference template \(D(i)\). Since there is no simple linear relationship between \(\text{SNR}_{\text{loss}}\) and \(w(i)\), it is not possible to directly determine the SNR loss caused by changes in \(D(i)\) using (42). However, the SNR loss corresponding to all possible \(D(i)\) values under the same parameterscan be provided as the main sidelobe level over a specified region (MSLSR) changes [11].