|
Error Analysis The error analysis of the system showed how small errors within the system affected the results of the resolution for source localization. By doing this, we gained a complete understanding of the behavior of our system and more accurately predicted the resolution. Furthermore, we knew the extent of accuracy in the mechanical devices would have an overall affect on the total resolution of the system, and in addition to resolution response, we gained a better perspective on how we wanted to control the microphone array. To begin analyzing the error, a complete understanding of the system’s behavior must be reached. For a given operational point or distant and with constant input factors such as the speed of sound and sampling frequency, a graph of the angles of resolution could be generated. These angles are given by:
where vs is the speed of sound, 340 m/s, fs is the sampling frequency, and d is the operational distance. m’ establishes the total number of angles on discrete intervals. This interval is defined on the following range.
Where gamma is given by,
The graph below shows the relationship between gamma and the operational position on the discrete intervals for m’.
The graph also shows how the different sampling frequencies affect the size of each interval, meaning for higher frequencies the operational point must lie on a short distance range to produce a given number of resolution angles. This becomes even more important when error is introduced in the system because if the desired operational point is not reached, either too short or too long, then this could result in a change in the resolution angles produced. Moreover, only error (ε) defined by the following equation will have a dramatic effect on the number of resolution angles.
A critical error of this type will significantly affect the total number of resolution angles. Notice, error less than the step size would not be significant enough to change to the total number of angles produced. By looking at the three possible cases – the desired operational point, an overshoot, and an undershoot, the behavior of the resolution angles can be determined.
The graph above shows the behavior of the resolution angle when error is introduced with the red data points representing no error, the blue data points representing an overshoot and the green data is an undershoot. It can be seen how error changes the total number of resolution angles at a particular operational distance. Furthermore, if the critical error gives an undershoot the total number of angles will decrease by 2; likewise, for a critical error which provides overshoot the total number of angles will increase by 2. The location of the operational distance on the step size range – defined by [xo, xo + stepsize] – has a direct impact on the resolution angles at a given discrete interval. The resolution angles on a discrete interval, defined by the distances between 0.043 – 0.050m for the next graph, show the how total resolution becomes more focused or dense as the operational distance moves toward the interval’s extreme. The resolution angles are widely spaced at a distance of 0.043m; however, as the operational point closes upon 0.050m the resolution angles begin to centralize and become more focused. This general analysis of error with respect to resolution becomes more important when considering the error produced by the mechanical robot. Based on the results of this analysis, a significant amount of error has been introduced into the system by way of the robotic devices. The small error of the mechanical robots will have a dramatic impact on the number of resolution angles for the acoustic sensors. From these results, it was recommended that the distance between pairs of sensors be held fix and the sampling frequency be fixed while moving forward with this project. |
Washington University in St. Louis
Department of Electrical and Systems Engineering
|
|
Robotic Microphone Sensing: Optimizing Source Estimation and Algorithms for Adaptive Control |
|
Chase LaFont - Undergraduate Research Summer/Fall ‘09 |
