Probability Bound of ML DOA Spectrum at Specific Search Azimuth Being the Largest of the ML Spectra at All Search Azimuths

For estimation of azimuth using the ML DOA estimation algorithm, the bound of the probability that the spectrum at specific search angle is greatest of the spectra at all search angles is derived. A random variable is defined as the difference of the two ML spectra at two different angles, and the probability density function (PDF) of the random variable is evaluated via the characteristic function of the random variable. The probability that the random variable is positive is calculated from the numerical integration of the PDF of the random variable. Finally, these probabilities for appropriate two search angles are combined to get the lower and the upper bound that the ML spectrum at specific grid is the greatest of spectra at all search angles. Since the probability itself can be evaluated via the Monte-Carlo simulation, the derived expression can be validated by checking that the probability itself from the Monte-Carlo simulation is actually between the derived lower bound and the upper bound of the probability.