Sharpened raised-cosine FIR filters

Shaping of transition bands in the ideal frequency response allows the analytic design of least-squares FIR filters with a direct control of the transition-band edges. The basic least-squares approximation is obtained by the Fourier series method. In this paper, we use it to develop a straightforward method for the design of steep roll-off FIR filters. The method is based on the frequency response obtained by the polynomial sharpening of raised-cosine transition band. The corresponding impulse response is well localized in time, what enables the design of low-order filters without additional window. Apparently, such an approach is suitable for the design of filters with wide transitions. However, for the design of filters with narrow transitions, composite systems are more appropriate. A computationally efficient composite system is frequency-response-masking (FRM) filter. Therefore, to obtain narrow transition filters the sharpened raised-cosine filters are incorporated into the FRM structure. The features of the presented single and composite filters are illustrated by examples which include the design of spatial filters and Hilbert transformers.