Elimination of limit cycles due to two's complement quantization innormal form digital filters

Normal form digital filters are attractive due to their desirable properties when implemented in finite wordlength arithmetic. These filters are free from all overflow limit cycles and quantization limit cycles when magnitude truncation is used, However, when two's complement truncation (TCT) quantization is used, limit cycles can still exist. In this paper, it is shown that when block structures are used, normal form digital filters can be made free of limit cycles due to TCT quantization. It is shown that this can be done with a small block size. An algorithm is also presented to find the minimum block size required for a given filter. Some examples are given to illustrate the results     

Two's complement quantization in periodic digital filters

The stability of periodically shift variant (PSV) filters are studied when implemented with two's complement truncation (TCT) quantization. Block form and standard (nonblock) form implementations are considered, and two sufficient conditions are established for stability. As a special case, second-order coupled-form PSV filters are then investigated under TCT quantization. Stability regions are established within the parameter space for block implementations. Examples are given to illustrate the results     

Determination of quantization error in two-dimensional digital filters

The evaluation of the quantization error in two-dimensional (2-D) digital filters involves the following computation J = Σm=0^∞Σn=0^∞y²(m, n). In this paper general method for the evaluation of J based on Laurent expansion of the integrand is presented. Illustrative example for such a computation is given. Notations: We denote with V^-2= {(z1, z2): |z1| ≤ 1, |z2| ≤ 1} the closed unit bidisc, with V²= {(z1, z2): |z1| < 1, |z2| < 1} the open unit bidisc, and with T2= {(z1, z2): |z1| = 1, |z2| = 1} the distinguished boundary of the unit bidisc.     

On the realization of two's complement overflow limit cycle free state-space digital filters

A frequency-domain criterion for the overflow stability of a class of state-space digital filters is presented. The result is derived using a Lyapunov technique and the associated MKY machinery. Some consequences of the present derivation are discussed.     

Two's complement quantization in orthogonal biquad digital filters

A special class of complex biquad digital filters called orthogonal filters are investigated for stability under two's complement quantization. A sufficient condition is derived for the asymptotic stability of the nonlinear filter. Bounds on the possible limit cycles are also obtained. Using these bounds, any given filter can be tested for stability. The stability triangle is then scanned using a dense grid, and each point on the grid is tested for stability/limit cycles. By this method, the stability region given by the sufficient condition is extended. Regions within the linear stability triangle where various types of limit cycles are possible are also identified.     

Asymptotic stability of fixed-point state-space digital filters with combinations of quantization and overflow nonlinearities

A novel criterion for the global asymptotic stability of fixed-point state-space digital filters under various combinations of quantization and overflow nonlinearities is presented. The criterion is compared with a previously reported criterion.     

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。     

Decoupling of the overflow and quantization phenomena in orthogonal biquad recursive digital filters

In digital communication networks, a special class of complex biquad recursive digital filters called orthogonal filters is increasingly being used. The separate effects of the overflow and quantization nonlinearities on these orthogonal filters' responses have been investigated [5], [6]. In this paper we examine the zero-input stability properties of the actual orthogonal filter having both overflow and quantization nonlinearities. The overflow nonlinearities considered include saturation, bit-by-bit inversion, zeroing, and modulo 2 arithmetic. The quantization techniques used may be roundoff, magnitude, or value truncation. An example demonstrates the adverse coupling effect between the overflow and quantization nonlinearities. Two criteria are therefore derived to ensure asymptotic overflowstability of the filter in the presence of quantization. These criteria have been translated to the coefficient plane; various regions corresponding to different minimum wordlengths required to ensure decoupling of the overflow and quantization phenomena have been derived.This work was supported in part by a Monash Graduate Scholarship and Telecom Australia under Contract No. 64515.     

On fast FIR filters implemented as tail-canceling IIR filters

We have developed an algorithm based on synthetic division for deriving the transfer function that cancels the tail of a given arbitrary rational (IIR) transfer function after a desired number of time steps. Our method applies to transfer functions with repeated poles, whereas previous methods of tail-subtraction cannot. We use a parallel state-variable technique with periodic refreshing to induce finite memory in order to prevent accumulation of quantization error in cases where the given transfer function has unstable modes. We present two methods for designing linear-phase truncated IIR (TIIR) filters based on antiphase filters. We explore finite-register effects for unstable modes and provide bounds on the maximum TIIR filter length. In particular, we show that for unstable systems, the available dynamic range of the registers must be three times that of the data. Considerable computational savings over conventional FIR filters are attainable for a given specification of linear-phase filter. We provide examples of filter design. We show how to generate finite-length polynomial impulse responses using TIIR filters. We list some applications of TIIR filters, including uses in digital audio and an algorithm for efficiently implementing Kay's optimal high-resolution frequency estimator     

Design criterion for zero-input asymptotic overflow-stability of recursive digital filters in the presence of quantization

This paper investigates the zero-input stability properties of the exact second-order recursive digital filter having both overflow and quantization non-linearities. Two examples demonstrate the adverse influence of quantization on the overflow-stability property of the filter. Three sets of conditions are presented to ensure asymptotic overflow-stability in the presence of quantization. Using these criteria, various regions in the coefficient plane corresponding to different minimum internal wordlengths required to ensure the non-interaction of overflow and quantization are derived. These results thus form a useful design criterion.This work was supported in part by a Monash Graduate Scholarship (awarded to P.K. Sim) and Telecom Australia under Contract No. 64514.     

