Sharpening the response of a symmetric nonrecursive filter by multiple use of the same filter

When processing data by filters we often find it necessary to improve the performance of the filter, either by increasing the out-of-band rejection (loss) or by decreasing the error in the passband, or both. A first approach is to process the data by repeated passes through the same filter. Each pass, while increasing the out-of-band loss, also increases the passband error, often to an undesirable level. It also increases the length (order) of the equivalent filter. How can we do a better job of filtering by suitably combining the results of several passes through the same filter? By "better" we mean both less passband error and greater out-of-band, or stopband, loss. This process is called filter sharpening. A simple, powerful method for doing this is described in detail, and its computational efficiency is compared to the best possible filter designs meeting the same specifications. The design method, based on the idea of the amplitude change function, is restricted to symmetric nonrecursive (finite impulse response) filters with piecewise constant pass and stopbands. Several illustrative examples are given.