# Performance Analysis of Amplitude Comparison Monopulse Direction-of-Arrival Estimation
## Appendix A The Explicit Expression of the MSEs Based on the Taylor Series  
In this appendix, the received noise of four antenna-\(n_{A}\), \(n_{B}\), \(n_{C}\) and \(n_{D}\)--are defined as a normal distribution random variable. Generally, random variables \(n_{A}\), \(n_{B}\), \(n_{C}\) and \(n_{D}\) are distributed normally with mean \(\mu\) and variance \(\sigma^{2}\)[22]. \[ \begin{array}{l}n_{A}\sim N\left(\mu,\sigma^{2}\right)\\ n_{B}\sim N\left(\mu,\sigma^{2}\right)\\ n_{C}\sim N\left(\mu,\sigma^{2}\right)\\ n_{D}\sim N\left(\mu,\sigma^{2}\right).\end{array}  \tag{10}\]  
Based on (10), various statistics in (11) and (12) can be obtained:  
\[E\left[n_{k}n_{l}\right]=\left\{ \begin{array}{cc}E\left[n_{k}{}^{2}\right]&k =l\\ E\left[n_{k}\right]E\left[n_{l}\right]&\text{otherwise}\end{array} \right. \tag{11}\]  
\[ \begin{array}{l}E\left[n_{A}\right]=\mu\\ E\left[n_{A}{}^{2}\right]=\mu^{2}+{\sigma_{A}}^{2}\\ E\left[n_{A}{}^{3}\right]=\mu^{3}+3\mu{\sigma_{A}}^{2}\\ E\left[n_{A}{}^{4}\right]=\mu^{4}+6\mu^{2}{\sigma_{A}}^{2}+3{\sigma_{A}}^{4} \end{array}  \tag{12}\]  
For brevity, \(\alpha_{A}\), \(\alpha_{B}\), \(\alpha_{C}\), \(\alpha_{D}\), \(\beta_{A}\), \(\beta_{B}\), \(\beta_{C}\), \(\beta_{D}\), \(\beta_{AB}\), \(\beta_{AC}\), \(\beta_{AD}\), \(\beta_{BC}\), \(\beta_{BD}\) and \(\beta_{CD}\) are defined from the expressions in (28):  
\[ \begin{array}{l}\alpha_{A}=\alpha_{D}=\frac{\left(\theta_{3 dB}/k_{m}\right)- \theta_{\text{T}}}{4 AC_{0}},\\ \alpha_{B}=\alpha_{C}=\frac{-\left(\left(\theta_{3 dB}/k_{m}\right)+\theta_{ \text{T}}\right)}{4 AC_{0}},\\ \beta_{A}=\beta_{D}=\frac{-2\left(\left(\theta_{3 dB}/k_{m}\right)-\theta_{ \text{T}}\right)}{\left(4 AG_{0}\right)^{2}},\\ \beta_{B}=\beta_{C}=\frac{2\left(\left(\theta_{3 dB}/k_{m}\right)+\theta_{ \text{T}}\right)}{\left(4 AG_{0}\right)^{2}},\\ \beta_{AB}=\beta_{AC}=\beta_{BD}=\beta_{CD}=\frac{2\theta_{\text{T}}}{\left(4 AG _{0}\right)^{2}},\\ \beta_{AD}=\beta_{A},\;\beta_{BC}=\beta_{B}.\end{array}  \tag{13}\]  
\[\begin{array}{l}E\left[\left(\hat{\phi}^{(approrai=2)}-\phi\right)^{2}\right]\\ =\left(\mu^{2}+{\sigma_{A}}^{2}\right){\alpha_{A}}^{2}+\left(\mu^{2}+{\sigma_{ D}}^{2}\right){\alpha_{B}}^{2}+\left(\mu^{2}+{\sigma_{C}}^{2}\right){\alpha_{C}}^{2}+ \left(\mu^{2}+{\sigma_{B}}^{2}\right){\alpha_{D}}^{2}\\ +\frac{\left(\mu^{4}+6\mu^{2}{\sigma_{A}}^{2}+3{\sigma_{A}}^{4}\right)}{4}{ \beta_{A}}^{2}+\frac{\left(\mu^{4}+6\mu^{2}{\sigma_{D}}^{2}+3{\sigma_{D}}^{4} \right)}{4}{\beta_{B}}^{2}+\frac{\left(\mu^{4}+6\mu^{2}{\sigma_{C}}^{2}+3{ \sigma_{C}}^{4}\right)}{4}{\beta_{C}}^{2}+\frac{\left(\mu^{4}+6\mu^{2}{\sigma _{B}}^{2}+3{\sigma_{B}}^{4}\right)}{4}{\beta_{D}}^{2}\\ +\left(\mu^{2}+{\sigma_{A}}^{2}\right)\left(\mu^{2}+{\sigma_{D}}^{2}\right){ \beta_{AB}}^{2}+\left(\mu^{2}+{\sigma_{A}}^{2}\right)\left(\mu^{2}+{\sigma_{ C}}^{2}\right){\beta_{AC}}^{2}+\left(\mu^{2}+{\sigma_{A}}^{2}\right)\left(\mu^{2}+{ \sigma_{B}}^{2}\right){\beta_{AD}}^{2}\\ +\left(\mu^{2}+{\sigma_{D}}^{2}\right)\left(\mu^{2}+{\sigma_{C}}^{2}\right){ \beta_{BC}}^{2}+\left(\mu^{2}+{\sigma_{B}}^{2}\right)\left(\mu^{2}+{\sigma_{D }}^{2}\right)\beta_{BD}{}^{2}+\left(\mu^{2}+{\sigma_{C}}^{2}\right)\left(\mu^ {2}+{\sigma_{B}}^{2}\right){\beta_{CD}}^{2}\\ +2\mu^{2}{\alpha_{A}}{\alpha_{B}}+2\mu^{2}{\alpha_{A}}{\alpha_{C}}+2\mu^{2}{ \alpha_{A}}{\alpha_{D}}+2\mu^{2}{\alpha_{BC}}+2\mu^{2}{\alpha_{B}}{\alpha_{D} }+2\mu^{2}{\alpha_{C}}{\alpha_{D}}\\ +\frac{1}{2}\left(\mu^{2}+{\sigma_{A}}^{2}\right)\left(\mu^{2  
* (14) [PERSON] Minimum variance monopulse technique for an adaptive phased array radar. _IEE Proc. Radar Sonar Navig._**1998**, _145_, 374-380. [CrossRef]
* (15) [PERSON] Monopulse estimation with adaptive arrays. _IEE Proc. F (Radar Signal Process. )_**1993**, _140_, 303-308. [CrossRef]
* (16) [PERSON] Monopulse estimation with subarray-adaptive arrays and arbitrary sum and difference beams. _IEE Proc. Radar Sonar Navig._**1996**, _143_, 232-238. [CrossRef]
* (17) [PERSON] Performance Analysis Angle Tracking Using Adaptive Kalman Filter in Monopulse Radar.