# Precoder and Decoder Co-Designs for Radar and Communication Spectrum Sharing +
## 2 Generalized Multicarrier Radar-Communication Coexistence Model
### Multiuser Radar-Communication Spectrum Sharing Scenario  
In this subsection, we consider a simplified \(K+1\) user radar-communication spectrum sharing scenario, which consists of \(K\)SISO communication users and a SISO multicarrier monostatic radar user. They share \(N_{\mathrm{sc}}\) subcarriers and the same frequency band. Consequently, the sum of the numbers of possible transmitter and receiver (TX-RX) pairs (including the signal channel and interference channel) is \(K(K+1)\). Each user is considered to be an intended TX-RX pair (matched link) transmitting a useful signal. Hence, the sum of useful signal channels is \(K\). Interference occurred in unintended TX-RX pairs (mismatched link), which may be from radar TX to one communication RX, from one communication TX to radar RX or from communication TX to another unintended communication RX. The interference channels correspond to \(K^{2}\) unintended links. A monostatic radar is colocated with one of these communication nodes. The joint radar communication system is considered to be a radar and a communication user separately, as illustrated in Figure 2.  
In this network configuration, we denote by \(\mathcal{A}\) the set of communication users and by \(\mathcal{B}\) the set of radar users, where \(|\mathcal{A}|=K,|\mathcal{B}|=1\). Here, \(|\mathcal{A}|\) denotes the cardinality of set \(\mathcal{A}\). The colocated multicarrier monostatic radar illuminates one target in the direction \(\theta\). Note that radar is likely forming narrow beams or has a highly directional antenna such that it can only interfere with part of the communication nodes. Recall that the signals are experiencing a block fading channel. Before decoding, the received multicarrier signal at the \(i\)th receiver in the \(K+1\)-user radar-communication coexistence scenario may be written as follows:  
Figure 2: A practical vehicle networking example, in which the colocated radar-communication system coexists with a multiuser communication network  
\[\mathbf{Y_{R}} =\underbrace{\mathbf{H}_{[R]}(\theta)(\mathbf{\Omega}_{[R]}\circ \mathbf{I}_{N_{Re}\times N_{Re}})\mathbf{P}_{[R]}\mathbf{C}_{[R]}\mathbf{S}_{[R ]}}_{\text{Radar Signal}}\] \[+\underbrace{\sum_{j\in\mathcal{A}}\mathbf{H}_{[ij]}(\mathbf{ \Omega}_{[j]}\circ\mathbf{I}_{N_{Re}\times N_{Re}})\mathbf{P}_{[j]}\mathbf{C}_ {[j]}\mathbf{S}_{[j]}}_{\text{Communication Signal}} \tag{5}\] \[+\mathbf{W}_{[i]}\]  
where \(\mathbf{Y}_{\mathbf{R}[i]}\) is the received signal for the \(i\)th user. Here, the \(i\)th receiver may refer to all \(K\) communication receivers and radar receiver \(i\in\mathcal{A}\cup\mathcal{B}\). \(\mathbf{\Omega}_{[R]}\) denotes the selection matrix associated with adaptive waveform design, e.g., pseudo-random pulse or multi-pulse radar. Due to the block fading channel assumption, the channel coherence interval is \(L\) pulses/resource blocks. Channel coding matrix \(\mathbf{C}\) will remain unchanged because it is designed based on \(\mathbf{H}\) in both communication and radar subsystems. Consequently, \(\mathbf{\Omega}_{[R]}\), \(\mathbf{\Omega}_{[j]}\), \(\mathbf{P}\) and \(\mathbf{S}_{[j]}\) may vary over \(L\) pulses/resource blocks. To describe the mode in detail, some key conditions are stated as follows:  
* Channel State Information (CSI): The channels \(\mathbf{H}_{[ii]}\) and \(\mathbf{H}_{[ij]}\) and the target response matrix \(\mathbf{H}(\theta)_{[ii]}\) are considered to be perfectly estimated at the communication and radar transmitters, respectively. CSI estimation has been applied in communication systems for a long time. For radar, one feasible method is to treat a known radar waveform as a shared pilot in both radar and communication systems. The pilot-aided approach can estimate all the channel information between radar and communication users [38]. Another approach is to embed the same pilot signal in both the radar coherence interval and communication frames [7]. The benefit of the distributed IA method is that CSI is available locally.