Peng-Hua Wang

Closed-form design of maximally flat FIR Hilbert transformers, differentiators, and fractional delayers by power series expansion

In this paper, novel closed-form designs of the FIR Hilbert transformers, maximally flat digital differentiators and fractional delayers are proposed. The transfer functions of these filters are analytically obtained by expanding some suitable functions into power series. Efficient implementations can be derived from the resultant transfer functions. The weighting coefficients and the building blocks of these filters are explicitly expressed in closed form. The proposed filter structures are more robust to the coefficient quantization than the direct form.

Closed-form design of all-pass fractional delay filters

In this paper, we propose a novel allpass (AP) fractional delay (FD) 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 ω = 0.

Analytical design of maximally flat FIR fractional Hilbert transformers

Recently, the classical Hilbert transformer is generalized into the fractional Hilbert transformer which could be implemented optically. This modification of the Hilbert transform adds an additional degree of freedom on the Hilbert transformer and improves the performance of the transform. In this paper, the design of FIR filters fractional Hilbert transformer is proposed. The FIR filters are designed in the maximally flat sense. The impulse responses of the filters are uniquely solved and expressed in simple analytic forms. The impulse responses can be exactly expressed as fixed point binary values. The resulting frequency responses approximate the ideal one very well in the middle-frequency band. Efficient hardware realization structures are obtained based on the symmetric properties of the impulse responses. Several design examples with various transform parameters and various filter orders are presented. Some examples of 1-D/2-D edge detection are given. The examples show that the proposed FIR filter can enhance the selected edges very efficiently.

Design of Fractional Delay Filter, Differintegrator, Fractional Hilbert Transformer, and Differentiator in Time Domain With Peano Kernel

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.

Maximally flat allpass fractional Hilbert transformers

Recently, a generalization of the Hilbert transformer, the fractional Hilbert transformer, was defined and developed. In this paper, we propose a design of the allpass filter to realize the fractional Hilbert transformer based on the maximally flat approximation to the desired phase response. The coefficients are solved analytically for the traditional Hilbert transformer which is a special case of the fractional Hilbert transformer. Based on the closed-form coefficients, we show that the maximally flat allpass Hilbert transformers are stable. Design examples indicate that the proposed filters exhibit good approximation to the desired frequency response.

Closed-form design and efficient implementation of generalized maximally flat half-band FIR filters

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.

Design of equiripple FIR filters with constraint using a multiple exchange algorithm

We propose a method of designing equiripple linear-phase FIR filters with linear constraint by using the Remez exchange algorithm. A novel technique is derived to convert a linearly constrained problem into an equivalent unconstrained one. We proposed a technique to modify the original desired frequency response so that the original linear constraint can be reduced to a simpler one (the null constraint) for the new target frequency response. The filter with null constraint can be designed without constraint by transformation of the original basis functions. The transformation is represented by a basis for the null space of the constraint. In this paper, we show that the transformed basis set also forms a Tchebycheff set. This fact indicates the proposed design is optimal in the Tchebycheff sense. The optimal filter is deigned by the Remez method according to the new target frequency response in transformed basis. Design examples suggest that the proposed algorithm converges fast and stably.

Design of a class of IIR eigenfilters with time- and frequency-domain constraints

An effective eigenfilter approach is presented to design special classes of infinite-impulse response (IIR) filters with time- and frequency-domain constraints is presented. By minimizing a quadratic measure of the error in the passband and stopband, an eigenvector of an appropriate real symmetric and positive-definite matrix is computed to get the filter coefficients. Several IIR filters such as notch filters, Nyquist filters and partial response filters can be easily designed by this approach. Some numerical design examples are illustrated to show the effectiveness of this approach.

Closed-form design of maximally flat R-regular Mth-band FIR filters

In this paper, we derive some properties of maximally flat R-regular Mth-band FIR filters. We show that the R-regularity implies maximally flat frequency response at /spl omega/=0. The R-regular constraints are a set of linear equations with complex coefficients. We can convert these complex-value equations to equivalent ones with only real coefficients. We also show that it is possible to completely determine the filter coefficients by R-regularity. Design examples are presented to illustrate the R-regularity properties and the effectiveness of the proposed approach.

General expressions of derivative-constrained linear-phase type-I FIR filters

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.