一维线性相位FIR数字滤波器的频域最小平方误差设计方法

本文提出了一种设计一维线性相位FIR数字滤波器的新方法。该方法采用频域的最小平方误差函数使所求的增益特性逼近所希望的增益特性 ,其计算公式非常简单 ,并可设计任意形状增益特性的FIR数字滤波器。设计实例表明提出的方法非常有效     

一种设计对数FIR数字滤波器的方法

本文提出了一种设计具有等波纹对数幅度响应的线性相位ＦＩＲ数字滤波器的方法,该设计方法以多次交换算法为基础。在给定通带与阻带误差比、通带误差和阻带误差三种情况下讨论该设计方法。介绍几个低通对数ＦＩＲ滤波器的设计例子,来说明该设计方法的效率。     

Least-squares design of IIR filters with prescribed magnitude andphase responses and a pole radius constraint

This paper presents a method for the frequency domain design of infinite impulse response (IIR) digital filters. The proposed method designs filters approximating prescribed magnitude and phase responses. IIR filters of this kind can have approximately linear-phase responses in their passbands, or they can equalize magnitude and phase responses of given systems. In many cases, these filters can be implemented with less memory and with fewer computations per output sample than equivalent finite impulse response (FIR) digital filters. An important feature of the proposed method is the possibility to specify a maximum radius for the poles of the designed rational transfer function. Consequently, stability can be guaranteed, and undesired effects of implementations using fixed-point arithmetic can be alleviated by restricting the poles to keep a prescribed distance from the unit circle. This is achieved by applying Rouche's theorem in the proposed design algorithm. We motivate the use of IIR filters with an unequal number of poles and zeros outside the origin of the complex plane. In order to satisfy simultaneous specifications on magnitude and phase responses, it is advantageous to use IIR filters with only a few poles outside the origin of the z-plane and an arbitrary number of zeros. Filters of this type are a compromise between IIR filters with optimum magnitude responses and phase-approximating FIR filters. We use design examples to compare filters designed by the proposed method to those obtained by other methods. In addition, we compare the proposed general IIR filters with other popular more specialized structures such as FIR filters and cascaded systems consisting of frequency-selective IIR filters and phase-equalizing allpass filters     

DESIGN OF POLYPHASE WAVE DIGITAL FILTERS WITH APPROXIMATELY LINEAR PHASE

It is well known that IIR digital filters require quite fewer computations,comparedwith FIR filters,in order to meet stringent magnitude specifications when the phase distortioncan be tolerated.An approximately linear phase,however,can be also obtained with the IIRfilter by making use of a technique without increasing the complexity.Based on a certain numberof attenuation zeros in the pass band,a new approach is developed for the design of polyphasewave digital filters with exact magnitude responses and Chebyshev approximation of the desiredphase responses.The minimum number of attenuation zeros is estimated,and some examples areincluded.     

具有近似线性相位特性的多相波数字滤波器的设计

不考虑相位失真,而要求满足比较苛刻的振幅特性的时候,和FIR滤波器比较起来,采用 IIR 滤波器,计算量要小很多.但对某些IIR滤波器,采取适当的方法,也可以使其相位失真变得很小.本文提出一种新的设计方法,用来设计振幅响应完全满足设计要求,而相位特性用Chebyshev 近似来实现的多相波数字滤波器.这种方法的关键在于确定通带中的若干衰减零点.文中对需要的最小衰减零点数作了估计.实例表明,用这种方法设计的滤波器,时延小,相位特性非常接近线性.     

Design of FIR digital filters with prescribed flatness and peak error constraints using second-order cone programming

This paper studies the design of digital finite impulse response (FIR) filters with prescribed flatness and peak design error constraints using second-order cone programming (SOCP). SOCP is a powerful convex optimization method, where linear and convex quadratic inequality constraints can readily be incorporated. It is utilized in this study for the optimal minimax and least squares design of linear-phase and low-delay (LD) FIR filters with prescribed magnitude flatness and peak design error. The proposed approach offers more flexibility than traditional maximally-flat approach for the tradeoff between the approximation error and the degree of design freedom. Using these results, new LD specialized filters such as digital differentiators, Hilbert Transformers, Mth band filters and variable digital filters with prescribed magnitude flatness constraints can also be derived.     

Design of IIR digital filters in the complex domain by transforming the desired response

In this paper, we present a new design method of infinite impulse response (IIR) digital filters with quasi-equiripple absolute error in the complex domain. This method is based on solving a least squares solution iteratively. At each iteration, the desired response for the least squares approximation is transformed to have equiripple error. This algorithm is efficient because there is no need for any initial value or complex optimization algorithm. By this method, a quasi-equiripple solution is obtained very quickly with less computational complexity. Moreover, by multiplying an arbitrary weighting function on the desired responses of passband and stopband, respectively, the error at the passband and stopband can be controlled. Finally, we show some examples to validate the proposed method.     

