Chebyshev approximation for linear phase nonrecursive digital filters

A procedure has been developed for the design of nonrecursive digital filters with prescribed passband and stopband amplitude characteristics. The proposed procedure is based on an efficient algorithm utilizing the simplex method of linear programming. The design algorithm yields equiripple approximation and it is an alternative to the one which is based on the Remz exchange algorithm. The design procedure allows exact specifications for arbitrary passband and stopband edges. Furthermore, no prior knowledge of the degree of the filter is required. To demonstrate the potential of the design algorithm, several examples with different requirements are worked out and a sample is presented. The obtained results show that the design procedure performed very well where the various parameters of the filter were taken into consideration.     

Coefficients of maximally flat low and high pass nonrecursive digital filters with specified cutoff frequency

A simple and general method is presented for determining the coefficients of a lowpass or highpass maximally flat nonrecursive digital filter with a specified cutoff frequency.     

Maximally selective generalized Chebyshev filters with asymmetrically located transmission zeros

An efficient design procedure for generalized Chebyshev filters with asymmetrically located transmission zeros is presented in this paper. The exact and simple procedure is given for the calculation of transmission zeros, for a class of generalized Chebyshev low-pass prototype filters with a maximum of three asymmetrically located transmission zeros of any multiplicity. This procedure is based on finding the frequencies of magnitude characteristic extreme values in closed form. New formulas for orders of zeros of maximally selective filters with equi-ripple characteristics in the pass-band and stop-band are deduced, and these are very useful in the design procedure. The obtained results are verified and illustrated by corresponding examples.     

基于Chebyshev正交多项式的通道相位估计算法

针对信号传输通道相位估计等问题,提出了一种通过利用Chebyshev正交多项式逼近函数计算反正切函数值求取通道相位的算法,该算法通过象限判别、区间划分的方法计算反正切函数的输入值,然后利用Chebyshev正交多项式逼近函数计算信号相位,最后,通过迭代算法简化计算。该算法具有算法简单,估计速度快、精度高的特点。     

Method for FIR filters design with compressed cosine using Chebyshev's norm

This paper considers a new method for FIR filters design. The method uses an L_∞ optimality norm. To achieve a better approximating effect, a new modulating function which compresses the oscillations of the cosine is proposed. A parameter sets the gradient of the modulating function, with respect to the oscillations’ compression. The approximating polynomial is carried out using Remez’ exchange algorithm. An optimal polynomial with lowest possible (four) degree, that approximates an ideal filter's response with high precision is proposed. With the proposed method a FIR filter with arbitrary specifications can be designed. Design examples of FIR filters with a minimization of calculation are performed. The obtained filter's responses are close to the ideal response. The design examples demonstrate that the proposed approach may be a good alternative in several applications.     

Chebyshev approximation problem of multidimensional IIR digital filters in the frequency domain

This paper investigates the problem of simultaneous approximation of a prescribed multidimensional frequency response. The frequency responses of multidimensional IIR digital filters are used as nonlinear approximating functions. Chebyshev approximation theory and the notion of line homotopy are used to reveal the approximation properties of this set of IIR functions. A sign condition is derived to characterize a convex stable domain in this set. This sign condition can be incorporated into the optimization of the Chebyshev simultaneous approximation. The generally sufficient global Kolmogorov criterion is shown to be a necessary condition, for the characterization of best approximation, in the considered set of approximating functions. Thus, it can be incorporated, as a stopping constraint, in the design of the optimal filter. Moreover, the local Kolmogorov criterion is shown to be also necessary for the set of approximating functions. Finally, it is proved that the best approximation is a global minimum.

     

Correction of phase-frequency characteristics of minimum-phase recursive digital low-pass filters

A technique and an example of solving the problem of PRC correction of minimum-phase recursive digital low-pass filters (RDF) have been considered using the minimax (Chebyshev) proximity criterion of approximable and approximating functions.     

基于巴特沃斯逼近的二维IIR数字滤波器的设计

本文给出了一种基于巴特沃斯逼近的二维ＩＩＲ数字滤波器的设计方法，得到了由基本的全通节级联，并联实现的各种二维滤波器函数，包括，镜象对称互补滤波器，扇形滤波器和具有任意矩形通、阻带的滤波器，结果表明，这种实现结构具有通有灵敏度低、滤波器系数少的优点，并且由于巴特沃斯逼近的最大平坦性，得到的滤波器具有良好的相位特性。     

1D and 2D economical FIR filters generated by Chebyshev polynomials of the first kind

Christoffel–Darboux formula for Chebyshev continual orthogonal polynomials of the first kind is proposed to find a mathematical solution of approximation problem of a one-dimensional (1D) filter function in the z domain. Such an approach allows for the generation of a linear phase selective 1D low-pass digital finite impulse response (FIR) filter function in compact explicit form by using an analytical method. A new difference equation and structure of corresponding linear phase 1D low-pass digital FIR filter are given here. As an example, one extremely economic 1D FIR filter (with four adders and without multipliers) is designed by the proposed technique and its characteristics are presented. Global Christoffel–Darboux formula for orthonormal Chebyshev polynomials of the first kind and for two independent variables for generating linear phase symmetric two-dimensional (2D) FIR digital filter functions in a compact explicit representative form, by using an analytical method, is proposed in this paper. The formula can be most directly applied for mathematically solving the approximation problem of a filter function of even and odd order. Examples of a new class of extremely economic linear phase symmetric selective 2D FIR digital filters obtained by the proposed approximation technique are presented.     

