## 1 Introduction
Although many large display vibrotactile systems attempt to effectively produce feedback throughout the display, there are often dead zones with no available feedback [18]. This is largely attributed to the edge boundary conditions used to secure touch displays. For example, a clamped boundary condition gives little control over the display's vibration mode shape without drastically changing the actuator's excitation frequency. Some of the most important factors that are desired for haptic localization are the ability to generate a response throughout the system, the ability to control the duration and magnitude level of the response, minimizing the number of actuators used, and having a quick response time. Currently, controlling localized vibrotactile sensations in displays is done primarily using wave focusing, eigenfunction superposition, and traveling wave control methods. A study by [PERSON] et al. showed that time-reversal wave focusing can be used to control displacement impulses at one or several locations simultaneously [18]. This control method produces impacts in the supersonic range, well out of the range of vibrotactile sensitivity. However, it was shown that finger stimulation is still achieved through the ejection of the finger from the screen surface, which results in a larger finger displacement and duration compared to the screen. Lastly, this technique requires a large number of actuators for proper implementation and can only control pulse displacements. An alternative method to wave focusing, called eigenfunction superposition, can also be used to create vibrotactile localization. This technique involves controlling modal coefficients and actuator weightings to render a targeted vibration distribution pattern. A study by [PERSON] et al. describes using this principle to control the vibration in targeted \"hotzones\" and suppress it in the other \"cold zones\" [19]. This method takes 18.2 ms to calculate the actuator weightings, showing a moderate response time. This study shows promise for the use of eigenfunction superposition for controlling vibration distribution patterns when dividing the screen up into large quadrants or grids but lacks the ability for high-resolution localization, which is the primary drawback of this control method. In addition, a large number of actuators were used to effectively utilize the eigenfunction superposition control method. Although eigenfunction superposition has important applications, the large number of actuators required and low-resolution localization make it not ideal for large touchscreen applications. Another method used to create haptic localization is using traveling waves. [PERSON] et al. also generated traveling waves as a means to create a targeted vibration distribution pattern [19]. They found that compared to eigenfunction superposition, using traveling waves was about 10% more successful (according to their metric) and had a 5.9 ms calculation time, roughly one-third as long. However, it was found that eigenfunction superposition uses approximately four times less power compared to the traveling wave control method. Using traveling wave control methods, it can be quite difficult to create long-duration vibrotactile feedback sensations. Similar to the superposition method, this method does not provide a high localization resolution despite also using a large number of actuators. A study by [PERSON] et al. showed that eigenfunction superposition could be used to create an ultrasonic traveling wave to create variable friction in a vibrotactile display using only two transducers [20]. Traveling wave control methods have a multitude of uses, but they often require too many actuators to be practical in all touchscreen systems, and it is often difficult to create long and steady vibrotactile responses. Most small modern touchscreen devices utilize vibrotactile feedback to communicate to users in a physical sense. However, this feedback is typically unavailable for devices with large screens.