# Dynamic Correction of the Influence of Long Measuring Path Irregularity in Antenna Tests
## 4 Errors in Measuring the Electrical Length
### Influence of Microwave Discontinuities a Long Transmission Path on the Node Shift of the Mixed Wave  
A long microwave path contains a number of elements that create stray reflections along it, and neither the true parameters of each irregularity, nor their exact location are known in advance. A statistical approach is used to study the effect of microwave discontinuities on the input reflection coefficient of the transmission path. Each \(i\)-th random element of the transmission path has a reflection coefficient \(\dot{\rho}_{i}=\rho_{i}e^{i\varphi_{i}}\) where \(\rho_{i}\) and \(\varphi_{i}\) are random values of the magnitude and phase of the reflection coefficient, respectively. The random nature of the input reflection coefficient \(\dot{\rho}\) is the result of cumulative effects of a large number of reflections from equivalent input irregularities of the path. When assessing the effect of microwave discontinuity on the input reflection coefficient of a path, only the magnitude of its reflection coefficient is essential. Each discontinuity along the path affects a node shift of the mixed standing and travelling wave \(\sin 180^{\circ}\frac{\Delta l_{i}}{\lambda}=\rho_{i}\), where \(\Delta l_{i}\) is the node shift of single irregularity; \(\lambda\) is an operating wavelength [20]. Thus,  
\[\Delta l_{i}=\frac{\lambda}{180^{\circ}}\text{arcsin}\rho_{i}, \tag{18}\]where \(\lambda\) is a deterministic, and \(\rho_{i}\) is a random quantity having the Rayleigh distribution [28; 29; 30]. Since \(\Delta l_{i}\) and \(\rho_{i}\) are linked through the inverse trigonometric function, the distribution law of \(\Delta l_{i}\) cannot be precisely determined. It is necessary to solve the problem using an approximate method. Let us analyze the resulting displacement of the mixed wave \(\Delta l\) at the path input. In this case, one can make the assumption that the \(\Delta l\) value has a probability distribution close to normal and verify this by constructing a quantile--quantile (q-q) plot. To obtain statistical data, the magnitude of the reflection coefficient \(\rho\) was measured at the long path input--a coaxial cable 20 m long, terminated with mismatched load. The measurements were made at frequencies of 1, 5, and 9 GHz. The conditions for the cable movement along with the turntable rotation were reproduced and 100 samples at each frequency were taken. Histogram plots of the \(\arcsin\rho=\frac{\Delta l}{\lambda}180^{\circ}\) value for each frequency were constructed (Figure 4a-c). The histogram presents density of \(\arcsin\rho\) values (i.e., number of hits \(r\) of the values in the corresponding interval; y-axis) over chosen intervals (x-axis). By the type of histogram dependence, one can make an assumption about the distribution law of the investigated quantity. In this case, the histograms in Figure 4a-c allow one to assume a normal law of \(\arcsin\rho\). Figure 4: Histograms of the mixed wave node shift: (**a**) 1 GHz, (**b**) 5 GHz, (**c**) 9 GHz. In order to prove this, it is necessary to construct q-q plots of the random variable arcsinp. The quantile plot is built on the normal probability paper and presents the investigated quantity arcsin\(\rho\) (y-axis) over the cumulative frequency (x-axis). The plotted data points (Figure 5a,c) are arranged so that a straight line can be drawn through them. A straight line drawn on normal probability paper represents a normal distribution. Therefore, considering the deviations of the scatter diagram from a straight line as a result of random influences, we can assume the distribution of arcsin\(\rho\) is close to normal. It can be seen that the deviations of the points from the theoretical line are less noticeable for the central frequencies (close to 0.5) than at the edges (close to 0 or 1). That is, when plotting a straight line, which is a theoretical estimate, the end points are less important than the mid-range ones. The q-q plot allows you to determine the main characteristics of the investigated probability distribution.