# Improved criterion for the elimination of overflow oscillations in digital filters with external disturbance

- Tao Li
^{1}Email author, - Ning Sun
^{1}, - Qiancheng Lin
^{2}and - Jianqing Li
^{3}

**2012**:197

https://doi.org/10.1186/1687-1847-2012-197

© Li et al.; licensee Springer 2012

**Received: **24 August 2012

**Accepted: **5 November 2012

**Published: **14 November 2012

## Abstract

In this paper, an improved criterion for the ${H}_{\mathrm{\infty}}$ elimination of overflow oscillations in digital filters with external disturbance is presented. Compared with some existing results in Ahn (AEU, Int. J. Electron. Commun. 65:750-752, 2011) and Kokil *et al.* (AEU, Int. J. Electron. Commun. doi:10.1016/j.aeue.2012.01.004, 2012), a distinct feature of the proposed criterion is that it can include the existing results as special cases or be less restrictive than them. Finally, an example is given to show this improvement over the existing conditions.

## Keywords

## 1 Introduction

When designing digital filters by using a fixed-point arithmetic, one encounters overflow nonlinearities [1, 2]. Such nonlinearities may lead to the instability and possibly the zero-input limit cycles of the designed filters. But the global asymptotic stability of the null solution guarantees the absence of limit cycles in the designed filter [1, 3]. Therefore, the problem of the global asymptotic stability of fixed-point digital filters using saturation arithmetic has received considerable attention from many researchers [3–18]. However, most existing criteria are not available under the unfavorable environments with external disturbance. Recently, [15, 16] have considered this case, but the method given in [15, 16] seems still restrictive. So, there is room for further investigation on this topic.

Motivated by the preceding discussion, in this paper, an improved criterion for the ${H}_{\mathrm{\infty}}$ elimination of overflow oscillations in digital filters with external disturbance is proposed. It is shown that the presented criterion is less restrictive or more general than those in [15, 16].

**Notation** Throughout this paper, *I* is used to denote an identity matrix with appropriate dimension. For a real symmetric matrix *P*, the notation $P>0$ (≥0) means that the matrix *P* is a positive definite (positive semi-definite), and $A>(\ge )\phantom{\rule{0.2em}{0ex}}B$ means $A-B>(\ge )\phantom{\rule{0.2em}{0ex}}0$. The superscript *τ* denotes the transpose of a vector or matrix.

## 2 Problem formulation and preliminaries

The nonlinearities given by (2) include two’s complement overflow arithmetic and saturation arithmetic as special cases.

The purpose of this paper is to develop an improved exponential stability with ${H}_{\mathrm{\infty}}$ performance *γ* under zero-initial conditions for all nonzero $\omega (r)$ for the system (1) satisfying conditions (2) and to compare it with the existing results in [15, 16]. To obtain our main results, we need the following lemma.

**Lemma 1**

*For the given system*(1)-(2),

*if there are a diagonal matrix*$L=diag\{{l}_{1},{l}_{2},\dots ,{l}_{n}\}\in {R}^{n\times n}$,

*a matrix*$N=[{n}_{ij}]\in {R}^{n\times n}$

*which satisfy*${l}_{j}\ge {\sum}_{i=1}^{n}|{n}_{ji}|$, $i,j=1,\dots ,n$,

*then the following inequality holds*:

*Proof*The inequality (3) can be rewritten as

When $-1\le {y}_{i}(r)\le 1$, it means ${y}_{j}(r)-{f}_{j}({y}_{j}(r))=0$. It is clear that (4) holds.

According to (6), hence, (4) holds. This completes the proof of Lemma 1. □

## 3 Improved stability criterion

First of all, we present the main result, *i.e.*, a new criterion for the ${H}_{\mathrm{\infty}}$ elimination of overflow oscillations in digital filters with external disturbance.

