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Signal Processing

Volume 89, Issue 2, February 2009, Pages 226-231
Signal Processing

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On normal realizations of digital filters with minimum roundoff noise gain

https://doi.org/10.1016/j.sigpro.2008.07.018Get rights and content

Abstract

It is well known that normal realizations are free of limit cycles and that a digital filter implemented with a state-space realization (A,B,C,d) has no limit cycles if there exists some diagonal matrix D>0 such that D-ATDA0. In this brief, a method is proposed to check the existence of such a D for any given realization. It is also shown that the normal realizations have a minimal error propagation gain. More interestingly, the normal realizations are characterized, the minimum roundoff noise normal realization problem is formulated and solved analytically. An example is presented to test the efficiency of the proposed method and to demonstrate the performance of the proposed optimal normal realizations.

Introduction

Consider an Nth order digital filter H(z) implemented with its state-space equations:x(n+1)=Ax(n)+Bu(n)y(n)=Cx(n)+du(n)where u(n) and y(n) are the input and output of the filter, respectively, while x(n) is the state vector of size N. (A,B,C,d) is called a state-space realization of H(z).

Denote SH as the set of all realizations of H(z). SH is characterized byA=T-1A0T,B=T-1B0,C=C0Twhere (A0,B0,C0,d)SH and TRN×N is any non-singular matrix.

For real-time applications, digital filters have to be realized ultimately with a digital device (say, a digital signal processor) of finite word length (FWL). Roundoff noise and limit cycles (also referred to as self-sustained/free oscillations) are two serious problems caused by the FWL errors [1], [2]. The optimal filter structure design has been considered as one of the most effective methods to reduce the FWL effects. The optimal roundoff noise realizations, denoted as RMRH, were derived by Mullis and Roberts [3] in 1976 and Hwang [4] in 1977, independently. Since then, many interesting results have been achieved [5], [6], [7], [8].

Pertaining to saturation arithmetic, it was shown in [9] that there is no overflow oscillations in the filter (A,B,C,d) if there exists a diagonally dominant positive definite matrix P such thatP-ATPA0where T is the transpose operator. A modified form of this result was given in [10]. Based on a linear matrix inequality  (LMI) approach, a criterion for nonexistence of saturation overflow oscillations was proposed in [11]. In [12], a necessary and sufficient condition for the absence of such oscillations was given for the controllable realization whose A-matrix is of companion form.

Under two's-complement overflow operation, it was shown in [13] that there are no self-sustained oscillations (because of either roundoff or overflow operation) in the filter if there exists some diagonal matrix D with positive diagonal entries such that the following matrix Q is positive semi-definite:QD-ATDA0This is an improved result of that reported in [14], [15]. Comparing (3) with (4), one can see that the former is in a more general form and that none of the two types of overflow oscillations mentioned above can occur if a realization satisfies (4). It should be pointed out that (4) is a sufficient condition for the absence of limit cycles and that for a given realization, as far as we know, whether a D-matrix exists such that (4) holds is still an open problem.

A realization (A,B,C,d) is said to be normal if A is normal, that is AAT=ATA. The normal realizations, denoted as Rn, and the optimal roundoff noise realizations RMRH are both free of limit cycles [15], [16]. Though RMRH yields a smaller roundoff noise gain, Rn does have some nice properties that RMRH does not possess such as having a minimal pole sensitivity [2] and as to be seen a minimum error propagation gain, which are related to the stability behavior of the filters. This is particularly important for narrow bandwidth filters which may become unstable easily when implemented with FWL devices [1], [2]. As to be seen, the normal realizations are not unique. Therefore, it is highly recommended to implement the filters using those normal realizations which minimize the roundoff noise.

An outline of the paper is as follows. In Section 2, based on an LMI approach a method is derived for checking the existence of a D-matrix in (4). The set of normal realizations is also characterized and compared with the realizations RMRH in this section. In Section 3 the minimum roundoff noise normal realizations are obtained analytically. A numerical example is presented in Section 4 to demonstrate the efficiency of the proposed method and the performance of the optimized normal realizations. Some concluding remarks are given in Section 5.

Section snippets

Computation of D-matrix and normal realizations

Using (4), one can show that some well known realizations are free from limit cycles. The input-balanced realizations, denoted as Rib, are those realizations (A,B,C,d) satisfyingI=AAT+BBTwhere I is the identity matrix. The above equation implies that I-AAT0 and hence I-ATA0. According to (4), the input-balanced realizations are free of limit cycles.

The question one may ask is, for a given realization whether there exists a D such that (4) holds; and if the answer is yes, how to find it.

Optimal roundoff noise normal realizations

Rounding errors are due to multiplications between signals and non-trivial parameters.1 Let τ be a non-trivial parameter (multiplier) in (1), ετ(n) be the roundoff noise due to this parameter and Δy(n) as the corresponding output deviation of the filter. The roundoff noise gain for the parameter τ is defined as GτE[(Δy(n))2]/E[ετ2(n)], where E[·] denotes the statistical averaging operation. Traditionally, for

A numerical example

Consider an eight-order band-pass elliptic filter generated with MATLAB command ellip(4,0.5,40,[0.250.30])The two poles closest to the unit circle are located at p1,2=0.7013±j0.6990, yielding |p1,2|=0.9901.

Starting from the l2-scaled controllable realization R0, an optimal normal realization Rn is computed using the procedure explained in the previous section. An input balanced realization Rib is obtained from R0 using the similarity transformation T=(Wc0)1/2, where Wc0 is the controllability

Conclusions

This paper deals with limit cycle free realizations with minimization of roundoff noise. A method has been proposed for checking if there exists a solution to (4) for any realizations that may be obtained based on other FWL error criteria than the limit cycles. The optimal roundoff noise normal realizations have been obtained analytically. Such optimal realizations are highly recommended for digital filter implementation as they are the most robust realizations having a minimal pole sensitivity

Acknowledgments

The authors would like to thank the reviewers for the constructive comments and suggestions which help improve the quality of this paper. This work was supported by the Qianjiang Chair Professorship Foundation of Zhejiang province.

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