Polynomial-Based Interpolation Filters—Part I: Filter Synthesis

This paper introduces a generalized design method for polynomial-based interpolation filters. These filters can be implemented by using a modified Farrow structure, where the fixed finite impulse response (FIR) sub-filters possess either symmetrical or anti-symmetrical impulse responses. In the proposed approach, the piecewise polynomial impulse response of the interpolation filter is optimized directly in the frequency domain using either the minimax or least mean square criterion subject to the given time domain constraints. The length of the impulse response and the degree of the approximating polynomial in polynomial intervals can be arbitrarily selected. The optimization in the frequency domain makes the proposed design scheme more suitable for various digital signal processing applications and enables one to synthesize interpolation filters for arbitrary desired and weighting functions. Most importantly, the interpolation filters can be optimized in a manner similar to that of conventional linear-phase FIR filters.     

Lost knowledge refound: sharpened FIR filters

A scenario is presented in which an engineer in the field finds that there is a problem with the system specifications and a symmetric finite impulse response (FIR) filter in the software does not do the job; it needs reduced passband ripple or, maybe, more stopband attenuation. We present a simple method for transforming an FIR filter into one with better passband and stopband characteristics, while maintaining phase linearity. While filter sharpening may not be used often, it does have its place in an engineer's toolbox. An optimal filter has a shorter impulse response than a sharpened filter with the same passband and stopband ripple, and thus is more computationally efficient. However, filter sharpening can be used whenever a given filter response cannot be modified, such as a software code that makes use of an unchangeable filter subroutine. The scenario described is hypothetical, but all practicing engineers have been in situations where a problem needs to be solved without the full arsenal of normal design tools. Filter sharpening could be used when improved filtering is needed, but insufficient ROM space is available to store more filter coefficients, or as a way to reduce ROM requirements. In addition, in some hardware applications using filter ASICs, it may be easier to add additional chips to a design than it is to design a new ASIC.     

B-spline design of maximally flat and prolate spheroidal-type FIRfilters

A digital FIR filter is described that offers excellent passband and stopband characteristics for general applications. Design formulae include parameters that adjust the magnitude response from one having characteristics like the maximally flat designs of Hermann (1971) and Kaiser (1975, 1979) to one having characteristics like the minimum-sidelobe energy approximations of Kaiser and Saramaki (1989). The impulse response coefficients are more straightforward to obtain than these filter designs while offering preferable response characteristics in many instances. Unlike FIR filters designed by window- or frequency-sampling methods, the filter coefficients are determined from the inverse Fourier transform in closed form once B-splines have been used to replace sharp transition edges of the magnitude response. Although the filters are developed in the frequency domain, a convergence window is identified in the convolution series and compared with windows of popular FIR filters. By means of example, adjustment of the transitional parameter is shown to produce a filter response that rivals the stopband attenuation and transition width of prolate spheroidal designs. The design technique is extended to create additional transitional filters from prototype window functions, such as the transitional Hann window filter. The filters are particularly suitable for precision filtering and reconstruction of sampled physiologic and acoustic signals common to the health sciences but will also be useful in other applications requiring low passband and stopband errors     

Optimal Design of 2D FIR Filters with Quadrantally Symmetric Properties Using Fractional Derivative Constraints

In this article, an optimal design of two-dimensional finite impulse response (2D FIR) filter with quadrantally even symmetric impulse response using fractional derivative constraints (FDCs) is presented. Firstly, design problem of 2D FIR filter is formulated as an optimization problem. Then, FDCs are imposed over the integral absolute error for designing of the quadrantally even symmetric impulse response filter. The optimized FDCs are applied over the prescribed frequency points. Next, the optimized filter impulse response coefficients are computed using a hybrid optimization technique, called hybrid particle swarm optimization and gravitational search algorithm (HPSO-GSA). Further, FDC values are also optimized such that flat passband and stopband frequency response is achieved and the absolute \(L_1\)-error is minimized. Finally, four design examples of 2D low-pass, high-pass, band-pass and band-stop filters are demonstrated to justify the design accuracy in terms of passband error, stopband error, maximum passband ripple, minimum stopband attenuation and execution time. Simulation results have been compared with the other optimization algorithms, such as real-coded genetic algorithm, particle swarm optimization and gravitational search algorithm. It is observed that HPSO-GSA gives improved results for 2D FIR-FDC filter design problem. In comparison with other existing techniques of 2D FIR filter design, the proposed method shows improved design accuracy and flexibility with varying values of FDCs.     

