## 3 Digital Beamforming General Considerations
### Amplitude and Phase Errors Estimation
\[\left\langle S_{REF}S_{REF}^{*}\right\rangle=2\left\langle\left\langle I_{ REF}I_{REF}\right\rangle-j\left\langle I_{REF}\mathcal{O}_{REF}\right\rangle \right\rangle=A_{REF}^{2}(T_{CNS}+T_{W-REF}+T_{receiver-REF})\,, \tag{4a}\]\[\left\langle S_{{}_{RUF}}S_{{}_{RUF}}^{{}^{*}}\right\rangle =2\left\langle\!\left\langle I_{{}_{RUF}}I_{{}_{RUF}}\right\rangle\!-j \left\langle I_{{}_{RUF}}Q_{{}_{RUF}}\right\rangle\!\right\rangle\!=A_{{}_{RUF}}^ {2}(T_{{}_{CNS}}+T_{{}_{W-RUF}}+T_{{}_{receiver-RUT}})\,, \tag{4b}\] \[\left\langle S_{{}_{REF}}S_{{}_{RUF}}^{{}^{*}}\right\rangle =2\left\langle\!\left\langle I_{{}_{REF}}I_{{}_{RUT}}\right\rangle\!-j \left\langle I_{{}_{REF}}Q_{{}_{RUF}}\right\rangle\!\right\rangle\!=A_{{}_{REF}} A_{{}_{RUF}}e^{j(\phi_{{}_{REF}}-\phi_{{}_{RUF}})}\,T_{{}_{CNS}}\,,\] (4c) \[A_{{}_{Cal-RUT}} =\sqrt{\left\langle\!\left\langle S_{{}_{REF}}S_{{}_{REF}}^{{}^{*}} \right\rangle\!\right\rangle\over\left\langle S_{{}_{RUF}}S_{{}_{RUF}}^{{}^{*} }\right\rangle}\,, \tag{5}\]  
and  
\[\mathrm{A}\,\phi_{{}_{Cal-RUT}}=\mathrm{atan}\frac{-\left\langle I_{{}_{REF}} Q_{{}_{RUF}}\right\rangle}{\left\langle I_{{}_{REF}}I_{{}_{RUT}}\right\rangle}\,, \tag{6}\]  
where:  
* \(\left\langle\,\right\rangle\) is the expected value operator,
* \(T_{{}_{CNS}}\) is the noise equivalent temperature of the _CNS_ signal injected at receivers' input;
* \(T_{{}_{W-REF}}\) and \(T_{{}_{W-RUT}}\)are the physical temperatures of the [PERSON] power splitter of the reference and the RUT receivers;
* \(T_{{}_{receiver-REF}}\) and \(T_{{}_{receiver-RUT}}\) are the receiver's noise temperatures of the reference and the RUT receivers;
* \(I_{{}_{REF}}\) is the real part of the analytic \(S_{{}_{REF}}\) signal (\(S_{{}_{REF}}\)= \(I_{{}_{REF}}\)+\(j\)\(Q_{{}_{REF}}\));
* \(I_{{}_{RUT}}\) and \(Q_{{}_{RUT}}\), are the real and imaginary parts of the analytic \(S_{{}_{RUT}}\)signal (\(S_{{}_{RUT}}\)= \(I_{{}_{RUT}}\)+\(j\)\(Q_{{}_{RUT}}\)),
* \(A_{{}_{Cal-RUT}}\) and \(\phi_{{}_{Cal-RUT}}\) are the estimated relative amplitude and phase between the reference receiver and the \(RUT\). However, the implementation of Equations (4-6) is prone to errors induced by:  
* the algorithm assumption that \(T_{{}_{W-REF}}\) = \(T_{{}_{W-RUT}}\) and \(T_{{}_{receiver-REF}}\) = \(T_{{}_{receiver-RUT}}\) to compute the amplitude which is not necessarily exactly true;
* correlated noise terms generated by the noise distribution network between the _CNS_ and the receivers' inputs that may introduce offset terms in Equations (4(a-c)); and
* phase and amplitude unbalances in the noise distribution network. A differential version of this technique was developed to minimize these effects. The cross-correlations are computed for two different power inputs \(T_{{}_{CNS}}\), which are subtracted before estimating the errors. The procedure results are shown in the following Equations (7-9):  
\[\left\langle S_{{}_{REF}}S_{{}_{REF}}^{{}^{*}}\right\rangle_{{}_{T_{max}}}=A_ {{}_{REF}}^{2}\,T_{{}_{CNS}}{}_{{}_{T_{max}}}+\mathit{offset}\,, \tag{7a}\] \[\left\langle S_{{}_{REF}}S_{{}_{RUT}}^{{}^{*}}\right\rangle_{{}_{T_{max}}}=A_ {{}_{REF}}\,A_{{}_{RUF}}e^{j(\phi_{{}_{REF}}-\phi_{{}_{RUF}})}\,T_{{}_{CNS}}{}_{{ }_{T_{max}}}+\mathit{offset}\,,\] (7b) \[A_{{}_{Cal-RUT}}=\sqrt{\left\langle S_{{}_{REF}}S_{{}_{REF}}^{{}^{*}}\right\rangle _{{}_{T_{max}}}-\left\langle S_{{}_{REF}}S_{{}_{REF}}^{{}^{*}}\right\rangle_{{}_{T_{ max}}}} \tag{8}\]  
and\[S_{RUT} =A_{RUT}\left(S_{CNS}+S_{W-RUT}\right)e^{j_{RUT}}+S_{receiver- RUT}\,, \tag{3}\]0  
where:  
* _offset\({}_{1}\)_ and _offset\({}_{2}\)_ include all the terms in Equations 4, different to \(T_{\mathit{CNS}}\) or any other possible offset due to unbalances or correlated noise. If the values of \(T_{\mathit{max}}\) and \(T_{\mathit{med}}\) are close, the estimated amplitude and phase are very sensitive to estimation errors (finite radiometric resolution). At the limit, when \(T_{\mathit{max}}=T_{\mathit{med}}\), the result is not determined.