**Theorem 1**

*For a given level*

*γ*

*and the system*(1)

*satisfying condition*(2),

*if there exist positive definite matrices*$P={P}^{\tau}\in {\mathbb{R}}^{n\times n}$, $S={S}^{\tau}\in {\mathbb{R}}^{n\times n}$,

*a positive diagonal matrix*$L=diag\{{l}_{1},{l}_{2},\dots ,{l}_{n}\}\in {R}^{n\times n}$,

*and a matrix*$N=[{n}_{ij}]\in {\mathbb{R}}^{n\times n}$

*such that*

*and*

*hold*, *then the system* (1) *is exponentially stable with* ${H}_{\mathrm{\infty}}$ *performance* *γ*.

*Proof*Consider the quadratic Lyapunov function

By summation of both sides of (10) from 0 to ∞ and computing $V(x(\mathrm{\infty}))\ge 0$ and $V(x(0))=0$, we have ${\sum}_{r=0}^{\mathrm{\infty}}{x}^{\tau}(r)Sx(r)<{\gamma}^{2}{\sum}_{r=0}^{\mathrm{\infty}}{\omega}^{\tau}(r)\omega (r)$.

According to (10) and Theorem 1 of [15], this guarantees the exponential stability. This completes the proof. □

## 4 Comparison with the existing results

In order to show the improvement of the proposed criterion over the existing results in [15, 16], we will first recall them.

**Lemma 2** (Ahn 2011 [15])

*For a given level*$\gamma >0$,

*if we assume that there exist symmetric positive definite matrices*

*P*,

*S*,

*a positive diagonal matrix*

*M*,

*and a positive scalar*

*δ*

*such that*

*then the system* (1) *with* (2) *is exponentially stable with* ${H}_{\mathrm{\infty}}$ *performance* *γ*.

**Lemma 3** (Kokil 2012 [16])

*For a given level*$\gamma >0$,

*if we assume that there exist symmetric positive definite matrices*

*P*,

*S*,

*and a positive scalar*

*δ*

*such that*

*where* *C* *is a row diagonally dominant matrix with positive diagonal elements*, *then the system* (1) *with* (2) *is exponentially stable with* ${H}_{\mathrm{\infty}}$ *performance* *γ*.

**Remark 1** As noted in [16], when *C* in Lemma 3 is identified as the positive definite diagonal matrix *M* in Lemma 2, Lemma 2 can be recovered as a special case of Lemma 3.

Comparing with Lemma 3, Theorem 1 may further relax the condition. We will show the improvement by using Remark 2.

**Remark 2**Letting $L-N=Z$ in Theorem 1, it is clear that

*Z*is a row diagonally dominant matrix with positive diagonal elements. Then LMI (7a) can be rewritten as

When $Z=-C$ and $2{l}_{1}=2{l}_{2}=\cdots =2{l}_{n}=\delta $ with *C* is defined in (13), it is worth pointing out that Lemma 3 can be recovered as a special case of Theorem 1.

## 5 Numerical examples

In this section, we use an example to illustrate the usefulness of our result.

**Example 1**Consider the second-order system (1) with

*γ*for this system as obtained via Theorem 1 is 12.9062. On the other hand, Lemma 2 and Lemma 3 give the minimum lower bound of

*γ*as 72.9310 and 35.1760. It is clear that the minimum lower bound of

*γ*in our paper is smaller than Lemma 1 and Lemma 2. In other words, Theorem 1 provides improved results over the previous results in [15, 16]. And a feasible solution is in the following:

## 6 Conclusion

In this paper, an LMI-based criterion for the ${H}_{\mathrm{\infty}}$ elimination of overflow oscillations in fixed-point state-space digital filters with external disturbance has been established. The obtained criterion has been shown to be less restrictive than some existing results or to cover them as special cases.

## Declarations

### Acknowledgements

This work was supported in part by the Major State Basic Research Development Program of China (973 Program) under Grant No. 2012CB720003, in part by the National Science Foundation of China under Grants No. 61125306, 60904025, 60904026.

