Weighted least squares near-equiripple approximation of variable fractional delay FIR filters

The weighted least squares (WLS) method is a well-known method for designing a finite impulse response (FIR) filter. And some authors have reported that if a suitable frequency response weighting function is used to design the filter, the WLS method can produce an equiripple result. However, the weighting function for minimax optimality of WLS design is hard to derive analytically. By an iterative method with an adjustable elaborately constructed weighting function, this idea is extended to design a near-equiripple variable fractional delay FIR filter. The proposed method is superior to the fixed-weighting WLS design in the peak absolute error by about 6.6874 dB. The algorithm converges very rapidly. From the simulation, it typically produces a design which is only about 1 dB away from the truly equiripple solution in two iterations and converges to within 0.0056 dB in eight iterations.     

Complex Chebyshev approximation for FIR digital filters

An adaptive approach to FIR (finite impulse response) filter design is applied to a fully general complex specification, showing it to be capable of giving an optimum Chebyshev solution. It is thought to be one of the first practical approaches to achieve this in the case of a complex function. Design examples include a bandpass filter with nearly linear phase and all-pass filters with general shaping phase. Comparisons are drawn with equivalent results from other algorithms     

A Fast FIR filtering technique for multirate filters

This paper presents a technique for applying Fast FIR Algorithms (FFAs) to interpolation and decimation filters. In the event that the prototype filter has a symmetric impulse response, it is shown that the subfilters which result from the application of the FFA will be jointly symmetric. This fact may be exploited in order to further reduce the computational complexity of the system. The effect of transposition upon the proposed structure is discussed, and it is shown that transposition yields a more economical structure in the case of a decimation filter. The computational complexity of the proposed schemes is shown to compare favorably with that of the standard parallel filtering approach.     

Optimal design of complex FIR filters with arbitrary magnitude and group delay responses

This paper presents a method for the frequency-domain design of digital finite impulse response filters with arbitrary magnitude and group delay responses. The method can deal with both the equiripple design problem and the peak constrained least squares (PCLS) design problem. Consequently, the method can also be applied to the equiripple passbands and PCLS stopbands design problem as a special case of the PCLS design. Both the equiripple and the PCLS design problems are converted into weighted least squares optimization problems. They are then solved iteratively with appropriately updated error weighting functions. A novel scheme for updating the error weighting function is developed to incorporate the design requirements. Design examples are included in order to compare the performance of the filters designed using the proposed scheme and several other existing methods.     

A simple and efficient design of variable fractional delay FIR filters

In this paper, a simple and efficient approach for designing one-dimensional variable fractional delay finite impulse response digital filters is proposed. Two matrix equations, based respectively on the weighted least-squares function of the optimum fixed fractional delay filter and the filter coefficient polynomial fitting, are formulated in tandem to form the design algorithm, which only has the computation complexity comparable with that of designing fixed finite impulse response digital filters. A design example is also given to justify the effectiveness and advantages of the proposed design method.     

Reduced order IIR approximation to FIR digital filters

An algorithm for designing an infinite-impulse-response (IIR) stable filter using a finite-impulse-response (FIR) given filter, with the objective of reducing the delay and order, is described. The design is in the time domain using the least-squares-inverse algorithm, which is briefly described. In this method, the numerator of the approximated filter is part of the FIR filter itself and no calculations and minimization are needed to find the numerator coefficients (except finding the FIR roots). An error analysis between the given FIR and approximated IIR filters is provided. This error analysis enables the designer to fix a design parameter, often unnoted, keeping the energies of the approximated and original filters equal. Results and two illustrative examples are presented     

Weighted least-squares approximation of FIR by IIR digital filters

This paper presents a method for the weighted least-squares approximation of finite impulse response (FIR) filters by infinite impulse response (IIR) filters. It is shown, how a solution to this approximation problem can be obtained by solving a related pure least-squares approximation problem. For the latter, we utilize a generalized version of a previously published technique with low computational complexity and guaranteed stability of the IIR filters. Unlike the well-established model-reduction approaches that are carried out in the state space, our method works directly with the numerator and denominator coefficients of the transfer functions. Thus, the influence of finite-precision arithmetic on the results is small. This makes our approach applicable for the approximation of large-order FIR filters and allows the usage of arbitrarily shaped weighting functions. It is shown that our method can successfully be employed to achieve a uniform approximation     

FIR digital filters with simultaneous conditions on amplitude and delay

A general analytic technique is given for the design of FIR digital filters with simultaneous conditions on both the amplitude and delay responses. It is shown that phase linearity can be maintained only within the passband, resulting in increased amplitude selectivity as compared with filters with exact linear phase at all frequencies.     

Eigenfilter approach for the design of variable fractional delay FIR and all-pass filters

An eigenfilter approach is presented for designing 1-D and 2-D variable fractional delay FIR and all-pass filters. First, the coefficients of filters are expressed as a polynomial of the fractional delay parameter. Then, the optimal polynomial coefficients are obtained from the elements of the eigenvector corresponding to the minimum eigenvalue of a real, symmetric and positive definite matrix. Finally, several design examples of 1-D and 2-D variable fractional delay FIR and all-pass filters are used to illustrate the effectiveness of the eigenfilter approach.     

