Characteristic function of least-mean-square passband filters with finite attenuation poles

Characteristic function of low-pass filters with finite attenuation poles approximating a zero passband loss in terms of a weighted least-mean-square norm is derived in explicit form. The number and locations of attenuation poles are subject only to the realizability conditions and can be adjusted by a simple pole placer program to meet either equal-ripple or general stopband requirements. The shape of the passband magnitude response is controlled by a variable parameter which appears in the weight function used to define the error integral. Filters with either Chebyshev, or maximally flat passband magnitude response are shown to be the special cases of the explicit formula derived.