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.     

一种Farrow结构数字延时滤波器的设计

普通数字延时滤波器虽然结构简单,但系数计算过程复杂,在延时参数快速变化时,系数更新速度无法满足实时性要求,在工程应用上受限制。采用Farrow结构数字延时滤波器能够更加灵活高效地进行分数延时滤波,延时参数改变时,无需重新计算滤波器系数,更容易在现场可编程门阵列(FPGA)上实现。介绍了一种Farrow结构数字延时滤波器,提出采用基于对称结构的滤波器系数求解方法,并经过加权优化,获得最终Farrow滤波器的系数。系数计算过程中,通过对设计所得Farrow滤波器延时精度和误差的分析,调整加权因子的取值和滤波器阶数,进而提高延时精度。计算机仿真结果证明了加权对称系数求解Farrow滤波器系数方法的有效性和实用性。     

软件无线电中用于采样速率转换的Farrow结构滤波器设计

首先对软件无线电中的信号重采样原理进行分析,然后对基于多项式冲激响应滤波器的输出进行推导,由此导出Farrow结构滤波器和转置Farrow结构滤波器的结构图.最后用DSP Builder进行了一个实例设计.     

A new closed form method for design of variable bandwidth linear phase FIR filter using different polynomials

In this paper, a new method for the design of variable bandwidth linear-phase finite impulse response (FIR) filters using different polynomials such as shifted Chebyshev polynomials, Bernstein polynomials and shifted Legendre polynomials is proposed. For this purpose, the transfer function of a variable bandwidth filter, which is a linear combination of fixed-coefficient linear-phase filters and the above polynomials are separately exploited as tuning parameters to control bandwidth of the filter. In order to determine the filter coefficients, mean squared difference between the desired variable bandwidth filter and the practical filter is minimized by differentiating it with respect to its coefficients leading to a system of linear equations. The matrix elements can be expressed in form of Toeplitz-plus-Hankel matrix, which reduces the computational complexity. Several examples are included to demonstrate effectiveness of the proposed method in terms of passband error (e_p), stopband error (e_s) and stopband attenuation (A_s).     

Closed-form design and efficient implementation of variable digital filters with simultaneously tunable magnitude and fractional delay

This paper proposes a closed-form solution for designing variable one-dimensional (1-D) finite-impulse-response (FIR) digital filters with simultaneously tunable magnitude and tunable fractional phase-delay responses. First, each coefficient of a variable FIR filter is expressed as a two-dimensional (2-D) polynomial of a pair of parameters called spectral parameters; one is for independently tuning the cutoff frequency of the magnitude response, and the other is for independently tuning fractional phase-delay. Then, the closed-form error function between the desired and actual variable frequency responses is derived without discretizing any design parameters such as the frequency and the two spectral parameters. Finally, the optimal solution for the 2-D polynomial coefficients can be easily determined through minimizing the closed-form error function. We also show that the resulting variable FIR filter can be efficiently implemented by generalizing Farrow structure to our two-parameter case. The generalized Farrow structure requires only a small number of multiplications and additions for obtaining any new frequency characteristic, which is particularly suitable for high-speed tuning.     

Analytical design method for optimal equiripple comb FIR filters

A novel analytical design method for highly selective digital optimal equiripple comb finite-impulse response (FIR) filters is presented. The equiripple comb FIR filters are optimal in the Chebyshev sense. The number of notch bands, the width of the notch bands and the attenuation in the passbands can be independently specified. The degree formula and the differential equation for the generating polynomial of the filter is presented. Based on the differential equation, a fast simple algebraic recursive procedure for the evaluation of the impulse response of the filter is described. Its arithmetic robustness outperforms, by far, the known analytical design method. Highly selective equiripple comb FIR filters with thousands of coefficients can be designed. One example demonstrates the efficiency of the filter design.     

