# Angle Of Arrival

Application of the combined near and far field models place the array in a position such that it is able to determine the degree of arrival (DOA) of the emitting source. With these angle, the array is capable of either, calculating the position of the source, or determining the angle of the position of source relative to the array. Both of these capabilities are advantageous to numerous applications and are the focus of this experiment. The combined model as mentioned in a previous section , found with the inequality, are shown in the figure below.

As the plane wave interacts with the two sensors comprising a sensor pair on the array, basic trigonometry can be utilized to find the angle to the normal of the interacting plane wave. This angle is the degree of arrival (DOA) angle the experiment is interested in. The concern in the calculation for this angle is the measurement of the time delay between the sensors. This estimate can be mathematically rigorous in derivation and is rooted in the statistical tools of Cross Correlation. Carful consideration must be maintained in that the system is of discrete time, based on samples of data and not a continues time measure. The full derivation of the angle is found here. Below is the final result of this derivation, note that the time delay is in discrete time pertaining to the number of samples occurring between successive points.

The previous calculations have been rooted in simple trigonometry. The delay estimate is obtained through the use of Generalized Cross Correlation(GCC). This is explained in more detail within the next section.