Performance Limit of Broadband Beamformer Designs in Space and Frequency

• Published in: Journal of optimization theory and applications Vol. 164; no. 1; pp. 316 - 341
• Main Authors: Feng, Zhi Guo, Yiu, Ka Fai Cedric, Nordholm, Sven Erik
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.01.2015; Springer Nature B.V
• Subjects: Applications of Mathematics; Applied mathematics; Beamforming; Broadband; Calculus of Variations and Optimal Control; Optimization; Communications equipment; Cost function; Design; Directivity; Engineering; Fittings; Mathematical models; Mathematics; Mathematics and Statistics; Microphones; Multimedia communications; Numerical analysis; Operations Research/Decision Theory; Optimization; Optimization techniques; Sensors; Studies; Theory of Computation; Filter length; 62P30; Number of microphones; 42A10; Optimal beamformer design; Performance limit
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-014-0543-5

The design problem of broadband beamformers is to choose the coefficients of the filters such that the difference between the actual spatial directivity pattern and a desired spatial directivity pattern is minimized. There are many factors which can influence the performance of the design. The prominent factors include the filter length and the number of microphones. In the literature, the condition in fitting a desired spatial directivity pattern totally has not been studied properly. We show here that it is essential to use both the filter length and the number of sensors in reducing the cost function to zero, subject to some conditions on the desired response. These conditions are investigated and developed. Numerical examples are used to verify the derived conditions and at the same time to present some optimal beamformer designs.