Maxflat Fractional Delay IIR Filter Design

Fractional delay (FD) filters are an important class of digital filters and are useful in various signal processing applications. This paper discusses a design problem of FD infinite-impulse-response (IIR) filters with the maxflat frequency response in frequency domain. First, a flatness condition of FD filters at an arbitrarily specified frequency point is described, and then a system of linear equations is derived from the flatness condition. Therefore, a set of filter coefficients can be easily obtained by solving this system of linear equations. For a special case in which the frequency response is required to be maxflat at omega = 0 or pi , a closed-form expression for its filter coefficients is derived by solving a linear system of Vandermonde equations. It is also shown that the existing maxflat FD finite-impulse-response (FIR) and IIR filters are special cases of the FD IIR filters proposed in this paper. Finally, some examples are presented to demonstrate the effectiveness of the proposed filters.     

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.     

具有任意频响的二维FIR数字滤波器设计的闭式最小二乘解

本文研究具有任意频响特性的二维ＦＩＲ数字滤波器的最小二乘设计问题。     

Design of two-channel low-delay IIR nonuniform-division filter banks using L/sub 1/ error criteria

The design of a two-channel nonuniform-division filter (NDF) bank with infinite impulse response (IIR) analysis/synthesis filters and low group delay in the sense of L/sub 1/ error criteria is considered. The problem formulation results in a nonlinear optimisation problem. Based on a variant of Karmarkar's algorithm, the optimisation problem is solved through a frequency sampling and iterative approximation technique to find the tap coefficients and the reflection coefficients for the numerator and the denominator of the IIR analysis filters. An efficient stabilisation procedure ensures that the reflection coefficients lie in (-1, 1). Simulation results are provided for illustration and comparison.     

Theory of Direct-Coupled-Cavity Filters

A new theory is presented for the design of direct-coupled-cavity filters in transmission line or waveguide. It is shown that for a specified range of parameters the insertion-loss characteristic of these filters in the case of Chebyshev equal-ripple characteristic is given very accurately by the formula P/sub 0/ / /P/sub L/ = 1+h/sup 2/T/sub n//sup 2/[/spl omega//sub 0/ / /spl omega/ sin(/spl pi/ /spl omega/ / /spl omega//sub 0/) / sin/spl theta//sub 0/'] where h defines the ripple level, T/sub n/ is the first-kind Chebyshev polynomial of degree n, /spl omega/ / /spl omega//sub 0/ is normalized frequency, and /spl theta//sub 0/' is an angle proportional to the bandwidth of a distributed lowpass prototype filter. The element values of the direct-coupled filter are related directly to the step impedances of the prototype whose values have been tabulated. The theory gives close agreement with computed data over a range of parameters as specified by a very simple formula. The design technique is convenient for practical applications.     

Generalized digital Butterworth filter design

This correspondence introduces a new class of infinite impulse response (IIR) digital filters that unifies the classical digital Butterworth filter and the well-known maximally flat FIR filter. New closed-form expressions are provided, and a straightforward design technique is described. The new IIR digital filters have more zeros than poles (away from the origin), and their (monotonic) square magnitude frequency responses are maximally flat at ω=0 and at ω=π. Another result of the correspondence is that for a specified cutoff frequency and a specified number of zeros, there is only one valid way in which to split the zeros between z=-1 and the passband. This technique also permits continuous variation of the cutoff frequency. IIR filters having more zeros than poles are of interest because often, to obtain a good tradeoff between performance and implementation complexity, just a few poles are best     

Weighted least-squares design and characterization of complex FIRfilters

The paper presents two novel weighted least-squares methods for the design of complex coefficient finite impulse response (FIR) filters to attain specified arbitrary multiband magnitude and linear or arbitrary phase responses. These methods are computationally efficient, requiring only the solution of a Toeplitz system of N linear equations for an N-length filter that can be obtained in o(N²) operations. Illustrative filter design examples are presented     

Digital filter design techniques in the frequency domain

Digital filtering is the process of spectrum shaping using digital components as the basic elements. Increasing speed and decreasing size and cost of digital components make it likely that digital filtering, already used extensively in the computer simulation of analog filters, will perform, in real-time devices, the functions which are now performed almost exclusively by analog components. In this paper, using the z-transform calculus, several digital filter design techniques are reviewed, and new ones are presented. One technique can be used to design a digital filter whose impulse response is like that of a given analog filter; other techniques are suitable for the design of a digital filter meeting frequency response criteria. Another technique yields digital filters with linear phase, specified frequency response, and controlled impulse response duration. The effect of digital arithmetic on the behavior of digital filters is also considered.     

