Temporal filters and spatial filters are widely used in many areas of signal processing. A number of optimal design criteria to these problems are available in the literature. Various computational techniques are also presented to optimize these criteria chosen. There are many drawbacks in these methods. In this paper, we introduce a unified framework for optimal design of temporal and spatial filters. Most of the optimal design problems of FIR filters and beamformers are included in the framework. It is shown that all the design problems can be reformulated as convex optimization form as the second-order cone programming (SOCP) and solved efficiently via the well-established interior point methods. The main advantage of our SOCP approach as compared with earlier approaches is that it can include most of the existing methods as its special cases, which leads to more flexible designs. Furthermore, the SOCP approach can optimize multiple required performance measures, which is the drawback of earlier approaches. The SOCP approach is also developed to optimally design temporal and spatial two-dimensional filter and spatial matrix filter. Numerical results demonstrate the effectiveness of the proposed approach.
An approach to designing time domain broadband frequency invariant beamformer via optimal array pattern synthesis and optimal FIR filters design is proposed. First, the working frequency band is decomposed into a number of narrow band frequency bins. The array weights at each frequency bin are designed via optimal array pattern synthesis methods to insure that the synthesized pattern approximates the desired one within the mainlobe area.Then, a bank of FIR filters corresponding to the input channels are designed to provide the frequency responses that approximate the array weights in the working frequency band for each sensor. Finally, each sensor feeds a FIR filter and the filter outputs are summed to produce the beam output time series. Both array pattern synthesis and FIR filters design problems are formulated as the second-order cone programming (SOCP), which can be easily solved using well-developed interior-point methods. Results of computer simulations and lake-experiment for a twelve-element semicircular array demonstrate satisfactory performance of the proposed approach.
