# Dynamic Correction of the Influence of Long Measuring Path Irregularity in Antenna Tests
## 2 Mathematical Model of a Long Microwave Path
The mathematical model of the passive signal routing unit of the source signal without phase stabilization can be written in matrix representation as:  
\[\mathbf{\Phi}_{p}(t)=\mathbf{\Phi}_{G}(t)\mathbf{\gamma}+\mathbf{L}(t)+ \mathbf{N}_{k}(t), \tag{4}\]  
where \(\mathbf{\Phi}_{p}(t)=\left[\Phi_{1}(t),\;\Phi_{2}(t),\ldots\Phi_{n}(t)\right]^ {T}\) is a column vector of the output phase process; \(n\) is the total number of output channels of the system; \(\mathbf{\Phi}_{G}(t)\) is the phase process at the output, determined by (2); \(\mathbf{\gamma}=\left[\underbrace{I,I,\ldots}_{n}\right]^{T}\) is identity column vector,  
\[\mathbf{L}(t)=\left[L_{1}(t),L_{2}(t),\ldots L_{n}(t)\right]^{T}\text{, } \mathbf{N}_{k}(t)=\left[N_{k1}(t),N_{k2}(t),\ldots N_{kn}(t)\right]^{T}.\]  
Expression (4) shows that the phase process at each channel output of a passive long signal path without phase stabilization is determined by the phase process of the signal source, the channel electrical length and its equivalent thermal noise. The second class of signal paths includes transmission systems with a driving oscillator, the parameters of which are stabilized either by correction or by auto-tuning for each channel separately [19]. The phase process at the output of the \(v\)-th channel of such a system in generalized form is written as:  
\[\dot{\Phi}_{v}(t)=\dot{\Phi}_{G}(t)+\sum_{j=1}^{k}a_{vj}\Big{\{}K_{vj}Y_{vj}(p )\Big{[}g_{vj}\Big{[}\dot{\Phi}_{vj}(t)\Big{]}+\varepsilon_{vj}(t)\Big{]} \Big{\}}, \tag{5}\]  
where \(k\) is the number of steps (hierarchy levels) for distributed transmission systems; \(K_{vj}\) represents gain of the correction loop for \(v\)-th channel at \(j\)-th step; \(Y_{vj}\) denotes the transfer function of correction loop; \(g_{vj}[\ldots]\) is phase discriminator frequency response; \(\alpha_{vj}\) stands for weight coefficients of adder (in the self-correction system, \(\alpha_{vj}=I\)); \(\varepsilon_{vj}\) is the equivalent phase noise of the channel. A mathematical model of the system with a driving oscillator and a correction can be written in the form:  
\[\mathbf{\Phi}_{S}(t)=\mathbf{\Phi}_{G}(t)\mathbf{\gamma}+\mathbf{A}(t), \tag{6}\]  
where \(\mathbf{A}(t)=[A_{1}(t),A_{2}(t),\ldots A_{N}(t)]^{T}\) is a column vector containing the diagonal of a square matrix \(\mathbf{D}(t)\) of \(N\times N\) size: \(A_{v}(t)=\left.D_{vj}(t)\right|_{v=j}\). Matrix \(\mathbf{D}(t)\) is defined by the expression:  
\[\mathbf{D}  
of the correction loop, the discriminator frequency response, and the phase noise of the system, as shown through expressions (6)-(10). Comparing mathematical models of two classes of excitation systems (4) and (6), we can conclude that the accuracy of phase stabilization of the output signal in first class systems is determined by difference in electrical lengths of the channels, the phase noise of the source, as well as channels themselves. Thus, total phase error in such a system can significantly exceed tens of degrees due to unregulated changes in the path's electrical length (for example, due to changing environmental conditions or mechanical impact). In second class systems, the total error is determined by the parameters of the correction system; therefore, it is possible to minimize it using a dynamic calibration.