## 3 Simulation Setup
### Implementation-Oriented Model (R-Model)  
The simulation setup under the realistic implementation-oriented model (_R-model_) has been built to mimic the system diagram shown in Figure 1, in which the direct conversion to baseband is assumed ideal, as also implicitly assumed in the conventional model. Matrices **X**, **H**, **V**, and **V**IN under the _R-model_ are generated as follows: To simulate a Gaussian distributed noise-like transmitted signal with controllable time correlation at the receiver side, **X** is formed by filtering i.i.d. zero mean complex Gaussian samples with a length-\(L\) moving average (MA) filter with no quantization (using floating-point computations). This type of filter was chosen for reasons of simplicity; any other low-pass filter could be adopted as well. The memory elements in the structure of this and subsequent MA filtering processes are assumed to have zero initial value before the first valid sample is applied to their inputs. As a result, the first \((L-1)\) samples resulting from the MA filtering, out of \((n+L-1)\), are discarded before subsequent operations. As in the case of the _C-model_, the Gaussian distribution for the entries of **X** is adopted because it accurately models several modulated signals. The time correlation introduced by the filter models the limited bandwidth of the transmitted signal, which is proportional to the symbol rate. The elements in the channel matrix **H** are zero mean i.i.d. complex Gaussian variables that simulate a flat Rayleigh fading channel between each primary transmitter and sensor (CR), assumed to be constant during a sensing period and independent from one period to another. To take into account the effect of the CR receive filters on the thermal and impulsive noises the entries in **V** and **V**IN are MA-filtered complex Gaussian variables that represent, respectively, the colored additive thermal and the impulse noise at the output of the receive filters. A normalization of filtered samples was done to guarantee the desired received signal-to-noise ratio (SNR), in dB, and the desired average IN power. Specifically, \(\textbf{X}\leftarrow\textbf{X}/\sqrt{{{{{{{\rm{}}}}}{{{{\rm{}}}}{{{\rm{}}}}{{{ \rm{}}}}{{{\rm{}}}}{{{\rm{}}}}{{{\rm{}}}}{{\rm{}}}{{\rm{}}}{{\rm{}}}{{\rm{}} }}{{\rm{}}}{{\rm{}}}{{\rm{}}}{{\rm{}}}{{\rm{}}}{\rm{}}}}}\) for unitary average received signal power, \(\mathbf{V}\leftarrow\left(\mathbf{V}/\sqrt{P_{\text{V}}}\right)\sqrt{10^{-\mathrm{ SNR}/10}}\) for an SNR-dependent average thermal noise power, and \(\mathbf{V}_{\text{IN}}\leftarrow\left(\mathbf{V}_{\text{IN}}/\sqrt{P_{\text{V}_{ \text{IN}}}}\right)\sqrt{K10^{-\mathrm{SNR}/10}}\) for an average IN power \(K\) times the thermal noise power, where the symbol \"\(\leftarrow\)\" represents the normalization process, \(P_{\text{X}}\), \(P_{\text{V}}\), and \(P_{\text{V}_{\text{IN}}}\) are the average time-series powers in \(\mathbf{X}\), \(\mathbf{V}\), and \(\mathbf{V}_{\text{IN}}\) before normalization, respectively. Moreover, to guarantee the desired received SNR, matrix \(\mathbf{H}\) is normalized so that \((1/mp)\|\mathbf{H}\|_{F}^{2}=(1/mp)\text{tr}(\mathbf{H}^{\dagger}\mathbf{H})=1\). The effect of the LNA and the AGC on the samples processed by the \(i\)-th CR, \(i=1,2,\ldots,m\), is given by the gain  
\[g_{i}=\frac{f_{\text{od}}D\sqrt{2}}{6\sqrt{\frac{1}{n}\mathbf{y}_{i}!\,\mathbf{ y}_{i}}}=\frac{f_{\text{od}}D\sqrt{2n}}{6\|\mathbf{y}_{i}\|_{2}} \tag{7}\]  
where \(\mathbf{y}_{i}\) is the \(i\)-th row of \(\mathbf{Y}\), _i.e._, the set of \(n\) samples collected by the \(i\)-th CR, and \(\|\mathbf{y}_{i}\|_{2}\) is the Euclidean norm of \(\mathbf{y}_{i}\). The reasoning behind proposing these gains is explained as follows: The combined gains of the LNA and the AGC are those that maintain the signal amplitude at the inputs of the in-phase and quadrature ADCs within their dynamic ranges \(D\).