An analytical design for an equiripple FIR digital filter by transforming error response

Design methods for one-dimensional FIR digital filters can be separated into equiripple optimal approximations such as the Remez algorithm, and weighted least-square methods where the weight function is combined with the simple square error in the algorithm. For two-dimensional FIR digital filters, on the other hand, there is no known optimal approximation method, and a weighted least-squares method is generally used. This paper proposes an analytical design method for the one- and two-dimensional digital filters based on the least-squares method and for which matrix inversion and other manipulations are not required. Using this approach, the filter coefficients for a higher-order digital filter can be derived in a computation time similar to that for the lower-order case. We show that by transforming the error obtained for the response of the filter toward the equiripple response, an equiripple solution can be obtained by iteration. This idea is extended to the two-dimensional case, and a design method is offered for a two-dimensional FIR digital filter with a quasi-equiripple response. Finally, design examples are presented, and we show that an equiripple frequency response can be realized in a short time, independently of the filter order.