# Parameter Estimation and Error Calibration for Multi-Channel Beam-Steering SAR Systems
## 4 Experiments and Discussion
### Point Targets Simulation Experiments  
The parameters in Table 1 are used in the simulation experiment. The DRR error, CDC error, and phase inconsistency errors are shown in Table 2. The effects of these errors are first simulated. The SNR is set as 20 dB in this experiment. The Doppler spectrum of multi-channel TOPSAR after pre-processing with different kinds of errors are shown in Figure 10 while the Doppler spectrum of multi-channel sliding spotlight is similar and omitted for briefness. However, the estimation results for both modes are shown in Table 2. Then the imaging results for both modes are presented in Table 3.  
From Figure 10, it can be seen the azimuth spectrum with DRR error is slightly widened since the energies from targets located at different azimuth positions are slightly separated due to the'symmetric shift' effect. As a result, some energy leaks out of the azimuth bandwidth and the amplitude near zero frequency is lower than that without error. The azimuth spectrum with CDC error is shifted because the constant Doppler error leads to a residual shift in the Doppler domain. The effect of phase inconsistency errors is more serious: the amplitudes of different sub-bands \(U_{a}(m+n\cdot f_{prf};l)\) are increased or decreased and the spectrum is discontinuous at the boundaries of adjacent sub-bands. The spectrum power outside the 3 dB bandwidth also clearly increases. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Errors**} & \multicolumn{4}{c}{**Multi-Channel TOPS**} & \multicolumn{4}{c}{**Multi-Channel Sliding Spotlight**} \\ \cline{2-7}  & **True Value** & **Estimated Value** & **Error** & **True Value** & **Estimated Value** & **Error** \\ \hline DRR error & \(530.8\) Hz/s & \(531.15\) Hz/s & \(0.35\) Hz/s & \(70.6\) Hz/s & \(70.1\) Hz/s & \(0.5\) Hz/s \\ CDC error & \(100\) Hz & \(100.24\) Hz & \(0.24\) Hz & \(100\) Hz & \(100.32\) Hz & \(0.32\) Hz \\ PI error (Channel 1) & \(0^{\circ}\) & \(0^{\circ}\) & \(0^{\circ}\) & \(0^{\circ}\) & \(0^{\circ}\) & \(0^{\circ}\) \\ PI error (Channel 2) & \(30^{\circ}\) & \(29.89^{\circ}\) & \(0.11^{\circ}\) & \(30^{\circ}\) & \(29.87^{\circ}\) & \(0.13^{\circ}\) \\ PI error (Channel 3) & \(-24^{\circ}\) & \(-23.85^{\circ}\) & \(0.15^{\circ}\) & \(-24^{\circ}\) & \(-24.05^{\circ}\) & \(0.05^{\circ}\) \\ PI error (Channel 4) & \(24^{\circ}\) & \(23.92^{\circ}\) & \(0.08^{\circ}\) & \(24^{\circ}\) & \(23.89^{\circ}\) & \(0.11^{\circ}\) \\ PI error (Channel 5) & \(-10^{\circ}\) & \(-9.98^{\circ}\) & \(0.02^{\circ}\) & \(-10^{\circ}\) & \(-9.97^{\circ}\) & \(0.03^{\circ}\) \\ PI error (Channel 6) & \(5^{\circ}\) & \(4.89^{\circ}\) & \(0.11^{\circ}\) & \(5^{\circ}\) & \(5.12^{\circ}\) & \(0.12^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experiments results of CDC error, DRR error, and the phase inconsistency errors. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Multi-Channel**} & \multicolumn{4}{c}{**Azimuth**} & \multicolumn{2}{c}{**Range**} & \multirow{2}{*}{**Normalized**} \\ \cline{2-2} \cline{4-7}  & **Resolution (m)** & **PSLR (dB)** & **ISLR (dB)** & **Resolution (m)** & **PSLR (dB)** & **ISLR (dB)** & **Amplitude** \\ \hline No Error & \(3.35\) & \(-13.27\) & \(-10.06\) & \(3.33\) & \(-13.27\) & \(-10.04\) & \(1.00\) \\ DRR Error & \(3.46\) & \(-13.24\) & \(-10.67\) & \(3.33\) & \(-13.26\) & \(-10.04\) & \(0.98\) \\ CDC Error & \(4.40\) & \(-12.68\) & \(-15.07\) & \(4.24\) & \(-24.61\) & \(-23.06\) & \(0.41\) \\ PI errors & \(3.35\) & \(-11.26\) & \(-8.49\) & \(3.33\) & \(-13.28\) & \(-10.04\)These impacts are more obvious after antenna correction (Figure 10b). The Doppler spectrum is deformed by DRR error and becomes tilted due to CDC error. Please note that the shape of the overall spectrum with DRR error is not the same as for arbitrary targets because the spectra of point targets located at different positions with DRR error are not located in same spectral interval.