Lowpass delay filters with flat magnitude and group delay constraints

This paper describes the design of finite impulse response (FIR) delay filters that minimize a squared error and have prescribed number of zeros at /spl omega/=/spl pi/ and prescribed magnitude and group delay flatness at /spl omega/=0. An important special case is the design of least squared error lowpass filters with prescribed flatness constraints and zeros at /spl omega/=/spl pi/. Even though the flatness constraints are in general nonlinear functions of the filter coefficients, we show the remarkable fact that for a subclass of the filters a simple orthogonal projection of least squared error filters onto a special linear subspace determined via Baher (1982) filters gives the solution. The paper also introduces the notion of delay filters that are high-order approximations to the ideal delay and establishes their equivalence to Baher filters. This connection gives novel elementary derivations of Baher filters and their properties. Matlab programs are provided at the end of the paper for the design of filters described in this paper.     

Selective filters using least-mean-square approximation technique

A new clan of all-pole lowpass filters having a flat magnitude characteristic at the origin and at a frequency ?0 in the pass-band is presented. The remaining degree of freedom is used to minimise the characteristic function over the passband using least-mean-square norm. It is shown that the magnitude characteristic of new filters is superior to those for the Butterworth and transitional Butterworth-Cheby?shev filters, both in the passband and in the stopband.     

General expression for the group delay of digital filters

An expression for the group delay of a digital filter is derived in terms of the numerator and denominator polynomials of the transfer function H(z). The application of this expression to three cases is shown, and an important theorem relating to the exact realisability of linear phase digital filters is given.     

Quasi-equiripple approximation of minimum phase FIR filters by updating desired response

The authors present a numerical method for the Chebyshev approximation of minimum phase FIR digital filters. This method is based on solving a least squares (LS) problem iteratively. At each iteration, the desired response is transformed so as to have an equiripple magnitude error. This method makes it possible to design minimum phase FIR filters whose magnitude error is quasi-equiripple. Using this method, a quasi-equiripple solution is obtained very quickly. Since the proposed methods do not require any time-consuming optimisation procedure, they require less computational complexity than conventional methods. Finally, some examples to illustrate the advantage of the proposed methods are shown.     

Uniform FIR filterbank optimization with group delay specifications

We characterize the delay properties of uniformly modulated finite impulse response (FIR) filterbanks. Conditions on the permissable delay are developed for this class of filterbanks when the perfect reconstruction condition is relaxed. Accurate linear approximations for the phase and the group delay of the total filterbank are derived. These approximations allow linear phase or group delay constraints to be introduced in the filter optimization problem. A tractable quadratic optimization problem for the design of optimal analysis and synthesis filter prototypes is proposed. The problem involves the minimization of the aliasing distortion while constraining group delay and amplitude distortion. Thus, a new algorithm is presented to solve this optimization problem for the analysis and synthesis filterbanks simultaneously. Numerical examples are presented that confirm the theoretical results and verify that the approximations used are highly accurate.     

Broadband negative group delay networks for compensation ofmicrowave oscillators and filters

The design of distributed broadband negative group delay circuits operating from 3 to 6 GHz with negative group delays of around 50 ps are described. These circuits are designed using LCR prototypes which are then converted to printed transmission line structures. The parameters affecting the value and bandwidth of the group delay are discussed     

A fast matrix iterative technique for the WLS design of 2-D quadrantally symmetic FIR filters

High computational complexity is a major problem encountered in the optimal design of two-dimensional (2-D) finite impulse response (FIR) filters. In this paper, we present an iterative matrix solution with very low complexity to the weighted least square (WLS) design of 2-D quadrantally symmetric FIR filters with two-valued weighting functions. Firstly, a necessary and sufficient condition for the WLS design of 2-D quadrantally symmetric filters with general nonnegative weighting functions is obtained. Then, based on this optimality condition, a novel iterative algorithm is derived for the WLS design problem with a two-valued weighting function. Because the filter parameters are arranged in their natural 2-D form and the transition band is not sampled, the computation amount of the proposed algorithm is reduced significantly, especially for high-order filters. The exponential convergence of the algorithm is established, and its computational complexity is estimated. Design examples demonstrating the convergence rate and solution accuracy of the algorithm, as well as the relation between the iteration number of the algorithm and the size and transition-band width of the filter are given.     

Splitting the unit delay [FIR/all pass filters design]

A fractional delay filter is a device for bandlimited interpolation between samples. It finds applications in numerous fields of signal processing, including communications, array processing, speech processing, and music technology. We present a comprehensive review of FIR and allpass filter design techniques for bandlimited approximation of a fractional digital delay. Emphasis is on simple and efficient methods that are well suited for fast coefficient update or continuous control of the delay value. Various new approaches are proposed and several examples are provided to illustrate the performance of the methods. We also discuss the implementation complexity of the algorithms. We focus on four applications where fractional delay filters are needed: synchronization of digital modems, incommensurate sampling rate conversion, high-resolution pitch prediction, and sound synthesis of musical instruments