The design and multiplier-less realization of software radio receivers with reduced system delay

This work studies the design and multiplier-less realization of a new software radio receiver (SRR) with reduced system delay. It employs low-delay finite-impulse response (FIR) and digital allpass filters to effectively reduce the system delay of the multistage decimators in SRRs. The optimal least-square and minimax designs of these low-delay FIR and allpass-based filters are formulated as a semi-definite programming (SDP) problem, which allows zero magnitude constraint at /spl omega/=/spl pi/ to be incorporated readily as additional linear matrix inequalities (LMIs). By implementing the sampling rate converter (SRC) using a variable digital filter (VDF) immediately after the integer decimators, the needs for an expensive programmable FIR filter in the traditional SRR is avoided. A new method for the optimal minimax design of this VDF-based SRC using SDP is also proposed and compared with traditional weight least squares method. Other implementation issues including the multiplier-less and digital signal processor (DSP) realizations of the SRR and the generation of the clock signal in the SRC are also studied. Design results show that the system delay and implementation complexities (especially in terms of high-speed variable multipliers) of the proposed architecture are considerably reduced as compared with conventional approaches.     

Digital Laguerre filter design with maximum passband-to-stopband energy ratio subject to peak and Group delay constraints

In this paper, we formulate a general design of transversal filter structures with maximum relative passband-to-stopband energy ratio subject to complex frequency response constraints in the passband and the stopband as well as additional constraints such as constraints. These constraints are important for applications where the suppression of noise at certain frequencies are important. Additional constraints are introduced allowing approximately linear phase and constant group delay in the passband. For a given set of basis functions, the design problem can be formulated as a semi-infinite quadratic optimization problem in the filter coefficients, which are the decision variables to be optimized. In this paper, we focus on the design of digital Laguerre filter and digital finite impulse response (FIR) filter structures. A modified bridging algorithm is developed for searching for the optimum pole of the Laguerre filters. Design examples are given to demonstrate the effectiveness of the proposed algorithm.     

Optimal design of two-channel nonuniform-division FIR filter bankswith -1, 0, and +1 coefficients

This paper deals with the optimal design of two-channel nonuniform-division filter (NDF) banks whose linear-phase FIR analysis and synthesis filters have coefficients constrained to -1, 0, and +1 only. Utilizing an approximation scheme and a weighted least squares algorithm, we present a method to design a two-channel NDF bank with continuous coefficients under each of two design criteria, namely, least-squares reconstruction error and stopband response for analysis filters and equiripple reconstruction error and least-squares stopband response for analysis filters. It is shown that the optimal filter coefficients can be obtained by solving only linear equations. In conjunction with the proposed filter structure, a method is then presented to obtain the desired design result with filter coefficients constrained to -1, 0, and +1 only. The effectiveness of the proposed design technique is demonstrated by several simulation examples     

Minimax design of two-channel IIR QMF banks with arbitrary groupdelay

The paper deals with the minimax design of two-channel infinite impulse response (IIR) QMF banks with arbitrary group delay, for which the IIR analysis filters and the resulting filter bank possess the frequency response optimal in the minimax (L_∞) sense. Utilising a lattice structure for the denominators of the IIR analysis filters, a design technique is presented based on an approximation scheme and a weighted least-squares (WLS) algorithm, previously developed by one of the authors for solving the resulting design problem that is basically a nonlinear optimisation problem. During the design process, this technique finds the tap coefficients for the numerator and the reflection coefficients for the denominator of the prototype IIR analysis filter simultaneously. The stability of the designed prototype IIR analysis filter is ensured by incorporating an efficient stabilisation procedure to make all of the reflection coefficient values fall between -1 and +1. Computer simulations show the effectiveness of the proposed design technique     

A 16-Channel Low-Power Nonuniform Spaced Filter Bank Core for Digital Hearing Aids

We describe a 16-channel critical-like spaced, high stopband attenuation ($geqslant60$dB, 109th$,times,$16-order), micropower (247.5$mu$W@1.1 V, 0.96 MHz), small integrated circuit (IC) area (1.62 mm$^2$@0.35-$mu$m CMOS) finite impulse response filter bank core for power-critical hearing aids. We achieve the low-power and small IC area attributes by our proposed common pre-computational unit to generate a set of pre-calculated intermediate values that is shared by all 16 channels. We also take advantage of the consecutive zeros in the coefficients of the filter channels, allowing the multiplexers therein to be simplified. We show that our design is very competitive compared to reported designs, and with the advantages of higher stopband attenuation and linear phase frequency response. Compared to a design using the usual approach, our design features 47% lower power dissipation and 37% smaller IC area.     

