Magnitude approximation of digital filters with specified degrees of flatness and constant group delay characteristics

New techniques to approximate the magnitude response of infinite-impulse response digital filters with maximally flat or equiripple delay characteristics are presented. A polynomial with maximally flat or equiripple group delay characteristics is chosen to be the denominator of the filter transfer function. The coefficients of the numerator polynomial are found such that the group delay characteristic is unaffected, and the magnitude response either has desired degrees of flatness at ω=0 and/or ω=π or is flat in the passband and equiripple in the stopband. The magnitude response can also be matched to prescribed values at a number of frequencies between ω=0 and ω=π. In the first case, the solution for the coefficients of the numerator polynomial is shown to be purely analytical. In the second case, the well-known Remez Exchange Algorithm is used to obtain the desired passband and stopband magnitude characteristics simultaneously. It is also shown that the proposed design methods can be used to obtain linear phase finite-impulse response filters with a magnitude response that is flat in the passband and flat or equiripple in the stopband