Zero-input Limit Cycles Research Articles

Tighter limit cycle bounds for digital filters

A procedure for obtaining tighter bounds on zero-input limit cycles is presented. The proposed new bounds are applicable to digital filters of arbitrary order, described in state-space formulation and implemented with fixed-point arithmetic. For the most part, we obtain smaller bounds than those reported in the literature, using a computationally efficient algorithm that is easy to implement and has a comparatively short execution time. Simulation results show the validity of the proposed theory.

Suppression of overflow limit cycles in LDI all-pass/lattice filters

In this paper it is shown that zero-input overflow limit cycles can be suppressed in a class of lossless digital integrator (LDI) all-pass filters. This holds for any filter order, provided that saturation overflow characteristics are used at the input of each delay and for certain restrictions for the multiplier values. The results are shown to apply also to lossless digital differentiator (LDD) filters. The restrictions of multiplier values have the effect of excluding certain combinations of poles within the unit circle, most of which are in the left half circle where corresponding LDD filters can be used. Asymptotic stability can be guaranteed for all second-order LDI and LDD filters.

New low roundoff noise realizations of second-order digital filter sections

New structures for second-order filter sections are proposed that yield lower output roundoff noise than many well known structures and are free of zero-input limit cycle oscillations. These new low-roundoff structures are based on these suggested by Bomar (1989) and are obtained by putting the constraint that some of the state-space coefficients be sum-of-two power-of-two terms. Numerical results are included to illustrate the performance of the proposed structures for narrow-band filters and to compare them with other structures.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Elimination of limit cycles in floating-point implementations of direct-form recursive digital filters

This paper focuses on the limit cycle analysis of floating-point implementations of direct form recursive digital filters. A sufficient criterion for the absence of zero-input limit cycles is derived for a direct-form implementation with a single quantizer in the recursive loop. Both the unlimited exponent range (UER) limit cycles and those due to exponent underflow are considered. The main result of the paper is that the limit cycle behaviour of the filter is directly related to the maximum gain of the recursive loop. The higher the recursive loop gain, the longer mantissa wordlength is required to eliminate the possible large-amplitude UER limit cycles. Also the root-mean-square (RMS) bound for the underflow limit cycles is directly proportional to the recursive loop gain. The error feedback technique can be used to reduce the peak gain and thus to save bits in the mantissa wordlength in critical applications, as shown by examples.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Elimination of zero-input and constant-input limit cycles in single-quantizer recursive filter structures

A general criterion for the absence of zero-input limit cycles in digital filter structures that can be implemented with a single quantizer in the recursive loop is presented. The criterion can be applied to a rounding or magnitude truncation quantizer, and it allows the use of error feedback as well. Additionally, a method for the elimination of constant-input limit cycles is presented, provided that the implementation is already free from zero-input limit cycles. The criteria are applied to several well-known filter structures, and new results on their limit cycle properties in the single-quantizer configuration are obtained. It is shown that certain structures are always free from all zero-input and constant-input limit cycles when magnitude truncation is used for quantization. The single-quantizer structures can always be provided with appropriate error feedback, not only to achieve the elimination of limit cycles but also to reduce the roundoff noise, resulting in excellent overall performance.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Elimination of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic

An unconventional approach for guaranteeing the nonexistence of zero-input limit cycles in fixed-point state-space digital filters using saturation arithmetic is presented. The approach is based on a 'passivity' property associated with the multiple saturation nonlinearities. Under certain conditions, the approach specializes to the known approach based on the sector information of the nonlinearities.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Zero-input limit cycles and stability in second-order fixed-point digital filters with two magnitude truncation quantizers

This paper is concerned with the stability analysis of second-order direct-form digital filters with two magnitude truncation quantizers. Zero-input stability is proved for the parameter regions where no conclusion can be drawn using the methods previously suggested in the literature. The areas of the parameter plane in which only limit cycles of period 1 or 2 exist, are determined and related to the cycle amplitudes. Finally, a transition graph is suggested to study the convergence patterns of the filter output.

Constant-input stability of some known digital biquads

Several second-order digital filter structures have been proposed recently, which possess excellent sensitivity characteristics and which are free from zero-input limit cycles. It is shown in this letter that by making some simple modifications, these filter structures can be stabilised against constant-input limit cycles as well.

