# Precoder and Decoder Co-Designs for Radar and Communication Spectrum Sharing +
## Appendix A  
A multicarrier radar and communication signal model with \(N_{sc}\) available subcarriers may be described as in (1). The coded data/waveforms are mapped to subcarriers after precoding and channel coding. Pulse multicarrier radar signals and continuous multicarrier communication signals are employed in this model.  
\[\begin{array}{ll}Y_{T}(k)=\begin{cases}2R\epsilon\left\{\sum_{n=0}^{N_{sc}-1 }\sum_{m=0}^{N_{p}-1}\sum_{m=0}^{N_{s-1}}\sum_{m=0}^{N_{s-1}}\sum_{m=0}^{N_{s-1 }}p_{m}\epsilon_{m}\epsilon_{m}\epsilon_{m}\epsilon_{m}\epsilon_{m}\epsilon_{ m}\epsilon_{m}\epsilon_{m}\epsilon_{m}\epsilon_{m}\epsilon_{m}\epsilon_{m} \epsilon_{m}\epsilon_{m}\epsilon_{m}\epsilon_{m}\epsilon_{m}\epsilon_{m* \(k=\{0,1,2,\ldots,MN_{sc}-1\}\) denotes the sample index.
* \(a\) is a multiplicity factor between communication symbol duration \(T_{c}\) and radar subpulse duration \(T_{p}\), where \(T_{p}\) is larger than \(T_{c}\) considering zero-padding in pulse radar. Then, the radar subpulse duration is given by \(T_{p}=aT_{c}\).
* \(c_{n,y}\) is a scalar with respect to the \(n\)th subcarrier and \(y\)th data symbol, \(c_{n,y}\) denotes the coding operator for radar signal and the channel coding operator in the communication signal; e.g., it will be \(\exp(j\phi)\) for P3 or P4 radar code and an element of the Turbo generator matrix for Turbo code.
* \(s_{y,m}\) is scalar data corresponding to the \(y\)th data symbol and \(m\)th subpulse, which is random for the payload communication signal and known in radar applications because the transmitted waveform information is known by its receiver.
* \(p_{k,n}\) is a scalar that denotes the precoding coefficient with respect to the \(k\)th subcarrier and \(n\)th subpulse before the signal emitted.  
A detailed derivation of (A1) is given in the following. Considering an OFDM signal with \(N_{sc}\) subcarriers and \(\Delta f\) Hz bandwidth, the continuous-time complex envelope of the baseband OFDM signal can be written for any instant \(t\) as  
\[s_{m}(t)=\sum_{x=0}^{N_{sc}-1}w_{m,x}\ rect[t-xT_{c}]\ e^{2\pi x\Delta ft}\] (A2)  
where \(rect[\cdot]\) is a discrete rectangular window, \(T_{c}\) is the symbol duration, and the constraint \(T_{c}=\frac{1}{\Delta f}\) satisfies the spectrum selection channel condition. \(w_{x}\) indicates the amplitude weighting of the \(x\)th subcarrier. Additionally,  
\[rect(t)=\begin{cases}1&\text{if }0\leq t\leq T_{c},\\ 0&\text{otherwise}.\end{cases}\] (A3)  
The broadcast multicarrier communication signal is considered to be continuous:  
\[y_{T,c}(t)=\sum_{m=-\infty}^{\infty}s_{m}(t-mT_{c}).\] (A4)  
In general, a multicarrier radar signal can be written as follows:  
\[y_{T,r}(t)=\sum_{m=0}^{M-1}s_{m}(t-mT_{p}),\] (A5)  
where \(M\) is the number of OFDM subpulses, \(T_{p}=aT_{c}\), \(a\in\mathbb{R}^{+}\) is the radar pulse duration. As both radar and communication signals coexist, are designed jointly and are postprocessed simultaneously in this paper, time synchronization is employed. The transmitted signal is a real part of the baseband complex envelope; then, (A5) can be rewritten as follows:  
\[\begin{split}\widehat{y}_{T,r}(t)&=2 Re\{\sum_{m=0}^ {M-1}s_{m}(t-mT_{p})\,\}\\ &=2 Re\{\sum_{m=0}^{M-1}\sum_{x=0}^{N_{sc}-1}w_{m,x}\ rect[t-xT_{c}-mT_{p}]\,\}\\ & e^{2\pi x\Delta ft}\,\}\\ &=2 Re\{\sum_{m=0}^{M-1}\sum_{x=0}^{N_{sc}-1}w_{m,x}\ rect[t-(x+am)T_{c}]\\ & e^{2\pi x\Delta ft}\,\}\end{split}\] (A6)According to the Nyquist-Shannon sampling theorem, the sampling rate is \(N_{sc}\Delta f\) samples/second. Additionally, replacing \(w_{m,x}\) with \(p_{k,n}\), \(c_{n,y}\) and \(s_{y,m}\), the radar transmitted signal can be rewritten as follows:  
\[\begin{split}\widehat{y}_{T,\mathcal{V}}(k)=& 2 Re\{\sum_{x=0}^{N_{sc}-1}\sum_{y=0}^{ N_{p}-1}\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}p_{k,n}c_{n,y}s_{y,m}\\ rect[\frac{kT_{c}}{N_{sc}}-(x+am)T_{c}]\ \times e^{2\pi(\int_{c}+x \Delta f)\frac{kT_{c}}{N_{sc}}}\}\end{split} \tag{10}\]  
The transmitted communication signal can also easily be derived.