Design of recursive variable digital filters with theoretically guaranteed stability

Stability is one of the most concerned issues in designing a recursive variable digital filter (VDF). This is because the coefficients of a recursive VDF constantly vary in the tuning process, and updating the coefficients may incur instability. Thus, an appropriate measure needs to be taken for ensuring its stability. This paper presents a new coefficient transformation (CT) method for transforming the coefficients of a recursive transfer-function denominator into a set of new coefficients. From the viewpoint of conventional constant-coefficient filter (constant filter) design, the new coefficients can take arbitrary values without incurring instability. For designing a stable VDF, we apply this CT to the variable case and approximate each transformed coefficient as a distinct polynomial in the tuning parameter. Thus, we can change the filter coefficients by changing the value of the tuning parameter, and thus tune the magnitude response. Thanks to the proposed CT, updating the filter coefficients will never incur instability. This is the core part of the CT-based design approach. In this paper, we utilise a weighting function to ignore the transition-band errors and thus enhance the design accuracy of important frequency bands (passband and stopband). Moreover, the polynomials use different degrees so as to reduce the VDF complexity. Two design examples (lowpass VDF and bandpass VDF) are provided for verifying the design accuracy and checking the stability.     

Design and multiplier-less implementation of a class of two-channel PR FIR filterbanks and wavelets with low system delay

A new method for designing two-channel PR FIR filterbanks with low system delay is proposed. It is based on the generalization of the structure previously proposed by Phoong et al. (1995) Such structurally PR filterbanks are parameterized by two functions (/spl beta/(z) and /spl alpha/(z)) that can be chosen as linear-phase FIR or allpass functions to construct FIR/IIR filterbanks with good frequency characteristics. The case of using identical /spl beta/(z) and /spl alpha/(z) was considered by Phoong et al. with the delay parameter M chosen as 2N-1. In this paper, the more general ease of using different nonlinear-phase FIR functions for /spl beta/(z) and /spl alpha/(z) is studied. As the linear-phase constraint is relaxed, the lengths of /spl beta/(z) and /spl alpha/(z) are no longer restricted by the delay parameters of the filterbanks. Hence, higher stopband attenuation can still be achieved at low system delay. The design of the proposed low-delay filterbanks is formulated as a complex polynomial approximation problem, which can be solved by the Remez exchange algorithm or analytic formula with very low complexity. In addition, the orders and delay parameters can be estimated from the given filter specifications using a simple empirical formula. Therefore, low-delay two-channel PR filterbanks with flexible stopband attenuation and cutoff frequencies can be designed using existing filter design algorithms. The generalization of the present approach to the design of a class of wavelet bases associated with these low-delay filterbanks and its multiplier-less implementation using the sum of powers-of-two coefficients are also studied.     

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     

以ⅡR滤波器作补偿的CIC滤波器设计

为解决积分梳状(CIC)滤波器通带失真大和阻带衰减小对其应用的限制,在分析传统CIC滤波器幅频特性的基础上,给出一种用二阶ⅡR滤波器作为补偿滤波器级联CIC滤波器的改进方法。仿真结果表明,它与同级数规模的内插二阶多项式CIC滤波器、锐化CIC滤波器(SCIC)相比,通带和阻带的性能得到较大改善,实现复杂度较低。因此,它适用于对通带、阻带性能和实现复杂度要求较高的多采样率转换系统。     

An improved method for the design of FIR quadrature mirror-imagefilter banks

A new method for the design of general finite-duration impulse response (FIR) quadrature mirror-image filter (QMF) banks that eliminates the computation of large matrices is proposed. The design problem is formulated to include low-delay QMF banks, which are highly desirable in some applications. The paper concludes with design results and comparisons that show that conventional QMF banks can be designed with only a fraction of the computational effort required by a method due to Chen and Lee (1992). On the other hand, in the case of low-delay QMF banks, the proposed method can increase the stopband attenuation substantially compared with what can be achieved by existing methods     

SVD-based design and new structures for variable fractional-delay digital filters

This paper shows that the problem of designing one-dimensional (1-D) variable fractional-delay (VFD) digital filter can be elegantly reduced to the easier subproblems that involve one-dimensional (1-D) constant filter (subfilter) designs and 1-D polynomial approximations. By utilizing the singular value decomposition (SVD) of the variable design specification, we prove that both 1-D constant filters and 1-D polynomials possess either symmetry or anti-symmetry simultaneously. Therefore, a VFD filter can be efficiently obtained by designing 1-D constant filters with symmetrical or antisymmetrical coefficients and performing 1-D symmetrical or antisymmetrical approximations. To perform the weighted-least-squares (WLS) VFD filter design, a new WLS-SVD method is also developed. Moreover, an objective criterion is proposed for selecting appropriate subfilter orders and polynomial degrees. Our computer simulations have shown that the SVD-based design and WLS-SVD design can achieve much higher design accuracy with significantly reduced filter, complexity than the existing WLS design method. Another important part of the paper proposes two new structures for efficiently implementing the resulting VFD filter, which require less computational complexity than the so-called Farrow structure.     

