## 4 Mathematical Representation of the Proposed System  
The technical design of the proposed SU-TX tower, which covers an area of 500 \(\times\) 500 m, is shown in Figure 5. The upper part of Figure 5 within the proposed SU-TX block diagram consists of the spreading encoder with a total of seven orthogonal codes, with 8 bits each for model 1, so that the seven input message signals will spread. Then, the multiplexer rearranges the data in one line for the OFDM process. The same previous explanation is applicable for model 2 but with a spreading encoder with a total of 15 orthogonal codes, with 16 bits each. The output of the OFDM process passes through the shaping filter (S.F), whose output mixes with the assigned propagation frequency carrier. The SU-RF signal is amplified through the upper power amplifier (PA) with a gain value equal to \(1-\gamma\) of the total transmission power of the SU-TX tower, where \(\gamma\) is the power incentive factor. The bottom part of Figure 5 within the proposed SU-TX block diagram consists of the RF wide-band filter, which allows the PU system RF signal to be listened to without any knowledge or changing of the PU signal structure. Also, the filtered PU-RF signal is  
Figure 4: Topology and spectrum representation of model 2. (**a**) Model 2 topology of the proposed DSCA-MC-CR using OMNeT. (**b**) Spectrum representation of model 2. Figure 5: The block diagram design of the proposed DSCA-MC-CR SU-TX tower. amplified through the power amplifier (PA) with a value equal to \(\gamma\) of the total SU-TX transmission power. Thus, the outputs of the two power amplifiers are summed, and the RF signal propagates from the transmitter system to cover the required coverage area. Table 1 denotes all the parameters, functions, and concepts that will be used in the next mathematical representation sections. The mathematical representation of the proposed model 1 and model 2 starts by defining the transmission process. In this regard, the SU-TX data messages are expressed as \(m_{j}\), where \(j\in\{1,2,\ldots,7\}\). So, the SU-TX encoder spreading codes are defined as \(c_{j+1}\), where, in model 1, code \(c_{1}\) is reserved for the PU-RF boosted signal, and codes \(c_{2}\) up to \(c_{8}\) are reserved for the SU transceivers within the coverage area. Hence, for simplicity, the Hadamard matrix \(H_{8}\) of model 1, which includes eight total orthogonal codes, is defined as  
\[H_{8}= \begin{bmatrix}c_{1}\\ c_{2}\\ \vdots\\ c_{8}\end{bmatrix}  \tag{1}\]  
The derivations of the transmission process assume asynchronous reception at the SU-RX side due to the channel impairments, i.e., in a practical case, each carrier undergoes a shift of \(\phi_{j}\) and a delay of \(\tau_{j}\). User \(i\)'s encoded message \(y_{i}(t)\) signal can be represented as  
\[y_{i}(t)=\sqrt{\frac{2E_{p}}{T_{p}}}m_{i}(t-\tau_{i})c_{i}(t-\tau_{i})\cos( \omega_{c}(t-\tau_{i})+\phi_{i}). \tag{2}\]  
The overall received signal \(r(t)\), which suffers from carrier shift and channel delay and assumes \(U\) users within the system coverage area, can be expressed as  
\[r(t)=\sum_{u=1}^{U}\sqrt{\frac{2E_{p}}{T_{p}}}m_{u}(t-\tau_{u})c_{u}(t-\tau_{ u})\cos(\omega_{c}(t-\tau_{u})+\phi_{u})+n(t)=\sum_{u=1}^{U}y_{u}(t)+n(t), \tag{3}\]