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.本文第一部分探讨了低通滤波器（lowpass filter）的设计方案，该滤波器可同时实现通带内恒定群时延与陡峭截止斜率，这是连续时间信号处理领域极具挑战性的难题之一。进一步地，研究通过全通滤波器（allpass filter）级联的方式，完成了切比雪夫滤波器（Chebyshev filter）的群时延失真均衡。该均衡过程采用牛顿-拉夫逊法（Newton–Raphson method）求解非线性方程组，以计算能够在等波纹（equiripple）意义下实现恒定群时延均衡的全通滤波器参数。此方法可使最终滤波系统获得近似对称的冲激响应（impulse response），且拥有更长的响应时长。本文第二部分探讨了两类多项式滤波传递函数的综合方案：一类为具备单调幅频特性的滤波器（LSM），另一类为采用切比雪夫多项式的滤波器。为开展对比研究，需调整切比雪夫滤波器的通带波纹因子，使两类滤波器具备相同的阻带插入损耗、群时延与瞬态响应。结果表明，切比雪夫滤波器更具优势，因其通带幅频失真与对元件偏差容限的灵敏度均显著更优。