数字混响效果器的设计研究

混响效果器的冲激响应常常具有很长的长度,应用目前常规滤波器的设计方法很难在实时处理的条件下使误差、存储量和延迟较小.根据冲激响应的频谱能量基本上集中在低频这一特点,本文提出一种将高低频段分别处理的设计方法.首先对冲激响应和信号的低频段进行抽取,抽取后的冲激响应构成FIR滤波器,抽取后的信号通过这个滤波器后再进行零值内插,然后通过一个低通滤波器输出.冲激响应的高频部分在时间较长时渐趋于零.因而将其截断后再采用变换域处理方法获得信号高频部分的输出.这一方法对于TMS320C5402 DSP芯片来说,能够在使误差、存储量和延迟较小的条件下达到实时处理.     

A new closed form method for design of variable bandwidth linear phase FIR filter using Bernstein multiwavelets

In this paper, a new method for the design of variable bandwidth linear-phase finite impulse response filters using Bernstein polynomial Multiwavelets is proposed. In this method, approximation has been achieved by linearly combining the fixed coefficient linear phase filters with Bernstein multiwavelets, which are used to tune bandwidth of the filter. Optimisation has been achieved by minimising the mean square error between the desired and actual filter response which leads to a system of linear equations. The matrix elements can be expressed in form of Toeplitz-plus-Hankel matrix, which reduces the computational complexity. The simulation results illustrate significant improvement in errors in passband (e_p), and stopband (e_s) as compared to earlier published work.     

A modified Chebyshev bandpass filter with attenuation poles in thestopband

This paper describes a design method of a modified Chebyshev bandpass filter with attenuation poles in the stopband. The insertion of attenuation poles into resonators in the authors' bandpass-filter design is accomplished by connecting a lumped inductor or capacitor in series with a shunt-type coaxial transmission-line resonator. The inserted poles which are distributed over the stopband can be chosen such that the insertion loss of the filter has equiripple characteristic and maximum attenuation in the stopband with the given number of attenuation poles. The modified Chebyshev bandpass filter designed by this method can be effectively used in diplexer design     

Adaptive Complex-Coefficient 2D FIR Trapezoidal Filters for Broadband Beamforming in Cognitive Radio Systems

A novel design method is proposed for an adaptive discrete-domain beamformer for the beamforming of temporally broadband-bandpass signals in cognitive radio (CR) systems. The method is based on a complex-coefficient 2D finite impulse response (FIR) filter having a trapezoidal-shaped passband. The temporally broadband-bandpass signals are received by a 1D uniformly distributed antenna array (1D UDAA), where the outputs of the antennas are complex-quadrature sampled by the front end of the CR system. This CR system is based on a software defined radio (SDR) architecture and can be instantly reconfigured by the control system to select the appropriate frequency band and the required sampling rate. The subsequent beamforming enhances the spectral components of the desired temporally broadband-bandpass signals by arranging for the asymmetric trapezoidal-shaped passband of the 2D filter transfer function to closely enclose the region of support (ROS) of the spectrum of the desired signal, whereas the ROSs of the spectral components of the interfering signals are enclosed by the stopband. The proposed novel closed-form design method facilitates instant adaptation of the shape and orientation of the passband of the beamforming 2D FIR trapezoidal filter in order to match the time-varying frequency band and the time-varying bandwidth of the signal, as well as to track and enhance received signals with time-varying directions of arrival (DOAs). Simulated results confirm that, compared with previously reported methods, the proposed method achieves the best overall tradeoff with respect to the instantaneous adaptations of the operating frequency band, the bandwidth, and the time-varying DOAs, the distortion of the desired passband signal, and the stopband attenuation of interfering signals.