Observations
During the course of the experiment, several key ideas were observed. Originally the experiment was unable to estimate any positions accurately. The first attempts at determining an angle always found the angle to be at one of three positions. The angle to these positions was consistently 180, 0, or 90 degrees. At first we had no explanation to this. As we modified the system we started to find new angles however, the results were very strange. We would run the experiment and calculate a position. Then we would move the source to a new position and the array would calculate the exact same position as before. At first this was very frustrating. We had no viable explanation as to why this was occurring. In order to get to the root of the problem we began to use the simulation feature that we designed so as to understand what was happening on the fundamental level. This trip back to the root of the concepts was exactly what the experiment needed. What we discovered was that there was an effect due to the discrete time delay that we had previously overlooked. In order to understand this effect we turn back to the equation derived for the degree of arrival (DOA).
We define the following value as:
So that,
The boundary for inverse cosine is defined as,
Additionally, since both the sample rate and 'd' are finite values we can say that alpha must be greater than 0. We then observe the boundary for the inverse cosine to be:
Since the number of samples is an integer quantity, we only consider the lowest integer values of the right and left sides of the inequality. This leads to a finite number of possible angles that the system can calcuate based on given parameters. Each point position estimation is found at a point of intersection between two angles:
Making the total number of possible points for a given configuration:
