# Joint Transmit Waveform and Receive Mismatched Filter Design to Suppress Range Sidelobe
## 2 Design of NLFM-CPM Integrated Waveform and Signal Processing Methods
The discrete baseband echo signal \(s_{r}(n)\) is defined as  
\[s_{r}(n)=\sigma_{t}s(n-n_{t})\cdot e^{-2\pi f_{c}\tau}+\sum_{k=1}^{K}\sigma_{k }s(n-n_{k})\cdot e^{-\beta 2\pi f_{c}\tau_{k}}+c(n),n=0,1,\ldots,N-1 \tag{10}\]  
where \(\sigma_{t}\) denotes the scattering coefficient of the target point, \(n_{t}=\lfloor\tau_{t}\cdot f_{s}\rfloor\) represents the discrete form of \(\tau_{t}\), and \(\tau_{t}=\frac{2R_{b}}{c}\) signifies the time delay of the target, with \(c\) being the speed of light. Similarly, \(\sigma_{k}\) is the scattering coefficient for the \(k\)-th clutter point, \(n_{k}=\lfloor\tau_{k}\cdot f_{s}\rfloor\) is the discrete representation of \(\tau_{k}\), and \(\tau_{k}=\frac{2R_{k}}{c}\) indicates the time delay of the \(k\)-th clutter point. The term \(c(n)\) denotes the additive noise component, and \(K\) is the total number of clutter points. Thus, the discrete Fourier transform (DFT) of (10) can be computed as  
\[ \begin{split} S_{r}(f)=\sigma_{t}S(f)\exp(-j2\pi f_{c}\tau_{t}) \cdot\exp\Bigl{(}-j2\pi f\frac{n_{t}}{N}\Bigr{)}+\sum_{k=1}^{K}\sigma_{t}S(f )\exp(-j2\pi f_{c}\tau_{k})\cdot\exp\Bigl{(}-j2\pi f\frac{n_{k}}{N}\Bigr{)}+ C(f),\\ f=0,1,\ldots,N-1\end{split}  \tag{11}\]  
where \(S(f)\) denotes the DFT of \(s(n)\), \(C(f)\) represents the DFT of \(c(n)\). The MF \(H_{\text{MF}}(f)\) in the frequency domain and \(S(f)\) are conjugate transpose  
\[H_{\text{MF}}(f)=S^{*}(f) \tag{12}\]  
By performing an inverse discrete Fourier transform (IDFT) on the product of the input signal and the filter's frequency response, the output of MF is obtained as  
\[ \begin{split} s_{\text{MF}}(n)=&\text{IDFT}\{S_{r} (f)S^{*}(f)\}\\ =&\text{IDFT}\Biggl{\{}\sigma_{t}|S(f)|^{2}e^{-j2\pi f _{c}\tau_{t}}e^{-j2\pi f\frac{n_{t}}{N}}+\sum_{k=1}^{K}\sigma_{k}|S(f)|^{2}e^ {-j2\pi f_{c}\tau_{k}}e^{-j2\pi f\frac{n_{k}}{N}}+W(f)S^{*}(f)\Biggr{\}}\\ =&\sigma_{t}e^{-2\pi f_{c}\tau_{t}}\cdot\text{psf}_ {\text{w}}(n-n_{t})+\sum_{k=1}^{K}\sigma_{k}e^{-j2\pi f_{c}\tau_{k}}\cdot \text{psf}_{\text{w}}(n-n_{k})+\text{IDFT}\{W(f)S^{*}(f)\}\\ &\text{where }\text{psf}_{\text{w}}(n)=\text{IDFT}\{S(f)W(f)\} \text{ represents the calculation of the system's point spread}\\ &\text{function }\text{psf}(n)\text{ using the signal's frequency domain power spectral density }|S(f)|^{2}\text{[28]}\text{.}\text{ In general, when the signal power is significantly higher than the noise, the influence of }\\ &\text{IDFT}\{W(f)S^{*}(f)\}\text{ on the MF result can be ignored.}\end{split}  \tag{13}\]To illustrate the impact of near-range cells on filtering, the linear convolution [12] of all range cells \(\mathbf{x}_{i}(i=-N+1,\cdots,N-1)\) can be represented as  
\[ \begin{split}\mathbf{\Lambda}(s)=&\begin{bmatrix} \mathbf{x}_{-N+1}&\mathbf{x}_{-N+2}&\cdots\mathbf{x}_{0}&\cdots&\mathbf{x}_{N -2}&\mathbf{x}_{N-1}\end{bmatrix}^{T}\\ =&\begin{bmatrix}s_{0}&s_{1}&\cdots&s_{N-1}&0&\cdots&\cdots&0\\ 0&s_{0}&\cdots&s_{N-2}&s_{N-1}&0&\cdots&0\\ \vdots&\vdots&\ddots&\vdots&s_{N-2}&s_{N-1}&0&\vdots\\ \vdots&\ddots&\vdots&s_{0}&\vdots&\vdots&\ddots&0\\ 0&\cdots&0&0&s_{0}&s_{1}&\cdots&s_{N-1}\end{bmatrix}^{T}\\ \end{split}  \tag{14}\]  
where \(\mathbf{\Lambda}(s)\) is the linear convolution matrix constructed from the signal \(\mathbf{s}\), the \(P\)-dimensional column vector \(\mathbf{x}_{0}= \begin{bmatrix}\mathbf{0}_{z}^{T}s_{0}\cdots s_{N-1}\mathbf{0}_ {z}^{T}\end{bmatrix} ^{T}\) is formed by zero-padding both ends of the received signal, and \(\mathbf{0}_{z}\) represents a column vector of length \(z\) filled with zeros. If \(\mathbf{x}_{0}\) represents the target information that needs to be preserved, then \(\mathbf{x}_{i}\) (\(i=-N_{1}+1,\cdots,-1,1,\cdots,N_{1}-1\)) represents the information that is not needed. This also explains the occurrence of sidelobes in the time-domain response. Figure 1 illustrates the arrangement of signal samples received by a radar receiver from different range cells.