# Hybrid FSK-FDM Scheme for Data Rate Enhancement in Dual-Function Radar and Communication
## 4 Receiver Design
### Communication Receiver  
There are \(K\) communication receivers in the far field, each having an array of \(N_{R}\) elements. For ease and convenience and a priory communication, the lookup table containing the dictionary of the orthogonal JWF made using FSK modulation and FDM symbols at the dual-function transmitter is known to each communication receiver. Assume that the \(k\)th communication receiver, equipped with \(N_{R}\) antenna elements arranged uniformly in a linear shape, receives the following FSK-FDM composite signal. \[\mathbf{y}_{k_{fdm}}(t,\tau)=\sqrt{\frac{M_{T}}{L_{B}}}\alpha_{k}(\mathbf{a}^{ T}(\phi_{l})\mathbf{s}(t,\tau))\mathbf{c}_{k}(\phi_{l})\mathbf{x}_{fdm}(t, \tau)+\mathbf{n}_{k_{fdm}}(t,\tau) \tag{5}\]  
Figure 2: The proposed composite information embedding methodology using frequency division multiplexing at the DFRC transmitter. where \(\sqrt{\frac{M_{T}}{L_{B}}}\) is the received power at the communication receiver, \(\alpha_{k}\) is the channel coefficient constant from the transmitter array towards the \(k\)th communication receiver which summarize the propagation environment, \(\mathbf{a}(\phi_{i})\) is the steering vector in the direction \(\theta_{m}\) from the dual-function transmitter, \(\mathbf{s}(t,\tau)\) is the base band signal, \(\mathbf{c}_{k}(\phi_{k})\) is the steering vector from the receive array in the direction \(\phi_{k}\) from the communication receiver, \(\mathbf{x}_{fdm}\) is the composite FSK-FDM signal, \(\mathbf{n}_{k}(t,\tau)\) is the AWG noise vector zero mean with variance \(\sigma^{2}\mathbf{I}\) at the communication receiver, and \((\phi_{k})\) is the direction of the \(k\)th communication receiver. As a first step, the beam-forming operation is applied to the received signal. The steering vector is separated from it by multiplying the beam-forming weights at the communication receiver. This allows for the extraction of the desired signal from the received signal, allowing the receiver to focus on the direction of the signal and ignore signals from other directions. This reduces interference from other sources and improves the signal-to-noise ratio (SNR) at the receiver. The beam-forming operation is mathematically expressed by Equation (6). \[\mathbf{g}_{k_{fdm}}(t,\tau)=\mathbf{c}_{k}^{H}(\phi_{j})\mathbf{y}_{k_{fdm}} (t,\tau) \tag{6}\]  
The next step in this process is to apply bandpass filtering techniques to the received FDM composite signal. The bandpass filter is implemented as a digital finite impulse response (FIR) filter and configured to have a passband with the desired bandwidth. The filtered signal is then demodulated to recover the original transmitted information. A mathematical description is given in Equation (7) and a graphic representation is shown in Figure 3.  
\[\mathbf{r}_{k}(t,\tau)=\mathbf{v}(\mathbf{\Omega})\mathbf{g}_{k_{fdm}}(t,\tau) \tag{7}\]  
where \(\mathbf{v}(\mathbf{\Omega})\) is the bandpass filtering coefficients at the \(k\)th communication receiver. Matched filtering is then applied to the signal, Equation (7), to identify the actual binary information transmitted. This involves multiplying the signal by a reference signal delayed by the same amount of time as the original signal. The multiplication results are then accumulated over a period of time. The accumulated signal can be used to identify  
Figure 3: The proposed composite information decoding methodology at the communication receiver. the binary information transmitted. Matched filtering is mathematically described by Equation (8). \[\mathbf{y}_{k,l}(t,\tau)= \begin{cases}\sqrt{\frac{M_{T}}{L_{B}}}\mathbf{r}_{1}(t, \tau)+\mathbf{n}_{1}(t,\tau),&if\quad B_{l}=00\\ \sqrt{\frac{M_{T}}{L_{B}}}\mathbf{r}_{2}(t,\tau)+\mathbf{n}_{2}(t,\tau),&if \quad B_{l}=01\\ \sqrt{\frac{M_{T}}{L_{B}}}\mathbf{r}_{3}(t,\tau)+\mathbf{n}_{3}(t,\tau),&if \quad B_{l}=10\\ \sqrt{\frac{M_{T}}{L_{B}}}\mathbf{r}_{4}(t,\tau)+\mathbf{n}_{4}(t,\tau),&if \quad B_{l}=11\end{cases}