A practical method of coefficient computation for a microprocessor-controlled variable recursive digital filter

The synthesis of general-type recursive digital filters from analog prototypes may require complicated computations not suitable for filter structures where the coefficients are to be evaluated by a microprocessor control unit. A general synthesis procedure suitable for the application to such structures is described and some design examples are reported.     

Optimal polygonal approximation of digital curves

An algorithm is described in this paper which finds the optimal polygonal approximation of a digital curve i.e. an approximation with the minimum possible number of line segments. The algorithm can be used either for digital curves in 2-space or 1-space, i.e. in waveforms.     

Design of a tunable frequency CMOS fully differential fourth-order Chebyshev filter

A new fully differential amplifier and a fully differential R-MOSFET-C fourth-order Chebyshev active lowpass filter employing passive resistors and current-steering MOS transistors as variable resistors are proposed. The implementation relies on the tunability of current-steering MOS transistors operating in the triode region which counteract the deviation of resistors in integrated circuit manufacturing technology in order that the cutoff frequency of Chebyshev active filter can be realized accurately tunable. The amplifier is not only with voltage common-mode negative feedback (VCMFB), but also with current common-mode negative feedback (CCMFB), which will benefit for the stability of its DC operating point. A cutoff frequency of 138 kHz fourth-order Chebyshev lowpass filter was designed and fabricated using 3.3 V power supply and 0.35 μm CMOS technology. Chip test results demonstrate better than −68 dB THD with 70 kHz, 2.0V_pp signal, frequency turning range of more than 14,000 from 3 Hz to 420 kHz, chip area of 0.36 mm² and power consumption of 16 mW.     

变频长自适应数字滤波器的稳定条件

变步长自适应数字滤波器的权值随信号和噪声的变化自动调整时可能出现不稳定。为解决这一问题,本文提出变步长自适应数字滤波器的权值收敛条件,该条件具有较少保守性,对输入信号的统计特性没有严格限制,本文证明自适应数字滤波的稳定性可由其权值收敛条件确定。利用该权值收敛条件,可设计确保变步长自适应数字滤波器的稳定的变收敛因子。     

Linear time-invariant space-variant filters and the parabolic equation approximation

Wave propagation in a random, inhomogeneous ocean is treated as transmission through a linear, time-invariant, space-variant, random communication channel. Using the parabolic equation approximation of the Helmholtz wave equation, a random transfer function of the ocean volume is derived. The ocean volume is characterized by a three-dimensional random index of refraction which is decomposed into deterministic and random components. Two additional calculations are performed using the transfer function. The first involves the derivation of the equations for the random, output electrical signals at each element in a receive planar array of complex weighted point sources in terms of the frequency spectrum of the transmitted electrical signal, the transmit and receive arrays, and the transfer function of the ocean medium. The second involves the derivation of the coherence function.     

Coefficients of maximally flat nonrecursive digital filters

A general method is presented for determining the coefficients of maximally flat nonrecursive digital filters. Recursive relations with integer coefficients are derived for these coefficients by explicit calculations imitating Crout's method for solving the linear equations obtained by setting the derivatives of the symmetric, linear phase transfer function to zero at w = 0 and w = π. For a given order of the filter, the amount of computation required in this method is uniformly the same for any combination of permissible degrees of flatness at w = 0 and w = π.     

On the use of continued fraction filter structures for the design of 2-D digital filters

In this paper a method for the realization of 2-D recursive digital filters of second order by means of continued fraction filter structures is presented. The proposed method is based on the interconnection of basic filter structures of first order and gives also the set of all 2-D transfer functions that are realizable by these structures. The general rule for interconnecting these basic structures is described. It is proved that, on the set of second order fraction filter structures, some classes of structures and ordering relations can be defined, so that the set of all the classes form a partially ordered set (poset). Those structures which least constrain the coefficients of the transfer function are defined maximum elements of the poset. It is shown that among all possible continued fraction structures of second order only six are suitable for realization of two dimensional filters. Some results following from the use of spectral transformations are discussed.     

Frequency and time domain comparison of selective polynomial filters with corrected phase characteristics

Two solutions to the polynomial filter’s transfer function synthesis problem are considered for comparison in the frequency and time domain: the broad class of filters with a critical monotonic amplitude characteristic (CMAC) in the passband and filters which use Chebyshev (C) polynomials. To complete the synthesis procedure for linear phase applications, group delay correctors are considered, for which a convenient approximation procedure is proposed here. Comparisons of the original functions and the corrected ones are performed in the frequency and time domain. It is shown that when CMAC and C are compared as such, the latter is by no means preferable from the selectivity point of view, while the opposite stands when the comparison is based on passband amplitude distortions. When phase-corrected filtering functions are compared, based on circuit complexity and time domain performance, the CMAC are shown to be preferable.     

多相滤波数字信道化的FPGA实现

以多相滤波为基础,结合均匀滤波器组,采用50％重叠的子信道划分,提出了一种数字信道化实现方法,解决了高速实时处理与FPGA处理速度之间的矛盾,克服了信道化接收机的接收盲区.基于FPGA,提出了短波宽带数字信道化的设计思路和实现方法.仿真结果表明,该设计有较强的实用性和通用性.     

A new approach to the design of complex all-pass IIR digital filters

A new method is proposed for designing complex all-pass IIR filters, the all-pass IIR filters with complex coefficients, in this paper. By minimizing the integration of certain square phase error over interested frequencies, an eigenvector of an appropriate real, symmetric and positive-definite matrix is computed to get the filter coefficients. The stability is achieved by specifying properly the desired phase specifications. If an appropriate iterative process is used, equiripple complex all-pass filter design can be obtained. The method is simple and the performance is comparable to the existing methods. Several examples are presented to demonstrate the effectiveness of the approach.     

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.