Approximation of Linear Phase Filters with Gaussian Damped Impulse Response

In the first part of this paper, a family of Gaussiansmoothed rectangle functions is discussed as an aid to bridging the gap between certain system requirements and approximation programs in the synthesis of linear filters. These functions can be used as desired frequency response magnitudes when low-pass filters having small in-band distortion, sharp frequency cutoff, and rapidly damped transients are desired. They depend upon only two parameters: bandwidth and a shape parameter
\omega_0
. Therefore, search-evaluation algorithms are relatively easy to apply. The second part of the paper deals with the role of even numerator polynomials in the rational approximation of a filter amplitude response with linear phase. A FORTRAN computer program for least squares amplitude approximation is described and a filter approximation example given.