## Authors’ Affiliations

## References

- Singh V: Stability analysis of a class of digital filters utilizing single saturation nonlinearity.
*Automatica*2008, 44: 282–285. 10.1016/j.automatica.2007.04.015View ArticleMATHMathSciNetGoogle Scholar - Ooba T: Stability of discrete-time systems joined with a saturation operator on the state-space.
*IEEE Trans. Autom. Control*2010, 48: 2153–2155.MathSciNetView ArticleGoogle Scholar - Shen T, Wang XH, Yuan ZG: Stability analysis for a class of digital filters with single saturation nonlinearity.
*Automatica*2010, 46: 2112–2115. 10.1016/j.automatica.2010.09.008MathSciNetView ArticleMATHGoogle Scholar - Kar H, Singh V: Elimination of overflow oscillations in digital filters employing saturation arithmetic.
*Digit. Signal Process.*2005, 15: 536–544. 10.1016/j.dsp.2005.02.001View ArticleGoogle Scholar - Kar H: An LMI based criterion for the nonexistence of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic.
*Digit. Signal Process.*2007, 17: 685–689. 10.1016/j.dsp.2006.11.003View ArticleGoogle Scholar - Kar H: An improved version of modified Liu-Michel’s criterion for global asymptotic stability of fixed-point state-space digital filters using saturation arithmetic.
*Digit. Signal Process.*2010, 20: 977–981. 10.1016/j.dsp.2009.10.020View ArticleGoogle Scholar - Liu D, Michel AN: Asymptotic stability of discrete-time systems with saturation nonlinearities with applications to digital filters.
*IEEE Trans. Circuits Syst. I*1992, 39: 798–807. 10.1109/81.199861View ArticleMATHGoogle Scholar - Shen T, Yuan ZG: Stability of fixed-point state-space digital filters using two’s complement arithmetic: further insight.
*Automatica*2010, 46: 2109–2111. 10.1016/j.automatica.2010.08.017MathSciNetView ArticleMATHGoogle Scholar - Singh V: A new realizability condition for limit cycle free state-space digital filters employing saturation arithmetic.
*IEEE Trans. Circuits Syst.*1985, 32: 1070–1071. 10.1109/TCS.1985.1085612View ArticleGoogle Scholar - Singh V: Elimination of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic: an LMI approach.
*Digit. Signal Process.*2006, 16: 45–51. 10.1016/j.dsp.2005.04.003View ArticleGoogle Scholar - Singh V: Modified form of Liu-Michel’s criterion for global asymptotic stability of fixed-point state-space digital filters using saturation arithmetic.
*IEEE Trans. Circuits Syst. II, Express Briefs*2006, 53: 1423–1425.View ArticleGoogle Scholar - Singh V: A new frequency-domain criterion for elimination of limit cycles in fixed-point state-space digital filters using saturation arithmetic.
*Chaos Solitons Fractals*2007, 34: 813–816. 10.1016/j.chaos.2006.03.075View ArticleGoogle Scholar - Ahn CK, Lee YS:Induced ${H}_{\mathrm{\infty}}$ stability of fixed-point digital filters without overflow oscillations and instability due to finite word length effects.
*Adv. Differ. Equ.*2012, 51: 1–7.MATHMathSciNetGoogle Scholar - Ahn CK: A new condition for the elimination of overflow oscillations in direct form digital filters.
*Int. J. Electron.*2012. doi:10.1080/00207217.2012.680185Google Scholar - Ahn CK: Criterion for the elimination of overflow oscillations in fixed-point digital filters with saturation arithmetic and external disturbance.
*AEÜ, Int. J. Electron. Commun.*2011, 65: 750–752. 10.1016/j.aeue.2010.12.001View ArticleGoogle Scholar - Kokil P, Kandanvli VKR, Kar H: A note on the criterion for the elimination of overflow oscillations in fixed-point digital filters with saturation arithmetic and external disturbance.
*AEÜ, Int. J. Electron. Commun.*2012. doi:10.1016/j.aeue.2012.01.004Google Scholar - Mills WL, Mullis CT, Roberts RA: Digital filter realizations without overflow oscillations.
*IEEE Trans. Acoust. Speech Signal Process.*1978, 26: 334–338. 10.1109/TASSP.1978.1163114MathSciNetView ArticleMATHGoogle Scholar - Singh V: Modified criterion for global asymptotic stability of fixed-point state-space digital filters using two’s complement arithmetic.
*Automatica*2010, 46: 475–478. 10.1016/j.automatica.2009.11.022View ArticleMATHMathSciNetGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.