Stability test for two-dimensional recursive digital filters using inner determinants

A recently introduced stability criterion for two-dimensional digital filters is implemented by decision algebra. The proposed test takes the form of a local positivity test applied to a two-variable polynomial with real coefficients. Two non-trivial examples are given to illustrate the procedure.     

Stability regions for second-order fixed-point digital filters in coupled form

The influence of the asymmetry parameterk on the extent and form of the stability region for second-order fixed-point coupled-form digital filters is analyzed. In particular, by extending the results of a previous paper by the same authors, the stability regions for filters with roundoff quantizers are first shown. Then, the corresponding results for filters with either two or four value-truncation quantizers are derived. It turns out that the shape of the stability areas is rather odd because of the asymmetry of the quantization characteristic.     

A comparison of LTE channelizers implemented with linear phase m-path recursive filters and FIR filters

Cell sites repeaters may receive a composite signal containing a mix of long term evolution channels with bandwidths of 5, 10, 15, and 20 MHz and be required to rearrange the frequency plan of the channels or to drop and insert specific channels prior to transmitting the altered composite signal. The straight forward approach to this task is to down-convert, and down-sample each channel in the mix and then up-sample and up-convert and merge the new traffic mix. The filters applied to the up and down conversion task as well as the up and down sampling task would likely be linear phase finite impulse response (FIR) filters because of the ease with which the resampling task can be embedded in the filtering task. We present an alternate filter structure formed from linear phase recursive filters and compare their performance and computational complexity with their FIR filter counterparts. We show that the recursive filter version of the channel extractor requires significantly few arithmetic operations and actually outperforms the non-recursive version as demonstrated by the error vector magnitude of the two options.     

Lowpass digital filters with linear phase

A method of synthetising lowpass recursive digital filters is given in the letter. The method relies on the properties of orthogonal functions on the z plane and the impulse response of an ideal lowpass linear phase filter. Considerable saving in computation and storage appears to have been achieved by this method compared with the nonrecursive (with window functions) truncation method.     

Two-dimensional digital filters with sparse coefficients

Is sparsity an issue in 2-D digital filter design problems to explore and why is it important? How a 2-D filter can be designed to retain a desired coefficient sparsity for efficient implementation while achieving best possible performance subject to that sparsity constraint? These are the focus of this paper in which we present a two-phase design method for 2-D FIR digital filters in two most common design settings, namely, the least squares and minimax designs. Simulation studies are presented to illustrate each phase of the proposed design method and to evaluate the performance of the filters designed.     

Nonrecursive digital filters with crescent ripple

The letter describes the results obtained on designing nonrecursive digital filters having crescent ripple. It is shown that the geometric mean of the maximum passband ripple and the maximum stopband ripple, when plotted against the normalized transition bandwidth, give a similar curve to that obtained for equiripple designs.     

Time-domain synthesis with multiple-shift-sequence digital filters

A method is given for implementing weighted polynomials p(kT)f(kT) by means of digital filters with multiple-shift sequences. The digital structure is obtained by adding to that realising the weight function f(kT) appropriate fractional delays and cyclically variable multipliers of integer values.     

Error-feedback digital filters

It is shown that the recursive digital filter1 employing the residue retention techniques of digital differential analysers and error-feedback digital filters2.3 are functionally equivalent. However, the error-feedback formulation can be readily extended to higher-order filters and clearly shows when error feedback should be used.     

Elliptic digital filters

A synthesis procedure is given for elliptic digital filters of order 2q, q=1, 2, 3?. The pulse transfer function is obtained in the form of biquadratic products in z?1. An example illustrates how this method is used to synthetise an elliptic digital filter.     

Two's complement quantization in two-dimensional state-spacedigital filters

Two-dimensional (2-D) state-space digital filters are investigated for stability with and without quantization. A sufficient condition is obtained for the stability of the linear Fornasini-Marchesini (1976) state-space model. The same model is then studied for stability when implemented using two's complement truncation quantization, and a sufficient condition for asymptotic stability of the nonlinear filter is obtained. In the process, a theorem is proved which gives a sufficient condition for the stability of a one-dimensional (1-D) state-space digital filter under the same type of quantization