## 3 Full-Wave Validation and Discussion
Therefore, we observe and compare the amplitudes of the spectral coherence of the source (i.e., matrix \(\mathbf{\hat{S}}\mathbf{\hat{S}}^{*}\)) and the mutual coherence of the measurement modes (i.e., matrix \(\mathbf{\hat{H}}\mathbf{\hat{H}}^{*}\)). The results are shown in Figure 3, where (a) is the amplitude of \(\mathbf{\hat{S}}\mathbf{\hat{S}}^{*}\), (b-d) are of the respective \(\mathbf{\hat{H}}\mathbf{\hat{H}}^{*}\) at the imaging range of 1 m, 2 m and 4 m.  
Figure 3 shows that the most energy of \(\mathbf{\hat{S}S}^{*}\) concentrates on the main diagonal, implying that only the measurement modes with adjacent modulation frequencies are coherent. This phenomenon proves that the spectral coherences of the source is reduced by the spatial modulation that avoids spectrum overlapping. With the linear spatial frequency modulation, the source possesses ideal coherence property. \(\mathbf{\hat{H}H}^{*}\) at different imaging ranges resembles to \(\mathbf{\hat{S}S}^{*}\), while side lobes appear symmetrically along the main diagonal. There are significant upsurges in the mutual coherences of the measurement modes with high spatial frequency modulation when the imaging range is increased. The result suggests that the mutual coherences in the measurement matrix is dependent on the spectral coherences of the sources, while the high spatial frequency components' attenuation that is imposed by shrinking low-pass area in the k-space increases the coherences. Figure 2: The low-pass filter demonstrated by the comparison in the k-space: (**a**) \(|\widetilde{J}_{\mathrm{m}}|\) and corresponding spectrum of measurement modes at the imaging range of (**b**) 1 m, (**c**) 2 m and (**d**) 4 m.  
In previous simulations, only one pair of the matrix \(\mathbf{H}\) and \(\mathbf{S}\) (namely one sample) is generated for validation. In the next, we need to obtain massive samples in which the spectrum of the source goes through the entire low-pass area in the k-space in order to prove that the proposed coherence relationship is still valid in general situation. Therefore, we perform the third numerical simulation in which a set of samples is obtained. In this simulation, the mutual coherences in \(\mathbf{H}\) and \(\mathbf{S}\) are evaluated by observing the average mutual coherence \(\mu^{\mathrm{T}}\) that is calculated in rows of the matrix (see Equation (15)). For comparison, the average mutual coherence \(\mu\) that is calculated in columns [25]:  
\[\mu(\mathbf{H})=\frac{\sum_{i\
eq j}\left|\left[\mathbf{\hat{H}^{*}\hat{H}} \right]_{ij}\right|^{2}}{N(N-1)} \tag{25}\]  
is also observed. \(\mu^{\mathrm{T}}\) and \(\mu\) represent the average level of the coherences in a matrix and thus allow evaluating the coherences more comprehensively. In this simulation, the source is modulated by the random spatial frequencies that are varying in the k-space. In order to obtain sufficient samples, 200 groups of modulation frequencies are generated via Monte Carlo method, while in each group the modulation frequencies are randomly distributed in a low-pass area of which the cut-off spatial frequency is also randomly selected between 0 and the one of the low-pass area at the imaging range of 1 m. If the proposed coherence relationship in matrix form (i.e., Equation (23)) holds for these samples, the average mutual coherences of matrix \(\mathbf{H}\) and matrix \(\mathbf{S}\) would be in positive relationship. The results are shown in Figure 4, where (a) demonstrates the relationship \(\mu^{\mathrm{T}}(\mathbf{H})\leftarrow\mu^{\mathrm{T}}(\mathbf{S})\) and (b) \(\mu(\mathbf{H})\leftarrow\mu(\mathbf{S})\):  
Figure 3: Comparison of mutual coherence: (**a**) \(\left|\mathbf{\hat{S}S}^{*}\right|\) and corresponding \(\left|\mathbf{\hat{H}H}^{*}\right|\) at the imaging range of (**b**) 1 m, (**c**) 2 m and (**d**) 4 m. Linear spatial frequency modulation (from \(-30\pi\) to \(30\pi\) rad/m) is applied to the source.