Design: Further Choices

Further Choices to Investigate
In order to optimize the performance of our project, we used directivity patterns to investigate the effects of three microphone array parameters: number of array elements, element spacing, and input frequency. Before reading the rest of this section, it is important to note that narrow main lobes and attenuated side lobes are characteristics well-suited for precise, direction signal processing.

Number of Array Elements
We first tested how the number of array elements affects the entire array’s directivity, and thus the array’s ability to “listen” to a particular direction. To do so, we held element spacing and input frequency constant, while calculating directivity for 2, 4, 8, and 16 microphones.

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Clearly, the width of the main lobe as well as the side lobes decreased as the number of array elements increased.

Element Spacing
Secondly, we tested how the element spacing affects the entire array’s directivity pattern. To do so, we held the number of array elements and input frequency constant, while calculating directivity for element spacings of 1.7, 3.4, and 5.1 cm.

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Clearly, the width of the main lobe as well as the magnitude of the side lobes decreased as the element spacing increased. Furthermore, apart from a few regions ([~70, ~110] and [~250, ~290]), the magnitude of the side lobes decreased as the element spacing increased.

Input Frequency
Finally, we tested how the input (sound signal) frequency affects the entire array’s directivity pattern. To do so, we held the number of array elements and element spacing constant, while calculating directivity for input frequencies of 4, 6, and 8 kHz.

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Clearly, the width of both the main lobe and side lobes decreased as the number of array elements increased. Apart from a few regions, centered around ~90 and ~270, the magnitude of the side lobes decreased as input frequency increased.

Although we were pleased with the results shown in the above figure, there were two additional factors that we used to judge the optimal input frequency: audible frequency ranges and spacial aliasing.The audible frequency range for humans is between 20 Hz and 20 kHz. After using the spatial sampling theorem, we determined the maximum spatial frequency for an array with element spacing equal to 1.7 cm is 10 kHz. Therefore, we determined that the input frequency must be between 20 Hz and 10 kHZ in order to both be both useful and accurate.

Decisions

We decided to use the maximum number of microphone elements available to us, instead of using less microphones and increasing element spacing. We also decided to use frequencies as close to 10 kHz as possible without exceeding 10 kHz.