# 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 System  
Based on the multi-symbol OFDM IRC waveform, the signal processing flow in IRC systems is shown in Figure 2. At the transmitter, the binary data are modulated by a quadratic-amplitude-modulated (QAM) modulator to generate \(\mathbf{X}\). The communication waveform \(\mathbf{s}_{X}\ =\ \mathbf{\Phi}\mathbf{X}\) is then passed through the waveform optimization module proposed in Section 4 to generate the IRC waveform \(\mathbf{s}\). The CP is then inserted into \(\mathbf{s}\) for transmission via the PA. It is noted that the CP duration is related to the multipath delay, i.e., it needs to be longer than the maximum multipath delay, otherwise, inter-symbol interference (ISI) will be caused [33]. If the ISI is eliminated by CP, then at the receiver, the received vector in the discrete-time domain is obtained after CP removal.  
For communication processing, the received vector in the discrete-time domain is processed symbol by symbol after serial-parallel conversion. Firstly, it is first converted to the discrete frequency domain by DFT calculation. Then, the data in the communication subcarriers is extracted, and the binary data is obtained after QAM demodulation, so far the information reception is completed.  
For radar processing, assuming that the IRC signal is reflected by \(L\) objects, ignoring noise, the received echo is written as [32]  
\[s_{r}(t)\ =\ \sum_{l=0}^{L-1}A_{l}s(t-\tau_{l}) \tag{4}\]  
where \(A_{l}\) is the attenuation factor, \(\tau_{l}=2R_{l}/c\), \(R_{l}\) is the range from the \(l\)-th object to radar, and \(c\) is speed of light.  
The range profile is obtained by echo pulse compression [36]. The impulse response of the matched filter is expressed as  
\[H(t)\ =\ s^{*}(-t) \tag{5}\]  
where \"\(()^{*}\)\" is the complex conjugate operation. Thus, the matched filter output of the echo in (4) is given by  
\[y(t)=s_{r}(t)\otimes H(t)=\int_{-\infty}^{\infty}s_{r}(t)s^{*}(\tau-t) \tag{6}\]  
where \(\bigotimes\) is convolution computation.  
Assuming the Fourier transform of \(s(t)\) and \(s_{r}(t)\) are \(S(f)\) and \(S_{r}(f)\), respectively, then the Fourier transform of \(y(t)\) can be achieved as  
\[Y(f)=S_{r}(f)S^{*}(f)=\sum_{l=0}^{L-1}A_{l}|S(f)|^{2}e^{-j2\pi\tau_{l}} \tag{7}\]  
Figure 2: Signal processing flow chart of the proposed multi-symbol OFDM-based IRC system. The communication symbol vector, the optimal reserved symbol vector, and the IRC waveform in the discrete-time domain are \(\mathbf{X}\), \(\mathbf{C}_{opt}\), and \(\mathbf{s}\). S/P denotes the serial to parallel conversion.  
Furthermore, the range profile obtained by an inverse Fourier transform of \(Y(f)\) is  
\[y(t)=\bar{R}(t)\otimes\sum_{l=0}^{L-1}A_{l}\delta(t-\tau_{l})=\sum_{l=0}^{L-1}A_{ l}\bar{R}(t-\tau_{l}) \tag{8}\]  
where \(\bar{R}(t)\) is the inverse Fourier transform of \(\left|S(f)\right|^{2}\), and is also the autocorrelation function of \(s(t)\). Therefore, the radar signal processing algorithm is implemented based on the Hadamard product between the discrete received vector \(\mathbf{S}_{r}\) and the discrete reference vector \(\mathbf{S}^{*}\). The range can be obtained from the IDFT results of this Hadamard product.