Treffer: Numerical testing of minimum-delay, positive-real, and positive-definite digital filters

Various types of digital filters are characterized by their phase spectra. The phase spectrum of a symmetric filter can only take on values that are integral multiples of π. The phase spectrum of a positive-definite filter is zero for all frequencies. The phase spectrum of a minimum-delay filter must have the same value at the negative Nyquist frequency as at the positive Nyquist frequency, so that its net phase change over the Nyquist frequency range is zero. A positive-real filter in addition to having a zero net phase change over the Nyquist frequency range must have a phase spectrum that is less than π 2 in magnitude. A reflection coefficient theorem is established which states that a filter is minimum-delay if and only if its associated reflection coefficients are less than one in magnitude. A positive-definite theorem is established which states that a filter is positive-definite if and only if a Chebyshev-related polynomial has no real roots in magnitude less than or equal to one. Numerical tests for minimum-delay, positive-definite, and positive-real are given based on these two theorems.