# Integrated Radar and Communications Waveform Design Based on Multi-Symbol OFDM
## 2 Multi-Symbol OFDM-Based IRC Systems and Signal Model
### Multi-Symbol OFDM-Based IRC Signal Model  
In this paper, it is assumed that there is an arbitrary number of (potentially non-contiguous) subcarriers for communication purposes lying within one OFDM symbol, which corresponds to the TR technique in conventional OFDM systems. Assume that the set of OFDM subcarriers is \(\{-N/2,\ldots,N/2-1\}\), where \(N\) denotes the number of subcarriers. Let \(\mathcal{R}=\{i_{0},i_{1},\ldots,i_{N-1}\}\) be the index set of reserved subcarriers over which the waveform is optimized, where \(N_{r}\) denotes its cardinality. Additionally, the complementary set \(\mathcal{R}^{c}\) of \(\mathcal{R}\) is the index set of communication subcarrier with \(N-N_{r}\) as its cardinality. TRR is defined as \(R_{N}=N_{r}/N\) (\(1\leq N_{r}<N\)). Therefore, each OFDM symbol computed by the inverse discrete Fourier transform (IDFT) of the \(N\)-point complex modulation sequence is  
\[s(q)\ =\ \sum_{n=-N/2}^{N/2-1}(X_{n}+C_{n})e^{j2\pi n\frac{q}{N}} \tag{1}\]  
where \(q\ =\ 0,1,\ldots,N-1\). \(X_{n}\) and \(C_{n}\) represent the \(n\)-th modulated complex element in the communication symbol vector \(\mathbf{X}\in\mathbb{C}^{N}\) and the reserved symbol vector \(\mathbf{C}\in\mathbb{C}^{N}\), respectively. It should be noted that at any subcarrier position, there must be a zero value between \(\mathbf{X}\) and \(\mathbf{C}\), i.e., \(X_{n}=0\) for \(n\in\mathcal{R}\) and \(C_{n}=0\) for \(n\in\mathcal{R}^{c}\).  
The IRC waveform with \(M\) consecutive OFDM symbols can be generated according to (1) as  
\[s(q)\ =\ \sum_{m=0}^{M-1}\sum_{n=-N/2}^{N/2-1}(X_{n,m}+C_{n,m})\phi_{m}(q,n) \tag{2}\]  
where \(q\ =\ 0,1,\ldots,MN-1\), and \(\phi_{m}(q,n)=\begin{cases}e^{j2\pi n\frac{q-N}{N}}&mN\leq q\leq(m+1)N-1\\ 0&elsewhere\end{cases}\).  
From (2), we can see that the IRC waveform is the splicing of the \(M\) OFDM symbols in the time domain. Thus, denoting \(\mathbf{s}=\left[\mathbf{s}_{0}^{T},\mathbf{s}_{1}^{T},\ldots,\mathbf{s}_{M- 1}^{T}\right]^{T}\in\mathbb{C}^{MN}\), where \(``()^{T}\) denotes the matrix transpose, and \(\mathbf{s}_{m}^{T},m=0,1,\ldots,M-1\) is the \(m\)-th OFDM symbol. Let \(\mathbf{X}=\left[\mathbf{X}_{0}^{T},\mathbf{X}_{1}^{T},\ldots,\mathbf{X}_{M- 1}^{T}\right]^{T}\in\mathbb{C}^{MN}\) and \(\mathbf{C}=\left[\mathbf{C}_{0}^{T},\mathbf{C}_{1}^{T},\ldots,\mathbf{C}_{M- 1}^{T}\right]^{T}\in\mathbb{C}^{MN}\), where \(\mathbf{X}_{m}^{T}\) and \(\mathbf{C}_{m}^{T}\) are the communication symbol vector and the reserved symbol vector of the \(m\)-th OFDM symbol, respectively. Then, (2) is expressed in matrix form as  
\[\mathbf{s}\ =\ \mathbf{\Phi}(\mathbf{X}+\mathbf{C}) \tag{3}\]  
where the IDFT matrix \(\mathbf{\Phi}=\left[\phi_{0},\phi_{1},\ldots,\phi_{M-1}\right]_{MN\times MN}\).  
At the transmitter, the IRC waveform with \(M\) consecutive OFDM symbols is generated before CP insertion, and the waveform structure in discrete time-frequency domain is shown in Figure 1. Here, one common CP is used for multiple OFDM symbols [33].  
Figure 1: IRC waveform structure in the discrete time-frequency domain.