Optimum Chebyshev filter with an equalised group delay response

The first part of the paper discusses designing of the lowpass filter that gives both a constant group delay over the passband and a sharp cut-off slope, which is one of the most difficult challenges in continuous-time signal processing. Furthermore, equalisation of the group delay distortion of the Chebyshev filter has been performed through a cascade connection of the allpass filters. The Newton–Raphson method is applied for solving the system of nonlinear equations that calculate the allpass filter parameters for equalising the constant group delay in an equiripple sense. This procedure gives nearly symmetric impulse response of the resulting filtering system, as well as a longer duration. In the second part of the paper, two solutions for synthesising polynomial filter transfer functions are discussed: filter with monotonic amplitude characteristic (LSM) and filter using Chebyshev polynomial. For the purpose of comparison, the passband ripple factor of the Chebyshev filter is necessary to be adjusted so that both filters have the same stopband insertion loss, group delay and transient response. It is shown that Chebyshev filter is preferable since the passband magnitude distortions and sensitivity to the element deviation tolerance are significantly superior.