# A Study on the Amplitude Comparison Monopulse Algorithm
## 3 Least-Squares-Based Amplitude Comparison Monopulse Algorithm
### Sum-Difference Measurement-Based Algorithm  
We can solve the estimation problem of azimuth and elevation angle deviations using the sum (\(\Sigma\)), horizontal difference (\(\Delta_{az}\)), and vertical difference (\(\Delta_{el}\)) signals, which are used in the conventional monopulse algorithm. Utilizing these signals is more useful because the same circuitry as the conventional one can be used. The measurement equations can be constructed as follows:  
\[ \begin{array}{c}\Sigma=A[g_{1}+g_{2}+g_{3}+g_{4}]+n_{1}+n_{2}+n_{3}+n_{4}\\ \Delta_{az}=A[g_{1}+g_{4}-(g_{2}+g_{3})]+n_{1}+n_{4}-n_{2}-n_{3}\\ \Delta_{el}=A[g_{1}+g_{2}-(g_{3}+g_{4})]+n_{1}+n_{2}-n_{3}-n_{4}\end{array}  \tag{26}\]  
Let the vector of the measured signals be \(Z=\left[ \begin{array}{cc}\Sigma&\Delta_{az}&\Delta_{el}\end{array} \right]^{T}\); then, Equation (26) can be written in matrix form compactly as follows:  
\[Z=Ah_{sd}(\theta_{az},\theta_{el})+Bn \tag{27}\]  
where  
\[h_{sd}=\left[ \begin{array}{c}g_{1}+g_{2}+g_{3}+g_{4}\\ g_{1}+g_{4}-(g_{2}+g_{3})\\ g_{1}+g_{2}-(g_{3}+g_{4})\end{array} \right],\quad B=\left[ \begin{array}{cccc }1&1&1&1\\ 1&-1&-1&1\\ 1&1&-1&-1\end{array} \right] \tag{28}\]  
Similar to Section 3.1, by Taylor series expansion about some nominal values (\(\theta_{az0},\theta_{ed0}\)), the linearized measurement equation becomes  
\[\begin{array}{c}\left[ \begin{array}{c}\delta\Sigma\\ \delta\Delta_{az}\\ \delta\Delta_{el}\end{array} \right]\approx B\left[ \begin{array}{c}n_{1}\\ n_{2}\\ n_{3}\\ n_{4}\end{array} \right]\\ +\frac{4\log 2}{\theta_{aa}^{3}}A\left[ \begin{array}{c}\Big{(}\frac{ \partial_{z}}{\partial z}-\partial_{az}\Big{)}(g_{1}+g_{4})-\Big{(}\frac{ \partial_{z}}{\partial z}+\partial_{az}\Big{)}(g_{2}+g_{3})&\Big{(}\frac{ \partial_{z}}{\partial z}-\partial_{a}\Big{)}(g_{1}+g_{2})-\Big{(}\frac{ \partial_{z}}{\partial z}+\partial_{az}\Big{)}(g_{3}+g_{4})\\ \Big{(}\frac{\partial_{z}}{\partial z}-\partial_{az}\Big{)}(g_{1}+g_{4})+ \Big{(}\frac{\partial_{z}}{\partial z}+\partial_{az}\Big{)}(g_{2}+g_{3})& \Big{(}\frac{\partial_{z}}{\partial z}-\partial_{a}\Big{)}(g_{1}-g_{2})+ \Big{(}\frac{\partial_{z}}{\partial z}+\partial_{az}\Big{)}(g_{3}-g_{4})\\ \Big{(}\frac{\partial_{z}}{\partial z}-\partial_{az}\Big{)}(g_{1}-g_{4})- \Big{(}\frac{\partial_{z}}{\partial z}+\partial_{az}\Big{)}(g_{2}-g_{3})& \Big{(}\frac{\partial_{z}}{\partial z}-\partial_{a}\Big{)}(g_{1}+g_{2})+ \Big{(}\frac{\partial_{z}}{\partial z}+\partial_{az}\Big{)}(g_{3}+g_{4})\\ \end{array} \right] \begin{array}{c}\delta\theta_{az}\\ \delta\theta_{d}\end{array}  \tag{29}\]  
which can be written in symbolic form as  
\[\delta Z\approx H_{sd}\delta\theta+Bn \tag{30}\]The least-squares solution can be found by minimizing the following functional:  
\[J\Big{(}\delta\hat{\theta}^{sd}\Big{)}=\frac{1}{2}\Big{(}\delta Z-H_{sd}\;\delta \hat{\theta}^{sd}\Big{)}^{T}\Big{(}\delta Z-H_{sd}\;\delta\hat{\theta}^{sd}\Big{)} \tag{31}\]  
The solution is  
\[\delta\hat{\theta}^{sd}=\Big{(}H_{sd}^{T}H_{sd}\Big{)}^{-1}H_{sd}^{T}\delta Z \tag{32}\]  
Similar to the previous method, iteration can be used to estimate the unknown target location. **4. Relationships with the Conventional Monopulse Algorithm**  
If we assume that the unknown target location is near the center of track axis, i.e., \(\theta_{az}^{2}+\theta_{cl}^{2}\approx 0\), then matrix \(\Big{(}H_{full}^{T}H_{full}\Big{)}^{-1}\) of Equation (24) is approximately computed by  
\[\Big{(}H_{full}^{T}H_{full}\Big{)}^{-1}\approx\frac{\theta_{3 dB}^{4}}{32{(\log 2 )}^{2}\;\theta_{s}^{2}\;{(g(\theta_{s}))}^{2}A^{2}}\;I_{2} \tag{33}\]  
where \(I_{2}\) is a \(2\times 2\) identity matrix. Therefore, with the nominal values of \(\theta_{az0}=0\) and \(\theta_{el0}=0\), the full measurement-based algorithm given by Equation (24) becomes approximately  
\[\delta\hat{\theta}^{full}\approx\frac{\theta_{3 dB}^{2}}{8\sqrt{2}\log 2\; \theta_{s}\;{g(\theta_{s})}A}\left[ \begin{array}{ccc}1&-1&-1&1\\ 1&1&-1&-1\end{array}