# Dynamic Correction of the Influence of Long Measuring Path Irregularity in Antenna Tests
## 2 Mathematical Model of a Long Microwave Path  
When investigating the output signal of various transmission systems, it is necessary to take into account all the factors affecting their stability over time. In general, errors occur as a result of combining two types of processes, namely slow and fast. A slow process is associated with a variation in electrical length of the signal path due to both environmental conditions and the electrical parameters drift of its component parts. In turn, fast process is associated with thermal, shot and flicker fluctuations. Let us compare errors at the output of a long microwave path based on the theory of network synchronization before and after applying the correction [15]. To do that, we write output signal of the signal source as:  
\[\dot{S}(t)=A(t)sin\ \dot{\Phi}(t),\]  
\[A(t)=A+\delta A(t),\]  
where \(\delta A(t)\) is an amplitude variation of the signal source output, relative to some fixed value \(A\) (amplitude fluctuations); \(\dot{\Phi}(t)\) is a phase at the signal source output. Since the phase at the output of the signal source changes with time, we will denote the function describing the behavior of the phase in time by the term \"phase process\". Next, we write the function of the instantaneous angular frequency of the signal source [16]. \[\dot{\omega}_{G}(t)=\omega_{0}+\sum_{k=0}^{M-1}\frac{Q(k)}{k!}t^{k}+\ \dot{ \dot{\xi}}(t), \tag{1}\]  
where \(\omega_{0}\) is the nominal value of the natural angular frequency of the signal source; \(Q(0)\) is a random value with a zero mean that expresses the initial frequency error. This error occurs due to uncertainty in the initial setting of the source's natural frequencies (setting error); \(Q(k)\), (\(k=1,\dots,\ \mathrm{M-I}\)) is a set of time-independent random variables that simulate frequency drifts of \(k\)-th order (long-term instability); \(\dot{\xi}(t)\) is a stationary random process with zero mean, describing short-term instabilities of the signal source. The phase process of the signal source is found as a result of integration (1) on interval [0, \(t\)]:  
\[\dot{\Phi}_{G}(t)=\Phi(0)+\omega_{0}t+\sum_{k=1}^{M}\frac{Q(k-1)}{k!}t^{k}+ \left[\dot{\xi}(t)-\dot{\xi}(0)\right], \tag{2}\]  
where \(\left[\dot{\xi}(t)-\ \dot{\xi}(0)\right]\) is generally a non-stationary stochastic process that describes short-term instabilities of the reference signal source (phase noise). Consider two types of long signal paths--without error correction and with full error correction. The first type includes, for example, passive systems for signal routing in the form of a series-parallel combination of power dividers and transmission lines [17], as well as test cables between the measuring equipment and AUT during antenna tests [18]. Based on (2), let us write the phase process at the output of any \(v\)-th channel of such a system:  
\[\dot{\Phi}_{v}(t)=\Phi(0)+\omega_{0}t+\sum_{k=1}^{M}\frac{Q(k-1)}{k!}t^{k}+ \left[\dot{\xi}(t)-\dot{\xi}(0)\right]+\beta l_{v}(t)+N_{kv}(t), \tag{3}\]  
where \(\beta l_{v}(t)=L_{v}(t)\) is electrical length of a given channel; \(N_{kv}\)--additive thermal noise of \(v\)-th channel.