Using fractional delay to control the magnitudes and phases of integrators and differentiators

The use of fractional delay to control the magnitudes and phases of integrators and differentiators has been addressed. Integrators and differentiators are the basic building blocks of many systems. Often applications in controls, wave-shaping, oscillators and communications require a constant 90deg phase for differentiators and -90deg phase for integrators. When the design neglects the phase, a phase equaliser is often needed to compensate for the phase error or a phase lock loop should be added. Applications to the first-order, Al-Alaoui integrator and differentiator are presented. A fractional delay is added to the integrator leading to an almost constant phase response of -90deg. Doubling the sampling rate improves the magnitude response. Combining the two actions improves both the magnitude and phase responses. The same approach is applied to the differentiator, with a fractional sample advance leading to an almost constant phase response of 90deg. The advance is, in fact, realised as the ratio of two delays. Filters approximating the fractional delay, the finite impulse response (FIR) Lagrange interpolator filters and the Thiran allpass infinite impulse response (IIR) filters are employed. Additionally, a new hybrid filter, a combination of the FIR Lagrange interpolator filter and the Thiran allpass IIR filter, is proposed. Methods to reduce the approximation error are discussed.     

FIR数字滤波器幅频响应约束最大加权相位误差最小化设计

多数信号滤波应用,对滤波器幅频响应的要求高于相频响应.本文研究了满足幅频响应约束的有限脉冲响应（Infinite Impulse Response,FIR）数字滤波器设计,提出了最大加权相位误差最小化方法.用凸的椭圆误差约束代替非凸的幅值误差约束,将设计问题转化为凸问题;通过与二分技术结合,提出了给定权函数的幅值误差约束最大加权相位误差最小化设计的求解算法.以此算法为核心,构建了迭代重加权最大加权相位误差最小化算法,其中的权函数不再固定,而是基于修改的群延迟误差包络线在迭代中不断更新.权函数收敛后,所得滤波器具有近似等纹波的群延迟误差,最大群延迟误差得到了有效减小.仿真实验表明,与现有相位误差约束最大幅值误差最小化方法相比,得到的FIR滤波器具有更小的最大相位误差和最大群延迟误差.     

Design of linear phase FIR filters using fractional derivative constraints

In this paper, the designs of linear phase FIR filters using fractional derivative constraints are investigated. First, the definition of fractional derivative is reviewed briefly. Then, the linear phase FIR filters are designed by minimizing integral squares error under the constraint that the ideal response and actual response have several same fractional derivatives at the prescribed frequency point. Next, the fractional maximally flat FIR filters are designed by letting the number of fractional derivative constraints be equal to the number of filter coefficients. Finally, numerical examples are demonstrated to show that the proposed method has larger design flexibility than the conventional integer derivative constrained methods.     

Low power and high-speed implementation of fir filters for software defined radio receivers

The most computationally intensive part of the wideband receiver of a software defined radio (SDR) is the intermediate frequency (IF) processing block. Digital filtering is the main task in IF processing. The computational complexity of finite impulse response (FIR) filters used in the IF processing block is dominated by the number of adders (subtracters) employed in the multipliers. This paper presents a method to implement FIR filters for SDR receivers using minimum number of adders. We use an arithmetic scheme, known as pseudo floating-point (PFP) representation to encode the filter coefficients. By employing a span reduction technique, we show that the filter coefficients can be coded using considerably fewer bits than conventional 24-bit and 16-bit fixed-point filters. Simulation results show that the magnitude responses of the filters coded in PFP meet the attenuation requirements of wireless communication standard specifications. The proposed method offers average reductions of 40% in the number of adders and 80% in the number of full adders needed for the coefficient multipliers over conventional FIR filter implementation methods     

A new method for higher-order linear phase FIR digital filter using shifted Chebyshev polynomials

In this work, a new method for the design of linear phase finite impulse response (FIR) filters using shifted Chebyshev polynomial is proposed. In this method, magnitude response of FIR filter is approximated with the help of shifted Chebyshev polynomials. The number of polynomials used for approximation depends upon the order of filter. Design problem of filter is constructed as minimization of integral mean-square error between the ideal response and actual response through differentiating it with respect to its coefficients, which leads to a system of linear equations. The simulation results included in this paper show the efficiency of proposed method. It is also evident from the results that the proposed method is suitable for higher filter taps.     

Time-domain design of FIR multirate filter banks

A new time-domain methodology for designing FIR multirate filter banks is proposed. The conditions for perfect reconstruction systems can only be met by a limited number of systems, and consequently one of the major problems is to design analysis and synthesis filters which reduce the reconstruction error to a minimum. A recursive technique is proposed which uses the synthesis filters from one iteration to update the analysis filters for the next. The Letter shows that this is computationally simpler than previously proposed time-domain methods and produces filter banks in which the reconstruction error is reduced to practically acceptable levels.<>     

用复系数数字FIR滤波器实现超宽带正交解调接收系统的幅相误差校正

本文首先对影响正交解调结构超宽带接收系统性能的幅相误差作了简要分析,随后提出了用复系数FIR滤波器实现整个系统幅相误差校正的方案,并推导出一种设计幅相校正FIR滤波器的算法,最后给出了一个用此方案实现超宽带系统幅相误差校正的实例.     

