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In this paper, a general structure of linear-phase finite impulse response filters, whose frequency responses satisfy given derivative constraints imposed upon an arbitrary frequency, is proposed. It is comprised of a linear combination of parallelly connected subfilters, called the cardinal filters , with weighted coefficients being the successive derivatives of the desired frequency response at the constrained frequency. An advantage of such a cardinal filters design is that only the weighted coefficients are relevant to the desired frequency response but not the cardinal filters; hence, a dynamic adjustment of the filter system becomes feasible. The key to derive the coefficients of cardinal filters is the determination of the power series expansion of certain trigonometric-related functions. By showing the elaborately chosen trigonometric-related functions satisfy specific differential equations, recursive formulas for the coefficients of cardinal filters are subsequently established, which make efficient their computations. At last, a simple enhancement of the cardinal filters design by incorporating the mean square error minimization is presented through examples.

In this paper, a general structure of type-III FIR filters with derivative constraints is proposed. It consists of a linear combinations of basic sub-filters, called cardinal filters, with weighting coefficients equal to derivatives of the target frequency response at ω = 0. Closed-form expressions for these cardinal filters are derived and their filter coefficients are shown to have a simple recursive relation. Numerical experiments confirm that the proposed design is much more numerically stable than the solution obtained by solving the derivative constraints.

Design of the finite impulse response (FIR) filters by three-point Taylor interpolation is studied in this paper. It is shown that the filter can be constructed by sum of weighted sub-filters which weighting gains are exactly the derivatives at interpolating frequencies. Recursive relations among the coefficients of the sub-filters are derived and used for designing the maximally flat fractional delay filters as an illustrating example. It shows that the proposed solution leads to numerical stable implementation in terms of the normalized root mean squared error.

This paper presents the closed-form designs of an infinite-impulse-response (IIR) Hilbert transformer with an integer delay. The maximally flat criterion is applied at the midband frequency π/2. These designs are further categorized into eight types according to the filter orders and the delay values being even or odd. Their coefficients can be explicitly solved in closed form. A recursive relation also exists, facilitating the computation of these coefficients. Moreover, under the suggested relations and formula for the design parameters based on the Enestrom-Kakeya theorem, we can obtain a satisfactory stable IIR Hilbert transformer.

In this paper, we propose a fractional delay filter, an integer-order differintegrator, a fractional Hilbert transformer, and a fractional differintegrator. Through the time-domain analysis on the desired input and output signals of a linear time-invariant system, we derive a set of linear equations, which can be solved to obtain the coefficients of the desired filter. We also show that the difference between the desired output signal and the actual output of the system can be represented as the convolution of the derivative of the input signal and the Peano kernel.

In this paper we provide some analytical results of maximally flat fractional delay finite impulse response filters. First, we represent their transfer functions in terms of the Gauss hypergeometric function. Secondly, we derive the generating function of these transfer functions. Finally, we provide recursive relations among transfer functions. The recursions are derived from the method of undetermined coefficients. In the appendix, we list some three-term recursive relations.

Eight types of maximally flat finite impulse response (FIR) fractional delay filters are proposed. They are designed by separately approximating the real part and the imaginary part of the frequency response. Among these eight types of filters, two of them are equivalent to the existing design results in the open literature. Therefore, a more general closed-form design of the maximally flat FIR fractional delay filters is described in this letter, and some design examples of FIR fractional delay filters will be illustrated

In this letter, we propose a novel all-pass (AP) fractional delay filter whose denominator polynomial is obtained by truncating the power series of a certain function. This function is derived from the frequency response of the denominator whose magnitude response is related to the desired phase response through the Hilbert transform since the denominator of a stable AP filter is of minimum phase. The target function and corresponding power series are calculated analytically and expressed in closed form. The closed-form expressions facilitate the analysis of stability. According to the properties for the coefficients of the denominator polynomial, we show that the proposed AP filter is stable for positive delay. Numerical examples indicate that the phase delays of the proposed filters are flat around DC.

In this paper, we propose a design of two-dimensional (2-D) linear-phase, diamond-shaped (DS) finite impulse response (FIR) filters by using Bernstein polynomials. Although the one-dimensional (1-D) FIR filter designed by the Bernstein polynomials has been well investigated, this approach has not been broadly applied to 2-D filter design yet. We present a novel method of designing the 2-D FIR DS filter. In order to be approximated by a 2-D Bernstein polynomial, the 2-D symmetrical frequency response is transformed into a new domain. The key observation is that the region of support of the transformed frequency response is not diamond-shaped. The boundary of the new region of support represents an ellipse, a circle, or a line, and is analytically derived. The resultant magnitude responses are flat in the passband and stopband.

In this paper, we propose a novel structure of linear-phase type-I FIR filters. The structure consists of a linear combination of some basic filters, called the cardinal filters. The weighting coefficients are exactly the derivatives of the amplitude response at ω = 0. We solve a closed-form recurrence relationship between the filter coefficients. Implementation of the cardinal filters is discussed.

In this letter, a closed-form expression for the impulse response of the generalized half-band (HB) maximally flat (MF) FIR filters is obtained by solving the linear equations of the MF conditions. Based on the resultant impulse responses, an efficient implementation structure is derived. The dynamic range of the multipliers of the new structure is shown to be greatly reduced in comparison to the one of the direct form impulse response.

A discrete fractional differentiator is proposed in this paper. Through a time domain analysis of the ideal input and output signal of a LTI system, we derive a set of linear equations which can be solved numerically for the filterpsilas impulse response. We also show that the difference between the designed system and the ideal system can be represented by the Peano kernel of the system. Some designing examples of the proposed filter are also demonstrated.

We propose a design of discrete Fractional Hilbert transformer. Through the time domain analysis of the ideal input and output signal of a LTI system, we derived equations which can be solved numerically for the filter's impulse response. The conventional Hilbert transformer is derived as a special case of the fractional Hilbert transformer. We also show that the difference between the designed system and the ideal system can be represented by the Peano kernel of the system. Some designing examples of the proposed filter are also illustrated.

The paper describes a cepstrum-based approach to design the maximally flat allpass fractional delay filter and the fractional Hilbert transformers. The maximal flatness criteria on the phase/group delay responses are formulated as a system of linear equations in terms of cepstral coefficients. The solution to the cepstral coefficients has closed-form expression. Moreover, it's very attractive that the resultant cepstral coefficients are directly proportional to the design parameters: the fractional delay and phase shift in the delay filters and Hilbert transformers, respectively. This virtue implies a simple scheme for updating new filter coefficients corresponding to various delay value or phase shift amount. Only one set of cepstral coefficients needs to be designed and stored and the new set is obtained simply by multiplying the old set with a ratio. A fixed module is used to update the filter coefficients from the cepstral coefficients.