Zero-input limit cycles due to rounding in digital filters

A necessary condition for the existence of zero-input limit cycles of period T due to rounding in general digital filters is established. Also, geometric sufficient conditions are presented for the existence of limit cycles of period one, of period two and of period T due to rounding. Some illustrative examples are included. >

Limit-cycle free complex biquad recursive digital filters

A novel type of complex biquad recursive digital structure, based on the concept of a generalized immittance converter, is discussed. Orthogonal filters which are widely used in time-division multiplexing-frequency-division multiplexing (TDM-FDM) transmultiplexers can be obtained as a special class of these complex biquads. It is shown that the proposed structure is completely free from zero-input limit cycles under linear conditions, and further, that the stability under finite arithmetic conditions can be achieved using the technique of magnitude truncation. It is also shown that for this structure, the stability of the forced response can be ensured for certain types of overflow nonlinearities.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Resonator-based digital filters

A digital filter structure that is structurally passive and can suppress all zero-input limit cycles, and if rounding is applied provides minimum roundoff noise is presented. These properties are due to the fact that this structure generates its output as a linear combination of orthogonal signal components, thus internally implementing an orthogonal realization of a lossless transfer function. >

New limit cycle bounds for digital filters

A period-independent bound, for zero and constant-input limit cycles in fixed-point digital filters, is developed. The bound is applicable to systems of arbitrary order, provided that all nonideal operations can be consolidated as single nonlinear operations applied at the input to each delay element. Three common types of nonlinearity (i.e. signal quantization) are considered, and a geometric interpretation is used to substantially tighten the bound for sign-magnitude signal quantization. Analytic expressions for the reduced bound are obtained for second-order sections, and an algorithm for computing the bound is presented for higher-order sections. Several numerical examples are presented for second-order sections and compared with previously published results. Two fifth-order low-pass filters (with elliptic and Chebyshev magnitude response characteristics, respectively) and one sixth-order elliptic-bandpass-filter are considered and shown to be free of zero-input limit cycles of amplitude greater than 33.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An improved sufficient condition for absence of limit cycles in digital filters

It is known that if the state transition matrix A of a digital filter structure is such that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D - A^{\dagger}DA</tex> is positive definite for some diagonal matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</tex> of positive elements, then all zero-input limit cycles can be suppressed. This paper shows that positive semidefiniteness of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D - A^{\dagger}DA</tex> is in fact sufficient. As a result, it is now possible to explain the absence of limit cycles in Gray-Markel lattice structures based only on the state-space viewpoint.

Subharmonics and other quantization effects in periodically excited recursive digital filters

Periodic state variations in digital filters have been extensively studied in the context of zero-input limit cycles. Besides such spontaneous oscillations, forced oscillations under periodic excitations also deserve consideration, among which are those for constant input (with unity period). It is observed that, in contrast with its ideal linear counterpart, an actual filter with limited signal wordlength reaches the steady (periodic) state after a finite time with a waveform that depends on the initial conditions, potentially applied in an infinitely remote past. Moreover, the period need not be the same as that of the excitation; subharmonics can occur with periods equal to integer multiples of those of the excitation. In this contribution, the above phenomena are considered as instabilities induced by the nonlinearities due to finite wordlength restrictions, with emphasis on quantization effects. Compared with overflow effects, these effects are relatively small in amplitude and decrease with decreasing quantization steps.

An extension of Jury - Lee's criterion for the stability analysis of fixed-point digital filters designed with two's complement arithmetic

Jury-Lee's criterion [1]-[3] for the absence of zero-input limit cycle oscillations in a class of fixed-point digital filters is considered. An extension of the criterion to permit the use of two's complement arithmetic for the filter design is carried out.

Formulation of a criterion for the absence of limit cycles in digital filters designed with one quantizer

In a recent paper [1], Chang presented a criterion for the absence of zero-input limit cycles in error feedback digital filters employing one quantizer. He derived his result using the idea of circulants. In the present paper, a correction to Chang's work [1] is pointed out. It is further shown how his criterion could be arrived at using a Lyapunov technique.

A new realizability condition for limit cycle-free state-space digital filters employing saturation arithmetic

A criterion for the absence of zero-input limit cycles in state-space digital filters employing saturation arithmetic is presented. An example is given, which brings out the novelity of the present approach.

Passivity properties of low-sensitivity digital filter structures

A general representation of a class of low passband sensitivity digital filter structures is proposed. The proposed representation for a transfer function of order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> consists of an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(N + 1)</tex> -pair memoryless system terminated at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> -pairs by delays. The <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(N + 1)</tex> -pair system contains only adders and multipliers, and is described by an orthogonal transfer matrix. The set of terminating delays can be looked upon as an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> -pair system with transfer matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z^{-1}{\bf 1}</tex> . Certain wave digital filter structures, Gray-Markel lattice structures and the coupled-form biquadratic section belong to the general form advanced here. Several properties satisfied in these special cases are derived in a unified manner using the generalized representation. Also, a quantization scheme that makes the structure free from zero-input limit cycles even under time-varying conditions is advanced, unifying similar such results independently reported for the above well-known structures.

On the elimination of constant-input limit cycles in digital filters

A theorem is proved which establishes sufficient conditions that will ensure freedom from constant-input limit cycles in a general digital-filter structure in which zero-input limit cycles can be eliminated. The theorem is then applied to a specific digital biquad which realizes simultaneously all the standard second-order transfer functions.

Maximum amplitude zero-input limit cycles in digital filters

An algorithm is developed for finding maximum amplitude zero-input limit cycles in recursive digital filters. Computer results are given which establish the feasibility of our algorithm and determine the tightness of previously published limit-cycle bounds. It is also shown that the maximum amplitude limit-cycle problem can be formulated as an integer linear program.