Two-channel IIR QMF banks with approximately linear-phase analysisand synthesis filters

Perfect linear-phase two-channel QMF banks require the use of finite impulse response (FIR) analysis and synthesis filters. Although they are less expensive and yield superior stopband characteristics, perfect linear phase cannot be achieved with stable infinite impulse response (IIR) filters. Thus, IIR designs usually incorporate a postprocessing equalizer that is optimized to reduce the phase distortion of the entire filter bank. However, the analysis and synthesis filters of such an IIR filter bank are not linear phase. In this paper, a computationally simple method to obtain IIR analysis and synthesis filters that possess negligible phase distortion is presented. The method is based on first applying the balanced reduction procedure to obtain nearly allpass IIR polyphase components and then approximating these with perfect allpass IIR polyphase components. The resulting IIR designs already have only negligible phase distortion. However, if required, further improvement may be achieved through optimization of the filter parameters. For this purpose, a suitable objective function is presented. Bounds for the magnitude and phase errors of the designs are also derived. Design examples indicate that the derived IIR filter banks are more efficient in terms of computational complexity than the FIR prototypes and perfect reconstruction FIR filter banks. Although the PR FIR filter banks when implemented with the one-multiplier lattice structure and IIR filter banks are comparable in terms of computational complexity, the former is very sensitive to coefficient quantization effects     

最小平方误差受限的FIR滤波器设计

在设计FIR滤波器时,往往会指定过渡带大小,但过渡带的引入只是为了便于滤波器的设计,而并不是物理上的需要,所以在设计中仅需指定截至频率。这是第一个设计理念。此外,在FIR滤波器的设计中存在两种准则:一是等波纹设计准则 (即最大误差最小化或者Chebyshev准则 ),另一种是平方误差最小准则。但在现实中两种准则往往要同时兼顾,所以仅基于其中一种准则来设计不能得到最佳结果。这是第二个设计理念。基于上述两种设计理念,提出了一种新的FIR滤波器设计算法。该算法采用最陡梯度下降法来对平方误差最小化下的最佳滤波器系数进行迭代修正,得到最佳结果。     

METEOR: a constraint-based FIR filter design program

It is proposed to specify a filter only in terms of upper and lower limits on the response, find the shortest filter length which allows these constraints to be met, and then find a filter of that order which is farthest from the upper and lower constraint boundaries in a minimax sense. The simplex algorithm for linear programming is used to find a best linear-phase FIR filter of minimum length, as well as to find the minimum feasible length itself. The simplex algorithm, while much slower than exchange algorithms, also allows the incorporation of more general kinds of constraints, such as concavity constraints (which can be used to achieve very flat magnitude characteristics). Examples are given to illustrate how the proposed and common approaches differ, and how the proposed approach can be used to design filters with flat passbands, filters which meet point constraints, minimum phase filters, and bandpass filters with controlled transition band behavior     

Minimax design of two-channel low-delay perfect-reconstruction FIRfilter banks

The authors deal with the design problem of low-delay perfect-reconstruction filter banks for which the FIR analysis and synthesis filters have equiripple magnitude response. Based on the minimax error criterion, the design problem is formulated in such a manner that the coefficients for the FIR analysis filters can be found by minimising the weighted peak error of the designed analysis filters, subject to the perfect-reconstruction constraints. A design technique based on a modified dual-affine scaling variant of Karmarkar's (1989) algorithm, in conjunction with approximation schemes, is then developed for solving the resulting nonlinear optimisation problem. The effectiveness of the proposed design technique is demonstrated by several simulation examples     

Design of smallest size two-dimensional linear-phase FIR filters with magnitude error constraint

This paper presents an online procedure that produces the smallest feasible size of two-dimensional FIR filters with prescribed magnitude error constraint. The procedure uses the mean square normalized error of constrained and unconstrained least-square filters to produce the initial and the subsequent sizes that converge to the smallest feasible one in a few iterations, where the constrained least-square filters are defined as the least-square filters satisfying the magnitude error constraint. The procedure finally returns a smallest size filter that satisfies the magnitude error constraint and has least total squared magnitude error. Design examples of diamond-shaped, rectangular, and elliptic filters are provided, and comparisons with an exhaustive search are given.     

Constrained least squares design of 2-D FIR filters

We consider the design of 2-D linear phase finite impulse response (FIR) filters according to the least squares (LS) error criterion subject to equality and/or inequality constraints. Since we use a frequency domain formulation, these constraints can be used to explicitly prescribe (frequency-dependent) error tolerances, the maximum, minimum, or fixed values of the frequency response at certain points and/or regions. Our method combines Lagrange multiplier and Kuhn-Tucker theory to solve a much wider class of problems than do standard methods. It allows arbitrary compromises between the LS and the equiripple design