Simulation Results

$\parbox{120mm}{ \centering \setlength{\unitlength}{1mm} \begin{pictu... ...pproximation) CDFs of the test-statistic $V$ in a linear scale. \vspace{5mm} }$

To ensure a very high accuracy of zero approximation the CDFs of the test-statistic $V$ were constructed for a 5-element antenna array with 5, 10 and 15 samples, by means of numerical simulation experiment and the suggested approximation method. Figure 1 shows the CDFs calculated on the base of zero approximation ($F_V^0(x)$-dashed curves) and the experimental CDFs based on 100,000 random realizations ($F_V^{exp}(x)$-solid curves). As seen in Figure 1a, in a linear scale the curves are practically indistinguishable. Representation of data in a logarithmic scale (see Figure 1b) allows to estimate the high accuracy of the zero approximations of the CDFs. The detailed investigation of the CDF approximation accuracy was carried out by numerical simulation for various numbers $n = 3,4,5,6,7$ of terms in the expansion 13. Figure 2 depicts the absolute errors of the CDF approximation $\Delta F_V (x)=F_V^{exp} (x)-F_V ^{app} (x)$ for the 5-element antenna array with number of samples $N = 10, 15$.

$\parbox{120mm}{ \centering \setlength{\unitlength}{1mm} \begin{pictu... ...imation) CDFs of the test-statistic $V$ in a logarithmic scale. \vspace{5mm} }$

$\parbox{120mm}{ \centering \setlength{\unitlength}{1mm} \begin{pictu... ...ions ($n=0,3,4,5,6,7$) for CDFs of the test-statistic $V$ for $p=5$, $N=10$. }$

$\parbox{120mm}{ \centering \setlength{\unitlength}{1mm} \begin{pictu... ...,5,6,7$) for CDFs of the test-statistic $V$ for $p=5$, $N=15$. \vspace{5mm} }$

By comparison of Figure 1 and Figure 2, it is seen that the zero approximation allow the evaluation of the CDFs with the absolute accuracy  $\Delta F_V (x) < 0.003$ in the whole definition interval [0,1].

The threshold $V_{th}$ calculation accuracy of the suggested approximation method was investigated too. Firstly, on the experimental CDF $F_V^{exp}(x)$ the threshold values $V_{th}$ of the test-statistic $V$ for the given probabilities of false alarm $P_{FA}\simeq 0,01; 0.05; 0.1$ had been found. Then, for this threshold values $V_{th}$ the false alarm probabilities $P_{FA}^{app}$ were analytically calculated by employing the approximating series 13 of various lengths $n = 0,3,4,5,6,7$. The errors of false alarm probability approximation $\Delta P_FA=P_{FA}-P_{FA} ^{app}$ are represented in the two tables (Table I for $N=10$ and Table II for $N=15$) placed below. By comparing these two tables, it is seen that errors of approximation decrease as the number of samples N increases.

For the worst case $N=10$, it is seen (from Table I) that the zero approximation allows to determine $P_{FA}$ with an accuracy better then $22\%$. The using of the approximating series 13 with four terms (taking into account four test-statistic moments) and six terms (taking into account six test-statistic moments) allows to calculate $P_{FA}$ with accuracy better then $5.5\%$ and $0.4\%$ respectively.

$\parbox{135mm}{ Table I. Errors of $P_{FA}$ approximation for N=10... ...6 &\scriptsize1.7 &\scriptsize0.36 &\scriptsize-0.13 \\ \hline \end{tabular}}$

 

$\parbox{135mm}{ Table II. Errors of $P_{FA}$ approximation for N=1... ...\scriptsize0.84 & \scriptsize0.84 & \scriptsize0.896 \\ \hline \end{tabular}}$