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A novel analytic technique is given, for the derivation of finite duration impulse response transfer functions which satisfy arbitrary attenuation characteristics, with or without exact linear phase at all frequencies. The method uses a parameter for the determination of the stopband attenuation at the outset, and the rate of cutoff and the passband performance can be improved by increasing the order of the filter. The resulting solutions provide a marked improvement over the available analytic solutions, particularly where monotonic passband behaviour is required. The technique is also characterised by its simplicity and gives explicit expressions for the coefficients of the transfer function in the z-domain. This obviates the need for the various transformations commonly employed, and consequently improves the accuracy in determining the filter coefficients. The realisation of the resulting functions can be accomplished in either digital or charge-coupled device form.