# Precoder and Decoder Co-Designs for Radar and Communication Spectrum Sharing +
## 2 Generalized Multicarrier Radar-Communication Coexistence Model
### Transmitter  
The discrete-time signal model in Appendix A can be rewritten in a compact matrix form,  
\[\mathbf{Y_{T}}=(\mathbf{\Omega}\circ\mathbf{B})\mathbf{PCS} \tag{1}\]  
where \(\circ\) is the Hadamard product. \(\mathbf{Y_{T}}\) is an \(N_{sc}\times M\)-dimensional sample matrix that occupies \(M\) time slots and \(N_{sc}\) subcarriers for \(k=\{0,1,2,\ldots,MN_{sc}-1\}\) samples, denoted as a resource block for communication or pulse for radar. The matrix \(\mathbf{P}\in\mathbb{C}^{N_{sc}\times N}\) denotes the linear frequency precoding matrix that contains subcarrier weights. Matrix \(\mathbf{C}\in\mathbb{C}^{N\times N_{p}}\) represents a coding matrix, where \(N_{p}\) denotes the length of the uncoded data sequence. Matrix \(\mathbf{S}\in\mathbb{C}^{N_{p}\times M}\) contains transmitted data \(\mathbf{S}=[\mathbf{s}(1)\ldots\mathbf{s}(M)]\), where \(\mathbf{s}(M)\) is an \(N_{p}\) data vector for \(M\) time slots. A block diagram of a multicarrier system for radar and communication coexistence is shown in Figure 1. This figure illustrates how the generalized multicarrier radar-communication signal model can be applied to generate signals at the transmitter and process them at the receiver. The input data will be processed by the channel coder, precoder, IDFT, subcarrier selection, parallel-to-serial conversion and digital-to-analog converter to generate a multicarrier radar communication signal. After decoding, the channel information is estimated in the communication subsystem, or target information is estimated in the radar subsystem. The matrix \(\mathbf{\Omega}\) is an \(N_{sc}\times N_{sc}\)-dimensional selection matrix, in which the elements of \(\mathbf{\Omega}\) are either 1 or 0 to indicate whether subcarriers in matrix \(\mathbf{Y_{T}}\) are active and to manage the pilot symbol. For example, a widely used radar pulse and communication signal can be controlled by deactivating resource block elements of \(\mathbf{\Omega}\). In practice, the selection matrix selects carriers to sharpen the waveform patterns, various waveforms can be generated by enabling or disabling its elements, such as a step approximation of the up-chirp pulse, a pseudo-random frequency-hopping pulse and a multicarrier communication signal with a Com  
Figure 1: Multicarrier radio system for both multicarrier communication and multicarrier radar systems or a colocated radar-communication system. In the case of a radar or communication user, the dashed boxes of communication or radar in the block diagram are ignored. P/S (S/P) denotes parallel-to-serial or serial-to-parallel conversion, and DAC/ADC represents a digital-to-analog converter or analog-to-digital converter. Matrix \(\mathbf{B}\in\mathbb{C}^{N_{bc}\times N_{bc}}\) in (1) is considered to be a modulation matrix. In an OFDM system, matrix \(\mathbf{B}\) is a Vandermonde matrix, and its elements are associated with subcarriers as follows [20]:  
\[\mathbf{B}=\frac{1}{\sqrt{N_{sc}}} \begin{pmatrix}1&1&\cdots&1\\ 1&\beta&\cdots&\beta^{N_{bc}-1}\\ 1&\beta^{2}&\cdots&\beta^{2(N_{bc}-1)}\\ \cdots&\cdots&\cdots&\cdots\\ 1&\beta^{N_{bc}-1}&\cdots&\beta^{(N_{bc}-1)(N_{bc}-1)}\end{pmatrix}  \tag{2}\]  
where  
\[\beta=e^{2\pi\Delta f(\frac{T_{c}}{N_{sc}})},\text{for both radar and communication} \tag{3}\]  
denotes baseband subcarriers without a carrier/sampling index. \(T_{c}\) is the symbol duration of the multicarrier system, and \(\Delta f\) is the subcarrier spacing. The time duration between row elements is (\(k\) mod \(N_{sc}\)) sample periods. The same active/inactive subcarrier pattern is applied for the time duration of each radar subpulse/communication symbol. This pattern justifies the (\(k\) mod \(N_{sc}\)) time duration above. In the special case of an OFDM signal, matrix \(\mathbf{B}\) is an inverse discrete Fourier transform (IDFT) matrix, \(\mathbf{B}^{H}\mathbf{B}=\mathbf{I}_{N_{sc}\times N_{sc}}\), where \(T_{c}=\frac{1}{\Delta f}\).