Chebyshev optimization of beamformers and FIR filters having failed elements

Graceful degradation can be achieved in operational beamformers and some types of FIR (finite impulse response) filters by optimally adjusting the functional elements to compensate for hydrophones or multipliers that may have failed. A linear programming (LP) design approach is investigated and compared with two other possible alternatives which maintain linear phase. For the cases considered, the LP result reduces the sidelobe level or increases the stopband suppression by as much as 20 dB, depending on which element has failed. For the cases studied, there were some instances where the linear program did not produce a normalized solution with the specified ripple ratio. For applications where this might cause a problem, the authors have suggested a technique which may lead to a more suitable design