Design of stable IIR digital filter based on least P-power error criterion

In this paper, the least p-power error criterion is presented to design digital infinite impulse response (IIR) filters to have an arbitrarily prescribed frequency response. First, an iterative quadratic programming (QP) method is used to design a stable unconstrained one-dimensional IIR filter whose optimal filter coefficients are obtained by solving the QP problem in each iteration. Then, the proposed method is extended to design constrained IIR filters and two-dimensional IIR filters with a separable denominator polynomial. Finally, design examples of the low-pass filter are demonstrated to illustrate the effectiveness of the proposed iterative QP method.     

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.     

The Design and Synthesis of a Class of Microwave Bandpass Linear Phase Filters

This paper is concerned with the design procedure and synthesis of a class of microwave bandpass linear phase filters which simultaneously exhibit a maximally flat amplitude and delay response about band center. In the first part of the paper a systematic procedure is developed for the construction of a nonminimum phase transfer function which exhibits a maximally flat delay and maximally flat amplitude characteristic. In the second part, a synthesis procedure is presented for the realization of the general nth-ordered transfer function by a generalized interdigital network. To simplify the design and construction of this filter, typical characteristics for filters of degree n = 3,4,5,6,7 are graphically presented together with a tabular representation of the polynomials which are required to design the filter. Finally, the results of an experimental filter of degree 3 are incorporated to illustrate that this class of nonminimum phase filters may readily be constructed in practice.     

Windowing Design Method for Polynomial-Based Interpolation Filters

An efficient implementation for finding digitally the interpolated samples is the Farrow structure. It mimics digitally a hybrid system where a continuous-time (CT) signal is reconstructed using an analog reconstruction filter having a piecewise-polynomial impulse response. The interpolated samples are obtained by sampling reconstructed signal. This paper introduces a generalized design method for polynomial-based interpolation filters and Farrow structure. The proposed method also can be used to calculate the coefficients of Selva interpolator. In this approach, the ideal CT impulse response is truncated by using CT window functions. The obtained windowed impulse response is then approximated using the piecewise Taylor polynomial approximation. Length of the impulse response and degree of the approximating polynomial can be arbitrarily selected, and in this way the transition band width can be controlled. However, if CT fixed-window functions are used, the stopband attenuation is determined by window type and remains approximately constant with increase of length and order of the impulse response. The stopband attenuation can be controlled by using CT dynamic windows such as Kaiser window. The presented windowing design method is an effective tool for calculation of the Farrow structure coefficients, with filter performance that is comparable to the frequency domain design.     

Design of two-channel linear-phase QMF banks based on real IIR all-pass filters

The design of two-channel linear-phase quadrature mirror filter (QMF) banks constructed by real infinite impulse response (IIR) digital all-pass filters is considered. The design problem is appropriately formulated to result in a simple optimisation problem. Using a variant of Karmarkar's algorithm, the optimisation problem can be efficiently solved through a frequency sampling and iterative approximation method to find the real coefficients for the IIR digital all-pass filters. The resulting two-channel QMF banks possess an approximately linear phase response without magnitude distortion. The effectiveness of the proposed technique is achieved by forming an appropriate Chebyshev approximation of the desired phase response and then finding its solution from a linear subspace in a few iterations. Finally, several simulation examples are presented for illustration and comparison     

On Parallelepiped-Shaped Passbands for Multidimensional Nonseparable ${bf M}$th-Band Low-Pass Filters

The “shape” of the desired frequency passband is an important consideration in the design of nonseparable multidimensional ($M$ -D) filters in $M$-D multirate systems. For $M$-D ${bf M}$th-band filters, the passband shape should be chosen such that the ${bf M}$th-band constraint is satisfied. The most commonly used shape of the passband for $M$-D ${bf M}$ th-band low-pass filters is the so-called symmetric parallelepiped (SPD) ${rm SPD}(pi {bf M}^{- {rm T}})$ . In this paper, we consider the more general parallelepiped passband ${rm SPD}(pi {bf L} ^{rm T})$, and derive conditions on $ {bf L} $ such that the ${bf M}$ th-band constraint is satisfied. This result gives some flexibility in designing $M$-D ${bf M}$th-band filters with parallelepiped shapes other than the commonly used case of $ {bf L} = {bf M}^{- 1}$. We present design examples of 2-D ${bf M}$th-band filters to illustrate this flexibility in the choice of $ {bf L} $.      

具有任意幅度频响的二维线性相位FIR数字滤波器的设计

本文提出了设计任意幅度频响的二维线性相位ＦＩＲ数字滤波器的解析最小二乘方法，通过最小化频域平方误差函数得到了滤波器系数的闭式解，运用导出的闭式式，可根据给定的任意幅度频响指标直接计算滤波器的系数，从而简化了滤波器设计程序，并大大降低了运算量。