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Finite impulse response (FIR) filters are very important in digital signal processing. Aspects of their design can be optimized by solving quadratic integer programs.

Most existing FIR filter design methods are based on infinite precision arithmetic and thus lead to filters which cannot be readily implemented with microprocessors. Such infinite precision filters have to be transformed into finite precision filters using techniques such as coefficient quantization and rounding before they can be implemented with hardware. This leads to two problems: (i) the likely sacrifice of filter performance because of the deviation from the infinite precision filter (which can be partially overcome by increasing the level of precision, although practical applications have a limit to the allowed wordlength); (ii) the use of more bits tends to exacerbate the cost and complexity of hardware implementation. In relation to (ii), a special class of finite precision filters has been found particularly attractive. The class consists of FIR filters whose coefficients are restricted to a sum of signed powers of two. This feature allows the filters to be implemented with simple adders and shifters only, eliminating the need to use any multipliers whose contribution to the cost and complexity is often great. Members of the team have considered a specific hardware-efficient structure based on an FIR filter with coefficients that consists of two powers-of-two followed by a first order digital integrator. This structure has been found to have a respectable low-pass frequency response. A design problem, which includes a frequency domain least squares criterion with an arbitrary frequency weighting and a scaling factor, is considered. The corresponding optimisation problem can be formulated as a quadratic integer programming problem.

from [here].