# Parameter Estimation and Error Calibration for Multi-Channel Beam-Steering SAR Systems
## 3 Estimation Methods for DRR, CDC Error, and Phase Inconsistency Errors
### Weighting for Estimation  
In the previous discussion, it was seen that the shape of the pre-processed spectrum is important in estimating DRR error, CDC error, and phase inconsistency errors. In practice, however, thermal noise may affect the pre-processed spectrum. Reference [12] demonstrates that the output noise power after reconstruction can be amplified for the case of highly non-uniform sampling. The uniformity factor \(Fu\) can be defined as  
\[F_{u}=\frac{f_{prf}}{f_{prf,uni}}{=}N\cdot d\Big{/}\left(v\Big{/}f_{prf}\right), \tag{48}\]  
where \(d\) is the effective distance between adjacent channels. In [7], the signal-to-noise ratios (SNR) scaling factor, which quantifies the amplified degree of output noise power due to filter reconstruction, is defined as  
\[\Phi_{bf}=\frac{\frac{SNR_{ht}}{SNR_{out}}}{\frac{SNR_{in}}{SNR_{out}}|_{PRF_{ uni}}}=\frac{1}{\sigma_{1}^{2}}+\frac{1}{\sigma_{2}^{2}}+\cdots+\frac{1}{\sigma_{N}^{2}}, \tag{49}\]  
Figure 8: Flowchart of the method to estimate phase inconsistency errors.  
where \(SNR_{in}\) and \(SNR_{out}\) are the SNR before and after filter reconstruction, respectively, and \(PRF_{uni}\) represents the PRF corresponding to uniform space sampling. Here \(\sigma_{1}\), \(\sigma_{2}\), \(\cdots\), \(\sigma_{N}\) are the singular values of matrix **H**(see (11)). For the uniform case (\(Fu=1\)), the SNR scaling factor is 1. When the system works with non-uniform PRF, the correlation between the adjacent row vectors in **H** will increase, which results in small singular values, so the SNR scaling factor will increase. More information on the SNR scaling factor can be found in [7].  
An example of an amplified noise spectrum is shown in Figure 9a. It can be seen the noise spectrum components at the edge of the bandwidth dominate the degradation of SNR. When the SNR is low, the noise power may exceed the signal power at the edge of the bandwidth, which will degrade the performance of the proposed estimation methods. According to [31], the normalized noise spectrum can be got by reconstruction matrix, given as  
\[S_{noise}(f+k\cdot f_{prf})=\left\|P_{k}\left(f\right)\right\|_{2}^{2}. \tag{50}\]  
Then a natural weighting function can be formulated to suppress the noise spectrum components as  
\[F_{1}\left(f\right)=\frac{1}{S_{noise}\left(f\right)}. \tag{51}\]  
Evidently, the noise spectrum can be flattened with a uniform floor by using the weighting filter in Equation (51). Since the estimation of the DRR and CDC errors is based on the maximum of the spectrum, the weighted spectrum will became more concentrated (see Figure 9b) and the effect of noise will be significantly reduced. For the phase inconsistency errors, however, the noise spectrum at the edge of the bandwidth will still reduce the estimation performance and even cause the iteration to converge to the wrong value. To tackle this problem, we modify Equation (51) as  
\[F_{2}\left(f\right)=\frac{win\left(f\right)}{S_{noise}\left(f\right)}, \tag{52}\]  
where \(win\left(f\right)\) is the Hanning window, Blackman window, or another commonly used window function. A criterion for choosing the window is that the edge sections of the spectrum cannot be weighted to 0 because all the spectrum components should be included when calculating the signal entropy. In this paper, Hamming window is chosen. The weighted spectrum becomes smoother after weighted by \(F_{2}(f)\), as shown in Figure 9c, and the noise is suppressed further, especially in the edge  
Figure 9: Weighting of spectrum for estimation. (**a**) shows the spectrum of signal, noise and noised signal after pre-processing without weighting; (**b**) is the spectrum of signal, noise and noised signal after pre-processing weighted by \(F_{1}(f)\); (**c**) is the spectrum of signal, noise and noised signal after pre-processing weighted by \(F_{2}(f)\).  
segment of the spectrum. After using the weighting function in (52), the estimation method for phase inconsistency errors will get better performance.