Analysis of Memory Effects in Digital Filters with Overflow Arithmetic

This paper deals with the problem of undesired memory effects in nonlinear digital filters owing to the influence of past excitations on future outputs. The nonlinearities under consideration cover the usual types of overflow arithmetic employed in practice. Based on the Hankel norm performance, a new criterion is proposed to ensure the reduction of undesired memory effects in digital filters with overflow arithmetic. In absence of external input, the nonexistence of overflow oscillations is also confirmed by the proposed criterion. A numerical example together with simulation result showing the effectiveness of the criterion is given


Introduction
Digital filters are linear time-invariant systems that perform mathematical operations on discrete-time signals to enhance or reduce certain features of such signals. They found wide applications in many fields such as speech and image processing, communications, noise suppression, control systems, spectrum and vibration analyses, etc. [1]. During the implementation of recursive digital filters on a fixed-point digital signal processor, the generation of nonlinearities due to quantization and overflow is a common phenomenon [2]. Due to the presence of such nonlinearities, the implemented filter may exhibit undesired zero-input oscillations [2][3]. The problem of stability of digital filters with finite wordlength nonlinearities has been studied extensively [2][3][4][5][6][7][8][9][10][11][12][13][14]. Several stability results to reduce the effects of nonlinear phenomenon have appeared recently [10][11][12][13][14][15][16].
If an excitation of finite duration is applied to a digital filter, ringing may occur after the withdrawal of the excitation [16]. The stored energy within the digital filter due to past excitations may produce undesired memory effects or ringing. Ringing may lead to noise, undesired oscillations, echoes etc. and degrade the performance of the system [16][17]. The Hankel norm performance can be used to measure the undesired memory effects of past excitations on future outputs [16][17][18][19][20][21][22].
The analysis of undesired memory effects in digital filters with finite wordlength nonlinearities is an important problem. The criteria in [3][4][5][6][7][8][9][10][11][12][13][14][15] cannot be used to measure the memory effects in digital filters. Recently, the Hankel norm performance criterion for the reduction of memory effects of digital filters with saturation arithmetic has been established in [16]. However, this problem has not been fully investigated in the literature so far. This paper deals with the problem of memory effect analysis for digital filters with overflow nonlinearities. The paper is organised as follows. In Section 2, a description of the system under consideration is presented and the problem is formulated. Section 3 proposes a new linear matrix inequality (LMI)-based criterion for the Hankel norm performance of digital filters with overflow arithmetic. Unlike [16], the proposed criterion can be used to affirm the reduction of undesired memory effects in digital filters employing any kind of overflow arithmetic used in practice (e.g., saturation, zeroing, two's complement and triangular). A numerical example is given in Section 4 to illustrate the applicability of the presented criterion. Finally, the paper is concluded in Section 5.

System Description and Problem Formulation
Consider a fixed-point state-space digital filter given by: is the state vector, () n rR  u denotes the external input or interference, is a known constant matrix and superscript T denotes the transpose. It is assumed that () r u acts as an excitation over the time period 0 rT  for the system with zero initial condition and the excitation disappears if is a linear combination vector of the states representing ring of the system for .
rT  It is assumed that the nonlinearities under consideration satisfy.
which include all kinds of overflow arithmetic employed in practice.
Given a scalar level and a time the purpose of this paper is to establish a new condition such that the digital filter (1) and (2) satisfies.
under zero initial condition. If (3) is satisfied, then the digital filter has the Hankel norm performance . The Hankel norm performance can be interpreted as a measure of the effects of past excitations on the future outputs.

Remark 1.
In view of (9), 0 Therefore, the zero-input digital filter is globally asymptotically stable under the conditions stated in Theorem 1. Remark 2. The approach in [16] focuses on saturation overflow arithmetic. By contrast, the present approach is applicable not only to saturation, but also to two's complement, zeroing, triangular overflow nonlinearities.
Remark 3. The conditions given by (4) are LMIs. Therefore, they can be easily tested by using MATLAB LMI toolbox [23,24].  (16) is not capable of measuring the memory effects of the digital filter precisely. Clearly, ̃ in view of (3) and (16).
Pertaining to the present system with zero initial condition, the effects of () r u  (20) Note that * 0.35.

 
This confirms the Hankel norm performance of the present system. Further, it can also be seen from Figures 1 and 2

Conclusion
A new LMI-based criterion (Theorem 1) for the Hankel norm performance of digital filters employing overflow arithmetic has been established. Unlike the approach in [16], the presented approach is applicable to saturation, zeroing, two's complement and triangular overflow nonlinearities. The criterion obtained in this paper is useful to check the reduction of undesired memory effects in digital filters to past inputs. Without external input, the nonexistence of overflow oscillations is also guaranteed by the presented criterion. An example showing the effectiveness of the criterion has been provided.