Insights Into the MMSE-Based Frequency-Invariant Beamformers for Uniform Circular Arrays

Abstract

This letter provides new insights into the minimum mean square error (MMSE)-based frequency-invariant beamformer with the uniform circular array (FIB-UCA). By exploiting the symmetry of the UCA, we derive an explicit form of the white noise gain (WNG), the directivity factor (DF) and the MSE, and also derive the corresponding approximate expressions at low frequencies. We then investigate the impact of the look direction on the performance of the MMSE-based FIB-UCA. Interestingly, we further prove that for a UCA with the fixed aperture, the MMSE-based FIB with 2 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> microphones achieves a similar WNG as that with 4 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> microphones at low frequencies if the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> th-order beampattern is steered to microphone angles. We also derive a unified explicit form of weighting vector and reveal that the MMSE-based FIB-UCA reduces to Jacobi-Anger expansion-based differential beamformer with the UCA at low frequencies.