Fast communicationOn normal realizations of digital filters with minimum roundoff noise gain
Introduction
Consider an Nth order digital filter implemented with its state-space equations:where and are the input and output of the filter, respectively, while is the state vector of size . is called a state-space realization of .
Denote as the set of all realizations of . is characterized bywhere and 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 , 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 if there exists a diagonally dominant positive definite matrix P such thatwhere 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:This 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 is said to be normal if A is normal, that is . The normal realizations, denoted as , and the optimal roundoff noise realizations are both free of limit cycles [15], [16]. Though yields a smaller roundoff noise gain, does have some nice properties that 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 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 , are those realizations satisfyingwhere I is the identity matrix. The above equation implies that and hence . 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 be a non-trivial parameter (multiplier) in (1), be the roundoff noise due to this parameter and as the corresponding output deviation of the filter. The roundoff noise gain for the parameter is defined as , where denotes the statistical averaging operation. Traditionally, for
A numerical example
Consider an eight-order band-pass elliptic filter generated with MATLAB command The two poles closest to the unit circle are located at , yielding .
Starting from the -scaled controllable realization , an optimal normal realization is computed using the procedure explained in the previous section. An input balanced realization is obtained from using the similarity transformation , where 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.
References (19)
- et al.
Digital Signal Processing
(1987) - M. Gevers, G. Li, Parametrizations in Control, Estimation and Filtering Problems: Accuracy Aspects, Communication and...
- et al.
Synthesis of minimum roundoff noise fixed-point digital filters
IEEE Trans. Circuits Systems
(September 1976) Minimum uncorrelated unit noise in state-space digital filtering
IEEE Trans. Acoust. Speech Signal Process.
(August 1977)Improved -sensitivity for state-space digital system
IEEE Trans. Signal Process.
(April 1997)- et al.
Delta operator realizations of direct-form IIR filters
IEEE Trans. Circuits Systems II
(January 1998) - et al.
A generalized direct-form delta operator-based IIR filter with minimum noise gain and sensitivity
IEEE Trans. Circuits Systems II
(April 2001) - et al.
On the generalized DFIIt structure and its state-space realization in digital filter implementation
IEEE Trans. Circuits Systems I
(April 2004) - et al.
Asymptotic stability of discrete-time systems with saturation nonlinearities with applications to digital filters
IEEE Trans. Circuits Systems I: Fundam. Theory Appl.
(October 1992)