Classical designs of maximally flat finite impulse response digital filters need to perform inverse discrete Fourier transformation of the frequency responses, in order to calculate the impulse response coefficients. Several attempts have been made to simplify the designs by obtaining explicit formulas for the impulse response coefficients. Such formulas have been derived for digital differentiators having maximal linearity at zero frequency, using different techniques including interpolating polynomials and the Taylor series etc. We show that these formulas can be obtained directly by application of maximal linearity constraints on the frequency response. The design problem is formulated as a system of linear equations, which can be solved to achieve maximal linearity at an arbitrary frequency. Certain special characteristics of the determinant of the coefficients matrix of these equations are explored for designs centered at zero frequency, and are used in derivation of explicit formulas for the impulse response coefficients of digital differentiators of both odd and even lengths.

New designs of MAXFLAT discrete and differentiating Hilbert transformers are presented using their interrelationships with digital differentiators. The new designs have the explicit formulas for their tap-coefficients, which are further modified to obtain a new class of narrow transition band filters, with a performance comparable to the Chebyshev filters.

New explicit formulas for tap-coefficients of halfband low/high pass MAXFLAT non-recursive filters are presented by using their relationship with already presented maximally linear type IV differentiators. These formulas are modified to give a new class of narrow transition band filters, with a performance comparable to that of optimal filters.

In order to ensure the reliability and safety of equipment installed in process lines, it is important that maintenance and management should make efficient use of machine diagnosis techniques. Machine diagnosis by means of acoustic signals has hitherto been beset with difficulty, but there is now a strong demand that new acoustic type diagnosis equipment (utilizing acoustic signals) be developed. In response to this demand, the authors recently conducted research on diagnosis of machine faults by means of the processing of acoustic signals. In this research they were able to develop new acoustic type machine diagnosis techniques, and, using these techniques, to develop acoustic diagnosis equipment for practical use.

Explicit formulas for the tap-coefficients of Taylor series based type III FIR digital differentiators have already been presented. However, those formulas were not derived mathematically from the Taylor series and were based on observation of different sets of the results. In this paper, we provide a mathematical proof of the formulas by deriving them mathematically from the Taylor series.

The explicit formula for the coefficients of maximally linear digital differentiators is derived by the use of Taylor series. A modification in the formula is proposed to extend the effective passband of the differentiator with the same number of coefficients.